40 research outputs found
Quasisplit Hecke algebras and symmetric spaces
Let (G,K) be a symmetric pair over an algebraically closed field of
characteristic different of 2 and let sigma be an automorphism with square 1 of
G preserving K. In this paper we consider the set of pairs (O,L) where O is a
sigma-stable K-orbit on the flag manifold of G and L is an irreducible
K-equivariant local system on O which is "fixed" by sigma. Given two such pairs
(O,L), (O',L'), with O' in the closure \bar O of O, the multiplicity space of
L' in the a cohomology sheaf of the intersection cohomology of \bar O with
coefficients in L (restricted to O') carries an involution induced by sigma and
we are interested in computing the dimensions of its +1 and -1 eigenspaces. We
show that this computation can be done in terms of a certain module structure
over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above.Comment: 46 pages. Version 2 reorganizes the explicit calculation of the Hecke
module, includes details about computing \bar, and corrects small misprints.
Version 3 adds two pages relating this paper to unitary representation
theory, corrects misprints, and displays more equations. Version 4 corrects
misprints, and adds two cases previously neglected at the end of 7.
Parameters for Twisted Representations
The study of Hermitian forms on a real reductive group gives rise, in the
unequal rank case, to a new class of Kazhdan-Lusztig-Vogan polynomials. These
are associated with an outer automorphism of , and are related to
representations of the extended group . These polynomials were
defined geometrically by Lusztig and Vogan in "Quasisplit Hecke Algebras and
Symmetric Spaces", Duke Math. J. 163 (2014), 983--1034. In order to use their
results to compute the polynomials, one needs to describe explicitly the
extension of representations to the extended group. This paper analyzes these
extensions, and thereby gives a complete algorithm for computing the
polynomials. This algorithm is being implemented in the Atlas of Lie Groups and
Representations software
On elliptic factors in real endoscopic transfer I
This paper is concerned with the structure of packets of representations and
some refinements that are helpful in endoscopic transfer for real groups. It
includes results on the structure and transfer of packets of limits of discrete
series representations. It also reinterprets the Adams-Johnson transfer of
certain nontempered representations via spectral analogues of the
Langlands-Shelstad factors, thereby providing structure and transfer compatible
with the associated transfer of orbital integrals. The results come from two
simple tools introduced here. The first concerns a family of splittings of the
algebraic group G under consideration; such a splitting is based on a
fundamental maximal torus of G rather than a maximally split maximal torus. The
second concerns a family of Levi groups attached to the dual data of a
Langlands or an Arthur parameter for the group G. The introduced splittings
provide explicit realizations of these Levi groups. The tools also apply to
maps on stable conjugacy classes associated with the transfer of orbital
integrals. In particular, they allow for a simpler version of the definitions
of Kottwitz-Shelstad for twisted endoscopic transfer in certain critical cases.
The paper prepares for spectral factors in twisted endoscopic transfer that are
compatible in a certain sense with the standard factors discussed here. This
compatibility is needed for Arthur's global theory. The twisted factors
themselves will be defined in a separate paper.Comment: 48 pages, to appear in Progress in Mathematics, Volume 312,
Birkha\"user. Also renumbering to match that of submitted versio
Hecke algebras with unequal parameters and Vogan's left cell invariants
In 1979, Vogan introduced a generalised -invariant for characterising
primitive ideals in enveloping algebras. Via a known dictionary this translates
to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it
is not a complete invariant, it is extremely useful in describing left cells.
Here, we propose a general framework for defining such invariants which also
applies to Hecke algebras with unequal parameters.Comment: 15 pages. arXiv admin note: substantial text overlap with
arXiv:1405.573
Dirac cohomology, elliptic representations and endoscopy
The first part (Sections 1-6) of this paper is a survey of some of the recent
developments in the theory of Dirac cohomology, especially the relationship of
Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology;
the second part (Sections 7-12) is devoted to understanding the unitary
elliptic representations and endoscopic transfer by using the techniques in
Dirac cohomology. A few problems and conjectures are proposed for further
investigations.Comment: This paper will appear in `Representations of Reductive Groups, in
Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa,
published by Springe
Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs
The complex analytic methods have found a wide range of applications in the
study of multiplicity-free representations. This article discusses, in
particular, its applications to the question of restricting highest weight
modules with respect to reductive symmetric pairs. We present a number of
multiplicity-free branching theorems that include the multiplicity-free
property of some of known results such as the Clebsh--Gordan--Pieri formula for
tensor products, the Plancherel theorem for Hermitian symmetric spaces (also
for line bundle cases), the Hua--Kostant--Schmid -type formula, and the
canonical representations in the sense of Vershik--Gelfand--Graev. Our method
works in a uniform manner for both finite and infinite dimensional cases, for
both discrete and continuous spectra, and for both classical and exceptional
cases
Unitary Shimura correspondences for split real groups
We find a relationship between certain complementary series representations for nonlinear coverings of split simple groups, and spherical complementary series for (different) linear groups. The main technique is Barbasch’s method of calculating some intertwining operators purely in terms of the Weyl group