(g, K)-module of O(p, q) associated with the finite-dimensional representation of sl(2)

The main aim of this paper is to show that one can construct (𝔤,K)-modules of O(p,q) associated with the finite-dimensional representation of 𝔰𝔩2 by quantizing the moment map on the symplectic vector space (ℂp+q) ℝ and using the fact that (O(p,q),SL2(ℝ)) is a dual pair. Then one obtains the K-type formula, the Gelfand–Kirillov dimension and the Bernstein degree of them for all non-negative integers m satisfying m + 3 ≤ (p + q)/2 when p,q ≥ 2 and p + q is even. In fact, one finds that the Gelfand–Kirillov dimension is equal to p + q − 3 and the Bernstein degree is equal to 4(m + 1)(p + q − 4)!/((p − 2)!(q − 2)!).