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

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.