Dual pairs and Kostant-Sekiguchi correspondence. I
Let \(G=GL(V)\) and \(G'=GL(V')\), where \(V\) and \(V'\) are finite-dimensional vector spaces over \({\mathbb R, C}\), or \({\mathbb H}\) and let \(W=\text{ Hom}(V,V')\oplus\text{ Hom}(V',V)\) with the skew-symmetric form defined by \(\langle(S_1,T_1),(S_2,T_2)\rangle=\text{ Re}\text{ Tr}(S_2T_1-S_1T_2)\). Then, \(G\) and \(G'\) are naturally embedded into \(Sp(W)\) and \((G,G')\) is an irreducible dual pair in \(Sp(W)\) of type \text{ II}. The moment maps \(\tau:W\to{\mathfrak g}\) and \(\tau':W\to{\mathfrak g}'\) are defined by \(\tau(S,T)=T\cdot S\) and \(\tau'(S,T)=S\cdot T\) respectively. Then the authors show that, for a nilpotent \(G\)-orbit \({\mathfrak g}_\lambda\) \(\subset {\mathfrak g}\), \(\tau'(\tau^{-1}(\overline{{\mathfrak g}_\lambda}))\) is the closure of a single nilpotent orbit \({\mathfrak g}'_\lambda\) \(\subset {\mathfrak g}'\): \[ \tau'(\tau^{-1} ( \overline{{\mathfrak g}_\lambda}))=\overline{{\mathfrak g}'_{\lambda'}}, \] where \(\lambda\) is a partition and \(\lambda'\) is explicitly determined by \(\lambda\). Let \({\mathfrak g}={\mathfrak p}\oplus{\mathfrak k}\) and \({\mathfrak g}'={\mathfrak p}'\oplus{\mathfrak k}'\) denote the Cartan decompositions of \({\mathfrak g}\) and \({\mathfrak g}'\) and \(K, K'\) the maximal compact subgroups of \(G,G'\) respectively. Let \(J\) be a compatible complex structure on \(W\) and \(W_{\mathbb C}^+\) the \(+i\)-eigenspace of \(J\) on \(W_{\mathbb C}\). Then the moment maps \(\tau_{c}:W_{\mathbb C}\to{\mathfrak g}_{\mathbb C}\) and \(\tau'_{c}:W_{\mathbb C}\to{\mathfrak g}'_{\mathbb C}\) map \(W_{\mathbb C}^+\) to \({\mathfrak p}_{\mathbb C}\) and \({\mathfrak p}'_{\mathbb C}\) respectively. They also show that the orbit structure of \(\tau'_{c}(\tau^{-1}_{c}(\overline{{\mathfrak p}_\lambda}))\) for a nilpotent \(K_{\mathbb C}\)-orbit \({\mathfrak p}_\lambda\subset {\mathfrak p}_{\mathbb C}\) has the same property: \[ \tau'_c(\tau^{-1}_c(\overline{{\mathfrak p}_\lambda})) =\overline{{\mathfrak p}'_{\lambda'}}. \] The first result for \(G\)-orbits follows from the same arguments in the authors' paper [J. Algebra 190, 518-539 (1997; Zbl 0870.22007)] and the second one for \(K_{\mathbb C}\)-orbits follows from the classification of nilpotent \(K_{\mathbb C}\times K_{\mathbb C}\)-orbits in \(W_{\mathbb C}^+\) in terms of \(ab\)-diagrams. As a consequence of these two results, the following conjecture is proved. Let \({\mathcal O}\) \(\subset {\mathfrak g}\) be a nilpotent \(G\)-orbit. Then \[ S'(\tau'(\tau^{-1}(\overline{\mathcal O}))) =\tau'_c(\tau^{-1}_c(\overline{S({\mathcal O})})), \] where \(S\) and \(S'\) are the Kostant-Sekiguchi correspondences for \(G\) and \(G'\) respectively.
scientific article; zbMATH DE number 1763539
scientific article