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HOWE CORRESPONDENCE AND SPRINGER CORRESPONDENCE A.-M. AUBERT (C.N.R.S. AND U.P.M.C., PARIS, FRANCE), W. KRASKIEWICZ (NICOLAS
 

Summary: HOWE CORRESPONDENCE AND SPRINGER CORRESPONDENCE
A.-M. AUBERT (C.N.R.S. AND U.P.M.C., PARIS, FRANCE), W. KRA´SKIEWICZ (NICOLAS
COPERNICUS UNIVERSITY, TORU´N, POLAND), T. PRZEBINDA (UNIVERSITY OF
OKLAHOMA, USA)
1. Introduction
Consider real reductive group G, as defined in [Wal88]. Let be an irreducible admis-
sible representation of G with the distribution character , [Har51]. Denote by u the
lowest term in the asymptotic expansion of , [BV80]. This is a finite linear combina-
tion of Fourier transforms of nilpotent coadjoint orbits, u = O cO ^µO. As shown by
Rossmann, [Ros95], the closure of the union of the nilpotent orbits which occur in this
sum is equal to WF(), the wave front set of the representation , defined in [How81].
Furthermore there is a unique nilpotent coadjoint orbit O in the complexification g
C
of the dual Lie algebra g of G such that the associated variety of the annihilator of the
Harish-Chandra module of in the universal enveloping algebra U(g) of g is equal to the
closure of O, [BB85]. Moreover, the closure of O coincides with the complexification
of WF(), see [BV80, Theorem 4.1] and [Ros95]. Given a Cartan subalgebra of gC we
have the corresponding Weyl group W. Springer correspondence associates an irreducible
representation of W to each complex nilpotent coadjoint orbit, assuming the group is
connected. See [Ros91] for a convenient geometric construction. We shall use this con-

  

Source: Aubert, Anne-Marie - Institut de Mathématiques de Jussieu
Przebinda, Tomasz - Department of Mathematics, University of Oklahoma

 

Collections: Mathematics