Invariant Hilbert spaces of distribution vectors in Lie group representations

TL;DR

This paper proves that for each unitary irreducible Lie group representation, the Hilbert space of the representation is uniquely the only nonzero invariant space of distribution vectors, establishing a fundamental uniqueness property.

Contribution

It establishes the uniqueness of the invariant Hilbert space of distribution vectors for all unitary irreducible Lie group representations.

Findings

The representation Hilbert space is the only nonzero invariant space of distribution vectors.

This result applies universally to all unitary irreducible representations of Lie groups.

Abstract

For every unitary irreducible representation of a Lie group we prove that the representation Hilbert space is the only nonzero invariant Hilbert space of distribution vectors.