Un cas simple de correspondance de Jacquet-Langlands modulo l

We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.

Supercuspidal characters of reductive p-adic groups

Let G be a general linear group over a p‐adic field and let D× be an anisotropic inner form of G. The Jacquet–Langlands correspondence between irreducible complex representations of D× and discrete series of G does not behave well with respect to reduction modulo ℓ≠p. However, we show that the Langlands–Jacquet transfer, from the Grothendieck group of admissible Q¯ℓ ‐representations of G to that of D× is compatible with congruences and reduces modulo ℓ to a similar transfer for F¯ℓ ‐representations, which moreover can be characterized by some Brauer characters identities. Studying this transfer more carefully, we deduce a bijection between irreducible F¯ℓ ‐representations of D× and ‘super‐Speh’ F¯ℓ ‐representations of G. Via reduction mod ℓ, this latter bijection is compatible with the classical Jacquet–Langlands correspondence composed with the Zelevinsky involution. Finally, we discuss the question whether our Langlands–Jacquet transfer sends irreducibles to effective virtual representations up to a sign. This presumably boils down to some unknown properties of parabolic affine Kazhdan–Lusztig polynomials. In the appendix, we reproduce Vignéras' first construction of Brauer characters for p‐adic groups. It follows Harish‐Chandra's classical approach while our construction uses resolutions on the building.

pdf/un-cas-simple-de-correspondance-de-jacquet-langlands-modulo-50vte3e63c.pdf

Un cas simple de correspondance de Jacquet‐Langlands modulo ℓ

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

On the local langlands correspondence for split classical groups over local function fields

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected, nor that the underlying field has characteristic zero

https://www.math.toronto.edu/jkorman/Articles/Loc-char-exp/loc-char-exp.pdf

The local character expansion near a tame, semisimple element

In their 1997 paper, Schneider and Stuhler gave a formula relating the value of an admissible character of a $p$-adic group at an elliptic element to the fixed point set of this element on the Bruhat-Tits building. Here we give a similar formula which works for compact elements. Elliptic elements have finitely many fixed facets in the building but compact elements can have infinitely many. In order to deal with the compact case we truncate the building so that we only look at a bounded piece of it. We show that for compact elements the (finite) information contained in the truncated building is enough to recover all of the information about the character. This works since the fixed point set of a compact (non elliptic) element is periodic. The techniques used here are more geometric in nature than the algebraic ones used by Schneider and Stuhler. We recover part of their result as a special case.

A character formula for compact elements (the rank one case)

The character of an irreducible admissible representation of a p- adic reductive group is known to be a constant function in some neighborhood of any regular semisimple element in the group. Under certain mild restric- tions on , we give an explicit description of a neighborhood of on which the character is constant. Mathematical Subject Classification:Primary 22E50; Secondary 22E35, 20G25. Keywords and Phrases: Characters, reductive p-adic groups, Bruhat-Tits build- ing.

/pdf/on-the-local-constancy-of-characters-1w2njdpj0z.pdf

On the local constancy of characters

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected nor that the underlying field has characteristic zero.

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Papers

The local character expansion near a tame, semisimple element

A character formula for compact elements (the rank one case)

On the local constancy of characters

The local character expansion near a tame, semisimple element