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Perverse Monodromic Sheaves

Abstract : The category P(G/B) of perverse sheaves on the flag variety G/B of a complex G reductive algebraic group is known to play a very important role in representation theory. R. Bezrukavnikov and S. Riche obtained in 2018 a completely general description of this category as an explicitly determined full subcategory in a category of modules over an explicitly determined ring (a “Soergel description”, as their arguments gives a geometric version of results obtained in the late 90’s by Soergel in his study of the principal bloc of the BGG category O of a semisimple complex Lie algebra). In order to establish their results, they crucially use a construction due to Verdier, called monodromy. This monodromy action gives a characterization of the perverse category as a category of perverse sheaves on the basic affine space G/U. Twisting this monodromy action, one then obtains a whole new bunch of categories, which can be viewed as deformation of P(G/B) along a parameter varying in an algebraic torus. Once we have these deformations (the categories of “monodromic perverse sheaves”) at hand, one may wonder if they share some of the known properties of the category P(G/B) itself. Following previous works of Bezrukavnikov--Yun, Bezrukavnikov--Riche and Lusztig--Yun, we give an extensive study of these categories. Eventually, we are able to show: 1) the monodromic category admits a natural highest weight structure 2) the monodromic category splits as a direct sum of “bloc subcategories” 3) for any bloc, we obtain a monodromic version of geometric Ringel duality relating subcategories of tilting and projective objects, and we obtain a Sorgel type description of these categories, as explicit full subcategories in categories of modules over an explicitly determined ring 4) there exists an equivalence of categories between the neutral block monodromic perverse subcategory and the category of perverse sheaves on the flag variety of an appropriate (“endoscopic”) complex algebraic group.
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https://tel.archives-ouvertes.fr/tel-03355535
Contributor : Valentin Gouttard Connect in order to contact the contributor
Submitted on : Monday, September 27, 2021 - 2:05:52 PM
Last modification on : Monday, December 13, 2021 - 9:16:42 AM
Long-term archiving on: : Tuesday, December 28, 2021 - 6:41:18 PM

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Valentin Gouttard. Perverse Monodromic Sheaves. Mathematics [math]. Université Clermont Auvergne, 2021. English. ⟨NNT : ⟩. ⟨tel-03355535⟩

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