Equivariant heat asymptotics on spaces of automorphic forms

Abstract

Let G be a connected, real, semisimple Lie group with finite center, and K a maximal compact subgroup of G. In this paper, we derive K-equivariant asymptotics for heat traces with remainder estimates on compact Riemannian manifolds carrying a transitive and isometric G-action. In particular, we compute the leading coefficient in the Minakshishundaram-Pleijel expansion of the heat trace for Bochner-Laplace operators on homogeneous vector bundles over compact locally symmetric spaces of arbitrary rank.

Global Automorphic Sobolev Theory and The Automorphic Heat Kernel

“…Automorphic heat kernels are important in physics and number theory: applications include multiloop amplitudes of certain bosonic strings [8], asymptotic formulas for spectra of arithmetic quotients [7,10,14], a relationship between eta invariants of certain manifolds and closed geodesics using the Selberg trace formula [30], systematic construction of zeta-type functions via heat Eisenstein series [23][24][25][26], sup-norm bounds for automorphic forms [2,3,13,21,22], limit formulas for period integrals and a Weyl-type asymptotic law for a counting function for period integrals, via identities that can be considered as a special cases of Jacquet's relative trace formula [36,37], and an average version of the holomorphic QUE conjecture for automorphic cusp forms associated to quaternion algebras [3]. Heat asymptotics on spaces of automorphic forms are of continuing interest; see [32] and its references.…”

Addendum to “Singular equivariant asymptotics and Weyl’s law”

“…Let M be a compact n-dimensional Riemannian manifold without boundary, carrying an isometric and effective action of a connected compact Lie group G with Lie algebra g. In the study of the spectral geometry of M one is led to an examination of the asymptotic behavior of oscillatory integrals of the form (1.1) I(µ) := T * U Ĝ e iµΦ(x,ξ,g) a µ (x, ξ, g) dg d (T * U ) (x, ξ), µ → +∞, where (γ, U ) denotes a chart on M , a µ ∈ C ∞ c (T * U × G) an amplitude that might depend on the parameter µ > 0, and the phase function is given by Φ(x, ξ, g) := γ(x) − γ(g • x), ξ , (x, ξ) ∈ T * x U, g ∈ G, see [7,6]. Here dg stands for the normalized Haar measure on G, and d(T * U ) for the canonical symplectic volume form of the co-tangent bundle of U , which coincides with the Riemannian volume form given by the Sasaki metric on T * U .…”

Singular equivariant asymptotics and the momentum map. Residue formulae in equivariant cohomology

“…By this we are able to obtain asymptotics for I 0 (µ) with remainder estimates in the case of singular group actions. This approach was developed first in [13,36] to describe the spectrum of an invariant elliptic operator on a compact G-manifold, where similar integrals occur, and used in the derivation of equivariant heat asymptotics in [35]. The asymptotic description of I ς (µ) in a neighborhood of ς = 0 then allows us to derive the In what follows, we assume that g is endowed with an Ad (G)-invariant inner product, which allows us to identify g * with g. Let further dX and dξ be corresponding measures on g and g * , respectively, and denote by F g : S(g * ) → S(g), F g : S ′ (g) → S ′ (g * ) the g-Fourier transform on the Schwartz space and the space of tempered distributions, respectively.…”

“…By this we are able to obtain asymptotics for I 0 (µ) with remainder estimates in the case of singular group actions. This approach was developed first in [13,36] to describe the spectrum of an invariant elliptic operator on a compact G-manifold, where similar integrals occur, and used in the derivation of equivariant heat asymptotics in [35]. The asymptotic description of I ς (µ) in a neighborhood of ς = 0 then allows us to derive the following residue formula.…”