Equivariant localization of \(\overline D\)-modules on the flag variety of the symplectic group of degree 4.
type: Item
instance of: scholarly article
title: Equivariant localization of \(\overline D\)-modules on the flag variety of the symplectic group of degree 4.
zbMATH Open document ID: 1133.20038
author: Masaharu Kaneda
author: Ye Jiachen
published in: Journal of Algebra
publication date: 2007-04-11T00:00:00Z
review text: 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 \(G=\text{SL}_3\).
reviewed by: Henning Haahr Andersen
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 5141716
zbMATH Keywords: crystalline differential operators
zbMATH Keywords: localization theorems
zbMATH Keywords: symplectic groups
zbMATH Keywords: flag varieties
zbMATH Keywords: Frobenius morphisms
zbMATH Keywords: invertible sheaves
zbMATH Keywords: simple Lie algebras
zbMATH Keywords: derived equivalences
zbMATH Keywords: enveloping algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.07.023
OpenAlex ID: W2065647309
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DOI: 10.1016/J.JALGEBRA.2006.07.023
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description: scientific article; zbMATH DE number 5141716