Representations with scalar \(K\)-types and applications
Let \(G\) be a noncompact connected semisimple (real) Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). Let \(\mathfrak{g}_0\) and \(\mathfrak{k}_0\) be, respectively, the Lie algebras of \(G\) and \(K\), let \(\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0\) be a Cartan decomposition of \(\mathfrak{g}_0\), and \(\mathfrak{g}\), \(\mathfrak{k}\), \(\mathfrak{p}\) the complexifications of \(\mathfrak{g}_0\), \(\mathfrak{k}_0\), \(\mathfrak{p}_0\), respectively. Given a vector space \(W\), denote by \(S(W)\) the symmetric algebra of \(W\), by \(S_k(W)\) the subspace of \(S(W)\) of the homogeneous elements of degree \(k\), where \(k \in \mathbb{N}^*\), and put \[ S^r(W) = \sum\limits_{0 \leq k \leq r}S_k(W) \qquad (r \in \mathbb{N}^*). \] The author proves the following theorem. Let \(\psi\) be the symmetrization mapping of the universal enveloping algebra \(\mathcal{U}(\mathfrak{g})\) of \(\mathfrak{g}\), i.e., the \(\mathbb{C}\)-linear canonical bijection of \(S(\mathfrak{g})\) onto \(\mathcal{U}(\mathfrak{g})\), suppose that \[ \psi(S^r(\mathfrak{g})^G) + \mathcal{U}(\mathfrak{g})\mathfrak{k} \supseteq \psi(S^r(\mathfrak{p})^K) \qquad (r \in \mathbb{N}^*) \] (true at least for all classical groups \(G\)), and let \(\rho\) be a one-dimensional element of the set of all equivalence classes of irreducible (unitary) representations of \(K\). Then, for each irreducible Harish-Chandra \((\mathfrak{g},K)\)-module \(V\) with nonzero \(K\)-isotypic component of \(V\) of type \(\rho\), the infinitesimal character of \(V\) determines \(V\) up to infinitesimal equivalence. This result is applied to the reductive dual pair \[ (O(p,q), Sp(2n,\mathbb{R})) \subseteq Sp(2N,\mathbb{R}), \] where \(N = 2(p+q)n\) [\textit{R. Howe}, Proc. Symp. Pure Math. Am. Math. Soc. 33, 1, 275--285 (1979; Zbl 0423.22016)], to obtain isomorphisms of certain representations of \(Sp(2n,\mathbb{R})\), for some \(n\), via the local \(\theta\)-correspondence [\textit{R. Howe}, J. Am. Math. Soc. 2, 535--552 (1989; Zbl 0716.22006)].
scientific article; zbMATH DE number 2053059