$(\mathfrak{g},K)$-module of $\mathrm{O}(p,q)$ associated with the finite-dimensional representation of $\mathfrak{sl}_2$

TL;DR

This paper constructs specific irreducible modules of the orthogonal group O(p,q) linked to finite-dimensional sl_2 representations, providing explicit formulas for their structure, dimensions, and degrees, extending known minimal representation results.

Contribution

It introduces a new family of irreducible (rak{g},K)-modules of O(p,q) associated with finite-dimensional sl_2 representations, including explicit K-type, Gelfand-Kirillov dimension, and Bernstein degree formulas.

Findings

Gelfand-Kirillov dimension equals p+q-3 for certain modules

Bernstein degree scales linearly with m

Special case m=0 recovers minimal representation

Abstract

The main aim of this paper is to construct irreducible $(\mathfrak{g},K)$-modules of $\mathrm{O}(p,q)$ corresponding to the finite-dimensional representation of $\mathfrak{sl}_2$ of dimension $m+1$ under the Howe duality, to find the $K$-type formula, the Gelfand-Kirillov dimension and the Bernstein degree of them, where $m$ is a non-negative integer. The $K$-type formula for $m=0$ shows that it is nothing but the $(\mathfrak{g},K)$-module of the minimal representation of $\mathrm{O}(p,q)$. One finds that the Gelfand-Kirillov dimension is equal to $p+q-3$ not only for $m=0$ but for any $m$ satisfying $m+3 \leq (p+q)/2$ when $p, q \geq 2$ and $p+q$ is even, and that the Bernstein degree for $m$ is equal to $(m+1)$ times that for $m=0$.