Spectral transfer morphisms for unipotent affine Hecke algebras

In this paper we will give a complete classification of the spectral transfer morphisms between the unipotent affine Hecke algebras of the various inner forms of a given quasi-split absolutely simple algebraic group, defined over a non-archimidean local field $\textbf{k}$ and split over an unramified extension of $\textbf{k}$. As an application of these results, the results of [O4] on the spectral correspondences associated with such morphisms and some results of Ciubotaru, Kato and Kato [CKK] we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on the formal degrees and adjoint gamma factors for all unipotent discrete series characters of unramified simple groups of adjoint type defined over $\bf{k}$.


Introduction
Recall from [54] that a normalized affine Hecke algebra H is essentially determined by a complex torus T and a meromorphic function μ on T . A spectral transfer morphism (see [54]) φ : H 1 ; H 2 between normalized affine Hecke algebras expresses the fact that μ 1 is equal to a residue of μ 2 along a certain coset of a subtorus of T 2 . This turns out to be a convenient tool to compare formal degrees of discrete series representations of different affine Hecke algebras.
The notion is based on the special properties of the μ-function of an affine Hecke algebra [52,53] which are intimitely related to its basic role in the derivation of the Plancherel formula for affine Hecke algebras via residues [21,52,55,56]. This approach to the computation of formal degrees has its origin in the theory of spherical functions for p-adic reductive groups [47], and was further inspired by early observations of Lusztig [35,38] and Reeder [59,60] on the behaviour of formal degrees within unipotent L-packets.
In the present paper we classify the spectral transfer morphisms (STMs in the sequel) between the unipotent affine Hecke algebras of the various inner forms of a given absolutely simple algebraic group G of adjoint type, defined and unramified over a non-archimedean local field k. In particular we will show, for any unipotent type τ = (P, σ ) of an inner form of G, existence and uniqueness (up to diagram automorphisms) of such STM of the Hecke algebra of τ to the Iwahori-Hecke algebra H I M (G) of G. The STMs of this kind turn out to correspond exactly to the arithmeticgeometric correspondences of Lusztig [40,43].
As an application of this classification, using the basic properties of STMs discussed in [54], we prove the conjecture [26,Conjecture 1.4] of Hiraga, Ichino and Ikeda expressing the formal degree of a discrete series representation in terms of the adjoint gamma factor of its (conjectural) local Langlands parameters and an explicit rational constant factor, for all unipotent discrete series representations of inner forms of G (where we accept Lusztig's parameters for the unipotent discrete series representations as conjectural Langlands parameters). It should be mentioned that it was already known from Reeder's work [59,60] (see also [26]) that this conjecture holds for the unipotent discrete series characters of split exceptional groups of adjoint type, and for some small rank classical groups. It should be mentioned that the stability of Lusztig's packets of unipotent representations was shown by Moeglin and Waldspurger for odd orthogonal groups [49] and by Moeglin for unitary groups [48].
Throughout this paper we use the normalization of Haar measures as in [17]. Let q = v 2 denote the cardinality of the residue field of k. The formal degree of a unipotent discrete series representation then factorizes uniquely as a product of a qrational number (which we define as a fraction of products of q-numbers of the form [n] q := (v n −v −n ) v−v −1 with n ≥ 2) and a positive rational number. Our proof of conjecture [26,Conjecture 1.4] involves the verification of the q-rational factors, which rests on the existence of the Plancherel measure preserving correspondences for STMs as discussed in [54], and the verification of the rational constants. The latter uses the knowledge of these rational constants from [60] for the case of equal parameter exceptional Hecke algebras, and continuity principles due to [11] and [56] (also [13]) which imply roughly that we can compute these rational constant factors in the formal degrees of discrete series of non-simply laced affine Hecke algebras at any point in the parameter space of the affine Hecke algebra once we know these rational constants in one regular point (in the sense of [56]) of the parameter space. In particular, for classical affine Hecke algebras of type C (1) n ; it was shown in [11] that at a generic point in the parameter space, the rational constants for all generic families of discrete series characters are equal. The constants at special parameters follow then by a continuity principle in the formal degree due to [56].
An alternative approach to the conjecture [26,Conjecture 1.4], restricted to the case of formal degrees of unipotent discrete series representations, was formulated in [12]. A conjectural formula for the formal degrees of unipotent discrete series characters is proposed in [12], which involves Lusztig's non-abelian Fourier transform matrix for families of unipotent representations [36,44,45] and a notion of the "elliptic fake degree" of a unipotent discrete series character in the unramified minimal principal series of G. In this approach the formula for the rational constant factors of the formal degrees appears in a very natural way from the basic properties of the non-abelian Fourier transform.
The notion of spectral transfer morphism is based on a certain heuristic idea on the behavior of L-packets under ordinary parabolic induction (see 3.1.3 for a more detailed discussion of this heuristic idea). The fact that this principle turns out to hold for all unipotent representations is striking. Also striking is the fact that the isomorphism class of the Iwahori-Hecke algebra H I M (G) of G is the least element in the poset of isomorphism classes of normalized affine Hecke algebras in the full subcategory of C es (G) whose objects are the Hecke algebras of unipotent types (P, σ ) of the inner forms of G, in the sense of [54,Paragraph 7.1.5]. Moreover, if H is such a unipotent affine Hecke algebra of an inner form of G, then the STM φ : H ; H I M (G) (which exists by the above) is essentially unique, and such STMs exactly match Lusztig's arithmetic/geometric correspondences. The proof of these statements reduces, as explaned in this paper, to the supercuspidal case [19] in combination with the above principle that one can parabolically induce unipotent supercuspidal STMs from Levi subalgebras to yield new STMs.
It is quite clear that the definition of the notion of STM could be generalized to Bernstein components [2,[23][24][25] in greater generality than only for the unipotent Bernstein components. It would be interesting to investigate the above mentioned induction principle in general. In view of our results, this could provide a clue how Lpackets are partitioned by the Bernstein center beyond Lusztig's unipotent L-packets for simple groups of adjoint type.
In the first section of this paper we will review the theory of unipotent representations of G with an emphasis on its harmonic analytic aspects. The results here are all due to [40,43,50,51] and [17]. This section serves an important purpose of reviewing the relevant facts on unipotent representations for this paper in the appropriate context of harmonic analysis, and fixing notations. We kept the setup in this section more general than necessary for the remainder of the paper, since this does not complicate matters too much and this may be useful for later applications. In the second section we will describe the structure of the STMs between the normalized unipotent Hecke algebras of the inner forms of G, and discuss the applications of this result. 1

Unipotent representations of quasisimple p-adic groups
The category of unipotent representations of inner forms of an unramified absolutely quasisimple p-adic group G is Morita equivalent to the category of representations of a finite direct sum of finitely many normalized affine Hecke algebras (called "unipotent Hecke algebras") in such a way that the Morita equivalence respects the tempered spectra and the natural Plancherel measures on both sides.
Therefore it is an interesting problem to classify all the STMs as defined in [54,Definition 5.1] between these unipotent normalized affine Hecke algebras. It will turn out that this task to classify these STMs essentially reduces to the task of finding all STMs from the rank 0 unipotent affine Hecke algebras to the Iwahori-Matsumoto Hecke algebra H I M (G ) of the quasisplit G such that G is an inner form of G . In turn this reduces to solving equation [54, equation (55)] where d 0 denotes the formal degree of a unipotent supercuspidal representation. The latter part of this task, the classification of the rank 0 unipotent STMs, will be discussed in a second paper (joint with Yongqi Feng [19]). It should be remarked that the results of the present paper, in which the existence of certain spectral transfer morphisms is established, plays a role in the proof of the classification result in [19].

Unramified reductive p-adic groups
Let k be non-archimedean local field. Fix a separable algebraic closure k of k, and let K ⊂ k be the maximal unramified extension of k in k. Let K = O/P be the residue field of K , and let p denote its characteristic. Let = Gal(k/k) denote the absolute Galois group of k, and let I = Gal(k/K ) ⊂ be the inertia subgroup. Let Frob be the geometric Frobenius element of Gal(K /k) = /I Ẑ , i.e., the topological generator which induces the inverse of the automorphism x → x q of K. Here q = p n denotes the cardinality of the residue field k := K Frob of k. We denote by v the positive square root of q. Let G be a connected reductive algebraic group defined over k, and split over K . We denote by G ∨ be the neutral component of a Langlands dual group L G for G (see [3]). The construction of L G presupposes the choice of a maximal torus S and a Borel subgroup B of G whose Levi-subgroup is S, and the choice of an épinglage for (G, B, S), in order to define a splitting of Aut(G). Let X * (Z (G ∨ )) be the character group of the center Z (G ∨ ) of G ∨ . The natural -action on this space factors through the quotient Gal(K /k) since we are assuming that G is K -split. Observe that the action of Frob on X * (Z (G ∨ )) is independent of the choice of a splitting of Aut(G).
We will always denote the group G(K ) of K -rational points of G by the corresponding non-boldface letter, i.e., G = G(K ). Kottwitz [31,Section 7] has defined a -equivariant functorial exact sequence In our situation there is a continuous equivariant action of the group /I on this sequence. We denote by F the action of Frob on G 1 and G, and by θ F the automorphism of X * (Z (G ∨ )) defined by F. This sequence has the property that the associated long exact sequence in continuous nonabelian cohomology yields an exact sequence and an isomorphism Now assume that G is semisimple. In this situation the above sequences simplify as follows. Let S be a maximal K -split torus of G, and let X := X * (S) be its cocharacter lattice. Let Q := X sc = X * (S sc ) be the cocharacter lattice of the inverse image of S in the simply connected cover G sc → G of G (hence Q ⊂ X is the coroot lattice of (G, S); we warn the reader that we call the roots of G ∨ "roots" and the roots of (G, S) "coroots". We apologize for this admittedly awkward convention). Let be the finite abelian group = X/Q. Then we may canonically identify X * (Z (G ∨ )) with . Hence (2) becomes (see [29,30]) and (3) becomes We remark that G der ⊂ G 1 ⊂ G, and that it can be shown that G der = G 1 if and only if p does not divide the order | | of . We will from now on always assume that G is absolutely quasisimple and K -split, unless otherwise stated.
The group W acts faithfully on the apartment A as an extended affine Coxeter group. We denote by S O = O × ⊗ X the maximal bounded subgroup of S. Then X = S/S O , and we define the associated F-stable apartment A = A(G, S) of the building of G by A(G, S) = R ⊗ X . As explained in [17,Corollary 2.4.3], [6, Section 3] the isomorphism (5) can be made explicit by a canonical bijection sending [ω] ∈ /(1 − θ) to the cohomology class of the cocycle z u which maps Frob to F u , where uS O = x ∈ X and x is a representative of ω ∈ X/Q. Let C be an F-stable alcove in A (such alcoves exist, see [67]). Let 1 → N → G sc → G → 1 be the simply connected cover of G, and let S sc be the inverse image of S. Proposition 2. 1 The image of G sc → G is equal to the derived group G der of G, and we have G/G der Proof Indeed, it is clear that the image is contained in G der because G sc is its own derived group [65]. The other inclusion follows by applying the long exact sequence in nonabelian cohomology to the central isogeny G sc → G and again appealing to Steinberg's Vanishing Theorem. It follows that the quotient of G by the image of G sc is the abelian group H 1 (K , N), whence the result. On the other hand, we have the obvious exact sequence which we can compare to the long exact sequence in cohomology (with respect to I) associated to the canonical exact sequence of diagonalizable groups 1 → N → S sc → S → 1.
We denote by W a C the F-stable normal subgroup of W generated by the reflections in the walls of C. This normal subgroup is independent of the choice of C and can be canonically identified with N G der (S)/S O ∩ G der ∼ −→ W a ⊂ W , the affine Weyl group of (G sc , S sc ).

Corollary 2.2 We have G
Proof Let B be the Iwahori subgroup of G associated with C [5, 5.2.6]. By [57, Appendix, Proposition 3] we have B = Fix(C) ∩ G 1 . In particular we have S O ⊂ G 1 , so that we have G der ⊂ G 1 := S O , G der ⊂ G 1 Hence by (4), the equality G 1 = G 1 is equivalent to showing that G/G 1 = G/G 1 = . By the previous proposition we have G/G der = K × ⊗ . Since Let C be the subgroup of W which stabilizes C. This subgroup may be identified with a subgroup of the group of special automorphisms (in the sense of [40, paragraph 1.11]) of the affine diagram associated with the choice of C. We have a semidirect product decomposition W = W a C , and thus a canonical isomorphism C ∼ −→ for any choice of C. Proof By Corollary 2.2 it follows that N G 1 (S) = N G der (S).S O . This implies the first assertion, since W a is the affine Weyl group of (G sc , S sc ) and G der is the homomorphic image of G sc . Since an Iwahori-subgroup of G sc is self-normalizing, we have similarly N G 1 (B) = N G der (B).S O = (B ∩ G der ).S O = B, proving the second assertion. For the third assertion, observe that C = (N G (B) ∩ N G (S))/S O . It is well known that B ∩ N G (S) = S O , hence C maps injectively into N G (B)/B. By the second assertion this group maps injectively into G/G 1 = . Since C are finite the two injective homomorphisms are in fact isomorphisms.
Since G is unramified there exist hyperspecial points in the apartment A [67]. A choice of a hyperspecial point a 0 ∈ A, induces a semidirect product decomposition W = W 0 X , where W 0 denotes the isotropy subgroup of a 0 in W . The k-structure of G defined by F is quasisplit, which implies that there exists a hyperspecial point a 0 ∈ A(G, S) which is F-fixed. In this case we denote by θ the automorphism of W (and of A) induced by F. We fix a 0 , an F-fixed hyperspecial point, and an Fstable alcove C having a 0 in its closure. Observe that the subgroup C depends on the choice of C, not of the hyperspecial point a 0 . Recall we have a canonical isomorphism C ∼ −→ = X/Q, which we will often use to identify these two groups. Observe that θ stabilizes the subgroups W a , C , X and W 0 of W .

Parahoric subgroups
Recall the explicit representation of pure inner forms G u as discussed in (7). Fix a representative u =ω ∈ N G (S) with ω ∈ C ⊂ W . Then F u acts on the apartment A(G, S) by means of the finite order automorphism ωθ. Since F u stabilizes C the Iwahori subgroup B is F u stable. Recall that the group C can be canonically identified with the group Following [57,Appendix] we may define a "standard parahoric subgroup of G" as a subgroup of the form Fix(F P ) ∩ G 1 where F P ⊂ C denotes a facet of C. By [57,Appendix,Proposition 3] this definition coincides with the definition in [5]. In particular, a standard parahoric subgroup of G is a connected pro-algebraic group. A parahoric subgroup of G is a subgroup conjugate to a standard parahoric subgroup.
It is well known (by "Lang's theorem for connected proalgebraic groups", see [40, 1.3]) that any F u -stable parahoric subgroup of G is G F u -conjugate to a "standard" F u -stable parahoric subgroup, i.e., an F u -stable parahoric subgroup containing B. It follows that the G F u -conjugacy classes of F u -stable parahoric subgroups are in oneto-one correspondence with the set of θ -orbits of ωθ-stable facets in the closure of C. Similarly, a parahoric subgroup P or a double coset of a parahoric subgroup is F u -stable iff it contains points of G F u . Let P be an F u -stable parahoric subgroup of G. We call P F u a parahoric subgroup of G F u .
We record two important properties of F u -stable parahoric subgroups which follow easily from Corollary 2.3. First of all, parahoric subgroups are self-normalizing in G 1 , i.e., Secondly, for an F u -stable standard parahoric subgroup P corresponding to a ωθ-stable facet C P of C, we have where P ⊂ C is the subgroup stabilizing C P , and P,θ ⊂ P its fixed point group for the action of θ (or F u = ωθ, which amounts to the same since C is abelian). We define an exact sequence where P,θ 1 is the subgroup of elements which fix the set of F u -orbits of vertices of C not in C P pointwise.

Normalization of Haar measures
Let G, F, and F u be as in the previous paragraph. Then G F u is a locally compact group. For any F u -stable parahoric subgroup P of G we denote by P F u the reductive quotient of P F u . This is the group of k-points of a connected reductive group over k. In particular this is a finite group.
Following [17, Section 5.1] we normalize the Haar measure of G F u uniquely, such that for all F u -stable parahoric subgroups P of G one has where a ∈ Z is equal to the dimension of P over K. It is well known that the right-hand side is a product of powers of v and cyclotomic polynomials in v.

The anisotropic case
It is useful to discuss the case where G F u is anisotropic explicitly. It is well known that an anisotropic absolutely simple group G F u is isomorphic to PGL 1 (D) := D × /k × , where D is an unramified central division algebra over k of degree m + 1, rank (m + 1) 2 (see for instance [16]). We choose a uniformizer π of k. D contains an unramified extension l of degree m + 1 over k, and we may choose a uniformizer of D which normalizes l, such that conjugation by restricted to l yields a generator for Gal(l : k), and such that m+1 = π . The group P := G 1 is the only F u -stable parahoric subgoup in this situation, and obviously G = N P. By (4) we have , a cyclic group of order m + 1. G F u contains a maximal prounipotent subgroup G + (denoted by V 1 in [16]) and we have G F u = C.G + , where C is generated by the anisotropic torus T F u := l × /k × and . We see that the reductive quotient P/(P ∩ G + ) is an anisotropic torus T of rank m over k, and that T F u can be identified with the group of roots of unity of order prime to p in l modulo the subgroup of those roots of unity in k. Hence

Unipotent representations and affine Hecke algebras
Let G be a quasisimple linear algebraic group, defined and quasisplit over k and K -split as above. Recall that the automorphism induced by the Frobenius F on the building of G was denoted by θ . Recall that the inner forms of G are canonically parameterized by the abelian group /(1 − θ) . Let u =ω ∈ N B be a representative of an elementω ∈ /(1 − θ) and let F u denote the corresponding pure inner twist of F. We denote by G u the pure inner form of G defined by this twisted k-structure (in particular G 1 = G).
A representation (E, δ) of a parahoric subgroup P F u of G F u (where P is an F ustable parahoric subgroup of G) is called cuspidal unipotent if it is the lift to P F u of a cuspidal unipotent representation of the reductive quotient P F u . An F u -stable parahoric subgroup is called cuspidal unipotent if it has cuspidal unipotent representations.
Lusztig [40] introduced the category R(G F u ) uni of unipotent representations of G F u . A smooth representation (V, π) of G F u is called unipotent if V is generated by a sum of cuspidal unipotent isotypical components of restrictions of (V, π) to various parahoric subgroups of G F u . As a generalization of Borel's theorem on Iwahori-spherical representations, R(G F u ) uni is an abelian subcategory of the category R(G F u ) of smooth G F u representations. It is central to the approach in this paper that this category is equivalent to the module category of an explicit finite direct sum of normalized Hecke algebras in the sense of paragraph [54, 3.1.2], in a way which is compatible with harmonic analysis. Let us therefore describe this in detail.
A cuspidal unipotent pair (P, δ) consists of an F u -stable parahoric subgroup P of G and an irreducible cuspidal unipotent representation δ of P F u . We say that According to [40], given two cuspidal unipotent pairs are either disjoint or equal, and this last alternative occurs if and only if the pairs (P i , δ i ) are G F u -conjugates (and not just associates). It follows that a smooth representation (V, π) is unipotent iff V is generated by ⊕(V | P Fu ) δ , where the direct sum is taken over a complete set of representatives (P, δ) of the finite set of θ -orbits of standard cuspidal unipotent pairs.
For each standard cuspidal unipotent pair s = (P, δ) we consider the algebra H u,s v of s-spherical End(E)-valued functions on G F u , equipped with a trace τ ( f ) := Tr V ( f (e)) and * defined by f * (x) := f (x −1 ) * . This algebra turns out to be the specialization at v of a finite direct sum of mutually isomorphic normalized (in the sense of [54, paragraph 3.1.2]) affine Hecke algebras (called unipotent affine Hecke algebras) defined over L = C[v ±1 ], and has been explicitly determined in all cases [40,51]. The following general result from the theory of types due to [7] (also see [22]) is fundamental to the approach in this paper: Let u =ω be a representative of an element ω ∈ /(1 − θ) . We see that the double B F u -cosets of G F u are parameterized by the ωθ fixed group W ωθ , and that ωθ = N G Fu B F u /B F u . Because is abelian we actually have ωθ = θ . We have W ωθ = W ωθ,a θ . By [40] this extended affine Weyl group is the underlying affine Weyl group of the Iwahori-Hecke algebra H u,I M := H u, (B,1) of the group G F u . When [ω] = 1 we denote this algebra simply by H I M , the generic Iwahori-Hecke algebra.
By (4), the Pontryagin dual ( θ ) * of θ can be viewed canonically as the group of (weakly) unramified complex linear characters X * un (G F u ) of G F u (i.e., the complex linear characters of G F u vanishing on G F u 1 ). This defines a natural functorial action of ( θ ) * on the category R(G F u ) uni (by taking tensor products). These functors are Plancherel measure preserving, as we will see, and play an important role.

Unramified local Langlands parameters
The based root datum of the connected component G ∨ of the Langlands dual group The dual Langlands group of G u is independent of u and defined by where θ denotes the outer automorphism of G ∨ arising from F.
We will denote by L Z the central subgroup L Z := Z (G ∨ ) θ ⊂ L G, so that L Z is canonically equal to ( * ) θ ⊂ S ∨,θ . It follows that we can canonically identify the group /(1 − θ) with the Pontryagin dual group of L Z , which is the version of Kottwitz's Theorem as explained in detail in [17].
Let us recall the space of unramified local Langlands parameters for G F u for later reference. Let W k denote the Weil group of k [66], with inertia subgroup I ⊂ W k , and let Frob denote a generator of W k /I. An unramified local Langlands parameter is a homomorphism λ : such that λ(Frob × id) = s × θ (with s ∈ G ∨ ) semisimple and such that λ is algebraic on the SL 2 (C)-factor. Given an unramified Langlands parameter λ we denote by [λ] its orbit for the action of G ∨ by conjugation. We will write for the set of orbits [λ] of unramified Langlands parameters. If λ is an unramified Langlands parameter, let A λ := π 0 (C G ∨ (λ)) be the component group of the centralizer of λ in G ∨ . We call λ elliptic (or discrete) if C G ∨ (λ) is finite, and denote by e the space of G ∨ -orbits of unramified elliptic Langlands parameters.
Let λ be an unramified elliptic Langlands parameter. Observe that The inner forms G u of G are canonically parameterized via Kottwitz's Theorem by the character group of L Z sc , the center of the L-group of G ad = G/Z (G). Given an inner form G u , we choose once and for all a character ζ u ∈ Irr(Z sc ) (with Z sc := Z (G ∨ ) sc )) which restricts to the character ω u ∈ ad /(1 − θ) ad of L Z sc that 2154 E. Opdam is represented by u =ω ∈ N G ad (B). Following [20, Section 7.2] (see also [1]) we consider the group A λ /Z (G ∨ ) ⊂ (G ∨ ) ad , and let A λ ⊂ (G ∨ ) sc be its full preimage in the simply connected cover (G ∨ ) sc of (G ∨ ) ad . Thus Z sc ⊂ A λ , and A λ is a central extension of A λ /Z (G ∨ ) by Z sc . We denote by Irr u (A λ ) the set of irreducible characters ρ of A λ on which Z sc acts by a multiple of ζ u .
The space of (G ∨ -orbits of) unramified discrete Langlands data for G u is defined by˜ and denote its elements by [λ, ρ]. For fixed λ with [λ] ∈ e we denote by˜ u λ the fiber of˜ u above [λ] (with respect to the projection of˜ u to the first factor). We will often simply write˜ if we refer to the space of (orbits of) unramified local Langlands data of the quasisplit group G = G 1 .
The isomorphism classes of pure inner forms G u are parametrized canonically by ω u ∈ Irr( L Z sc ). In the refined version of the local Langlands correspondence where we restrict ourselves to pure inner forms of G, it is therefore more natural to work with pairs (λ, ρ) with ρ ∈ Irr u (A λ ), the set of irreducible characters of A λ which restrict to a multiple of ω u on L Z (hence there is no need to make choices of the extensions ζ u in this case).
It is well known [3, Paragraph 6.7] that we have a canonical isomorphism Observe that the group ( θ ) * of unramified characters on G F u is exactly the "central subgroup" of the complex torus T v (C) := Hom(X θ , C × ), i.e., the subgroup of W θ 0invariant elements. Here we consider T as the diagonalizable group scheme with character lattice Z × X θ over the ring L = C[v ±1 ] and we use the notation T v to denote its fiber over v ∈ C × .
We have natural compatible actions of X * un (G F u ) = ( θ ) * on the sets and˜ u defined by ω[λ] = [ωλ] and ω[λ, ρ] = [ωλ, ρ] respectively, provided that we choose the extensions ζ u in a compatible way within each orbit under X * un (G F u ) (for pure inner forms we do not need to worry about this).
We remark that W 0 \T v (C) can be identified with the maximal spectrum  [39,40,43], and [56] on the Hecke algebra side). This bijection [λ] → W 0 r λ is defined by This map is equivariant with respect to the natural action of the group X * un (G F u ) of weakly unramified characters of G F u .

Unipotent affine Hecke algebras
According to [40, 1.15, 1.16, 1.17, 1.20] we can decompose for each cuspidal unipotent pair (P, δ) of G u the algebra H u,s of s-spherical functions on G u explicitly as a direct sum of mutually isomorphic extended affine Hecke algebras as follows.
Let us use the shorthand notation N B for N G (B) etc. Recall that, since Borel subgroups of a connected reductive group are mutually conjugate and self normalizing, the group P = N P/P is naturally a subgroup of the finite abelian group = N B/B [see (10)]. It is known that the group P,θ [see (10)] acts trivially on the set of irreducible unipotent cuspidal representations of P F u . Even more is true [40]: for every cuspidal unipotent representation (E, δ) of P F u there exists an extension (E,δ) of (E, δ) to the normalizer N P F u := N G Fu (P F u ) of P F u in G F u . We denote the group P,θ by s,θ to stress the invariance of the cuspidal pair s = (P F u , δ). One observes that the set of such extensions is a torsor for the group ( s,θ ) * of irreducible characters of s,θ = N P F u /P F u by tensoring. Hence the group X * un (G F u ) = ( θ ) * of weakly unramified characters of G F u acts transitively on the set of extensions of (E, δ) to N P F u , and the kernel of this action is equal to the subgroup ( θ / s,θ ) * of ( θ ) * of weakly unramified characters of G F u which restrict to 1 on N P F u .
Lusztig showed that the s-spherical Hecke algebra H u,s is of the form where H u,s,a is an unextended affine Hecke algebra associated with a certain affine Coxeter group (W s , S s ) and a parameter function m s S , all defined in terms of the pair s = (P F u , δ). In particular, they are independent of the chosen extensions of (P, δ) to N P F u ; for this reason we will often suppress the extension in the notation and write H u,s,a instead of H u,s,a .
In order to define a normalized affine Hecke algebra (in our sense) from these data one needs to choose a distinguished set S s,0 ⊂ S s . Although this is not canonically defined, different choices are related via admissible isomorphisms. Let s,θ 1 ⊂ s,θ be the subgroup which acts trivially on S s [see (11)]. Then the quotient s,θ 2 = s,θ / s,θ 1 acts faithfully on (W s , S s ) by special affine diagram automorphisms. Lusztig [39, 1.20] showed that H u,s is isomorphic to the tensor product of the group algebra C[ where χ denotes the characteristic function of P F u .
Fix an extensions of s as in the previous paragraph. By (10) the unit element e s can be decomposed as a sum of mutually orthogonal idempotents e s = λ∈( s,θ ) * esλ (21) where we view λ ∈ ( s,θ ) * as a linear character of N P F u and where By (19) (and the text just above it) we see that the unit element of H u,s,e is equal to In particular, the group ( s,θ ) * acts transitively on the set of idempotents es ,e obtained by choosing different extensionss of s, and the kernel of this action is the subgroup The other canonical basis elements of H u,s,e are supported on other double cosets of N P F u . In particular, the trace τ vanishes on those other basis elements. Hence τ is a multiple of the standard trace of the affine Hecke algebra H u,s,e , and the normalization factor is of the form d τ,s,e := τ (es ,e ) = | s,θ The rational number d τ,s,e is the evaluation of a Laurent polynomial in the square root v of the cardinality q of the residue field k. When we treat v and q as an indeterminate we will denote these as v and q respectively. By our normalization of the Haar measure the factor Vol(P F u ) in the denominator is equal to, up to a power of v, the cardinality of the group of k-points of the reductive group P with Frobenius action F u . Therefore all factors in d τ,s,e are explicitly known rational function in v (cf. [9, Section 2.9, Section 13.7]). The following property of d τ,s,e is very convenient: Proposition 2.5 Let T = T Z T S denote a maximal F u -stable, maximally k-split torus of P, with T Z the maximal central subtorus. Let V Z (resp. V S ) denote the rational vector space spanned by the algebraic character lattice L Z (resp. L S ) of T Z (resp. T S ), and let F Z (resp. F S ) be the automorphism of L Z (resp. L S ) induced by F u . Then we have where l is the semisimple rank of P over K, d i are the primitive degrees of the Weyl group invariants of the semisimple part of P, the i are the eigenvalues of F S acting on the co-invariant ring with respect to the Weyl group action on V S (certain roots of unity, see [9, Section 2.9]), and where a ∈ Z is such that  [54,Theorem 4.8] and Theorem 2.4 that the formal degree fdeg(π ) of a discrete series representation π of a unipotent Hecke algebra has a unique representation fdeg(π ) = λfdeg(π ) q ∈ K × where λ ∈ Q + , and fdeg(π ) q is a qrational number (by which we mean a fraction of products of q-integers [n] q : with n ∈ N). We call fdeg(π ) q the q-rational factor of fdeg(π ). 2 ) * ⊂ ( s,θ ) * is the kernel of the induced action on the set of these direct summands. (10). In particular this group acts naturally on the set of F u -stable standard cuspidal parahoric subgroups of G u . This action extends naturally to an action on the set of equivalence classes of standard cuspidal unipotent pairs s = (P, δ) by ω · (P, δ) = ( ω P, ω δ); as was remarked before, the isotropy group of s = (P, δ) is the same as that of its first component P. If ω · s 1 = s 2 , then conjugation by ω ∈ θ C gives rise to an isomorphism φ ω : H u,s 1 ∼ −→ H u,s 2 which maps the various normalized extended affine Hecke algebra summands of the form (H u,s 1 ,e , d τ,s 1 ,e ) in H u,s 1 to corresponding normalized extended affine Hecke algebra summands of H u,s 2 by essentially strict isomorphisms.
Given an orbit O of standard cuspidal unipotent pairs s of G u for action of the group θ C , one can form the crossed product algebra Then H u,O is Morita equivalent to the direct sum H u,s s,θ . If (V, π) is an object of is a representation of H u,O (see also paragraph 2.2.4). Here e s denotes the unit element of H u,s . The Pontryagin dual X * un (G F u ) = ( θ C ) * of θ C acts in a natural way on the algebra H u,O by automorphisms as follows. If χ ∈ ( θ C ) * , then the corresponding automorphism α χ acts as the identity on the subalgebra ⊕H u,s , while α χ (ω) = χ(ω)ω. If χ ∈ ( θ C / s,θ ) * (i.e., χ | s,θ = 1), then α χ is the inner automorphism obtained by conjugation with ω∈ θ / s,θ χ(ω)e ωs . In particular the subgroup ( θ In this section we will classify the STMs between unipotent affine Hecke algebras (which will be referred to as "unipotent STMs").

Restriction of STMs
Let (H, τ ) denote a normalized affine Hecke algebra, and let L denote a generic residual coset L ⊂ T for H. Then there exists a unique "parabolic subsystem" R P ⊂ R 0 such that L can be written in the form L = r T P with r ∈ L ∩ T P . After moving L with a suitable Weyl group element w ∈ W 0 , we may assume that R P is standard and associated with a subset P ⊂ F 0 . To this subset we may associate a subalgebra H P ("a standard Levi subalgebra") and its semisimple quotient algebra H P whose associated algebraic torus is the subtorus T P ⊂ T (cf. [52]). In this situation {r } ⊂ T P is a residual point for H P .
Definition 3. 1 We will normalize the affine Hecke algebra H P by the trace τ P defined by τ P (1) = τ (1). We normalize H P by the trace τ P defined by the property Suppose that φ : (H , τ ) ; (H, τ ) is a strict STM which is represented by φ T : T → L n with L = r T P a residual coset. By modifying the representing map φ T appropriately, we may assume that r K n L = φ T (e) and such that Dφ T (t ) = t P for some subset P ⊂ F m . It follows easily from Corollary [54,Corollary 5.7] and Corollary [54,Corollary 5.8] that for any inclusion P ⊂ Q ⊂ F m , after possibly modifying the representing morphism φ T by a Weyl group element again, the inverse image φ −1 T (K n L (L ∩ T Q )/K n L ) ⊂ T is a subgroup whose identity component is a subtorus of T with as Lie algebra a subspace of t := Lie(T ) of the form t Q for some standard parabolic subsystem Q ⊂ F m . Indeed, in Corollary [54, Corollary 5.7] we saw that Dφ T induces a bijective correspondence between parabolic subsystems R Q of R m and parabolic subsystems R Q of R m containing R P . By modifying φ T with an appropriate Weyl group element w ∈ W (R m ), we may assume that R Q = (Dφ T ) −1 (R Q ) is standard, associated to a subset Q ⊂ F m . By Definition 3.1 and the definition of an STM it is easy to see that in this context, φ T also defines an STM φ Q : H Q ; H Q , and that the restriction φ T,Q of φ T to T Q ⊂ T defines an STM φ Q : H Q ; H Q (Recall thatK n L = N W P (L), and so L n is also the image of φ Q . If T P Q denotes the identity component of T Q ⊂ T P , then L Q := r T P Q ⊂ T Q is a residual coset of T Q . Thus L Q,n = L Q /K n L ∩ T P Q so that L Q,n ⊂ L n . Hence L Q,n is the image of the restriction of φ Q ). Definition 3. 2 We call φ Q the restriction of φ to H Q , and we say that φ is induced from φ Q . In particular, φ is induced by the rank 0 STM φ P : L ; H P .

Induction of unipotent STMs
By the above, every STM is induced from a rank 0 transfer map. The converse is clearly not true: not every rank 0 STM of the form ψ : H = L → H P is the restriction of an STM : H ; H. Indeed, if Im(ψ) = r (a generic residual point of the subtorus T P ⊂ T ), then we should have Im( ) = L = r T P . But the spectral measure ν Pl on a component S (P,δ) , where δ is a discrete series representation of H P with central character W P r , is given (up to a rational constant depending on δ) by the restriction of the regularisation μ (L) | L temp of the μ-function to S (P,δ) = W 0 \W 0 L temp (cf. Theorem [54,Theorem 4.13]). This regularisation does in general not behave like a μ-function of an affine Hecke algebra, unless for every restricted root of R 0 \R P to L the appropriate cancellations occur. However, as we will see in 3.1.3., if H = H I M (G) for a quasisplit almost simple algebraic groups G, and H is the normalized Hecke algebra for a maximal cuspidal unipotent pair (P, σ ) of an inner form of the standard Levi subgroup of G associated to P ⊂ F 0 , then ψ will be the restriction of an STM : H ; H.

Induction and cuspidality of unipotent STMs
This brings us to an informal discussion of the heuristic ideas and surprising facts behind the notion of STMs between unipotent affine Hecke algebras (with their canonical normalizations as in 2.4.1). We refer to such STMs as "unipotent STMs". Let G be an absolutely quasisimple unramified group over k, and let G u denote a k-group in the same inner class. We fix a maximal K -split torus S ⊂ G defined over k. Let the automorphism induced by the action of the Frobenius F of G on the character lattice of S be denoted by θ . We will assume that u =ω ∈ N B is a representative of an elementω ∈ /(1 − θ) (as in paragraph 2.2.4). We choose a minimal F-stable parabolic subgroup A 0 ⊂ G.
These data give rise to the "local index" of G F , a (possibly twisted) affine Dynkin diagram which contains a hyperspecial node, whose underlying finite root system is the restricted root system of G F with respect to the k-split center of A F 0 (again, we apologize for denoting the restricted roots of G(k) as "coroots"). We can now "untwist" the affine diagram by doubling some of the restricted roots of G(k); the resulting root system is denoted by R ∨ 0 . We have thus associated a based root datum R := (1) , and u acts on this affine diagram via the action of ω as a special affine diagram automorphism. Notice that u acts naturally on the root system R 0 by an element w u ∈ W 0 (R 0 ). The local index comes equipped with integers m S (a i ) attached to the nodes a i , which we transfer unaltered to the nodes of the untwisted diagram. This is the arithmetic diagram a (R, m) of [54, Subsection 2.3] associated to H I M . The associated spectral diagram s (R, m) is an untwisted affine Dynkin diagram for the affine root system R m = R (1) m . Let T be the complex algebraic torus with character lattice X .
The first remarkable fact is that for a cuspidal unipotent representation σ u of G u , its formal degree equals (up to a rational constant) the formal degree of an Iwahorispherical unipotent discrete series representation δ of G, and the central character W 0 r of the corresponding discrete series representation δ σ of the Iwahori-Hecke algebra H I M := H I M (G, B) of G is uniquely determined by this formal degree, up to the action of X * un (G). Here r is a generic residual point. (This is the rank 0 case of Theorem 3.4 that we already mentioned above). Let us agree to call a residual point r of H I M cuspidal if the q-rational factor of the residue μ I M,({r }) (r ) equals the qrational factor of the formal degree of a cuspidal unipotent representation σ u for an inner form G u of G as above.
We claim that this is also true if we replace G u by a proper Levi subgroup M u = C G u (S u ) 0 of G u (with S u ⊂ G u the k-split part of the connected center of M u ) which carries a cuspidal unipotent representation σ u M , in the following sense. We may assume that S u ⊂ S, the subtorus of S defined by the vanishing of the K -roots of M u . Then S u ⊂ S also gives rise to a k- The cuspidal pair (M u , σ u M ) is associated to a unique "extended type" s := (N P F u ,δ) in the sense of [51] (also see paragraph 2.2.4), where P ⊂ G is an F ustable parahoric subgroup such that P F u ∩ M F u is a maximal parahoric of M F u , and such that the set of affine roots associated to the parahoric subgroup P has a basis given by a proper ω-invariant subset a (R, m). The Plancherel measure on the set of tempered representations which belong to the unipotent Bernstein component whose cuspidal support is the inertial equivalence class of the cuspidal pair (M u , σ u ) is given by the Plancherel measure of the normalized unipotent affine Hecke algebra H u,s,e (cf. e.g., [8,21,50,51,60]).
Let us now move M to its standard position, so that R ∨ M is replaced by a standard parabolic subsystem R ∨ Q of roots associated to a subset Q ⊂ F 0 . This corresponds to a standard Levi subgroup G Q = C G (S u ) of G which is conjugate to M. Suppose that σ Q is an Iwahori spherical representation of G Q which is tempered and L 2 modulo the center of G Q . The corresponding tempered representation π Q of H I M Q is then the form π Q = (π Q ) t for some Iwahori spherical discrete series representation π Q of H I M Q and some t ∈ T Q u . Assume now that the central character of π Q is a cuspidal residual point of T Q . This means by definition that there also exists a cuspidal unipotent representation σ u M of some Levi subgroup M u ss of some inner form G u as above, whose formal degree has the same q-rational factor as that of σ Q (or equivalently of π Q ). The second important heuristic ingredient we now apply is the general expectation that the q-rational factor of the formal degree of the members of a discrete series unipotent L-packet should be the equal for all members of the packet [60]. It then follows from the above uniqueness assertions that σ Q and a twist σ u Q of σ u M (i.e., σ u Q is obtained from σ u M by a pull-back via a k-isomorphism between G u Q and M u ) must belong to the same L-packet of G Q .
Recall from paragraph [54, 4.2.5] how the Plancherel measure ν Pl | S (P,σ ) on the component S (P,σ ) of the tempered spectrum of H I M associated to a discrete series representation σ of H I M Q with central character W Q r Q is expressed in terms of the residue μ I M,(L) where L = r Q T Q (see paragraph [54, 4.2.5]). This implies that if discrete series representations σ 1 , σ 2 of H P are associated to the same central character W P r ∈ W P \T P , then the components S (P,σ i ) (i = 1, 2) are related to each other by a Plancherel measure preserving (up to a rational constant) correspondence as in Theorem [54, Theorem 6.1].
The third heuristic idea is that such a correspondence should exist for Plancherel measures on the tempered components determined by any two discrete series induction data (G Q , σ Q ) and (G u Q , σ u Q ) whenever σ Q and σ u Q belong to the same L-packet. But for the latter cuspidal unipotent pair, this Plancherel measure is computed as the most continuous part of the tempered spectrum of the normalized unipotent affine Hecke algebra H u,s,e . On the other hand, for the first pair it was already discussed above that the Plancherel measure can be computed essentially as the residue measure of the μ-function of H I M with respect to the tempered residual coset L temp = r Q T Q u . Thus these ideas suggest the existence of a unique STM H u,s,e ; H I M represented by a morphism φ T with image L = r Q T Q (or more precisely, the finite quotient L n = L/K n L of L, where K n L ⊂ K Q := T Q ∩ T Q is the subgroup of elements whose action on L can be represented by an element of W (R Q ) [see [54,Paragraph 5.1.1])] associated to the cuspidal pair (M u , δ u ) (or equivalently, to the extended types).
Recall from Proposition [54, Proposition 5.6] that any STM H ; H represented by an affine morphism φ T : T → T with image L = r T Q , the subtorus T Q ⊂ T is W 0 -conjugacte to a subtorus T J ⊂ T which is defined as above by an excellent subset J of the spectral diagram of H. Therefore it is clear that if there exists a unipotent STM φ : H u,s,e ; H I M as expected by the above discussion, then its image must be uniquely determined by the type of M ssa in combination with the qrational factor of the formal degree of σ u M , up to the action of X * un (G). By Proposition [54, Proposition 7.13] we see that φ itself is therefore determined up to the action of which is uniquely determined modulo the action of K Q /K n L (with K n L ⊂ K Q as above, hence the image of a representing morphism φ T for is L/K n L ), this is obvious. But by our discussion above we expect that: Conversely, any cuspidal unipotent STM φ : (L, τ 0 ) =: , is completely determined by the type of G Q and the q-rational factor of the degree τ 0 (1) of H 0 .
In the above arguments two important aspects of cuspidal residual points played a role. The first is that they can be defined by the property that the associated residue degree μ I M,({r }) (r ) has the same q-rational factor as that of the formal degree of a cuspidal unipotent representation of some inner form of G. The second is that a cuspidal residual point r Q of a semisimple quotient Hecke algebra H Q of H I M (where cuspidal means now that there exists a Levi subgroup M u of an inner form G u of G which carries a cuspidal unipotent representation σ u and which is isomorphic to an inner form of G Q , such that the q-rational factor of the formal degree of σ u is equal to that of the residue of μ I M Q at r Q ) is always the restriction of an STM H u,s,e ; H I M , for any inclusion of (R Q , m Q ) as a standard parabolic subsystem of the based root datum A priori the second property seems much more restrictive (except for the "final" exceptional groups E 8 , F 4 and G 2 ), but miraculously these properties lead to the same notion of cuspidality. The essential uniqueness part of Theorem 3.4 reduces to the rank 0 (or cuspidal) case in this way. The cuspidal case is done by direct inspection for the exceptional groups (most of the required results are in [21,59,60]). For the classical groups, the cuspidal case is treated in [19].
Of course the arguments above are only heuristic, but they tell us precisely where we should expect STMs, how these should be defined by induction from cuspidal ones, and what is necessary to check in order to prove that these maps really are STMs (thus providing a Proof of Theorem 3.4). In the remainder of this paper we will prove that indeed, any unipotent cuspidal pair (P u , σ ) of an inner form G u gives rise in this way to an essentially unique STM H u,s,e ; H I M , thereby proving Theorem 3.4 in full generality.
For exceptional groups the required verifications that induction of cuspidal STMs from Levi subgroups always give rise to STMs is based on the notion of a "transfer map diagram". This notion is defined and discussed in paragraph 3.1.4. One can also study more generally the STMs between two unipotent affine Hecke algebras, not just the ones with H I M as a target. This is interesting in itself, since in several cases the "unipotent spectral transfer category" is generated by very simple building blocks of this kind. Indeed, this is how we show the existence of STMs induced from cuspidal ones in the classical cases.

The transfer map diagram of a unipotent STM Such an expected unipotent STM
: H u,s,e ; H I M is determined (up to the action of Aut C (H u,s,e ) op ) by the image r Q = s Q c Q of φ T,Q , a cuspidal generic residual point for the Iwahori-Matsumoto Hecke algebra H I M Q of the quasisplit Levi subgroup G Q of G. We can choose the unitary part s Q = s(e Q ) ∈ T Q,u such that it corresponds to a vertex e Q ∈ C Q,∨ , the fundamental alcove for dual affine Weyl group i of the corresponding dual affine root a ∨ i (this value is a power of q). We may put c Q in a dominant position with respect to the roots Da ∨ i where i runs over the nodes of J Q . This is essentially the weighted Dynkin diagram of a linear generic residual point (in the sense of [56], but obviously restricted in our context of the fixed line in the parameter space defined by m ∨ R ) for the finite type root system defined by J Q with the parameters m ∨ Q | J Q (in a multiplicative notation). As was remarked above, if the rank of is positive, a finite type Dynkin diagram of type J Q fits uniquely as an excellent subdiagram J of the spectral diagram s (R, m) associated to G (this can be checked case-by-case), up to the action of X * un (G). Now we also assign weights to the nodes of K = I \J as follows. By modifying φ T (within its equivalence class) by an element of the Weyl group W = W (R 0 ) we may assume that via Dφ T the affine simple reflections of (R m ) (1) (relative to the base (F ) m of (R ) m ) correspond bijectively to the elements of K.
This allows us to use k ∈ K also to parameterize the elements of the base of (R m ) (1) . Let k 0 ∈ K be the vertex of the unique (dual) affine simple root which is not in F m . From (T2), Proposition [54, Proposition 5.6](4) and Corollary [54, Corollary 5.8] (applied to the case Q = ∅) we see that this is the unique element k 0 ∈ K for which the corresponding vertex D a φ T (0) = ω k 0 ∈ C ∨ has the shortest length. We interpret the gradient Da ∨ of a (dual) affine root a ∨ ∈ R (1) m as a character on T (and similarly for dual affine roots of (R m ) (1) on T ). By construction, the character lattice of L n is mapped injectively to a sublattice of X Q m and injectively to a sublattice of X m . From (T3), [54, equation (8)] and considering the numerator of the μ-function (see Definition ([54, Definition 3.2]) it is easy to see that Dφ * (1) . This sets up a bijection between the set of simple affine roots of (F m ) (1) and the set K = I \J , and using this we will parameterize the elements of (F m ) (1) also by the set K. By Proposition [54, Proposition 5.6](3) this bijection defines an isomorphism of affine reflection groups. By Proposition [54, Proposition 5.6](4) it is then clear that b ∨ k 0 has to be the extending affine root of the spectral diagram of H(R m , m ).
And we can say more precisely, by considering the formula of the μ function of the Hecke algebra and (T3), that f −1 k ∈ N, and that we can thus interpret the fraction itself is a coset of T Q with origin r Q = s Q c Q , and using the above remarks, it follows that for all k ∈ K, Da ∨ k lifts to a constant multiple of a character of a suitable covering of T (namely the fibered product of L and T over L n ). We call this lift of Da ∨ k , expressed as a radical of b ∨ k , the weight w k of k ∈ K. In view of (T3) and [54, equation (8)], and using Proposition [54, Proposition 5.6] we see that: k 0 -th primitive root of 1, and c k ∈ Z (which can be computed by evaluating Da ∨ k on c Q ). All this gives rise to the following notion: Observe that i∈I w n i i = 1, where 1 = i∈I n i a ∨ i is the decomposition of the constant function 1 as a linear combination of (dual) affine simple roots of R m in terms of the base of simple roots F (1) m . In particular, there exists a constant C such that for all k ∈ K, n k f k = Cñ k , where k∈Kñ k b ∨ k = 1 is the decompostion of the constant function 1 in terms of the (dual) affine simple roots (F m ) (1) = (F ) m of (R ) m . Clearly the value of Da ∨ k 0 on s Q = ω k 0 is a primitive n k 0 -th root of 1. Therefore we see that C = 1, and f −1 k = n k /ñ k (this integer is called z k by Lusztig [ We also note that, from the tables in [40,Section 7] and [44,Section 11], for all k ∈ K: n k /n k 0 ∈ Z. By the above it is clear that is completely determined by its transfer map diagram. The finite abelian group K n L ⊂ T W L L can be recovered from the transfer map diagram as the product over all k ∈ K\{k 0 } of cyclic groups C k of order z k = f −1 k (for those k ∈ K\{k 0 } for which n m (b k ) = 1) or of order z k /2 (for k ∈ K\{k 0 } such that n m (b k ) = 2 and z k is even). For classical groups K n L is always trivial.

Main theorem
We finally have everything in place to formulate the two main theorem of this paper. Let G be a connected absolutely quasisimple K -split, k-quasisplit linear algebraic group.
For simplicity we will assume that G is of adjoint type. Recall that H I M denotes the Iwahori-Matsumoto Hecke algebra of G = G(K ), i.e., the generic affine Hecke algebra H I M = H I M (G) such that H I M v is the Iwahori-Matsumoto Hecke algebra of G(k) = G F with respect to the standard cuspidal unipotent pair s 0 := (B, 1) where B denotes the Iwahori subgroup. Since s 0 is fixed for the action of where V = RY , θ denotes the action on Y of the outer automorphism of G ∨ corresponding to F, and  (4)]. In the proof of the theorem below, we will denote the unipotent normalized affine Hecke algebra of irreducible type H(R m , m), with X m the weight lattice of the irreducible reduced root system R m , as follows. If R m is simply laced and the parameters m + (α k ) are equal to b, we denote this unipotent Hecke algebra by R m [q b ]. If R m is not simply laced and not of type C d , then we will denote this algebra by R m (m where α 1 ∈ F m is long and α 2 is short, and q b is the base for the Hecke parameters (equivalently, we could write R m (bm + (α 1 ), bm + (α 2 ))[q]). If both parameters are equal to b, we may also simply write R m [q b ], this will not create confusion. For Theorem 3.4 Let G be a connected, absolutely simple, quasisplit linear algebraic group of adjoint type, defined and unramified over a non-archimedean local field k. Let C uni (G) be the full subcategory of the spectral transfer category C es (with essentially strict STMs as morphisms) whose set of objects is the set of normalized unipotent affine Hecke algebras H u,s,e associated with the various inner forms G u of G (where u ∈ Z 1 (F, G) runs over a complete set of representatives of the classes [u] ∈ H 1 (F, G)). Let H uni denote the direct sum of all the objects of C uni (G). Recall that there is a natural action of X * un (G) on H uni such that direct summands are mapped to direct summands, preserving the rank. In particular X * un (G) acts on the unique summand H I M (G) of H uni of largest rank.
There exists a X * un (G)-equivariant STM which is essentially unique in the sense that if is another such equivariant STM, then there exits a spectral transfer automorphism σ of (H uni (G), τ ) such that = • σ . The transfer map diagrams corresponding to the restrictions of to the various direct summands H u,s,e of H uni (G) are equal to the corresponding geometric diagrams of [40].

Corollary 3.5
Recall that the spectral isogeny class of an object of C uni (G) is equal to its isomorphism class [54,Proposition 8.3], and that these classes admit a canonical partial ordering as defined in [54,Definition 8.2].
Theorem 3.4 is a consequence of the combined results of the following subsections.

The case of G = PGL n+1
In this case, the only cuspidal unipotent representation comes from the anisotropic inner form G u = D × /k × (where D is an unramified division algebra over k of rank (n + 1) 2 ) and has a formal degree with q-rational factor given by fdeg : has only one orbit of residual points (up to the action of X * un (G)) and this has indeed the desired residue degree. Based on this it is easy to construct the general STM for the unipotent types for this G, and prove that these are unique. Suppose we have a factorization n + 1 = (d + 1)(m + 1). Consider an inner form G u of G such that u has order m + 1.
where D is an unramified division algebra over k of rank (m + 1) 2 . Then J is of type A d+1 m , which fits in a unique sense (up to diagram automorphisms as usual) as an essential subdiagram of the spectral diagram of H I M . The Hecke algebra H u,s,u is of type A d [q m+1 ]. For the unique strict STM we make sure that J does not contain a ∨ 0 . The weights for the vertices of J are equal to q, and for those of k ∈ I \J equal to q −m Db k . It is an easy check that this indeed The uniqueness of such strict STM up to Aut C (H u,s,e ) is clear as before: Any strict is obtained by induction of a cuspidal one for M ssa , which determines J and the underlying geometric diagram of φ as before. There assignment of weights to the vertices of K is dictated by the basic properties of an STM as explained above. Hence φ must be equal to the STM constructed above, up to a diagram automorphism of A d . By Theorem 2.8 the direct summands of H u,s form a torsor for ( s,e 1 ) * = * /( s,e 2 ) * ≈ C m+1 , and hence there is a unique way to write down a * -equivariant STM H u,s ; H I M , up to the action of Aut es (H u,s ). This completely finishes the proof for the case of PGL n+1 .

Existence and uniqueness of rank 0 STMs for exceptional groups
This is a case check (with some help of Maple, to simplify the product formulas for the q-factor of the formal degree as given in [56]), almost all of which has already been done in the existing literature. Let G be an k-quasisplit adjoint group over with k which is split over K , of type 3 D 4 , E 6 , 2 E 6 , E 7 , E 8 , F 4 , G 2 . One uses the classification of the residual points and the product formula for the q-rational factor of the formal degree from [56] to compute, for each orbit W 0 r of generic residual points (in the sense of the present paper), the q-rational factor of the residue degree μ I M,(r ) (r ) for the Iwahori-Matsumoto Hecke algebra H I M of G. (Many of these results are already in the literature; For E 8 this list was given in [21] using essentially the same method. For all split exceptional groups this list can be found in [60]. Note that the computations in [60] can be simplified a lot using the classifications and the product formula from [56], to just "clearing q-fractions", since our formal degree formula is already given in "product form" (as opposed to an alternating sum of rational functions as in [60]). Also note that we are for this list only interested in the Iwahori spherical case.) We note that these lists reveal that these residues of μ I M at distinct orbits W 0 r = W 0 r are distinct for all exceptional cases. Hence in the exceptional cases the uniqueness (up to diagram automorphisms) of rank 0 spectral transfer maps for irreducible unipotent Hecke algebras is guaranteed by this.
The existence of the desired cuspidal unipotent STMs is now an easy task; one considers the list of all cuspidal unipotent representations of all inner forms of G. This means that we need to make a list of all maximal F u -stable parahoric subgroups of the inner forms G u , consider their reductive quotients over k, and for those quotients which admit a cuspidal unipotent character, compute the normalization of the associated Hecke algebra H u,s,e := (L, τ s,e ) according to (25). Of course the main part of this formula is the degree of the unipotent cuspidal characters of the simple finite groups of Lie type, which is due to Lusztig [32][33][34] and conveniently tabulated in [9]. Finally we need to see if the q-rational parts of these expressions show up in our list of residues of the μ-function. This indeed leads to cuspidal transfer map diagrams with the same underlying sets J as listed by Lusztig in [40] and [43], and for each of those diagrams, there exists one generic linear residual point for J (in the form of the collection of weights assigned to the vertices of J ) producing the correct residue of μ and thus an STM.
Let us give the results for the two non-split quasisplit cases which were not yet treated in the existing literature. The unipotent Hecke algebra G 2 (3,1)[q] (for 3 D 4 ), and F 4 (2, 1)[q] (for 2 E 6 ). The first case G 2 (3,1)[q] has 4 residual points. The spectral diagram [54] of this Hecke algebra is the untwisted version of the Kac diagram [61,Subsection 4.4], with the equal parameters 3k attached to the nodes (a similar remark applies to all simple quasisplit unramified cases).
Let us use the maximal subdiagrams of this Kac diagram to name the various orbits of residual points. There are two orbits of residual points G 2 and G 2 (a 1 ) with positive central character. The corresponding groups A λ [where λ denotes the corresponding discrete unramified Langlands parameters via equation (17)] are 1 and S 3 respectively. There are two nonreal orbits A 2 (with A λ = 1) and A 1 × A 1 (with A λ = C 2 ). Looking at the q-rational factor of the residue of the μ function at these points, we find that the cuspidal (orbits of) residual points are G 2 (a 1 ) (matching the degree of 3 D 4 [1]) and A 1 × A 1 (matching the degree of 3 D 4 [−1]). Together with the Iwahori spherical unipotent discrete series these cases make up for the set of 7 unipotent discrete series (5 of which are Iwahori spherical, while the others are cuspidal).
A similar discussion for F 4 (2, 1)[q] shows the following. Again we use the Kac diagram [61,Subsection 4.5], this time with the constant parameters 2k attached to each node to indicate the various orbits of residual points. We have 9 orbits of residual points (in the notation of [56]). There are 4 orbits with positive real central character, Table 3] (or equivalently, the D i (x, x) are the weighted Dynkin diagrams F 4 , F 4 (σ 1 ), F 4 (σ 2 ), F 4 (σ 3 ) (in this order) of the distinguished nilpotent orbits of F 4 as denoted by [9])). The corresponding groups A λ are of the form 1, S 2 , S 2 , S 3 respectively. In addition there is 1 orbit corresponding to A 1 × B 3 with A λ = C 2 , 1 orbit for A 2 × A 2 with A λ = C 3 , 1 orbit for A 3 × A 1 with A λ = C 2 , and finally 2 orbits corresponding to C 4 with A λ = 1 (the regular orbit) and A λ = C 2 (the subregular orbit) respectively. The cuspidal orbits of residual points are in this case the ones corresponding to F 4 (σ 3 ) (matching the degree of 2 E 6 [1]) and the one of type A 2 × A 2 (matching the degree of 2 [40] and [43].

Existence of STMs for the exceptional cases
Let us now consider the existence of the positive rank STMs in the exceptional cases. Let S u be a k-spit torus. As always, we assume that S u ⊂ S, with S ⊂ G a fixed maximal k-split torus. Consider In particular, at most one of the irreducible components of Q will not be of type A, and possible irreducible factors of type A have to be in the anisotropic kernel of G u . By the results for type A and for rank 0 STMs for irreducible exceptional types, there exists a unique rank 0 STM ψ : L ; H I M (M ssa ) for the cuspidal unipotent representation σ u . Our task will be to see that this ST M map can be induced to H I M . As By what was said in the previous paragraph it follows that these diagrams are exactly the exceptional geometric diagrams of Lusztig, with weights attached to the vertices of the boxed set of vertices J . We remark that for all exceptional cases, the geometric diagrams with J such that |K| > 1 (i.e., of positive rank), the components of J are all of type A (this simplicity is in remarkable contrast with the classical cases). Therefore the weights w j with j ∈ J are simply equal to q m ∨ R (a ∨ j ) . If there would indeed exist a corresponding transfer map, then its transfer map diagram should be obtained by assigning in addition weights to the vertices in K = F (1) m \J , as described in Definition 3.3. These weights turn out to be uniquely determined by the basic property Proposition [54, Proposition 5.2] of spectral transfer maps (applied to the case of residual points), and this also enables us to find these weights w i easily (using the known classification of residual points of [22] and [56]). Our task is then to prove that these eligible diagrams thus obtained are indeed transfer map diagrams.
In order to do so, we need to first find k 0 . This has to be the unique vertex k ∈ K of the geometric diagram such that the corresponding vertex ω k ∈ C ∨ has the shortest length. It is easy to check in all exceptional geometric diagrams that this condition defines a unique vertex k 0 ∈ K. The cuspidal unipotent representation σ u of M ssa lifts to a cuspidal unipotent representationσ u of M, and the cuspidal pair (M,σ u ) is obtained by compact induction from a cuspidal unipotent type s := (P J , δ). The affine Hecke algebra H u,s,e of the cuspidal unipotent type is given, and let R m be the corresponding based root datum with multiplicity function m .
Next we need to determine the bijection between the affine simple roots b ∨ i of the spectral diagram of H u,s,e and of K. This was done by Lusztig: According to the main result of [40], there exists a matching such that k 0 corresponds to b 0 , and such that the underlying affine Coxeter diagram of the spectral diagram of H u,s,e matches the Coxeter relations of the reflections in the quotient roots α k = Da ∨ k |L (cf. [40, 2.11(c)]). (Here we identify L with T J ⊂ T , the maximal subtorus on which the gradients of the roots from J are constant, by choosing r M ∈ L as its origin (given by the weights of j ∈ J and s 0 corresponding to ω k 0 )). Since this matching is only based on the underlying affine Weyl groups, and by Proposition [54, Proposition 5.6], it is clear that a possible spectral map diagram has to provide the same matching. It is easy to check case-by-case that such a matching is unique up to diagram automorphisms preserving the parameters m ∨ R of the spectral diagram of H u,s,e . Thus we fix such a matching, and use this to also parameterize the (dual) affine simple roots of the spectral diagram of H u,s,e by k ∈ K .
Following notations as in 3.1.4, we need to assign an integer c k to each node k ∈ K, in order to define the weights w k for all k ∈ K. We define c k by the formula where a ∨ k denotes the (dual) affine root of R m associated with k, and b ∨ k the corresponding (dual) affine root of the spectral diagram of H u,s,e .
The diagram thus obtained defines a map φ from T to a suitable quotient of L ⊂ T . For each maximal proper subdiagram D of a spectral diagram of a semi-standard affine Hecke algebra H(R, m) there exists a generic residual point for all k ∈ D, and for k ∈ K\D, such that b ∨ k (r D R |v = 1) is a primitive root of 1 of order n k . The above assignment means that we require the alleged spectral transfer map φ to satisfy the property that φ(r D R ) = r D∪J R . We can check easily case-by-case that this map then also sends all other residual points of H u,s,e to residual points of H I M , and that these weights are the only possible weights defining a map with such properties.
Remark 3.6 Thus, the image under φ Z of the central character of the one-dimensional discrete series representation of H u,s,e which is the deformation of the sign character of its underlying affine Weyl group is equal to the central character of H I M (G) of the analogous one dimensional character. If H u,s,e is the Iwahori-Hecke algebra of an inner twist G u of G, then this is true in general, since we know that the formal degree of the Steinberg character is unchanged by inner twists. For exceptional groups it is true for all unipotent STMs of positive rank, which seems related to the fact that for these STMs, the subset J ⊂ F (1) m consists of type A components only. In classical cases STMs do not have this property in general.
Finally we need to check that the map φ we have thus defined indeed defines an STM. This amounts to applying φ * to (μ I M ) (L) , making the substitutions φ * (α i ) = w i for all i ∈ I \{0}, and checking that this equals the μ-function μ (u,s,e) of (H u,s,e , d τ,s,e ) up to a rational constant. Now this is already clear for the constant factor of d τ,s,e because of our choice of the weights of the j ∈ J and the fact that we started out from a cuspidal STM for σ u for M u ssa . Hence we only need to consider, for all k ∈ K, the cancellations in φ * ((μ I M ) (L) ) for the factors in the numerator and denominator which are of the form and ζ a root of unity). This is a tedious but simple task: We need to compile the table of all positive roots α ∈ R m,+ , consider those α such that α = α| L is a nonzero multiple of α k (upon ignoring the coefficients of α at the j ∈ J , and using the relation (with β = Db ∨ k ) remains, as desired. In this way we verify that all the diagrams so obtained are spectral map diagrams of spectral transfer maps, in all cases.
As a (rather complicated) example, let us look atẼ 8 /A 3 A 3 A 1 . This diagram arises by induction from the cuspidal pair (E 7 , σ u ), whose spectral map diagram is given byẼ 7 /A 3 A 3 A 1 (see the geometric diagram of [40, 7.14]). The spectral diagram of H u,s,e is of type C 1 (7/2, 4)[q]. The vertex k 0 is labelled by 1 in [40, 7.8]. We write the simple roots of C 1 (7/2, 4)[q] in the form b ∨ 1 = 1 − 2β, and b ∨ 2 = 2β. The weights of the roots a 1 := α 6 and a ∨ 2 := α 3 are w 1 = √ −1v −6 (−β/2) and w 2 = v −7 β/2. When α runs over the positive roots of E 8 such that φ * (α) is a nonzero multiple of β/2, the following lists of factors in front of β/2 appear: For ζ := ± √ −1, the following powers of v: v ±6 , 2 times v ±4 , 3 times v ±2 , and 4 times 1, and for ζ := ±1, the following powers of v: v ±5 , 2 times v ±3 , 3 times v ±1 . In addition the restricted root β appears, with factor 1. One easily checks that this indeed produces the μ function of C 1 (7/2, 4)[q]. The group K n L is isomorphic to C 2 (caused by taking the square root of β). Note that this is equal to the central subgroup T W L L with T L ⊂ T the subtorus whose cocharacter lattice is the coroot lattice of E 7 ⊂ E 8 .
As an example of a somewhat different kind, let us look at the unramified nontrivial inner form 2Ẽ 7 of type E 7 (cf. [40, 7.18]). This case is induced from the trivial representation σ u of the anisotropic kernel M u ssa of the group of type 2Ẽ 7 , which is an anisotropic reductive group of rank 3. The spectral diagram of H u,∅,e is of type F 4 (1, 2) (24)] the qrational factor of the formal degree of σ u is [2] −3 q . This corresponds to the residue degree of the μ-function of a Hecke algebra of type A 1 [q] 3 at its unique residue point. Hence we need to take J of type A 1 A 1 A 1 . This subdiagram fits in a unique way as an excellent subset in the spectral diagram of typeẼ 7 , up to the diagram autmorphism ofẼ 7 . However, we need to choose the unique such embedding of J such that the root a ∨ 0 does not belong to J (i.e., J ⊂ F m,0 here; it is easy to check that the other possibility does not lead to a strict STM (although it does lead to an essentially strict but non-strict STM, obtained by composing the strict STM we are about to construct by the nontrivial diagram automorphism of the spectral diagram of type E 7 , cf [54, Remark 6.2])). Since we know that a transfer map diagram which is induced from this cuspidal pair must have the property that J appears as an excellent subset of the diagram of, it is clear that Lusztig's geometric diagram for [40, 7.18] indeed should be the underlying geometric diagram of a spectral transfer map (if it exists), and k 0 is the vertex numbered by 5 in [40, 7.18]. The f k are all equal to 1, and (in the numbering of [40, 7.18]) we have w i = q λ i Db ∨ i with λ i = −1 for i = 1, 2 and 0 for i = 3, 4, 5. It is easy to check that this gives a spectral transfer map . All other examples are done similarly by executing this algorithm. We remark that z k ≤ 3 in all cases, except possibly when Db ∨ k is a divisible root of R m , when z k = 4 may occur (as in the above example). We leave it to the reader to check the remaining exceptional cases by him/herself.

The exceptional non-split quasisplit cases
For convenience we explicitly list the unipotent STMs for the non-split quasisplit cases 3 D 4 and 2 E 6 . Both groups do not have nontrivial inner forms. The rank 0 STMs were all described in paragraph 3.2.3. For the case 3 D 4 , up to G F -conjugacy, the only F-stable cuspidal unipotent pairs (P, σ ) are those with P, an F-stable Iwahori subgroup, and σ = 1, or with P maximal hyperspecial. Thus, the only nontrivial unipotent STMs are the rank 0 ones which were already described in paragraph 3.2.3 For 2 E 6 , besides the rank 0 cases already described in paragraph 3.2.3, we have the rank 1 STM which arises from the cuspidal unipotent pair (P, σ ) where P is of type 2 A 5 (and σ its unique cuspidal unipotent representation). This gives rise to a unipotent affine Hecke algebra of type C 1 (4,5)[q]. The unique STM : C 1 (4,5)[q] ; F 4 (2, 1)[q] maps the two central characters of the two discrete series of C 1 (4,5)[q] in a unique way to two orbits of residual points of F 4 (2, 1)[q]. Namely, q 5 maps to A 1 × B 3 , while −q 4 maps to A 3 × A 1 . More precisely, can be represented by a morphism φ : T 1 → L n of torsors of the algebraic tori. Here we consider the algebraic tori T i associated to the two relevant affine Hecke algebras (with T i of rank i, and with coordinates given by the simple roots β 1 for T 1 and α 1 , . . . , α 4 for T 4 , with α 3 , α 4 the short simple roots). Further L ⊂ T 4 is a rank 1 residual coset given by the equations Remark 3.7 In these two cases 3 D 4 and 2 E 6 we see that a parameterization of the unipotent discrete series representations is completely determined by the matching condition that the q-rational factor of the formal degree needs to equal the residue μ I M,({r }) (r ), together with the requirement that we assign the generic representation to the trivial representation of A λ (where λ is the unramified Langlands parameter which corresponds to W 0 r according to (17)). (To be precise, in the case 2 E 6 this fixes the parameterization except for the interchangeability of 2 E 6 [θ ] and 2 E 6 [θ 2 ].)

3.2.6
Unipotent affine Hecke algebras of type C d (m − , m + ) For an absolutely simple, quasisplit classical group G of adjoint type other than PGL n+1 , the proof of the essential uniqueness of an STM φ : H u,s,e ; H I M (G) for an affine Hecke H u,s,e of any unipotent type s for any inner form G u follows the same pattern as in the exceptional case, by reducing the statement to the essential uniqueness for cuspidal STMs. The proof of the existence of an STM φ as above is treated quite differently however, for most cases by generating φ as a composition of a small number of basic STMs which generate the spectral transfer category whose objects consist of all unipotent affine Hecke algebras of the form H u,s,e for all groups in certain classical families (containing G). It turns out that in essence there are only 2 types of basic building blocks generating almost all STMs between the unipotent affine Hecke algebras associated to the unitary, orthogonal and symplectic groups. Apart from the STMs built from these basic generators, there is one additional, very important type of basic STMs of the form φ : H u,s,e ; H I M (G) for the orthogonal and symplectic cases which we call extraspecial.
As mentioned above, we will now first define some basic building blocks of STMs between classical affine Hecke algebras which are associated to the unitary, orthogonal and symplectic groups. We define a category C class whose objects are normalized affine Hecke algebras of type Hence the objects of C class are in bijection with the set V of triples (d; (m − , m + )) as described above.
The trace τ = τ m − ,m + is normalized as follows. First, we decompose V in six disjoint subsets V X with X ∈ {I, II, III, IV, V, VI}, which are defined as follows. If m ± ∈ Z ± 1 4 , write |m ± | = κ ± + (2 ± −1) 4 with ± ∈ {0, 1} and κ ± ∈ Z ≥0 . Define δ ± ∈ {0, 1} by κ ± ∈ δ ± + 2Z. Then we define: Observe that the type X of (d; (m − , m + )) only depends on (m − , m + ); we will often simply write (m − , m + ) ∈ V X instead of (d; (m − , m + )) ∈ V X . We now normalize the traces τ m − ,m + as follows. These traces are of the form m + is independent of the rank d (and d is suppressed in the notation). Explicitly we define τ m − ,m + by is the q-rational part of the formal degree of a cuspidal unipotent character for the adjoint group G of type 2 A l which is compactly induced from the unique cuspidal unipotent character of a maximal parahoric subgroup whose reductive quotient is of type 2 A l (q 2 ), with l = 1 2 (s 2 +s)−1 (with s ∈ Z ≥1 ) (see Proposition 2.5; it is convenient to extend this to s = 0 by setting d τ,{ 2 A} 0 = 1); similarly d τ,B s (q) denotes the q-rational part of the formal degree of a cuspidal unipotent representation of G of type B l induced in this sense from B l (q) with l = s 2 + s (with s ∈ Z ≥0 ) (this degree covers the cuspidal character of the odd orthogonal and the symplectic groups); d τ,D s (q) denotes the q-rational part of the formal degree of a cuspidal unipotent representation of G of type D l induced from D l (q), with l = s 2 (with s ∈ Z ≥0 ) (this degree covers the cuspidal character of the even split orthogonal groups (s even) and of the even quasisplit orthogonal groups (s odd)). (Using [9, Section 13.7] and (25) it is easy to give explicit formulas for these formal degrees.) where the set {a, b} with a, b ∈ Z ≥0 is determined by the following equalities of sets: This determines a and b in case II, V, VI, and it determines a and b up-to-order in the other cases, so that the normalization (33) is always well defined. Now we define the building blocks of the STMs between these affine Hecke algebras. First, the group D 8 of essentially strict spectral isomorphisms as described in Remark [54, Remark 7.7] acts on the collection of objects of C class . This corresponds to the action of D 8 on the set V by preserving d, and on a pair (m − , m + ), the action is generated by the interchanging m − and m + and by sign changes of the m ± . Observe that these operations preserve the type X .
Then there exist additional basic STMs in C class of the types indicated below. (In these formulas we have used the notation (x) = x/|x| ∈ {±1} to denote the signature of a nonzero rational number x.) In the first 5 cases one of the parameters m − or m + is translated by a step of size 1 (if the translated parameter is half integral) or 2 (if the translated parameter is integral) in a direction such that its absolute value decreases. In these first 5 cases both parameters can be translated in this way, as long as the absolute value of this parameter is larger than 1 2 (in the half integral case) or 1 (in the integral case). A formula corresponds to an STM provided that this condition on the absolute value of the parameter which will be translated is satisfied.
We denote the first 5 cases of these STMs by , where the sign ± in the subscript indicates which of the parameters m − or m + will be translated. Notice that if we combine the basic STMs of the first 5 cases with the group D 8 of spectral isomorphisms of C class , then we are allowed for all objects X ∈ {I, II, III, IV} either one of m − and m + (by a step of size 1 or 2 depending on the residue modulo Z of the parameter to be translated) as long as the absolute value of this parameter can still be reduced by such steps. Observe that these steps preserve the type X .
Finally we are of course allowed to compose these basic STMs thus obtained with each other and with the group D 8 of spectral isomorphisms. The basic translation steps as above commute with each other and have the obvious commutation relations with the group D 8 of spectral isomorphisms (this also follows easily from the essential uniqueness of STMs discussed below, see Proposition 4.1). Observe that while the parameters are strictly decreasing with these basic translation steps, the rank is strictly increasing.
Among the objects of C X class of the types X ∈ {I, II, III, IV}, the minimal spectral isogeny classes of objects (in the sense of [54, Definition 8.1]) are of the form: Note that for all objects in C X class , the spectral isogeny class of an object is just its isomorphism class [54,Proposition 8.3]. By abuse of language we will sometimes call the objects in a minimal (least) spectral isogeny class in this sense also "minimal" (respectively "least"). Note that some of these minimal objects admit a group of order 2 of spectral automorphisms (the cases X = II or IV).
The cases X ∈ {V, VI} are of a different nature. There are no STMs between the different objects of these cases, as we will see below. But from each object of C V class , there is an essentially unique (i.e., unique up to spectral automorphisms) STM to the least object in C III class and from each object of C V class there is an essentially unique STM to one of the two types of minimal objects in C IV class . We call these STMs extraspecial. It is easy to give a representing morphism φ = φ defining the basic STMs of this kind. The first 4 cases, the building blocks of elementary translations in the parameters m − and m + , do in general not correspond to geometric diagrams as given in [40] and [43], since the image of the spectral transfer map is in general not a least object. However, as we will see below, these building blocks are quite simple and their existence can be established easily by a direct computation. The extraspecial cases correspond to the geometric diagrams [40, 7.51, 7.52] and to [43, 11.5] (in a way that will be made precise below).
The formula defining this morphim φ for the first 5 cases (thus a minimal translation step in one of the parameters of an object of type X ∈ {I, II, III, IV}) only depends on the value modulo Z of the parameter to be translated. Using the group D 8 of spectral isomorphisms it is enough to write down the formula for a basic translation in m + where m+ > 0.
Let by (t 1 , . . . , t d ). We consider X d as the root lattice of the root system of type B d . The Weyl group W 0 acts by signed permutations on X d . For m ± ∈ Z+1/2 we define a homomorphism φ  (t 1 , t 2 , . . . , t d , 1, q Finally, for the extraspecial cases X ∈ {V, VI}, we define, for m ± > 0, a morphism φ as follows. Observe that L = 2d + L − + L + where L ± := κ ± (2κ ± + 2 ± − 1)/2. We first define, for m ∈ Z ± 1 4 , residual points r e (m) recursively by putting, for m > 1, The proof of the fact that these formulas indeed define an STM is a straightforward computation in the cases X ∈ {I, II, III, IV}.
In the extraspecial case one notices first that this map for general d ≥ 0 is induced from the cuspidal map of this kind with d = 0. It is easy to verify that this map is an STM, by considering the set of positive roots of the root system R 0 of type B 2d+L − +L + which restrict to a given simple root α i of B d (this process is similar to what we did in the exceptional cases), provided that the inducing rank 0 map is indeed a cuspidal STM. The latter can be proved by induction on m ± , using the recursive definition of r e (m) and the formula (easily obtained from (33) applied to the two cases {V, VI}):

Existence of enough STMs for the classical cases
After having established the existence of these STMs between affine Hecke algebras of type C corresponding to an embedding of the right-hand side as an index two subalgebra of the left-hand side. Here the right-hand side is normalized as object in C I class . The representing morphism φ T has kernel ω ∈ T I M , the unique nontrivial W 0 (B n )-invariant element. We can identify ω with the nontrivial element of X * un (G) = ( θ C ) * , which equals C 2 in this case. It acts as a diagram automorphism on the geometric diagram via the simple affine reflection σ = s 1−2x 1 of the afine Weyl group of type C  [53]. This implies that we have a unique factorization of φ T as desired in all cases. Remark that s,θ 1 = 1 except if s is a supercuspidal unipotent type of G u of type 2 A 2n−1 or its non-quasisplit inner form, which is also θ C -invariant. In this case s,θ 1 = C 2 , and the supercuspidal STM φ T as above has image L (a residual point) which lifts to two residual points which are not conjugate under W (B n ). Hence in this case we obtain two STMs defined by the lifts of φ T , corresponding to the two summands of H s,u . If s,θ 1 = 1 but s,θ = θ C = C 2 , then X * un (G) acts nontrivially on the connected inverse image of L. This is precisely the case where one parameter of H is 0, and the rank is positive. In other cases H is itself already an object of C I class whose STM has a unique lift to H I M (G).
, and all unipotent affine Hecke algebras are objects of C II class ; hence this case is straightforward by the above. For G = P(CO 0 2n ), we have a non-semistandard STM H I M (G) ; C n (0, 0)[q] which is represented by a degree 2 covering of tori (essentially the "same" covering of tori as for the case PU 2n+1 , but this time equipped with the action of W (D n ) instead of W (B n )) (see [54, 7.1.4]). Here the right-hand side is normalized as an object in C IV class . All other direct summends H of H uni (G) either are objects of C IV class with both m − and m + even, or of C VI class with δ − = δ + = 0, or there exists a semi-standard covering STM H ; H arising from an index two embedding H ⊂ H of an object H of C IV class for which one of the parameters m − or m + equals 0. If both of m ± = 0, then s,θ = 1, and any composition of basic STMs or the extraspecial STM φ : H ; C n (0, 0)[q] admits a unique lift to an STM φ : H ; H I M (G) as before. If one of m ± equals zero, then s,θ = C 2 . As before, in the positive rank case we have s,θ 1 = 1, and X * un (G) acts non-trivially by spectral isomorphisms, via its quotient ( s,θ ) * = C 2 , on the connected inverse image of the residual coset L which is the image of φ. Finally if one of m ± = 0 and the rank of φ is 0, then s,θ = s,θ 1 = C 2 , and L has two lifts under the 2 : 1 covering which are not in the same W (D n )-orbit but which are exchanged by the action of X * un (G). In this case, H u,s decomposes as a direct sum of two copies of L, and we have still an essentially unique STM H u,s ; H I M (G).
. This case is similar to the previous case, except that this time the relevant objects from C IV class are those with m − and m + both odd, and those of C VI class , the objects with δ − = δ + = 1. Hence this case is easier, since the direct summands of H of H uni (G) are themselves already objects of C IV class and of C VI class , and no dicussion of lifting of STMs is required.  3.), and since we are clearly in the standard case, this is anti-isomorphic to * X Q Y Q 0 by Proposition [54,Proposition 3.4]. But we know (see [53], [52,Theorem A.14(3)]) that W Q r Q is fixed for the action of Y Q 0 , so that we need to consider only the orbit of W Q r Q for the action of * X Q := (X Q /ZR Q ) * . In the case at hand

Proof of Theorem 3.4 Suppose
But this is exactly equal to K Q , hence any STM which is induced from a cuspidal STM φ Q : (L, τ 0 ) := H 0 ; H I M Q has as its image L n . By the rigidity property Proposition [54,Proposition 7.13] we see that any two such STMs are equal up to the action of Aut C (H). But as was explained in 3.1.3., the subset Q ⊂ F 0 is itself completely determined by just the root system of M ssa , and this is determined by s. It follows that any other STM φ : H u,s,e (G) → H I M (G) can be represented by a φ T whose image is L n , with L a residual coset in the X * un (G)-orbit of L.
As to the possibility to define an equivariant STM for the action of X * un (G) = ( θ C ) * , that is an application of Theorem 2.8. Recall that X * un (G) acts on H u,O via its quotient ( s,θ ) * ; we need to check in all cases that the subgroup ( s,θ 2 ) * ⊂ ( s,θ ) * is the stabilizer of W 0 (L). For the classical cases this was discussed in the previous sections, and for the exceptional cases this is an easy verification. It follows that the direct sum H u,s of all summands of H uni (G) in the X * un (G)-orbit of H u,s,e can be mapped X * un (G)-equivariantly by an STM to H I M (G), and that such an equivariant STM is essentially unique up to the spectral automorphism group of H uni (G). Taking the direct sum over all orbits X * un (G)-orbits of unipotent types, we obtain the desired result.
Hence Theorem 3.4 is now reduced to the cuspidal case. For the exceptional cases we have already shown the essential uniqueness for cuspidal STMs, and for G = PGL n+1 this was obvious. Hence the Proof of Theorem 3.4 is completed by the following result, whose proof will appear in [19]: Proposition 3.8 [19] The essential uniqueness of Theorem 3.4 holds true for the cuspidal part (or rank 0 part) H uni,cusp (G) of H uni (G) for G of type PU n , SO 2n+1 , PCSp 2n , P(CO 0 2n ), and P((CO * 2n ) 0 ). Here we denote by H uni,cusp (G) the direct sum of all the cuspidal (or rank 0) normalized generic unipotent affine Hecke algebras associated to G and its inner forms. In other words, there do not exist other rank 0 STMs than the ones constructed above, and this yields a X * un (G)-equivariant STM cusp : (H uni,cusp (G), τ ) ; (H I M , τ I M ) which is essentially unique in the sense of Theorem 3.4.
The proof of this proposition reduces to the analogous statement for the spectral categories C X class . For X = I, II this is rather easy. When X = III, IV, V, VI the essential uniqueness proof for cuspidal STMs is based on the existence of the cuspidal extraspecial STMs of C V class and C VI class . It is easy to see that every generic residual point of C X class for X = III, IV is in the image of a unique extraspecial cuspidal STM, and this sets up a bijection between the set of generic residual points of the combined objects of C V, VI class and those of C III, IV class . If we impose the necessary condition for cuspidality, namely that the formal degree (in our normalization) has no odd cyclotomic factors, then one can show that the corresponding generic residual point of C V, VI class is given by a pair (ξ − , ξ + ) of partitions whose Young tableaux are of rectangular shape, and almost a square. After applying the extraspecial STM, the solutions correspond to a pair (u − , u + ) of unipotent orbits of G s ⊂ G, a semisimple subgroup of maximal rank, whose elementary divisors are both of the form (1, 3, 5, . . . ) or (2, 4, 6, . . . ), or are both of the form (1, 5, 9, . . . ) or (3, 7, 11, . . . ). These solutions thus correspond to the cuspidal local systems for the endoscopic groups G s ⊂ G (cf. [34,37]). Proof For any object H in C Y class (Y as in the Theorem) there exists an STM φ : H ; H min , where H min denotes a minimal object, and where φ is a translation STM or an extraspecial STM. By the essential uniqueness of Theorem 3.4 it follows that any STM ψ : H ; H min is of the form ψ = β • φ • α with α ∈ Aut es (H) and with β ∈ Aut es (H min ). In C Y class , the group Aut es (H) is trivial (if the parameters m − and m + are unequal) or C 2 (if the parameters are equal). If there exists a nontrivial α 0 ∈ Aut es (H), then m − = m + and it follows easily from the definitions that there also exists a nontrivial β 0 ∈ Aut es (H min ), and that φ is equivariant in the sense φ • α 0 = β 0 • φ. Hence, if ψ is also a composition of basic translation STMs or if ψ is an extra special STM, we see that ψ = φ. From the injectivity (obvious from the definitions) of the basic generating STMs it now follows that the basic translation STMs commute.

The classical case
We also conclude from the injectivity of the basic generating STMs that, up to spectral isomorphisms, there can exist at most one STM between any two objects of C Y class . For X ∈ {I, II, III, IV} it follows from a consideration of the spectral transfer map diagrams (Definition 3.3) of the (essentially unique, injective) STMs φ 1 : H 1 ; H min and φ 2 : H 2 ; H min that a possible factorizing STM φ : H 1 ; H 2 (uniquely determined if it exists) must be itself composed of basic translation STMs and spectral isomorphism itself. It is also easy to see in this way that there can not exist STMs between objects of C V∪VI class and non-minimal objects of C X class with X ∈ {I, II, III, IV}. Between objects of C X class for X ∈ {V, VI} there are no STMs. This again follows from the injectivity of the extra special STMs, in view of the fact that the images of two extraspecial STMs of the form φ 1 : H 1 ; H min and φ 2 : H 2 ; H min map to disjoint subsets of the spectrum of the center of H min , unless H 1 and H 2 are isomorphic (this follows from the "extra special bijection" proved in [19]).
As a consequence we obtain a general description of all spectral transfer maps between all unipotent affine Hecke algebras in the classical cases: It is not difficult to describe all STMs between the unipotent affine Hecke algebras for exceptional types as well, but we will not do this here.

Partitioning of unramified square integrable L-packets according to Bernstein components
Let G be connected, absolutely simple, defined and quasisplit over k, split over K , and of adjoint type. Let H be a unipotent affine Hecke algebra associated to a unipotent type of an inner form of G (hence, a summand of H uni (G)). By our Theorem 3.4 we know that there exists an essentially unique X * un G-equivariant STM φ : H uni (G) ; H I M (G), and we know that such a map is compatible with the arithmetic/geometric correspondence of diagrams of Lusztig [40,43]. By [54,Theorem 6.1], this STM φ gives rise to a correspondence between components of the tempered irreducible spectra of H and H I M (G) which preserves, up to rational constant factors, the Plancherel densities on these components, and which is compatible with the map φ Z on the level of central characters of representations. In particular for unipotent discrete series representations, given an orbit of residual points W 0 r L ∈ W 0 \T (L) for H I M (G) (these carry the discrete series representations, by [52]), we collect the irreducible discrete series characters of the various direct summands H = H u,s,e of H uni (G) whose central character W 0 r L satisfies φ Z (W 0 r L ) = W 0 r L .  There exists a natural bijection (cf. [52,Corollary B.5], and paragraph 2.3) e [λ] → W 0 r λ,L between orbits of discrete unramified Langlands parameters and orbits of residual points W 0 r L for H I M . Theorem 3.4 implies that the packets W 0 r λ,L defined by STMs, admit a classification in terms of local systems on the G ∨ -orbits of discrete unramified Langlands parameters: In [13,56] the discrete series characters of arbitrary affine Hecke algebra H are parameterized differently. This point of view will be quite fruitful for the applications we have in mind, especially for unequal parameter Hecke algebras, and this is what we will discuss next.
Let L be a the ring of complex Laurent polynomials over the natural maximal algebraic torus of (possibly unequal) Hecke parameters associated to the underlying root datum of H (this ring was denoted by in [56]). Explicitly, L is the ring of Laurent polynomials in invertible indeterminates v α,± (with α ∈ R 0 ) subject to the conditions v α,± = v w(α),± for all α ∈ R 0 and w ∈ W 0 , and v α,− = 1 iff 1 − α ∨ ∈ W α ∨ ). We give L the structure of a L-algebra by putting v α,± = v m ± (α) . Then we have a generic affine Hecke algebra H L defined over L, and H = L ⊗ L H L .
Let V be the space of points of the maximal spectrum of L such that for all v = (v α,± ) ∈ V, we have v α,± := v(v α ± ) ∈ R + for all α ∈ R 0 . Let V be the space of points v ∈ R+ of the maximal spectrum of L. Thus we have an embedding V → V, and v α,± = v m ± (α) .
It was shown in [56, Theorem 3.4, Theorem 3.5] that an irreducible discrete series character δ of H is the specialization δ = δ L at L of a generic family of irreducible discrete series characters δ of H L which is well defined in an open neighborhood of (v m ± (α) ). Each discrete series character of H can thus be locally deformed in the parameters m ± (α). We will write such deformation as m ± (α) = m ± (α) + ± (α), where ± (α) vary in a sufficiently small open interval (− , ) ⊂ R. The irreducible discrete series representations of affine Hecke algebras with arbitrary positive parameters (v m ± (α) ), so in particular of all affine Hecke algebras of the form H = H u,s,e , have been classified in [56] from the point of view of deformations over the ring L.
In the case of nonsimply laced irreducible root systems R u,s,e 0 , the classification of [56] is in terms of the generic central character map gcc which associates to any irreducible discrete series character a W 0 := W (R u,s,e to a semisimple subroot system R s(e),1 ⊂ R 1 , with the parameter function m ,e ± (α) obtained by restriction of the parameters m ± (α) to the subdiagram of the geometric diagram s (R m ) obtained by omitting the vertex e, and replacing the group of diagram automorphisms by the isotropy subgroup e ⊂ . Finally, ξ denotes a linear residual point (see [56,Section 6]) for the generic graded affine Hecke algebra defined by R s(e),1 and m ,e ± (α). Thus ξ depends linearly on parameters m ,e ± (α) of the graded affine Hecke algebra. The specialization W 0 r 0 of the orbit W 0 r at ± = 0 is a confluence of finitely many orbits of generic residual points W 0 r i , with i ∈ I W 0 r 0 (some finite set which one can explicitly determine, see [56,Section 6]) from the explicit classification of linear residual points). For each irreducible discrete series character δ with central character W 0 r 0 , its unique continuous deformationδ, locally in the Hecke parameters, has a central character ofδ equal to one of the orbits W 0 r i of generic residual points which specialize at ± = 0 to W 0 r 0 .
This defines [56] a unique "generic central character" map gcc" from the set of irreducible discrete series at central character W 0 r 0 to the set I W 0 r 0 turns out to be bijective with the single exception of the orbit of generic residual points denoted f 8 of F 4 (which is one of the three generic residual points which come together at the weighted Dynkin diagram of the minimal unipotent orbit of F 4 ). In this case, there are two generic discrete series associated to f 8 . The map gcc also works well for the affine Hecke algebras of type D n , by relating this case with affine Hecke algebras of type C n (0, 0)[q]. We refer to [56] for details. The cases of type E n have to be treated in a different way (classically as in [28], or see [13]).
We would like to match up these two ways of parameterizing the discrete series characters in the packet λ (with [λ] ∈ e ). This will be important for the purpose of proving Theorem 4.11. Indeed, recall that the formal degree ofδ was shown to be continuous in terms of ( ± (α)) [56, Theorem 2.60, Theorem 5.12], and that it was given explicitly by the product formula [56,Theorem 5.12]. In addition it is known [13] that the formal degree of a generic family of discrete series characters is a product of an explicitly known rational constant and an explicit rational function of the parameters v α,± . This enables us to compute the rational constants in the formal degree of any discrete series character δ of any normalized unipotent affine Hecke algebra H = H u,s,e by a limit argument, using the generic familyδ and its formal degree. Motivated by this, let us consider in more detail our parameterization with this comparison in mind.

Parameterization for classical types
For PGL n this was discussed in paragraph 3.2.2.
For classical groups (other than type A) everything is governed by Hecke algebras of the form C n (m − , m + )[q b ], via the spectral correspondences of certain spectral covering morphisms. These correspondences can be made explicit by restriction and induction operations with respect to subalgebras of equal rank, and this will be discussed in detail when treating the various cases of classical type. In this paragraph we will concentrate on the principles for Hecke algebras of type C n (m − , m + )[q b ].
The corresponding graded affine Hecke algebras have a root system of type R s(e),0 of type B n − × B n+ , where n − + n + = n, and graded Hecke algebra parameters (m − , m + ). A W s(e),0 -orbit of generic linear residual points is given by the W s(e),0 = W 0 (B n − ) × W 0 (B n + )-orbit of an ordered pair (ξ − , ξ + ), where ξ ± is a vector of affine linear functions of ± such that ξ ± ( ± ) is the vector of contents of the boxes of the "m ± ± -tableau" of a partition π ± of n ± with the property that at ± = 0, the extremities [63] of the resulting m ± -tableau are all distinct.
At the "special" parameter value m ± (integral or half-integral), the W 0 (B n ± )-orbit of the vector ξ ± (0) is an orbit of linear residual points at parameter m ± . By a result of Slooten [63] (also see [56]), the set of such orbits of linear residual points is in bijection with the set of "unipotent partitions" u ± = u ξ ± of length l ± ≥ m ± − 1 2 of N ± := 2n ± + (m ± − 1 2 )(m ± + 1 2 ), consisting of distinct even parts (if m ± is half integral), or a partition u ± of length l ± ≥ m ± of N ± := 2n ± + m 2 ± , having distinct odd parts (if m ± is integral). Let us call such a pair u = (u − , u + ) of partitions a distinguished unipotent partition of type m = (m − , m + ). This set of partitions π ± (and thus the set of W 0 (B n ± )-orbits of generic residual points W 0 (B n ± )ξ ± which are confluent at ± = 0 to the same orbit W 0 (B n ± )ξ ± (0)) was parameterized by Slooten in terms of the so-called m ± -symbols σ ± . These symbols are certain Lusztig-Shoji symbols with defect D ± := m ± (see [63], [56,Definition 6.9]). Slooten's symbols [56,Definition 6.11] attached to orbits W 0 (B n ± )ξ ± (0) all have the same parts, but they are distinguished from each other by the selection of the parts which appear in the top row. Let us call these Slooten's symbols associated to u ± at parameter ratio m ± the u ± -symbols of type m ± . The point of view in [63] is that of the deformation picture sketched above: The symbols are "confluence data", and each such symbol represents an orbit of generic residual points which evaluates to W 0 (B n ± )ξ ± (0) at the parameter ratio m ± . It is convenient to formulate the results of this "abstract" classification in terms of abstract packets of representations associated to central characters of the discrete series of the minimal objects of C I class , C II class , C III∪V class and C IV∪VI class : ] be an affine Hecke algebra which appears as an object of C X class for X = I, II, III or IV. Let u = (u − , u + ) be an ordered pair of distinguished unipotent partitions corresponding to a central character W 0 r of a discrete series character of H , with u ± of type m ± . Let φ : H ; H be the translation STM to a minimal object H of C X class . Let φ Z (W 0 r ) = W 0 r . Then the ordered pair of distinguished unipotent partitions corresponding to W 0 r is equal to u as well! The set of irreducible discrete series characters of H in W 0 r is parameterized by ordered pairs (σ − , σ + ), where σ ± is a u ± -symbol of type m ± . Let Y In the context of an unramified classical group of adjoint type G, the Hecke algebras of the form H u,s are direct sums of normalized extended affine Hecke algebras H u,s,e which are spectral coverings of objects of C class . In particular, an unramified discrete Langlands parameter λ for G determines (via the comparison of the Kazhdan-Lusztig classification and the classification of discrete series representations as in [56]) an orbit of residual points W 0 r for H I M (G). In turn, via the morphism [54, Corollary 5.5] associated to this spectral covering map, this determines a pair (u − , u + ) of distinguished unipotent partitions in the sense of Proposition 4.7, for an appropriate pair of parameters (m − , m + ) of the form (m − , m + ) = (0, 1 2 ), (1, 1 2 ), ( 1 2 , 1 2 ), (0, 0), (0, 1) or (1, 1) (see paragraph 3.2.6). Working this out amounts to determining the multiplicities and types of the normalized extended affine Hecke algebras H u,s,e , and the branching rules for the algebra inclusions associated to the spectral covering maps to the relevant objects of C class . This is not difficult, and we can check that in all classical cases the STM of Theorem 3.4 gives rise to packets W 0 r of discrete series characters whose members are parameterized by pairs of Slooten's symbols or come from an extraspecial STM (see paragraphs 4.6.1, 4.6.2, 4.6.3 for more details).
Slooten's symbols are known to correspond with Lusztig's symbols [37] if one uses Lusztig's arithmetic/geometric correspondences for the "geometric" graded affine Hecke algebras in the following sense. Let the central character W 0 r 0 of a discrete series character π of C n (m − , m + )[q b ] be given by the pair of unipotent partitions (u − , u + ) (where u ± has at least m ± parts). The set of discrete series characters with central character W 0 r 0 is parameterized by the set of generic central characters W 0 r (see [56]) which evaluate to W 0 r 0 , via the map gcc. In turn, these generic central characters are parameterized by pairs σ − , σ + ) of the Slooten symbols (with defects D ± = m ± ) covering (u − , u + ). By the results of [10,42], [27,Section 4], the top graded part with respect to Slooten's functions a m ± [63] of the corresponding graded Hecke algebra module is the irreducible W (C n − ) × W (C n + )-module corresponding to (σ − , σ + ), via the generalized Springer correspondence of [46]. Via Proposition 4.7, the spectral correspondences of the standard STMs to H I M (G) together exhaust the set of pairs of Slooten symbols (σ − , σ + ).
The same is known to be true for the additional contributions to packet Y W 0 r coming from the extraspecial STMs (see [10], [27,Section 4]). These remarkable facts should be considered as an aspect of Langlands duality. Slooten's symbols are defined entirely in terms of affine Hecke algebras (describing the set of orbits of generic residual points specializing to the central character of a discrete series representation), whereas Lusztig's symbols describe cuspidal local systems on an associated nilpotent orbit of L G. Comparing this with Theorem 3.4 and Proposition 4.7 we see that our parametrization of W 0 r matches with Lusztig's assignment [40,43] of unramified Langlands parameters to the members of W 0 r . We see that the defect (D − , D + ) of an unordered pair (σ − , σ + ) of u-symbols for a member of W 0 r (corresponding to a pair of distinguished unipotent partitions (u − , u + )) determines the parameters of the affine Hecke algebra from which it originates under the STM . This determines the Bernstein component to which the corresponding discrete series character belongs (up to the action of X * un (G)). The final statement of Proposition 4.7 will be proved in [19]. The additional contributions from the extraspecial STMs to the packets of unipotent discrete series of PCSp 2n , P(CO 0 2n ) and P((CO * 2n ) 0 ) correspond to the fact that one takes the centralizers of the discrete Langlands parameter in the Spin group. This gives rise to a nontrivial central extension by C 2 of the centralizer in SO 2n (or SO 2n+1 respectively). These can be be described in detail in terms of central products of groups of type D 8 (the dihedral group with 8 elements), Q 8 , C 2 2 or C 4 (see [37]) and among those groups we typically find extraspecial 2-groups. The precise type of the groups that arise is complicated, but we are merely interested the the number of their irreducible representations and their dimensions which is less difficult, following the description in [37] (and also using [61] for the twisted cases) one obtains: , with l ± the number of parts of u ± . Thus u ± is a partition with distinct odd parts, and |u| := |u − | + |u + | = 2n with l ± both even if G = P(CO 0 2n ); |u| = 2n + 1 with l − even and l + odd if G = PCSp 2n ; and |u| = 2n with l ± both odd if G = P((CO * 2n ) 0 ). Let us write 2 (2a)+b for a 2-group of size 2 2a+b which has 2 2a+b−1 one-dimensional irreducible representations, and 2 b−1 irreducibles of dimension 2 a .
If u − is the zero partition, then A λ (as defined in paragraph 2.3) is of type 2 (l + −1)+1 if l + is odd, and of type 2 (l + −2)+2 if l + is even. If u + and u − are both nonzero, then A λ is of type 2 (l − +l + −4)+3 if l ± are both even, A λ is of type 2 (l − +l + −3)+2 if l ± are unequal modulo 2, and A λ is of type 2 (l − +l + −2)+1 if l ± are both odd.

Parameterization for split exceptional groups
For split exceptional groups, the major work to match up the irreducible discrete series characters of affine Hecke algebra summands of H uni (G) with Lusztig's parameters has been done by Reeder in [60] by computing the W -types explicitly. With this parameterization, the main Theorem of [60] is known to be a special case of the conjecture [26, Conjecture 1.4] (as discussed loc. cit.), which takes a lot of work out of our hands.
For the types E 6 and E 7 we need in addition to discuss the contribution of the nontrivial inner forms, which we take up in the next two paragraphs. 6 The inner forms of G are parameterized by u ∈ ≈ C 3 . We have X * un (G) = * . For u = 1 we have the following X * un (G) orbits of unipotent types:

Inner forms of the split adjoint group G of type E
. For u = 1 we have the following orbits of unipotent types: s u ∅ , s u 3 D 4 [1] , s u 3 D 4 [−1] . The orbit of s is a torsor for ( s 1 ) * (a quotient of X * un (G)). By inspection we check:

Remark 4.9
In all the cases above, we have s 1 = u := u ⊂ .
We choose an equivariant bijection α → s α between * u and the orbit of s. Then H uni (G) is isomorphic to the direct sum of the extended affine Hecke algebras H u,s α ,e Table 1 The packets u λ for type E 6 and the contributing STMs where u ∈ , s runs over the orbits of unipotent types, and α ∈ * u . By Theorem 3.4 there exists an essentially unique * -equivariant collection of STMs u,α s : H u,s α ,e ; H I M . Assume that we have chosen such a collection of STMs.
The extended affine Hecke algebras H u,s α ,e of positive rank which appear as summand of H uni (G) are: . It turns out that for each u ∈ , G u has 21 unipotent discrete series representations. Table 1 displays for each X * un (G) = * -orbit of discrete unramified Langlands parameters: A representative λ, its isotropy group * λ , the group A λ , and for each u ∈ , the STMs u,α s which contribute to the corresponding packet u λ of unipotent discrete series of G u . The argument of the STM indicates the corresponding central character of H u,s α ,e , expressed in terms of central characters of graded Hecke algebras via [56,Theorem 8.7], using standard notations referring to distinguished nilpotent orbits for equal parameter cases, and notations for a corresponding generic linear central character as in [56, Section 6] otherwise.
We choose the packets u λ := u W 0 r λ ,L compatibly with respect to the X * un (G)action, but the precise composition of the u λ depends on the choices of the STMs u,α s . Recall from Sect. 2.3 that their parameterization by the elements of Irr u (A λ ) is chosen in a X * un (G)-invariant way. By this requirement it suffices to fix the parameterization of the u λ for a set of representatives λ of the X * un (G)-orbits of discrete unramified Langlands parameters. With the choices made above, the parameterization of the packets 1 λ is determined if we also agree that the generic member of λ corresponds to the trivial representation of A λ . For u = 1 and λ = A 1 A 5 or A 3 2 , more information is needed to determine the exact parameterization of the sets u λ (of size 2 and 3 respectively) by a local system as in [40]. Since A λ is abelian here, Theorem 4.11 is independent of such choices. Therefore, we ignore this issue here. 7 We use the same setup and notations as for the case of E 6 . The inner forms of G are parameterized by u ∈ ≈ C 2 . We have X * un (G) = * . For u = 1 we have the following X * un (G) orbits of unipotent types: s . The orbit of s is a torsor for ( s 1 ) * (a quotient of X * un (G)). By inspection we check that the analog of Remark 4.9 again holds.
The precise constituents of the packets u λ depend on the choices of the STMs u,α s . Again the exact parameterization of the packets by Irr u (A λ ) is not uniquely determined for all λ and u. If A λ is abelian, this does not affect the statement of Theorem 4.11, and we ignore this problem here (but: see [13]). But for λ = E 7 (a 5 ) and u = −1 we need to be more careful. This packet corresponds to the generic central character f 8 (notation as in [56,Section 7]) of F 4 (1, 2)[q]. As was explained in [60], [56,Section 7], [13, paragraph 3.5.2], there are two algebraic generic parameter families δ 8 and δ 8 of irreducible discrete series characters of F 4 (m 1 , m 2 )[q] which stay as irreducible discrete series for all m 1 , m 2 > 0 (and in particular the corresponding W 0 (F 4 )-types are independent of the parameters). One of these (δ 8 say) is 10-dimensional, and specializes at equal parameters for F 4 to the discrete series [60] with Langlands parameters (F 4 (a3), [4]). The other, δ 8 restricts to the discrete series with Langlands parameters (F 4 (a3), [22]). Comparing with the tables in [64], we see that δ 8 corresponds with (E 7 (a 5 ), − [3]), while δ 8 corresponds to (E 7 (a 5 ), − [21]). On the other hand, by [56] and [13] we conclude that fdeg(δ 8 ) = 2fdeg(δ 8 ), and this is also equal to 2fdeg( 2 E 6 [1]). In view of the above Langlands parameters, this is in accordance with the conjecture [26, Conjecture 1.4].

Parameterization for non-split quasisplit exceptional groups
The parameterization and the STMs for the remaining twisted exceptional cases were discussed in 3.2.5. By Corollary 4.4, Corollary 4.5 and Remark 3.7 it then follows that Lusztig's parameterization of W 0 r L is uniquely determined by this, so this gives rise to a canonical matching of Lusztig's parameterization and our parameterization.

Formal degree of unipotent discrete series representations
The application in this section is independent of the uniqueness result based on [19].
A general conjecture has been put forward by [26] expressing the formal degree of a discrete series character in terms of the adjoint gamma factor (also see [20]). Recall that our standing assumption is that G is a connected, absolutely simple algebraic group of adjoint type, defined and quasisplit over k, and split over K .
In order to formulate the conjecture in our setting, we first should note that the Haar measures in [26] are equal to those we have used (following [17]) times v −dim(G) . Hence the formal degrees in [26] are v dim(G) times the formal degree in our setting. Let G u be an inner form of G. Given a discrete unramified local Langlands parameter λ for G, we defined A λ (see 2.3). Suppose that for an irreducible representation ρ ∈ Irr(A u λ ) we have a corresponding unipotent (or unramified) discrete series representation π (λ,ρ) of G F u , satisfying the expected character identities as asserted in the local Langlands conjecture.
Then [26,Conjecture 1.4] (also see [20,Conjecture 7.1]) is equivalent to (with our normalization of Haar measures): where γ denotes the adjoint gamma factor of the discrete local Langlands parameter λ. Following [26,Lemma 3.4], it is easy to show that (using the notations of 2.3) where we should remind the reader that the normalization of the μ-function μ I M of H I M (G) is given by the trace τ I M such that τ I M (1) = Vol(B F ) −1 . It was verified in [26] that Reeder's results [60] for Iwahori spherical discrete series representations of adjoint, split exceptional groups over a non-archimedean field are compatible with the conjecture. We are now able to extend this result to arbitrary adjoint absolutely simple groups over a non-archimedean local field which split over an unramified field extension. Proof We need to consider the classical groups, the nontrivial inner forms of split exceptional groups, and the non-split quasisplit exceptional groups. The way in which we assign unramified discrete Langlands parameters to the members of the packets W 0 r λ,L of discrete series characters of Definition 4.3 for these cases was explained in Sects. 4.3, 4.4 and 4.5.
We know that π (λ,ρ) corresponds via Lusztig's arithmetic-geometric correspondences to an irreducible discrete series representation δ λ,ρ of an extended affine Hecke algebra of type H u,s,e for some cuspidal type s of G u . By our main Theorem 3.4, there exists an STM φ : H u,s,e ; H I M (G) such that φ Z (cc(δ λ,ρ )) = W 0 r , and we have fdeg(π (λ,ρ) ) = fdeg H u,s,e (δ (λ,ρ) for some rational constant c (λ,ρ) ∈ Q. Combining (37) and (38) we see that what is necessary to verify in order to prove the conjecture in these cases is that In [20,Section 5.1] it was shown that with γ (λ) q a q-rational number, N λ ∈ N (which is in fact always 0 with our definition of q-rational numbers, but this is not important here), and C F λ ⊂ A λ a normal subgroup such that is the group of F-fixed points in the component group of the centralizer M λ of λ| SL 2 (C) in G ∨ . (The group C F λ is the group of F-fixed points in the identity component M 0 λ of M λ (a torus).) With this notation we are reduced to proving that (where ∼ refers to asymptotic behavior if q tends to 0) for some N λ ∈ N. Let us write λ ad for the composition of λ with the canonical homomorphism of G ∨ to G ∨ ad . In the twisted cases it is helpful to note that A λ / L Z is the centralizer of λ ad | SL 2 (C) in the identity component C G ∨ ad (λ ad (F)) 0 , and realizing that λ ad (F) is a semisimple element of G ∨ ad θ of the form (s, θ), where s is a vertex of the alcove of the restricted root system R θ 0 consisting of roots of R 0 restricted to t θ , extended to an affine reflection group by the lattice of translations obtained from projecting the coweight lattice P(R ∨ 0 ) onto t θ (see [61]). The semisimple centralizers C G ∨ ad (λ ad (F)) are described by Reeder in [61].
This amounts to a long list of case-by-case verifications. The case of PGL n+1 is easy. For u of order (m + 1)|(n + 1) we have s,e 2.3 and Proposition 2.5). This yields n +1 unipotent discrete series characters, each with formal degree (n +1) −1 [n +1] −1 q . In total we thus obtain n + 1 packets of unipotent discrete series characters, each with n + 1 members (one element for each inner form).
The case of G = PU 2n or G = PU 2n+1 is easy too, since all unipotent affine Hecke algebras are in a generic parameter situation here, in the sense of [11]. It is shown in [11] that the rational constant in the formal degree is then independent of the particular discrete series we consider (of a given Hecke algebra of this type). Looking at the Steinberg character [52, Equation (6.26)] we easily check therefore that the rational constants for all unipotent discrete series are equal to | L Z | −1 (so 1 2 if n is odd, and 1 otherwise).
In the remaining classical cases, one also uses the results of [11], where it was shown that the rational constant factor of the formal degree of a generic discrete series representation of a generic multi parameter type C n affine Hecke algebra specialized at non-special parameters is equal for all generic discrete series. All rational constants for the discrete series representations of affine Hecke algebras with special parameters can subsequently be computed by this result by a limit procedure, since [56] shows that any discrete serie representation is the limit of a generic continuous family of discrete series representations in a small open set in the parameter space, and that the formal degree is locally continuous in the parameters. We will use Propositions 4.7 and 4.8 to build the packets from the various unipotent Hecke algebras, and compute the expected rational constants according to [26,Proposition 1.4]. On the spectral side, one again relies on the results of [11] and [56] to compute the rational constants.
For exceptional cases, the results of [60] prove the statement for the unipotent discrete series of all split adjoint groups G. For the non-split cases, and the nontrivial inner forms of E 6 and E 7 more work needs to be done, but this follows the same scheme as discussed above, with the help of [61], the tables in Sects. 4.4 and 4.5, and the results of [13,56]. That is, we need to compute the rational constants for the formal degrees of discrete series representations of the multi parameter affine Hecke algebras arising from these non-split cases. The classical Hecke algebras are treated as before, so that leaves the exceptional unequal parameter Hecke algebras which appear in this way.
We find that we need to compute the formal degrees of G 2 (3, 1) (for type 3 D 4 ), G 2 (1, 3) (for type 3 E 6 ), F 4 (2, 1) (for type 2 E 6 ), and F 4 (1, 2) (for type 2 E 7 ). The first main observation in this kind of computations is the fact [56] that any discrete series character δ defines a generic central character gcc(δ) = W 0 r (an orbit of generic residual points) and extends uniquely to a continuous family of discrete series characters on a connected component C of the open subset of the space of positive parameters of the Hecke algebra on which W 0 r is still residual. Moreover fdeg(δ) depends continuously on the parameters in such a continuous family of discrete series.
But there is a deeper fact which is very useful. The formal degree of a generic family of discrete series representations (in the sense of [56]) depends algebraically on the parameters, and this expression only depends on the elliptic class of the limit q → 1 of the discrete series representation (a representation of W ). This result follows essentially from [14] and the Euler-Poincaré formula in [56], using the argument of [12,Proposition 5.6] in the unequal parameter setting. This implies (see [13] for details) that the formal degree of generic families associated with the same generic central character W 0 r but defined on different connected components C and C of the open subset of the positive parameter space where W 0 r is residual is given by the same algebraic expression (provided the families define the same elliptic representation of W ), except possibly for a sign change. (This result generalizes the result of [11] to arbitrary Hecke algebras). This algebraic expression for the formal degree is a product formula (see [56]) of terms (1 ± M) ±1 where M is a monomial in the parameters, multiplied by a rational constant d (which only depends on an elliptic representation of W ), a monomial in the parameters, and a sign.
The upshot is that in order to compute fdeg(δ) it is sufficient to compute fdeg(δ ) for any discrete series δ with gcc(δ ) = W 0 r at any positive parameter q where W 0 r (q ) is residual, provided δ and δ define the same elliptic representation of W . Using the results of [60], we can find a δ and q where the constants are known for every generic family. Hence the generic rational constants d can be determined, and from this we can determine the formal degree at any singular parameter line in the parameter space by continuity.
See Example 4.6.3 for more details in the case 3 D 4 . For F 4 (2, 1) we have a similar situation, here we need to cross the singular lines k 1 k 2 = 6 5 , 4 3 , 3 2 , 5 3 in the parameter space. For this we need to know the confluence relations of the generic discrete series at these singular lines. This can be deduced from [56, Table 3]. The considerations are similar as in Example 4.6.3. Similarly for 3 E 6 and 2 E 7 .
4.6.1 Unipotent representations of inner forms of PCSp 2n , P(CO 0 2n ), P((CO * 2n+2 ) 0 ) In these cases, a unipotent affine Hecke algebra is always isomorphic to a direct sum of finitely many copies of a normalized affine Hecke algebra which is related to an object of C III∪V class or C IV∪VI class through a (finite) sequence of spectral covering maps. Let us first compute the rational factors appearing in the formal degrees of discrete series representations of a normalized affine Hecke algebra H of type C d (m − , m + ) [q] with m ± ∈ Z, normalized by τ (1) = 1. Using the group D 8 of spectral isomorphisms (see [54,Remark 7.7]) we may, without loss of generality, assume that 0 ≤ m − ≤ m + .
As described in Sect. 4.2, the discrete series of H are parameterized by ordered pairs (σ − , σ + ) of symbols associated to an ordered pair (u − , u + ) of distinguished unipotent partitions for the pair of parameters m = (m − , m + ) (so u ± is a partition of m 2 ± + 2d ± ). By Slooten's "joining procedure" [63,Theorem 5.27] (see also the explanation in Sect. 4.2), the set of symbols σ ± corresponds bijectively to the set of partitions π ± of d ± whose m ± -tableaux have distinct extremities in the sense of [63] and such that the corresponding orbit of linear residual points corresponds to u ± . Then the vector consisting of the contents of the boxes of this m ± := m ± + ± -tableau of π ± defines, for all ± sufficiently small, a linear residual point ξ ± (m ± + ± ) whose W n ± -orbit generically supports a unique discrete series character. We will denote the discrete series character by δ (π − ,π + ) ( − , + ).
In our present case, R 0, For a positive root α of type D, we have m − (α) = 0 and m + (α) = 1; for positive root β of type A d 1 , we have m − (β) = m − + − and m + (β) = m + + + . In the limit = ( − , + ) → 0, some of the factors which are generically nonzero tend to 0, but the number of those factors in the numerator and denominator is equal by [56,Corollary 4.4] (or [53]). This potentially produces rational factors in the limit, but actually all such factors (for type D roots as well as for type A d 1 roots) are of the form (1 − q ±2 − ), (1 − q ±2 + ), or (1 − q ±( − − + ) ). For each of these three types, the total number of these factors in the numerator and denominator has to be equal by the above regularity result. Hence altogether these factors yield at most a sign in the limit, and that does not contribute to fdeg Q (π (u − ,u + ),(σ − ,σ + ) ). In addition we have factors (1 + q l( ) ), with l( ) linear in , in the denominator and numerator. Each such factor yields a factor 2, regardless of the precise form of l( ). Let the total number of factors 2 thus obtained be denoted by M. In order to count M, let us write h m ± u ± (x) for the number coordinates of ξ ± (m ± ) which are equal to x (for x ∈ Z ≥0 ) (cf. [22], or [56, Proposition 6.6]). We also define H . It is straightforward to deduce from the above formula for m W 0 r that where J m u (0) = 1 if 1 is a part of u (equivalently, if 0 is a jump of ξ ), and J m u (0) = 0 otherwise (this value depends only on u (is independent of m)). Recall that ( [63], or [56, Proposition 6.6]) the number of jumps of the vector of contents ξ(m) of T m (π ) equals #(u), and that this is also equal to m + H m u (0). In the second equality above we used that δ( , 1} be such that δ ± ≡ m ± mod(2). There exist partitions π ± such that set of jumps of the vector ξ ± of contents of the δ ± -tableau T δ ± (π ± ) of π ± equals the set of jumps of ξ ± (cf. [56, Proposition 6.6]). By Proposition 4.7, the central character Then it follows from the definition of the jump vector at m ± and at δ ± that for x ≥ 1, denotes the indicator function of the interval [1, m ± − 1]. Let #(u − ∩ u + ) denote the number of parts that u − and u + have in common. Then we get finishing the proof. In the above computation we used at several steps that 0 ≤ m − ≤ m + , and that H m A similar but easier computation shows a similar result for Hecke algebras of unipotent representations of SO 2n+1 (cf. 4.6.2).
Finally consider the inner forms with u = ρ or u = ρη. These two inner forms are equivalent as rational forms, via the outer automorphism corresponding to θ , hence it suffices to consider the case u = ρ only. This time the cuspidal unipotent parahoric subgroups P ρ s,t are given by ρ-invariant subdiagrams of type P ρ s,t D s 2 ∪ D s 2 ∪ 2 A 1 2 (t 2 +t)−1 if n even 2 D s 2 ∪ 2 D s 2 ∪ 2 A 1 2 (t 2 +t)−1 if n odd such that d + 1 := 1 2 (n − 2s 2 − 1 2 (t 2 + t) + 2) ∈ Z >0 . This corresponds to a type s := s Using Theorem 4.14, we obtain discrete series representations π α (π − ,π + ),extra with rational parts of formal degrees equals fdeg Q (π α (π − ,π + ),extra ) = 2 1 2 (l − +l + −2)−#(u − ∪u + ) (in all cases), where α denotes an irreducible character of s 1 . In view of Proposition 4.8 and (47), this is indeed the rational factor of the formal degree of the two elements of the Lusztig packet attached to the Langlands parameter λ on which L Z ⊂ A λ acts by ρ times the identity, as predicted by (43). As to the numerology of counting the number of such irreducible representations in a Lusztig packet attached to a unipotent discrete Langlands parameter λ for G: Let us write (u − , u + ) for the (ordered) pair of unipotent partitions attached to λ (these are partitions with odd, distinct parts such that |u − | + |u + | = 2n). If u − = 0 and u − = u + , then we have two such packets (for the pairs (u − , u + ) and (u + , u − ) which contain a discrete series representation with the same q-rational factor. According to Proposition 4.8 both these packets contain 2 irreducibles on which L Z acts as a multiple of ρ (and also two where L Z acts as multiple of ρη) (together these are the four irreducibles in each of these packets which are not one-dimensional). This matches the "Hecke side", since we have (by (48)) that H ρ,s is either a direct sum of four copies of H ρ,s,e , each contributing one irreducible discrete series with the desired q-rational factor in the formal degree (if s = 0, or equivalently m − = m + ) or of two such copies (if s = 0, or equivalently m − = m + ). But in the latter case, each of these copies of H ρ,s,e contributes two such irreducible discrete series (whose central characters are mapped by the STM to (u − , u + ) and (u + , u − ) respectively). If u − = u + , then necessarily m − = m + , and the two copies of H ρ,s,e contribute each one discrete series to the packet associated to λ, corresponding to the two irreducibles of A λ on which L Z acts as ρ. Finally we have the case u − = 0. In this case there are four distinct discrete Langlands parameters λ 1 = λ, λ 2 , λ 3 , λ 4 which share the same q-rational factor in the formal degree, and each of the four corresponding Lusztig packets should have one member associated to the single irreducible of A λ i on which L Z acts by ρ (according to Proposition 4.8). Hence in all cases the Hecke algebra side and the L-packet side indeed match. This finishes the case G = P(CO 0 2n ). The last case to consider is the non-split quasisplit orthogonal group P((CO * ) 0 2n+2 ). Now we have u ∈ /(1 − θ) = / η ρ C 2 . We have H I M (G) = C n (1, 1)[q]. The conjugacy classes of parahoric subgroups P d,a,b which support a (unique) cuspidal unipotent representation are parametrized by ordered pairs (a, b) with a, b ∈ Z ≥0 , with a even and b odd, and such that d = n + 1 − a 2 − b 2 ≥ 0. The parahoric P d,a,b is of type D a 2 ∪ 2 D b 2 . The corresponding cuspidal unipotent type is denoted by s = s d,a,b . We have H ,s,e = C d (m − , m + )[q], with m + = a + b and m − = |a − b|. Furthermore, ,s,θ Let us now consider u = ρ. Now the cuspidal unipotent parahoric subgroups P ρ s,t are given by ρ-invariant subdiagrams of type P ρ s,t D s 2 ∪ D s 2 ∪ 2 A 1 2 (t 2 +t)−1 if n even 2 D s 2 ∪ 2 D s 2 ∪ 2 A 1 2 (t 2 +t)−1 if n odd such that d + 1 := 1 2 (n − 2s 2 − 1 2 (t 2 + t) + 3) ∈ Z >0 . This corresponds to a type s := s ρ d,s,t for G ρ which is completely determined by a pair of nonnegative integers (s, t) satisfying the above inequality, and the congruences: s ≡ n(mod 2), t ≡ 1, 2(mod 4) (if n even), and t ≡ 0, 3(mod 4) (if n odd). The corresponding affine Hecke algebra H ρ,s,e is of always type Hence, using Theorem 4.14, the extra special STM H ρ,s,e ; H I M (G) yields one additional discrete series representation π (π − ,π + ),extra added to the Lusztig packet associated to λ, whose formal degree satisfies fdeg Q (π (π − ,π + ),extra ) = 2 1 2 (l − +l + )−#(u − ∪u + ) , as desired in view of Proposition 4.8.

Unipotent representations for inner forms of SO 2n+1
In these cases, a unipotent affine Hecke algebra is always spectrally isomorphic to a direct sum of finitely many copies of objects of C II class . The treatment of these cases is analogous to the symplectic and even orthogonal cases discussed in the previous paragraph, but in all aspects much simpler (no branching phenomena, no extraspecial STM's). We will content ourselves to give the results only.
We have = { , η} C 2 , and H I M (G ) is of type C n ( 1 2 , 1 2 )[q]. The conjugacy classes of parahoric subgroups P d,a,b of G supporting a (unique) cuspidal unipotent representation are parametrized by ordered pairs (a, b) with a, b ∈ Z ≥0 , with a even, and such that d = n − a 2 − (b 2 + b) ≥ 0. The parahoric P d,a,b is a type D a 2 ∪ B b 2 +b . The corresponding cuspidal unipotent type is denoted by s = s d,a, (in all cases). For the nontrivial inner form G η of G, the formulas are the same except that now a is odd, and P d,a,b has type 2 D a 2 ∪ B b 2 +b . Now an orbit of discrete unipotent Langlands parameters λ for G corresponds to an ordered pair (u − , u + ) of unipotent partitions with u ± 2n ± such that n − + n + = n, where u ± consists of distinct, even parts.

4.6.3
The example of type 3 D 4 Let G be the group of type 3 D 4 defined over a non-archimedean local field k. The group G is quasisplit, and the dual L-group is isomorphic to L G := θ G ∨ where G ∨ = Spin(8) and where θ is an outer automorphism of order 3. Hence L Z = 1, and G has no nontrivial inner forms. There are two cuspidal unipotents called 3 D 4 [1] and 3 D 4 [−1] (cf. [9], section 13.7).
The image φ(F) = sθ of the Frobenius element under a discrete unramified Langlands parameter φ is an isolated semisimple automorphism. Via the action of Int(G ∨ ) it is conjugate to a semisimple element of the form θ s i with s i a vertex of C θ (cf. [20]). In the case at hand, we label the nodes of the twisted affine root diagram according to [20,Section 4.4], and we have to consider θ s 0 , θ s 1 and θ s 2 .
We have H I M (G) = G 2 (3,1)[q], normalized by τ (1) := [3] −1 q (v−v −1 ) −2 according to (25). The W 0 -orbit space of the character torus T of the root lattice X of type G 2 can be identified [3] with the space of Int(G ∨ )-orbits of semisimple classes of L G of the form θg, via the map T t → θ t. In this way we will identify, as usual, the space of central characters of affine Hecke algebra H I M (G) = G 2 (3, 1)[q] and the space of semisimple Int(G ∨ )-orbits of this form of L G. The Hecke algebra H I M (G) has two orbits of real residual points W 0 r 0,reg and W 0 r 0,sub , and two nonreal ones W 0 r 1 and W 0 r 2 (using the same numbering of the nodes of the diagram as before). At each residual point of G 2 (m l , m s )[q] at the parameter value (m l , m s ) = (3, 1), the number of irreducible discrete series characters supported at this point is equal to the number of generic residual points which specialize at (3, 1) to the given residual point. This number is always 1, except for W 0 r 0,sub , where it is equal to two [56].
We can and will baptise these orbits of generic residual points W 0 r , using Kazhdan-Lusztig parameters for the discrete series of G 2 (1, 1)[q], by an irreducible representation of A λ , where λ is the Langlands parameter of the split group of type G 2 . The subregular unipotent orbit of G 2 gives rise to a unipotent discrete Langlands parameter λ = λ sub of 3 D 4 with A λ = S 3 . Its "weighted Dynkin diagram" is r 0,sub . The two orbits of generic residual points of the generic Hecke algebra of type G 2 which are confluent at (1,1) are also confluent at (3, 1). By the above, we call these two orbits of generic residual points W 0 r sub,triv and W 0 r sub,σ , where σ is the two dimensional irreducible character of S 3 . The orbit of generic points W 0 r sub,triv represents a generic discrete series character of degree 3, which has generic formal degree with rational constant factor 1 2 . The other orbit of generic residual points W 0 r sub,σ has degree 1, and generic rational constant 1.
At the confluence of these two generic residual points at parameter (1, 1), we get in the limit an additional constant factor 3 1 for W 0 r sub,triv leading to the well known equal parameter case of Theorem 4.11 at the subregular unipotent orbit for split G 2 (cf. [59]). At the confluence point for the parameters (3, 1) the rational constants do not change, however. Thus together with the cuspidal character 3 D 4 [1] we get a packet λ for λ = λ sub consisting of three representations, naturally parameterized by the characters of A λ = S 3 , whose formal degrees have rational constant 1 2 (for the cuspidal 3 D 4 [1] corresponding to the "missing representation" sign of A λ = S 3 , and for the generic discrete series character associated to W 0 r sub,triv evaluated at the parameter value (3, 1)), and rational constant 1 (for the generic discrete series W 0 r sub,σ evaluated at (3, 1)).
For the regular parameter of G ∨ θs 1 , we get two discrete series characters, namely the cuspidal one 3 D 4 [−1] and the Iwahori spherical one. Both have 1 2 as a rational constant factor.
Finally, at the regular parameter of G ∨ θs 2 , we have one Iwahori spherical discrete series representation, with rational constant 1.
For a discrete Langlands parameters λ with θ s i = λ(F) and such that u := λ 1 1 0 1 is regular within the connected reductive group G ∨ θs i , this follows because C F λ = 1 in such a case (this is obvious for i = 0 and i = 2 since then A λ = 1, and for i = 1 we see that u is the distinguished element [5,3] in Spin(8) by the table of [9, p. 397], whence M 0 λ = 1), and hence A λ ≈ (π 0 (M λ )) F is isomorphic to Z (G ∨ θs i )/Z ( L G) = Z (G ∨ θs i ) (by [59,Section 6]). This yields the result for all cases except λ sub . For this case we remark that the image of the subregular unipotent of G 2 in Spin(8) is a unipotent class of Spin(8) with elementary divisors [1,1,3,3]. Hence M 0 λ is a two-dimensional torus, on which F acts as a rotation of order 3. Thus C F λ is cyclic of order 3, and (π 0 (M λ )) F ≈ C 2 . This indeed yields the constants we just computed.
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