Langlands Duality for the Affine Hecke Category

Explore a 51-minute lecture on Langlands duality for the affine Hecke category, delivered by Roman Bezrukavnikov at ICBS2024. Delve into the equivalence between affine and coherent categorifications of the affine Hecke algebra, examining its applications and generalizations as a special case of geometric local Langlands duality. Discover the substantial implications for representation theory across various mathematical structures, including semi-simple Lie algebras in positive characteristic, quantum groups at roots of unity, affine Lie algebras at critical levels, and reductive algebraic groups in positive characteristic. Investigate the role of the equivalence in local L-factors of p-adic group representations and character sheaves on loop groups. Gain insights into the perverse t-structure on the constructible side, described through the noncommutative Springer resolution. If time allows, explore potential generalizations to other symplectic singularities. This talk draws from collaborations with I. Mirkovic, Q. Lin, S. Riche, A. Okounkov, I. Losev, A. Braverman, M. Finkelberg, D. Kazhdan, Y. Varshavsky, and D. Ciubotaru.