Integrable representation

A continuous irreducible unitary representation $\pi$ of a locally compact unimodular group $G$ in a Hilbert space $H$ such that for some non-zero vector $\xi \in H$ the function $g\mapsto (\pi (g)\xi ,\xi )$, $g\in G$, is integrable with respect to the Haar measure on $G$. In this case, $\pi$ is a square-integrable representation and there exists a dense vector subspace $H^{\prime }\subset H$ such that $g\mapsto (\pi (g)\xi ,\eta )$, $g\in G$, is an integrable function with respect to the Haar measure on $G$ for all $\xi ,\eta \in H^{\prime }$. If $\{\pi \}$, the unitary equivalence class of the representation $\pi$, denotes the corresponding element of the dual space $\hat{G}$ of $G$, then the singleton set containing $\{\pi \}$ is both open and closed in the support ${\hat{G}}_r$ of the regular representation.

Comments[edit]

Instead of integrable representation one usually finds square-integrable representation in the literature. Let $\pi$ and ${\pi }^{\prime }$ be two square-integrable representations; then the following orthogonality relations hold:

$$\underset{G}{\int }(\pi (g)\xi ,\eta )\overline{({\pi }^{\prime }(g){\xi }^{\prime },{\eta }^{\prime })}dg=$$

$$=\{\begin{array}{ll}0& \text{ if }\pi \text{ and }{\pi }^{\prime }\text{ are notequivalent , }\\ d_{\pi }^{-1}(\xi ,{\xi }^{\prime })(\eta ,{\eta }^{\prime })& \text{ if }\pi ={\pi }^{\prime },\end{array}$$

where the integral is with respect to Haar measure. The scalar $d_{\pi }$ is called the formal degree or formal dimension of $\pi$. It depends on the normalization of the Haar measure $dg$. If $G$ is compact, then every irreducible unitary representation $\pi$ is square integrable and finite dimensional, and if Haar measure is normalized so that ${\int }_{G}dg=1$, then $d_{\pi }$ is its dimension.

The square-integrable representations are precisely the irreducible subrepresentations of the left (or right) regular representation on $L_2(G)$ and occur as discrete direct summands.

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