32 research outputs found
Characters of Springer representations on elliptic conjugacy classes
For a Weyl group W, we give a simple closed formula (valid on elliptic
conjugacy classes) for the character of the representation of W in each
A-isotypic component of the full homology of a Springer fiber. We also give a
formula (valid again on elliptic conjugacy classes) of the W-character of an
irreducible discrete series representation with real central character of a
graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin
double cover of W and the Dirac operator for graded affine Hecke algebras play
key roles.Comment: 15 pages, minor changes in exposition, corrected typo
Duality for nonlinear simply laced groups
Let G be a nonlinear double cover of the real points of a connected reductive
complex algebraic group with simply laced root system. We establish a uniform
character multiplicity duality theory for the category of Harish-Chandra
modules for G.Comment: 51 pages, 1 figur
Characters of Springer representations on elliptic conjugacy classes
Abstract. For a Weyl group W , we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of W in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W -character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of W and the Dirac operator for graded affine Hecke algebras play key roles
Algebraic and analytic Dirac induction for graded affine Hecke algebras
We define the algebraic Dirac induction map \Ind_D for graded affine Hecke
algebras. The map \Ind_D is a Hecke algebra analog of the explicit
realization of the Baum-Connes assembly map in the -theory of the reduced
-algebra of a real reductive group using Dirac operators. The definition
of \Ind_D is uniform over the parameter space of the graded affine Hecke
algebra. We show that the map \Ind_D defines an isometric isomorphism from
the space of elliptic characters of the Weyl group (relative to its reflection
representation) to the space of elliptic characters of the graded affine Hecke
algebra. We also study a related analytically defined global elliptic Dirac
operator between unitary representations of the graded affine Hecke algebra
which are realized in the spaces of sections of vector bundles associated to
certain representations of the pin cover of the Weyl group. In this way we
realize all irreducible discrete series modules of the Hecke algebra in the
kernels (and indices) of such analytic Dirac operators. This can be viewed as a
graded Hecke algebra analogue of the construction of discrete series
representations for semisimple Lie groups due to Parthasarathy and
Atiyah-Schmid.Comment: 37 pages, revised introduction, updated references, minor correction