Journal of Lie Theory
Vol. 8, No. 2, pp. 325-334 (1998)

Paires de Guelfand generalisees associees au groupe d'Heisenberg

K. Mokni and E. G. F. Thomas

Departement de Mathematiques,
Faculte des Sciences,
5019 Monastir,

University of Groningen,
Department of Mathematics,
P.O.Box 800, 9700 AV Groningen,
The Netherlands.

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.

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number]