Paires de Guelfand generalisees associees au groupe d'Heisenberg

Abstract: We consider the Heisenberg group $\scriptstyle H_{p+q}=\cb^{p+q}\times\rb$, and the semi-direct product ${\scriptstyle \U(p,q,\cb).H_{p+q}}$. It is known that the pair ${\scriptstyle (\U(p,q,\cb).H_{p+q},\U(p,q,\cb))}$ is a generalized Gelfand pair.We give a characterisation of subgroups $\scriptstyle K$ of ${\scriptstyle \U(p,q,\cb)}$, not necessarily compact, such that the pair ${\scriptstyle(K.H_{p+q},K)}$ is a generalized Gelfand pair. For this we establish a general theorem proving that: if ${\scriptstyle\gamma,\pi}$ are two unitary representations of a Lie group $G$, such that $\scriptstyle\gamma$ is irreducible and possesses a distribution character, then $\scriptstyle\gamma$ is a subrepresentation of $\scriptstyle\pi$ if and only if the tensor product, ${\scriptstyle\overline\gamma\otimes\pi}$, has a $\scriptstyle G$-fixed distribution vector.

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