Coincidence of algebraic and smooth theta correspondences

TL;DR

This paper proves that the algebraic and smooth local theta correspondences over the real numbers coincide for dual pairs without quaternionic type I factors, confirming a natural continuity question in representation theory.

Contribution

It establishes the equivalence of algebraic and smooth theta correspondences over  for a broad class of dual pairs, resolving an open automatic continuity problem.

Findings

Algebraic and smooth theta correspondences agree over  for certain dual pairs.

The result holds when the dual pair has no quaternionic type I irreducible factor.

This confirms a natural automatic continuity question in the theory of local theta correspondence.

Abstract

An "automatic continuity" question has naturally occurred since Roger Howe established the local theta correspondence over $\mathbb R$: does the algebraic version of local theta correspondence over $\mathbb R$ agrees with the smooth version? We show that the answer is yes, at least when the concerning dual pair has no quaternionic type I irreducible factor.