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\((\mathfrak{g},K)\)-module of \(\text{O}(p,q)\) associated with the finite-dimensional representation of \(\mathfrak{sl}_2\). (English) Zbl 1495.22007

Summary: The main aim of this paper is to show that one can construct \((\mathfrak{g},K)\)-modules of \(\text{O}(p,q)\) associated with the finite-dimensional representation of \(\mathfrak{sl}_2\) by quantizing the moment map on the symplectic vector space \((\mathbb{C}^{p+q})_{\mathbb{R}}\) and using the fact that \((\text{O}(p,q),\mathrm{SL}_2(\mathbb{R}))\) 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 \(m+3\le (p+q)/2\) when \(p,q\ge 2\) 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 \(4(m+1)(p+q-4)!/((p-2)!(q-2)!)\).

MSC:

22E46 Semisimple Lie groups and their representations
17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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