Noncommutative Resolutions and Canonical Bases
Vasily Krylov (Harvard/CMSA)
Thursday 16th July 15:00-16:00
Maths 311B
Abstract
Kazhdan and Lusztig identified the affine
Hecke algebra with the equivariant K-theory of the Steinberg variety for the
Langlands dual group. Bezrukavnikov categorified this identification and, in
particular, gave a geometric description of the Kazhdan–Lusztig canonical
basis in terms of the classes of simple perverse modules over the corresponding
noncommutative resolution.
We propose an analogous construction in a different setting: the resolution of an affine Schubert variety in type A. We construct the corresponding noncommutative resolution and describe the resulting basis via the action of the quantum loop group. Time permitting, we will emphasize a relation to categorical Howe duality, K-theoretic Satake, Lusztig's cell filtration, and affine Temperley–Lieb algebras. This talk will be based primarily on joint work with Dumanski (arXiv:2606.15981)
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