Semiclassical Asymptotics on Stratified Manifolds

We study the problem on semiclassical asymptotics for (pseudo)differential equations with singularities on a stratified manifold of a special form—the orbit space \(X\) of a smooth action of a compact Lie group \(G\) on a smooth manifold \(M\). The operators under consideration are obtained as the restriction of \(G\)-invariant operators with smooth coefficients on \(M\) to the subspace of \(G\)-invariant functions, naturally identified with functions on \(X\), and have singularities on strata of positive codimension. The asymptotics are associated with Lagrangian manifolds in the phase space defined by the Marsden–Weinstein symplectic reduction of the cotangent bundle \(T^*M\) under the action of the group \(G\); rapidly oscillating integrals defining the Maslov canonical operator on such manifolds contain exponentials as well as special functions related to representations of the group \(G\). For the simplest stratified manifold—a manifold with boundary obtained as the orbit space of a semi-free action of the group \( \mathbb{S} ^1\) on a closed manifold—the corresponding construction of semiclassical asymptotics was realized earlier. Note that, in this case, the class of equations under consideration on manifolds with boundary includes the linearized shallow water equations in a basin with a sloping beach. The present paper deals with the general case.