Spectral correspondences for affine Hecke algebras

Abstract

We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations.

Spectral transfer morphisms for unipotent affine Hecke algebras

“…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.…”

“…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 .…”

“…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.…”

“…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.…”

“…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.…”