# Dirac induction for Hecke algebras

Date(s) : 13/12/2018   iCal
14 h 00 min - 15 h 00 min

The Dirac operator for graded affine Hecke algebras was introduced by Barbasch, Ciubotaru and Trapa (2010). Based on this construction we defined Dirac induction for graded affine Hecke algebras (Ciubotaru-O.-Trapa). Using this, we (Ciubotaru-O.) showed that given a so-called “massive” elliptic character chi of Euler-Poincare norm 1 of an affine Weyl group, there exists a unique family Ind_D(chi) of virtual “generically discrete series” characters of the associated affine Hecke algebra over its space of Hecke algebra parameters, characterised by the property that its limit for q tends to 1 has elliptic class chi. For every specialisation of the Hecke algebra at a positive parameter, all its irreducible discrete series characters belong to such a unique parameter family of this type.

We show that the formal degree of Ind_D(chi) is a rational function m_\chi(q):=fdeg(Ind_D(chi,q)) of the Hecke parameters q, which is regular for positive parameters (this is quite striking, because the family itself is not continuous). This fact enables us to compute m_\chi(q) for every chi as above completely explicitly. We show that Ind_D(chi,q) is a discrete series character (up to a sign) iff m_chi(q) is nonzero, which gives a classification of the discrete series which is uniform in the Hecke algebra parameters q.

Based on joint work with Ciubotaru and Trapa (JIMJ 2014), and with Ciubotaru (ANT 2017).

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