Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Abstract

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let ρ _t be the heat kernel on G; that is, ρ _t is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from ρ _t and its derivatives, are the compact group analogs of the classical Hermite polynomials on $\mathbb{R}^n$. Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on $\mathfrak{g}\;\; = \;\;Lie(G)$ in the limit of small t, where the Hermite functions are viewed as functions on $\mathfrak{g}$ via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of $\sqrt t$. For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.