Perverse Coherent Sheaves in Representation Theory
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dumanskiy-dumanski-phd-math-thesis2026.pdf
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Author(s)
Dumanski, Ilya
Advisor(s)
Bezrukavnikov, Roman
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
A perverse coherent sheaf is a natural algebro-geometric concept arising in several core constructions related to geometric Langlands duality. However, some of the key tools developed for well-studied constructible perverse sheaves are not available in the coherent setting; thus the theory presents a new set of challenges. We develop the theory in three directions. First, we study the coherent Satake category by relating it to the category of modules over the quantum affine group, and construct short exact sequences which conjecturally are categorical cluster mutations. Second, we define the non-commutative affine Schubert varieties and the perversely-exotic t-structure on their resolutions in type A. We identify the corresponding basis of simples in equivariant K-theory with Lusztig’s canonical basis for quantum affine groups. The proof goes by establishing properties of parabolic Bezrukavnikov equivalences. Third, we propose the notion of perverse coherent sheaves on symplectic singularities and study its properties. We demonstrate the way in which this construction generalizes the two previously studied instances of perverse coherent t-structure.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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