Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

Details Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

2005-09-13

arxiv.org/abs/math/0509291v1

Mathematics

Operator Algebras

20 pages

Scientific paper

For discrete Hecke pairs $(G,H)$, we introduce a notion of covariant representation which reduces in the case where $H$ is normal to the usual definition of covariance for the action of $G/H$ on $c_0(G/H)$ by right translation; in many cases where $G$ is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of $c_0(G/H)$ which are multiples of the multiplication representation on $\ell^2(G/H)$, and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from $H$ to $G$.

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