2021-01-29 Prof. Harrison Chen (Cornell University)

Abstract:
The local Langlands correspondence roughly predicts a bijection between irreducible representations of p-adic or loop groups (on the "automorphic side") and certain Langlands parameters in the Langlands dual group (on the "spectral side"). There has been recent interest in a categorical form of this conjecture: on the spectral side, upgrading the set of Langlands parameters to the category of coherent sheaves on a moduli stack of parameters, and on the automorphic side, upgrading the set of irreducible representations to either Frobenius-twisted adjoint equivariant sheaves on the loop group, or sheaves on the moduli stack of principal bundles on the Fargues-Fontaine curve. Both of the automorphic categories contain the category of representations as a full subcategory (as sheaves on the closed and open orbit respectively).
Kazhdan and Lusztig proved a subcase of the local Langlands conjecture by Deligne-Langlands, between principal series irreducibles (i.e. those with Iwahori-fixed vectors) and "Springer block" unipotent Langlands parameters (i.e. certain q-commuting semisimple-nilpotent pairs). We lift this statement to a categorical one. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra.
Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category. This is a joint work with David Ben-Zvi, David Helm and David Nadler.