The behavior of the characters of the supercuspidal representations on the regular set

1970 ◽  
pp. 43-62
Author(s):  
Harish-Chandra
Keyword(s):  
Supercuspidal Representations

Symplectic supercuspidal representations and related problems

2010 ◽  
Vol 53 (3) ◽  
pp. 533-546 ◽  
Author(s):  
DiHua Jiang ◽  
ChuFeng Nien ◽  
YuJun Qin
Keyword(s):  
Supercuspidal Representations

scholarly journals The cohomology of semi-infinite Deligne–Lusztig varieties

2020 ◽  
Vol 2020 (768) ◽  
pp. 93-147
Author(s):  
Charlotte Chan
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Cohomology Group ◽  
Geometric Realization ◽  
Local Fields ◽  
Irreducible Representations ◽  
Division Algebras ◽  
Full Generality ◽  
Homology Groups ◽  
Supercuspidal Representations

AbstractWe prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

Globalization of Distinguished Supercuspidal Representations of GL(n)

2002 ◽  
Vol 45 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Jeffrey Hakim ◽  
Fiona Murnaghan
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Cuspidal Representation ◽  
Supercuspidal Representation ◽  
Local Component ◽  
Supercuspidal Representations

AbstractAn irreducible supercuspidal representation π of G = GL(n, F), where F is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup H of G and a quasicharacter χ of H if HomH(π, χ) ≠ 0. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to GL(n). Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

Supercuspidal representations and preservation principle of theta correspondence

2019 ◽  
Vol 2019 (750) ◽  
pp. 1-52
Author(s):  
Shu-Yen Pan
Keyword(s):  
Unitary Group ◽  
Orthogonal Group ◽  
Classical Groups ◽  
Theta Correspondence ◽  
Dual Pairs ◽  
Supercuspidal Representations ◽  
Explicit Characterization ◽  
A Chain

Abstract The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p-adic classical groups. In this paper, we give an explicit characterization of the chain starting from an irreducible supercuspidal representations of a unitary group of one variable or an orthogonal group of two variables. In particular, we define the Lusztig-like correspondence of generic cuspidal data for p-adic groups and establish its relation with local theta correspondence of supercuspidal representations for p-adic dual pairs.

Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group

2000 ◽  
Vol 52 (2) ◽  
pp. 306-331 ◽  
Author(s):  
Clifton Cunningham
Keyword(s):  
Lie Algebras ◽  
Symplectic Group ◽  
Orbital Integrals ◽  
Finite Linear Combination ◽  
Supercuspidal Representations ◽  
Rank 2 ◽  
The Fourier Transform ◽  
The Stability ◽  
Springer Correspondence ◽  
Finite Linear

AbstractThis paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of p-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

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