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 \(\mathbb H\).


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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