Dirac induction for Hecke algebras

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

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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).

http://staff.fnwi.uva.nl/e.m.opdam/

Olivier CHABROL
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