Symplectic and orthogonal Robinson-Schensted algorithms
Let \(G_{\mathbb{R}}\) be a linear reductive real Lie group with maximal compact subgroup \(K_{\mathbb{R}}\). Let \(K\) be the complexification of \(K_{\mathbb{R}}\) and let \(\mathcal{B}\) denote the flag variety of the complexified Lie algebra \(\mathfrak{g}\) of \(G_{\mathbb{R}}\). Let \(Q\) be a \(K\)-orbit on \(\mathcal{B}\) and denote by \(\mu\) the moment map from \(T^*\mathcal{B}\) to the nilpotent cone \(\mathcal{N}(\mathfrak{g}^*)\). The paper under review gives a simple algorithm to compute the image under \(\mu\) of the conormal bundle \(T_Q^*(\mathcal{B})\) explicitly when \(G_{\mathbb{R}}\) is \(\text{Sp}(2n,\mathbb{R})\) or \(O(p,q)\). Moreover, for a generic point \(\xi\) of the image of \(\mu\), the author obtains a new parametrization of the orbits of the component group of the centralizer of \(\xi\) in \(G\) on the irreducible components of the Springer fiber \(\mu^{-1}(\xi)\) in terms of the domino tableaux. This parametrization is closely related to the computation of annihilators of the derived functor modules for the groups under consideration. Similar questions were previously treated by A. Yamamoto, but her algorithms are different and not as well suited for the analysis of the components of the Springer fiber. The computations of this paper have very nice combinatorial interpretations as generalizations of the classical Robinson-Schensted algorithm.
scientific article; zbMATH DE number 2164788