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In this article we prove that if is a connected, simply connected, semisimple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra of the Lie algebra of can be endowed with a Koszul grading (extending results of Andersen, Jantzen, and Soergel). We also give information about the Koszul dual rings. In the case of the block associated to a regular character of the Harish-Chandra center, the dual ring is related to modules over the specialized algebra with generalized trivial Frobenius character. Our main tool is the localization theory developed by Bezrukavnikov, Mirković, and Rumynin
We introduce the concept of a geometric categorical action and relate it to that of a strong categorical action. The latter is a special kind of -representation in the sense of Lauda and Rouquier. The main result is that a geometric categorical action induces a strong categorical action. This allows one to apply the theory of strong actions to various geometric situations. Our main example is the construction of a geometric categorical action on the derived category of coherent sheaves on cotangent bundles of Grassmannians
Lusztig constructed a Frobenius morphism for quantum groups at an th root of unity, which gives an integral lift of the Frobenius map on universal enveloping algebras in positive characteristic. Using the Hall algebra, we give a simple construction of this map for the positive part of the quantum group attached to an arbitrary Cartan datum in the nondivisible case
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