Parity preservation of $K$-types under theta correspondence

TL;DR

This paper demonstrates that the parity of $K$-types is preserved under Howe's theta correspondence, and applies this to show parity preservation in all $K$-types of any irreducible $(rak{g},K)$-module in reductive dual pairs.

Contribution

It introduces a degree-parity preservation property for $K$-types under theta correspondence and extends this to all $K$-types in irreducible modules across reductive dual pairs.

Findings

Parity of $K$-types is preserved under theta correspondence.

All $K$-types in irreducible $(rak{g},K)$-modules maintain parity.

The result applies to arbitrary reductive dual pairs.

Abstract

This note shows a property of degree-parity preservation for $K$-types under Howe's theta correspondence. As its application, we deduce the preservation of parity of all $K$-types occurring in an arbitrary irreducible $(\mathfrak{g},K)$-module of any Lie group in reductive dual pairs.