Growth of heat trace coefficients for locally symmetric spaces

Abstract

We study the asymptotic behavior of the heat trace coefficients $a_n$ as n tends to infinity for the scalar Laplacian in the context of locally symmetric spaces. We show that if the Plancherel measure of a noncompact type symmetric space is polynomial, then these coefficients decay like 1/n!. On the other hand, for even dimensional locally rank 1-symmetric spaces, one has $|a_n|$ grows like C^n* n! for some C>0; we conjecture this is the case in general if the associated Plancherel measure is not polynomial. These examples show that growth estimates conjectured by Berry and Howls are sharp. We also construct examples of locally symmetric spaces which are not irreducible, which are not flat, and so that only a finite number of the heat trace coefficients are non-zero.