scholarly journals Duality for Ext-groups and extensions of discrete series for graded Hecke algebras

2016 ◽  
Vol 294 ◽  
pp. 410-453 ◽  
Author(s):  
Kei Yuen Chan
Keyword(s):  
Discrete Series ◽  
Hecke Algebras ◽  
Ext Groups ◽  
Graded Hecke Algebras

scholarly journals Twisted graded Hecke algebras

2007 ◽  
Vol 317 (1) ◽  
pp. 30-42 ◽  
Author(s):  
Sarah Witherspoon
Keyword(s):  
Hecke Algebras ◽  
Graded Hecke Algebras

Generalized Graded Hecke Algebras of Types B and D

2006 ◽  
Vol 34 (6) ◽  
pp. 2105-2128 ◽  
Author(s):  
Charlotte Dezélée
Keyword(s):  
Hecke Algebras ◽  
Graded Hecke Algebras

scholarly journals Algebraic and analytic Dirac induction for graded affine Hecke algebras

2013 ◽  
Vol 13 (3) ◽  
pp. 447-486 ◽  
Author(s):  
Dan Ciubotaru ◽  
Eric M. Opdam ◽  
Peter E. Trapa
Keyword(s):  
Weyl Group ◽  
Discrete Series ◽  
Hecke Algebras ◽  
Reductive Group ◽  
Dirac Operators ◽  
Hecke Algebra ◽  
Semisimple Lie Groups ◽  
Affine Hecke Algebra ◽  
Affine Hecke Algebras ◽  
Dirac Induction

AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{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 affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

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