Unipotent representations of complex semisimple groups
type: Item
instance of: scholarly article
title: Unipotent representations of complex semisimple groups
zbMATH Open document ID: 0582.22007
DOI: 10.2307/1971193
author: Dan Barbasch
author: David A. jun. Vogan
published in: Annals of Mathematics. Second Series
publication date: 1985-01-01T00:00:00Z
review text: The authors give a generalization of families of representations, which were introduced by \textit{J. Arthur} [Lect. Notes Math. 1041, 1-49 (1984; Zbl 0541.22011)], of a complex semisimple Lie group G. Especially, they give a character formula for a finite set of the resulting representations which is called ''special unipotent representation''. Let B and K be a Borel and a maximal compact subgroup of G, respectively, and put \(H=Z_ G(B\cap K)\). Suppose that \(\lambda,\mu\in {\mathfrak h}^*\) and \(\lambda-\mu\) is a weight of a finite dimensional holomorphic representation of G. Let \(\bar X(\lambda,\mu)\) be the Langlands subquotient of the K-finite part of \(Ind^ G_ B {\mathcal C}(\lambda,\mu)\), where \({\mathcal C}(\lambda,\mu)\) is the trivial extension on B of the character of H with differential \((\lambda,\mu)\). Then a nilpotent orbit \({\mathcal O}\in {\mathfrak g}^*\) is called ''special'' if there is \(X=\bar X(\lambda,\mu)\) such that (a) \(WF(X)={\mathcal O}\), (b) \(\lambda,\mu\) are integral, where WF(X) is the wavefront set of X. Let \(^ L{\mathfrak g}\) be the dual Lie algebra of \({\mathfrak g}\). Then there exists an order-reversing bijection \(\eta: {\mathcal O}\mapsto^ L{\mathcal O}\) between special nilpotent orbits in \({\mathfrak g}\) and \(^ L{\mathfrak g}\). Suppose that \({\mathcal O}\) is a special nilpotent orbit with \(^ L{\mathcal O}\) even. Then a ''special unipotent representation'' attached to \({\mathcal O}\) is an \(X=\bar X(\lambda,\mu)\) such that (a) \(WF(X)={\mathcal O}\), (b) \(\lambda,\mu\) are both conjugate to \(\lambda_{{\mathcal O}}\) under W, where W is the Weyl group of \((g,h)\) and \(2\lambda_{{\mathcal O}}\in {\mathfrak h}^*\) corresponding to the semisimple element \(^ Lh\) in \(^ L{\mathfrak h}\) attached to \(^ L{\mathcal O}.\) The main result can be stated as follows. First there exists a bijection \(\pi \mapsto X_{\pi}\) between the set of irreducible representations of the finite group \(\bar A({\mathcal O}…
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 3931302
zbMATH Keywords: complex semisimple Lie group
zbMATH Keywords: character formula
zbMATH Keywords: unipotent representation
zbMATH Keywords: Langlands subquotient
zbMATH Keywords: nilpotent orbit
zbMATH Keywords: wavefront set
zbMATH Keywords: Weyl group
zbMATH Keywords: Lusztig quotient
reviewed by: Takeshi Kawazoe
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971193
OpenAlex ID: W2331441710
description: scientific article; zbMATH DE number 3931302