Howe correspondence for real unitary groups
Let \((G,G')\) be a reductive dual pair in \(H\), i.e., a pair of real reductive Lie groups embedded in a large group \(H\) in such a way that they are each other's centralizers in \(H\). The restriction of a representation \(\rho\) of \(H\) to \(G\cdot G'\) may be regarded as a representation of \(G\times G'\) via the natural mapping \(G\times G'\to G\cdot G'\). Let \({\mathcal R}\) be the set of all equivalence classes of irreducible representations \(\pi\otimes\pi'\) of \(G\times G'\), where \(\pi\) and \(\pi'\) are, respectively, irreducible representations of \(G\) and \(G'\), such that, if \(Y\) is the carrier space of \(\rho\), there exists a \(G\times G'\)-invariant subspace \(X\) of \(Y\) with the action of \(\rho| G\times G'\) on the quotient space \(Y/X\) equivalent to \(\pi\otimes\pi'\). If the equivalence class of \(\pi\otimes\pi'\) belongs to \({\mathcal R}\), then \(\pi\) is said to correspond to \(\pi'\) and this dual pair correspondence for \((G,G')\) defines a relationship between some of the irreducible representations of \(G\) and \(G'\). If now \(H\) is the symplectic group \(Sp(2n,\mathbb{R})\) \((n\in\mathbb{N}^*)\), \(\widetilde{Sp}(2n,\mathbb{R})\) its nontrivial twofold covering, and \(\widetilde G,\widetilde G'\) the inverse images of \(G,G'\), respectively, under the covering mapping \(\widetilde{Sp}(2n,\mathbb{R})\to Sp(2n,\mathbb{R})\), then the dual pair correspondence for \((\widetilde G,\widetilde G')\) is a bijection between certain sets of irreducible representations of \(\widetilde G\) and \(\widetilde G'\) [\textit{R. Howe}, J. Am. Math. Soc. 2, 535-552 (1989; Zbl 0716.22006)]. In this paper, the author determines explicitly this bijection for the reductive dual pairs \((\widetilde U(p,q),\widetilde U(r, s))\) in \(\widetilde{Sp}(2(p+q)(r+s),\mathbb{R})\) with \(r+s=p+q\) and remarks that it may be interpreted as a bijection between representations of \(U(p,q)\) and \(U(r, s)\). She describes it in terms of Langlands parameters and shows that, if \(p,q\) are fixed and \(r,s\) vary with \(r+s=p+q\), then every irreducible admissible representation of \(U(p,q)\) occurs in a dual pair correspondence for \((U(p,q), U(r,s))\) with fixed \(r\) and \(s\). The correspondence for discrete series representations was determined by \textit{J.-S. Li} [Duke Math. J. 61, 913-937 (1990; Zbl 0723.22011)] and Li's result is combined with the induction principle of \textit{S. Kudla} [Invent. Math. 83, 229-255 (1986; Zbl 0583.22010)] in order to obtain the correspondence for induced representations. An example shows that unitarity is not always preserved by dual pair correspondence.
scientific article; zbMATH DE number 1268052
scientific article