83 research outputs found
One-W-type modules for rational Cherednik algebra and cuspidal two-sided cells
We classify the simple modules for the rational Cherednik algebra that are
irreducible when restricted to W, in the case when W is a finite Weyl group.
The classification turns out to be closely related to the cuspidal two-sided
cells in the sense of Lusztig. We compute the Dirac cohomology of these modules
and use the tools of Dirac theory to find nontrivial relations between the
cuspidal Calogero-Moser cells and the cuspidal two-sided cells.Comment: 16 pages; added references, corrected misprint
Dirac cohomology for symplectic reflection algebras
We define uniformly the notions of Dirac operators and Dirac cohomology in
the framework of the Hecke algebras introduced by Drinfeld. We generalize in
this way the Dirac cohomology theory for Lusztig's graded affine Hecke
algebras. We apply these constructions to the case of symplectic reflection
algebras defined by Etingof-Ginzburg, particularly to rational Cherednik
algebras for real or complex reflection groups with parameters t,c. As
applications, we give criteria for unitarity of modules in category O and we
show that the 0-fiber of the Calogero-Moser space admits a description in terms
of a certain "Dirac morphism" originally defined by Vogan for representations
of real semisimple Lie groups.Comment: 28 pages, expanded introduction, added an example at the end,
corrected formulas in sections 4.5 and 5.4, added reference
Multiplicity matrices for the affine graded Hecke algebra
In this paper, we look at the problem of determining the composition factors
for the affine graded Hecke algebra via the computation of Kazhdan-Lusztig type
polynomials. We review the algorithms of \cite{L1,L2}, and use them in
particular to compute, at every real central character which admits tempered
modules, the geometric parameterization, the Kazhdan-Lusztig polynomials, the
composition series, and the Iwahori-Matsumoto involution for the
representations with Iwahori fixed vectors of the split -adic groups of type
and .Comment: 30 page
Star operations for affine Hecke algebras
In this paper, we consider the star operations for (graded) affine Hecke
algebras which preserve certain natural filtrations. We show that, up to inner
conjugation, there are only two such star operations for the graded Hecke
algebra: the first, denoted , corresponds to the usual star operation
from reductive -adic groups, and the second, denoted can be
regarded as the analogue of the compact star operation of a real group
considered by \cite{ALTV}. We explain how the star operation appears
naturally in the Iwahori-spherical setting of -adic groups via the
endomorphism algebras of Bernstein projectives. We also prove certain results
about the signature of -invariant forms and, in particular, about
-unitary simple modules.Comment: 27 pages; section 3 and parts of sections 2 and 5 were previously
contained in the first version of the preprint arXiv:1312.331
Types and unitary representations of reductive p-adic groups
We prove that for every Bushnell-Kutzko type that satisfies a certain
rigidity assumption, the equivalence of categories between the corresponding
Bernstein component and the category of modules for the Hecke algebra of the
type induces a bijection between irreducible unitary representations in the two
categories. This is a generalization of the unitarity criterion of Barbasch and
Moy for representations with Iwahori fixed vectors.Comment: 21 pages; v2: 23 pages, introduced "rigid types
Hermitian forms for affine Hecke algebras
We study star operations for Iwahori-Hecke algebras and invariant hermitian
forms for finite dimensional modules over (graded) affine Hecke algebras with a
view towards a unitarity algorithm.Comment: 29 pages, preliminary version. v2: the classification of star
operations for the graded Hecke algebras and the construction of hermitian
forms in the Iwahori case via Bernstein's projectives have been removed from
this preprint and they will make the basis of a new pape
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