The spectrum of the Laplace operator on connected compact simple rank four Lie groups. I

TL;DR

This paper computes the Laplace operator spectrum for functions on all connected compact simple rank four Lie groups with biinvariant metrics, linking the results to number theory and quadratic forms.

Contribution

It provides explicit spectrum calculations for rank four Lie groups and connects these formulas to number theory and quadratic forms.

Findings

Explicit spectra for all rank four Lie groups with root systems B4, C4, D4.

Established connections between spectra and number theory.

Linked Laplace spectra to integer quadratic forms.

Abstract

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank four Lie groups with biinvariant Riemannian metric, corresponding to root systems $B_4$, $C_4$, $D_4$ and established a connection of obtained formulas with the number theory and integer quadratic forms from two, three and four variables.