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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.