Equivariant local scaling asymptotics for smoothed T\"{o}plitz spectral projectors

Abstract

Let $X$ be the unit circle bundle of a positive line bundle on a Hodge manifold. We study the local scaling asymptotics of the smoothed spectral projectors associated to a first order elliptic T\"{o}plitz operator $T$ on $X$, possibly in the presence of Hamiltonian symmetries. The resulting expansion is then used to give a local derivation of an equivariant Weyl law. It is not required that $T$ be invariant under the structure circle action, that is, $T$ needn't be a Berezin-T\"{o}plitz operator.