Generalised Whittaker models as instances of relative Langlands duality

TL;DR

This paper explores how generalized Whittaker models fit into the duality framework of the relative Langlands program, revealing new connections between branching problems through quantization and duality theories.

Contribution

It characterizes generalized Whittaker models within the duality theory for orthogonal and symplectic groups and provides examples satisfying conjectural duality expectations.

Findings

Characterization of generalized Whittaker models for specific groups.

Identification of an infinite family of duality examples.

Verification of duality predictions via the theta correspondence.

Abstract

The recent proposal by Ben-Zvi, Sakellaridis and Venkatesh of a duality in the relative Langlands program, leads, via the process of quantization of Hamiltonian varieties, to a duality theory of branching problems. This often unexpectedly relates two a priori unrelated branching problems. We examine how the generalised Whittaker (or Gelfand-Graev) models serve as the prototypical example for such branching problems. We give a characterization, for the orthogonal and symplectic groups, of the generalised Whittaker models possibly contained in this duality theory. We then exhibit an infinite family of examples of this duality, which, provably at the local level via the theta correspondence, satisfy the conjectural expectations of duality.