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Dualité de Koszul et algèbres de Lie semi-simples en caractéristique positive

Abstract : Recent works of Bezrukavnikov, Mirkovic and Rumynin obtain a good localization theory for Ug-modules in positive characteristic (where g is the Lie algebra of a connected, simply-connected, semi-simple algebraic group), yielding equivalences of derived categories between certain categories of g-modules and certain categories of coherent sheaves of Springer's variety. In this thesis we apply and extend some results of this theory. In chapter II, we give a geometric construction of an action of the extended affine braid group appearing in localization theory. Chapter III contains the main results of this thesis: we develop an appropriate version of a “linear Koszul duality”, which allows us to prove that certain blocks of Ug can be endowed with a Koszul grading, if the characteristic of the base field is sufficiently large. This generalizes previous results of Andersen, Jantzen and Soergel. In chapter IV, in collaboration with Mirkovic, we consider again “linear Koszul duality” in a slightly different, and more general, setting. Finally, chapter I (in collaboration with Bezrukavnikov) gives explicit computations in the case of SL(3) which were the starting point of this work.
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Contributor : Simon Riche <>
Submitted on : Monday, September 14, 2009 - 12:21:41 PM
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Simon Riche. Dualité de Koszul et algèbres de Lie semi-simples en caractéristique positive. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2008. Français. ⟨tel-00416471⟩

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