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Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of irreducible with finite multiplicity?

I know that this is true, for $\pi$ trivial and $H$ unimodular (=> G unimodular). Is unimodularity here necessary?

Please provide either a counterexample or a reference, that this does or does not hold in general.

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up vote 5 down vote accepted

Yes it is necessary. For instance take $G=GL(2, {\mathbb Q}_p )$ and $H$ the non-unimodular subgroup of upper triangular matrices. Then the (smoothly) induced representation ${\rm Ind}_H^G {\mathbf 1}$ is not semisimple. It is of length $2$. It has the trivial representation as a subrepresentation and the Steinberg representation as quotient. More generally if $G$ is a reductive $p$-adic group, reducible parabolically induced representations are not semisimple.

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Can also continuous spectrum apear for cocompact induced representations? – Marc Palm Jul 5 '11 at 8:36
I think you can adapt the standard proof of the discrete decomposition of $L^2 (\Gamma \backslash G )$, where $\Gamma$ is a discrete cocompact subgroup of a Lie group $G$. Cf. the book of Gelfand, Graev, Piateski-Shapiro. – Paul Broussous Jul 6 '11 at 7:16

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