Induced representations of infinite-dimensional groups

Abstract

The induced representation ${\mathrm{Ind}}_H^{G}S$ of a locally compact group $G$ is the unitary representation of the group $G$ associated with unitary representation $S:H\to U(V)$ of a subgroup $H$ of the group $G$. Our aim is to develop the concept of induced representations for infinite-dimensional groups. The induced representations for infinite-dimensional groups in not unique, as in the case of a locally compact groups. It depends on two completions $\tilde{H}$ and $\tilde{G}$ of the subgroup $H$ and the group $G$, on an extension $\tilde{S}:\tilde{H}\to U(V)$ of the representation $S:H\to U(V)$ and on a choice of the $G$-quasi-invariant measure $\mu$ on an appropriate completion $\tilde{X}=\tilde{H}\backslash \tilde{G}$ of the space $H\backslash G$. As the illustration we consider the "nilpotent" group $B_0^{\mathbb{Z}}$ of infinite in both directions upper triangular matrices and the induced representation corresponding to the so-called generic