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We study automorphic categories of nilpotent sheaves under degenerations of smooth curves to nodal Deligne-Mumford curves. Our constructions realize affine Hecke operators as the result of bubbling projective lines from marked points. We use this to construct a gluing functor from the automorphic category of a nodal Deligne-Mumford curve to the automorphic category of a smoothing.
We explore induced mappings between character varieties by mappings between surfaces. It is shown that these mappings are generally Poisson. We also explicitly calculate the Poisson bi-vector in a new case.
We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis
We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bod
We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $Sigma$ on a finite abelian category $mathcal{M}$, we introduce the notion of a $Sigma$-twisted trace on the cla
Finite group actions on free resolutions and modules arise naturally in many interesting examples. Understanding these actions amounts to describing the terms of a free resolution or the graded components of a module as group representations which, i