Differential invariants on symplectic spinors in contact projective geometry

TL;DR

This paper classifies and constructs symplectic spinor differential operators in contact projective geometry, revealing new equivariant operators related to the Segal-Shale-Weil representation and generalized Verma modules.

Contribution

It provides a complete classification and explicit construction of equivariant differential operators on contact projective spaces using advanced representation theory techniques.

Findings

Classification of homomorphisms of generalized Verma modules

Construction of equivariant differential operators

Extension of duality principles in representation theory

Abstract

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of the Segal-Shale-Weil representation twisted by a one-parameter family of characters. The proof is based on the classification of homomorphisms of generalized Verma modules for the Segal-Shale-Weil representation twisted by a one-parameter family of characters, together with a generalization of the well-known duality between homomorphisms of generalized Verma modules and equivariant differential operators in the category of inducing smooth admissible modules.