Equivariant localization of \(\overline D\)-modules on the flag variety of the symplectic group of degree 4.
Let \(G\) denote a semisimple algebraic group over a field of characteristic \(p>0\). Then \textit{R. Bezrukavnikov, I. Mirković}, and \textit{D. A. Rumynin} [Ann. Math. (2) (to appear)] have established a derived localization theorem for the sheaf \(\mathcal D\) of algebras of crystalline differential operators on the flag variety \(X=G/B\). Let now \(\overline{\mathcal D}=\mathcal End_{X^{(1)}}(\mathcal O_X)\) where \(X^{(1)}\) is the twist by the Frobenius morphism \(F\) on \(X\). Then \(\overline{\mathcal D}\) is a central reduction of \(\mathcal D\). The authors prove in this paper a derived localization theorem for \(\overline{\mathcal D}\) in the case where \(G=\text{Sp}_4\). They do so by checking via explicit calculations of the socle series for baby Verma modules that \(F_*\mathcal O_X\) is tilting (this requires \(p\geq 5\)). This generalizes earlier work of the first author with \textit{Y. Hashimoto} and \textit{D. Rumynin} [Contemp. Math. 413, 43-62 (2006; Zbl 1121.14041)] where the same result is proved for \(G=\text{SL}_3\).
scientific article; zbMATH DE number 5141716
scientific article