Orbit correspondences for real reductive dual pairs (Q611875)

To describe the main results of the paper under review, let \((V,\langle\cdot,\cdot\rangle)\) and \((W,(\cdot,\cdot))\) denote a finite-dimensional symplectic real vector space and a finite-dimensional Euclidean real vector space, respectively. Consider the linear isomorphisms \(\varphi_1\colon V\otimes W\to\text{Hom}(V,W)\), \(\varphi_1(v\otimes w)=\langle v,\cdot\rangle w\), and similarly, \(\varphi_2\colon V\otimes W\to\text{Hom}(W,V)\), \(\varphi_2(v\otimes w)=(w,\cdot)v\). Then define \(\tau_1\colon V\otimes W\to\text{End}(V)\), \(\tau_1(\gamma)=\varphi_2(\gamma)\circ\varphi_1(\gamma)\), and \(\tau_2\colon V\otimes W\to\text{End}(W)\), \(\tau_2(\gamma)=\varphi_1(\gamma)\circ\varphi_2(\gamma)\). One can show that the range of \(\tau_1\) is contained in \({\mathfrak s}{\mathfrak p}(V)\), and the range of \(\tau_2\) is contained in \({\mathfrak s}{\mathfrak o}(W)\). The main theorem proved in the present paper says that if \({\mathcal O}\) and \({\mathcal Q}\) are nilpotent orbits in \({\mathfrak s}{\mathfrak p}(V)\) and \({\mathfrak s}{\mathfrak o}(W)\) which are contained in the images of \(\tau_1\) and \(\tau_2\), respectively, then each of the sets \(\tau_2(\tau_1^{-1}(\overline{\mathcal O}))\) and \(\tau_2(\tau_1^{-1}(\overline{\mathcal Q}))\) is the union of at most two closures of nilpotent orbits in \({\mathfrak s}{\mathfrak o}(W)\) and \({\mathfrak s}{\mathfrak p}(V)\), respectively. The author also discusses the significance of this result in the representation theory of reductive Lie groups.