On the Schwartz estimate for Hodge Laplacians on semisimple Lie groups

TL;DR

This paper establishes Schwartz estimates for Hodge Laplacians and Dirac operators on semisimple Lie groups, extending previous results on symmetric spaces to facilitate heat equation analysis on vector bundles.

Contribution

It generalizes Schwartz estimates and Kuga lemma to semisimple Lie groups, enabling advanced Fourier analysis for heat problems on homogeneous spaces.

Findings

Proved Schwartz estimates for Hodge Laplacian and Dirac operators.

Extended Kuga lemma to Lie algebra cohomology.

Facilitated heat equation analysis on vector bundles.

Abstract

In this paper, we prove Schwartz estimates for Hodge Laplacian and Dirac operators on semisimple Lie groups. Alongside, we gives a version of Kuga lemma for its Lie algebra cohomology. This is a generalization of similar results on symmetric spaces. The main purpose of such estimates is to study the heat problem not only in the scalar case, but also for sections of vector bundles on homogeneous spaces using Fourier analysis.