Small-time asymptotics of Hermite functions on compact symmetric spaces (Q5955579)

Let \(M\) be a compact connected oriented Riemannian manifold, and \(\rho_t (\cdot,\cdot)\) be the heat kernel on \(M\). If \(D\) is a smooth differential operator on \(M\), then the Hermite function on \(M\) associated to \(D\) at time \(t\) is defined by \(K_D(\cdot,t)= {(D\rho_t) (\cdot) \over\rho_t (\cdot)}\). The author studies an asymptotic expansion of these functions (in powers of \(\sqrt t)\) for small time variables. It is proved that on any such manifold the Hermite function has an asymptotic expansion in powers of \(\sqrt t\), whose coefficients are functions related to polynomials on the tangent space via the exponential mapping. If \(M\) is a compact symmetric space, then one can regard special differential operators called symmetric derivatives and the related Hermite functions. It is proved that the asymptotic expansion of these Hermite functions is a series that contains integral powers of \(t\) only.