L-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction

The preservation principle of local theta correspondences of reductive dual pairs over a p-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams and Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

Correspondences of the Gelfand invariants in reductive dual pairs

For each complex reductive dual pair introduced by R. Howe, this paper presents a formula for the central elements of the universal enveloping algebras given by I. M. Gelfand. This formula provides an explicit description of the correspondence between the ‘centers’ of the two universal enveloping algebras.