(𝔤,K)-module of O(p,q) associated with the finite-dimensional representation of 𝔰𝔩2

Abstract

The main aim of this paper is to show that one can construct [Formula: see text]-modules of [Formula: see text] associated with the finite-dimensional representation of [Formula: see text] by quantizing the moment map on the symplectic vector space [Formula: see text] and using the fact that [Formula: see text] is a dual pair. Then one obtains the [Formula: see text]-type formula, the Gelfand–Kirillov dimension and the Bernstein degree of them for all non-negative integers [Formula: see text] satisfying [Formula: see text] when [Formula: see text] and [Formula: see text] is even. In fact, one finds that the Gelfand–Kirillov dimension is equal to [Formula: see text] and the Bernstein degree is equal to [Formula: see text].