Non-decreasable K-types are unitarily small

TL;DR

This paper proves that non-decreasable K-types in certain Lie groups are unitarily small, confirming a previously conjectured relationship between these concepts.

Contribution

It establishes that all non-decreasable K-types are unitarily small, resolving Conjecture 2.1 from prior research.

Findings

Confirmed that non-decreasable K-types are unitarily small.

Resolved Conjecture 2.1 in the literature.

Provides a link between K-type properties and unitarily small representations.

Abstract

Let $G$ be a connected simple non-compact real reductive Lie group with a maximal compact subgroup $K$. This note aims to show that any non-decreasable $K$-type (in the sense of the first named author) is unitarily small (in the sense of Salamanca-Riba and Vogan). This answers Conjecture 2.1 of \cite{D} in the affirmative.