Induced representations of infinite-dimensional groups

TL;DR

This paper extends the concept of induced representations to infinite-dimensional groups, highlighting their non-uniqueness and dependence on completions, measures, and extensions, with an example involving infinite upper triangular matrices.

Contribution

It develops a framework for induced representations of infinite-dimensional groups, addressing their non-uniqueness and providing concrete examples.

Findings

Induced representations for infinite-dimensional groups depend on completions and measures.

The framework generalizes classical induced representations to infinite-dimensional settings.

An example with infinite upper triangular matrices illustrates the theory.

Abstract

The induced representation ${\rm Ind}_H^GS$ of a locally compact group $G$ is the unitary representation of the group $G$ associated with unitary representation $S:H\rightarrow U(V)$ 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 $\tilde S:\tilde H\rightarrow U(V)$ of the representation $S:H\rightarrow U(V)$ 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