On Sobolev norms for Lie group representations

TL;DR

This paper introduces Sobolev norms of any real order for Lie group representations using a specific differential operator, establishing a spectral gap and employing a novel delta distribution factorization.

Contribution

It defines a new class of Sobolev norms for Lie group representations and proves a spectral gap result using a novel delta distribution factorization technique.

Findings

Established Sobolev norms of arbitrary real order for Lie group representations.

Proved a spectral gap for the differential operator on smooth vectors.

Developed a new factorization of the delta distribution on Lie groups.

Abstract

We define Sobolev norms of arbitrary real order for a Banach representation $(\pi, E)$ of a Lie group, with regard to a single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is a Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for $D$ on the space of smooth vectors of $E$. The main tool is a novel factorization of the delta distribution on a Lie group.