Non-decreasable K-types are unitarily small

TL;DR

This work proves that every non-decreasable $K$-type for a connected simple non-compact real classical group $G$ is unitarily small in the sense of Salamanca-Riba and Vogan, answering Conjecture 2.1 of [D] affirmatively. The authors develop an equal-rank framework using Vogan diagrams, and perform a detailed decomposition analysis of compact simple roots to constrain $K$-types. They establish the main result for equal-rank classical groups, then extend to non-equal-rank classical cases by case analysis of $\mathfrak{sl}(2n{+}1,\mathbb{R})$, $\mathfrak{sl}(2n,\mathbb{R})$, $\mathfrak{sl}(n,\mathbb{H})$, and $\mathfrak{so}(2p{+}1,2q{+}1)$, showing that non-decreasable $K$-types collapse to a trivial or very small set, all of which lie in the $u$-small convex hull. The results hinge on refining the Helgason–Johnson bound and applying SV’s dominance criterion to deduce $2\rho_{\rm n}-\mu$ is $\mathfrak g$-dominant, hence $\mu$ is $u$-small, thereby advancing the Dirac-series program for these groups.

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.

Figures (5)

• Figure 1: Non-decreasable $K$-types and u-small $K$-types for $U(2, 1)$
• Figure 2: Vogan diagram for $\mathfrak{su}(p, q)$
• Figure 3: Dynkin diagram for type $B_n$
• Figure 4: Dynkin diagram for type $C_n$
• Figure 5: Dykin diagram for type $D_n$

Theorems & Definitions (25)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof