Multiplicity on a Richardson variety in a cominuscule \(G/P\) (Q2847026)

Let \(G\) be a connected complex semisimple algebraic group and \(P\) be a parabolic subgroup of \(G\). Let \(T \subset P\) be a maximal torus and \(B\) be a Borel subgroup such that \(T \subset B \subset P\). This data determines a Weyl group \(W\), a system of positive roots \(R^+\) and a system of simple roots \(S\). Let \(W_P \subset W=W_G\) be the subgroup associated to \(P\) and \(W^P=W/W_P\) be the set of minimal representatives. For \(\nu, \omega \in W^P\), let \(X_{\omega}\) be the Schubert variety associated to \(\omega\) and \(X^{\nu}\) be the opposite Schubert variety associated to \(\nu\). The intersection \(X_{\omega}^{\nu}=X_{\omega} \cap X^{\nu}\) is called a \textit{Richardson variety}.NEWLINENEWLINEFor a maximal parabolic \(P\), let \(\alpha\) be a simple root \textit{associated} to \(P\) i.e., \(W_P\) is generated by reflections \(s_{\delta}\) where \(s_{\delta} \in S \setminus \{\alpha\}\). Then \(P\) (or \(\alpha\)) is said to be \textit{cominuscule} if \(\alpha\) occurs with coefficient \(1\) in the decomposition of the highest root of \(R^+\) and \textit{minuscule} if \(\alpha^{\vee}\) is cominuscule in the dual root system \(R^{\vee}\). A flag variety \(G/P\) is said to be \textit{cominuscule} if \(P\) is cominuscule.NEWLINENEWLINEFor Richardson varieties in a minuscule \(G/P\), the multiplicity of a \(T\)-fixed point has been determined by \textit{V. Kreiman} and \textit{V. Lakshmibai} [Contributions to automorphic forms, geometry, and number theory. Baltimore, MD: Johns Hopkins University Press. 573--597 (2004; Zbl 1092.14059)]NEWLINENEWLINEIn this paper the author determines the multiplicity of an arbitrary point on a Richardson variety in a cominiscule partial flag variety \(G/P\). The main result of the paper is:NEWLINENEWLINETheorem: Assume \(P\) is cominuscule. Let \(m \in X_{\omega}^{\nu}\) be an arbitrary point and denote by \(\mu_{\omega}\) (resp. \(\mu^{\nu}, \mu_{\omega}^{\nu}\)) the multiplicity of \(m\) on \(X_{\omega}\) (resp. \(X^{\nu}\), \(X_{\omega}^{\nu}\)). Then NEWLINE\[NEWLINE\mu_{\omega}^{\nu}=\mu_{\omega} \mu^{\nu}.NEWLINE\]