Abstract

We classify the spectral transfer morphisms (cf. Opdam in Adv Math 286:912–957, 2016) between affine Hecke algebras associated to the unipotent types of the various inner forms of an unramified absolutely simple algebraic group G defined over a non-archimedean local field k. This turns out to characterize Lusztig’s classification (Lusztig in Int Math Res Not 11:517–589, 1995; in Represent Theory 6:243–289, 2002) of unipotent characters of G in terms of the Plancherel measure, up to diagram automorphisms. As an application of these results, the spectral correspondences associated with such morphisms (Opdam 2016), and some results of Ciubotaru, Kato and Kato [CKK] (also see Ciubotaru and Opdam in A uniform classification of the discrete series representations of affine Hecke algebras. arXiv:1510.07274) we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on formal degrees and adjoint gamma factors in the special case of unipotent discrete series characters of inner forms of unramified simple groups of adjoint type defined over k.

Highlights

  • Which imply roughly that we can compute these rational constant factors in the formal degrees of discrete series of non- 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 of the parameter space

  • It will turn out that this task to classify these spectral transfer morphism (STM) essentially reduces to the task of finding all STMs from the rank 0 unipotent affine Hecke algebras to the Iwahori–Matsumoto Hecke algebra HI M (G ) of the quasisplit G such that G is an inner form of G

  • Notice that if we combine the basic STMs of the first 5 cases with the group D8 of spectral isomorphisms of Cclass, we are allowed for all objects X ∈ {I, II, III, IV} either one of m− and m+ as long as the absolute value of this parameter can still be reduced by such steps

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Summary

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. 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 HI 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 d0 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
Parahoric subgroups Recall the explicit representation of pure inner forms
Unramified local Langlands parameters
Unipotent affine Hecke algebras
Spectral transfer morphisms
Main theorem
D R such that
The classical case
Parameterization for classical types
Parameterization for split exceptional groups
Parameterization for non-split quasisplit exceptional groups
Formal degree of unipotent discrete series representations
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