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Dirac cohomology for graded affine Hecke algebras. (English) Zbl 1276.20004

Summary: We define an analogue of the Casimir element for a graded affine Hecke algebra \(\mathbb H\), and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology \(H^D(X)\) of an \(\mathbb H\)-module \(X\), and show that \(H^D(X)\) carries a representation of a canonical double cover of the Weyl group \(\widetilde W\). Our main result shows that the \(\widetilde W\)-structure on the Dirac cohomology of an irreducible \(\mathbb H\)-module \(X\) determines the central character of \(X\) in a precise way. This can be interpreted as \(p\)-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of \(\mathbb H\).

MSC:

20C08 Hecke algebras and their representations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B22 Root systems
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
58J05 Elliptic equations on manifolds, general theory
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