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Representation Theory

ISSN 1088-4165



Equivariant derived category of a complete symmetric variety

Author: Stéphane Guillermou
Journal: Represent. Theory 9 (2005), 526-577
MSC (2000): Primary 16E45; Secondary 55N91
Published electronically: October 19, 2005
MathSciNet review: 2176937
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Abstract: Let $ G$ be a complex algebraic semi-simple adjoint group and $ X$ a smooth complete symmetric $ G$-variety. Let $ L= \bigoplus_\alpha L_\alpha$ be the direct sum of all irreducible $ G$-equivariant intersection cohomology complexes on $ X$, and let $ \mathcal E= \operatorname{Ext}^\cdot_{\mathrm{D}_G(X)}(L,L)$ be the extension algebra of $ L$, computed in the $ G$-equivariant derived category of $ X$. We considered $ \mathcal E$ as a dg-algebra with differential $ d_\mathcal E =0$, and the $ \mathcal E_\alpha = \operatorname{Ext}^\cdot_{\mathrm{D}_G(X)}(L,L_\alpha)$ as $ \mathcal E$-dg-modules. We show that the bounded equivariant derived category of sheaves of $ \mathbf{C}$-vector spaces on $ X$ is equivalent to $ \mathrm{D}_\mathcal E\langle \mathcal E_\alpha \rangle$, the subcategory of the derived category of $ \mathcal E$-dg-modules generated by the $ \mathcal E_\alpha$.

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Stéphane Guillermou
Affiliation: Université de Grenoble I, Département de Mathématiques, Institut Fourier, UMR 5582 du CNRS, 38402 Saint-Martin d’Hères Cedex, France

Received by editor(s): March 28, 2005
Published electronically: October 19, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.