Unitary dual of GL(n) at archimedean places and global Jacquet–Langlands correspondence

AbstractIn a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

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Local Langlands Correspondence for Regular Supercuspidal Representations of GL(n)

International Mathematics Research Notices ◽

10.1093/imrn/rnaa197 ◽

2020 ◽

Author(s):

Masao Oi ◽

Kazuki Tokimoto

Keyword(s):

General Linear ◽

Linear Groups ◽

Langlands Correspondence ◽

Supercuspidal Representations ◽

General Linear Groups ◽

The Difference ◽

Local Langlands Correspondence

Abstract In this paper, we prove the coincidence of Kaletha’s recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris–Taylor’s one in the case of general linear groups. The keys are Bushnell–Henniart’s essentially tame local Langlands correspondence and Tam’s result on Bushnell–Henniart’s rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha’s and Tam’s $\chi $-data.

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CHARACTERIZATION OF THE PROJECTIVE GENERAL LINEAR GROUPS PGL(2,q) BY THEIR ORDERS AND DEGREE PATTERNS

International Journal of Algebra and Computation ◽

10.1142/s0218196709005433 ◽

2009 ◽

Vol 19 (07) ◽

pp. 873-889 ◽

Cited By ~ 8

Author(s):

LIANGCAI ZHANG ◽

XUEFENG LIU

Keyword(s):

Simple Groups ◽

General Linear ◽

Linear Groups ◽

Finite Simple Groups ◽

Degree Pattern ◽

General Linear Groups ◽

A Finite Group ◽

Projective General Linear Groups

Let G be a finite group and π(G) = {p1, p2,…,pk}. For p ∈ π(G), we put deg (p) := |{q ∈ π(G)|p ~ q}|, which is called the degree of p. We also define D(G) := ( deg (p1), deg (p2), …, deg (pk)), where p1 < p2 < ⋯ < pk, which is called the degree pattern of G. Using the classification of finite simple groups, we characterize the projective general linear group PGL(2,q)(q a prime power) by its order and degree pattern in the present paper.

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