Growth of heat trace coefficients for locally symmetric spaces

TL;DR

This paper investigates the asymptotic growth of heat trace coefficients in locally symmetric spaces, revealing polynomial decay or factorial growth depending on the Plancherel measure, and confirms conjectures on their sharpness.

Contribution

It establishes the relationship between Plancherel measure properties and heat trace coefficient growth, providing new examples and confirming conjectured growth estimates.

Findings

Coefficients decay like 1/n! when Plancherel measure is polynomial.

Coefficients grow like C^n * n! for certain symmetric spaces.

Examples of non-irreducible, non-flat spaces with finitely many non-zero heat trace coefficients.

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.