scholarly journals On Reducibility and Unitarizability for Classical p-Adic Groups, Some General Results

2009 ◽  
Vol 61 (2) ◽  
pp. 427-450 ◽  
Author(s):  
Marko Tadić
Keyword(s):  
Reductive Groups ◽  
Parabolic Induction

Abstract. The aim of this paper is to prove two general results on parabolic induction of classical p-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.

scholarly journals ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

2016 ◽  
Vol 17 (2) ◽  
pp. 453-488 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Category Theory ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Adjoint Functors ◽  
Left Action ◽  
Parabolic Induction ◽  
Left And Right ◽  
Restriction Functor ◽  
Tempered Representation ◽  
Standard Concept

Let$E$be a (right) Hilbert module over a$C^{\ast }$-algebra$A$. If$E$is equipped with a left action of a second$C^{\ast }$-algebra$B$, then tensor product with$E$gives rise to a functor from the category of Hilbert$B$-modules to the category of Hilbert$A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clareet al.[Parabolic induction and restriction via$C^{\ast }$-algebras and Hilbert$C^{\ast }$-modules,Compos. Math.FirstView(2016), 1–33, 2].

scholarly journals Parabolic induction and restriction via -algebras and Hilbert -modules

2016 ◽  
Vol 152 (6) ◽  
pp. 1286-1318 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Inner Product ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Parabolic Induction ◽  
Full Treatment ◽  
Reduced Group ◽  
The Right ◽  
Tempered Representation

This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

Symplectic slices for actions of reductive groups

2006 ◽  
Vol 197 (2) ◽  
pp. 213-224
Author(s):  
I V Losev
Keyword(s):  
Reductive Groups

scholarly journals Alvis–Curtis duality for representations of reductive groups with Frobenius maps

2020 ◽  
Vol 32 (5) ◽  
pp. 1289-1296
Author(s):  
Junbin Dong
Keyword(s):  
Irreducible Representations ◽  
Reductive Groups ◽  
Infinite Type ◽  
Abstract Representations

AbstractWe generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.

scholarly journals Geometric approach to parabolic induction

2016 ◽  
Vol 22 (4) ◽  
pp. 2243-2269 ◽  
Author(s):  
David Kazhdan ◽  
Yakov Varshavsky
Keyword(s):  
Geometric Approach ◽  
Parabolic Induction
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