Vanishing cycles on Grassmannians
Let X denote the flag variety associated to a semi-simple complex group G. For an element w of the Weyl group of G let \(L_ w\) denote the irreducible D-module with regular singularities on X supported on the Schubert variety \(\bar X_ w\). The following conjecture is due to \textit{Kazhdan-Lusztig}: ``For \(G=SL_ n\), the singular support of \(L_ w\) is an irreducible subvariety of \(T^*X''.\) The authors treat the case of Grassmannians (degenerate flag varieties for \(SL_ n)\) and prove the following theorem: ``Let \(G_{k,\ell}\) denote the Grassmannian of k-planes in \(k+\ell\) dimensional complex vector space. Let \(\bar X_{\lambda}\) be a Schubert variety in \(G_{k,\ell}\). Let \(L_{\lambda}\) denote the irreducible holonomic D- module with regular singularities on \(G_{k,\ell}\) supported on \(\bar X_{\lambda}\). Then the singular support of \(L_{\lambda}\) is an irreducible subvariety of \(T^*G_{k,\ell}''.\) The theorem is reduced to study of Zelevinsky's resolutions using the formula of Brylinski for calculation of multiplicities of components of the characteristic cycle of a D-module in terms of vanishing cycles. Then, the proof is based on a detailed analysis of the geometry of the resolutions of singularities of Schubert varieties in Grassmannians introduced by Zelevinsky.
multiplicities of components of the characteristic cycle of a D-module
scientific article; zbMATH DE number 4200385