Some results on \({\mathcal D}\)-modules on Borel varieties in characteristic \(p>0\)
Let \(G\) be a semisimple connected and simply connected group over an algebraically closed field \(k\) of characteristic \(p > 0\), let \(B = TU\) be a Borel subgroup of \(G\), and let \(X = G/B\) be the Borel variety. Let \({\mathcal D}_X\) denote the sheaf of differential operators on the structure sheaf \({\mathcal O}_X\). \({\mathcal D}_X\) has distinct right and left \({\mathcal O}_X\)-module structures and is not Noetherian. For an invertible sheaf \(L\) on \(X\), \({\mathcal D}_L \simeq L \otimes {\mathcal D}_X \otimes L^{-1}\) denotes the sheaf of differential operators on \(L \). In particular, one can take the invertible sheaf \(L = {\mathcal O} (\lambda)\) on \(X\) associated to a dominant weight \(\lambda\) of \(T\); the corresponding sheaf of differential operators is denoted by \({\mathcal D}_\lambda\) and its global sections by \(\Gamma_\lambda\). The paper under review studies certain modules over \(\Gamma_\lambda\) and \({\mathcal D}_\lambda\). In particular, it is shown that the Weyl module with highest weight \(\lambda\) is irreducible over \(\Gamma_\lambda\). Furthermore, up to isomorphism, \({\mathcal O}(\lambda)\) is the only indecomposable coherent \({\mathcal D}_\lambda\)-module (theorem 2.2). Another result of the paper is that there exists a unique \(B^-\)-equivariant \({\mathcal D}_\lambda\)-module with support on a given Schubert variety.
scientific article; zbMATH DE number 763965