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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Equivariant derived category of a complete symmetric variety
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by Stéphane Guillermou PDF
Represent. Theory 9 (2005), 526-577 Request permission


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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Additional Information
  • Stéphane Guillermou
  • Affiliation: Université de Grenoble I, Département de Mathématiques, Institut Fourier, UMR 5582 du CNRS, 38402 Saint-Martin d’Hères Cedex, France
  • Email:
  • Received by editor(s): March 28, 2005
  • Published electronically: October 19, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 9 (2005), 526-577
  • MSC (2000): Primary 16E45; Secondary 55N91
  • DOI:
  • MathSciNet review: 2176937