Weyl calculus and dual pairs

M. Mckee (1) , A. Pasquale (2) , T. Przebinda (3)

1 Department of Mathematics
2 IECL - Institut Élie Cartan de Lorraine
3 Department of Mathematics

We consider a dual pair (G,G'), in the sense of Howe, with G compact acting on L^2(R^n) for an appropriate n via the Weil Representation. Let G~ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation Π of G~ we compute the Weyl symbol of orthogonal projection onto L^2(R^n)_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 G'~ and to compute of the wave front set of Π' by elementary means.

Mots clés: Lie superalgebras; Reductive dual pairs; Weyl calculus; Howe duality

Domaines: Mathématiques [math]

M. Mckee, A. Pasquale, T. Przebinda. Weyl calculus and dual pairs. 2014. ⟨hal-01096778⟩