We 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... more
We 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 irred...
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Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameter family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a... more
Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameter family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the " rational Cherednik algebra " , and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups G(m, p, N), the set of singular parameters in the parameter family of these structures is described explicitly, using the theory of nonsymmetric Jack polynomials.
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The Iwahori-Hecke algebra has a canonicaltrace $\tau$. The trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group.... more
The Iwahori-Hecke algebra has a canonicaltrace $\tau$. The trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group. The Iwahori-Hecke algebra contains an important commutative sub-algebra ${\bf C}[\theta_x]$, that was described and studied by Bernstein, Zelevinski and Lusztig. In this note we compute the generating function for the value of $\tau$ on the basis $\theta_x$.
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... Consider the *-structure on NH(R+, k) defined by w* = w-1 for w C W and C* = -wo wo (() wo for ( CV and extended to all of NH(R+, k) as an anti-linear anti-involution. ... Z{Q(w)a(O)f ()Q(w)0(0) + Q(w)f (rq)Q(wwo)0(wo0)Q(wo)g(r) } ...... more
... Consider the *-structure on NH(R+, k) defined by w* = w-1 for w C W and C* = -wo wo (() wo for ( CV and extended to all of NH(R+, k) as an anti-linear anti-involution. ... Z{Q(w)a(O)f ()Q(w)0(0) + Q(w)f (rq)Q(wwo)0(wo0)Q(wo)g(r) } ... Q(w)f (?])Q(wwo)&(wof)Q(wow-1)Q(w)g(r,)} ...