Analytic factorization of Lie group representations

TL;DR

This paper proves a factorization theorem for analytic vectors in moderate growth representations of real Lie groups, showing they can be expressed via a specific algebra of superexponentially decreasing functions, with implications for Laplace--Beltrami operators.

Contribution

It introduces a Dixmier--Malliavin type factorization for analytic vectors in Lie group representations, connecting them to a natural algebra of functions and extending understanding of their structure.

Findings

Analytic vectors can be factorized using a natural algebra of functions.

E^{ ext{ω}} coincides with the space of analytic vectors for the Laplace--Beltrami operator.

The factorization extends the classical Dixmier--Malliavin theorem to analytic vectors.

Abstract

For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^{\omega}. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^{\omega} = A(G) * E^{\omega}. As a corollary we obtain that E^\omega coincides with the space of analytic vectors for the Laplace--Beltrami operator on G.