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Schieder Bialgebra and the Geometric Satake Correspondence

Student: Vasily Krylov

Supervisor: Mikhail V. Finkelberg

Faculty: Faculty of Mathematics

Educational Programme: Mathematics (Master)

Final Grade: 10

Year of Graduation: 2019

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $\Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\nvee)$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\gvee$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Full text (added May 31, 2019)

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