Weyl calculus and dual pairs

2014/05/10 by M. McKee, McKee, M., A. Pasquale +3
Mathematics · #22E30 #FOS: Mathematics #Primary: 22E45 #Representation Theory (math.RT) #math.RT #msc:22E30 #msc:22E45 #msc:22E46 #secondary: 22E46

paper · pdf · doi:10.48550/arxiv.1405.2431

arxiv created 2015/11/13 · arxiv updated 2015/11/16

Abstract

We consider a dual pair (G,G'), in the sense of Howe, with G compact acting on L²(\mathbb Rⁿ) for an appropriate n via the Weil Representation. Let \widetildeG be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation Π of \widetildeG we compute the Weyl symbol of orthogonal projection onto L²(\mathbb Rⁿ)_Π, the Π-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation Π' of \widetildeG' and to compute of the wave front set of Π' by elementary means.

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