Categorical Deligne-Langlands and coherent Springer theory

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 (formulated by Scholze, Zhu): 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.

We prove a subcase of this conjecture, i.e. a categorical version of Kazhdan and Lusztig's bijection 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). Namely, we define a coherent Springer sheaf on the stack of unipotent Langlands parameters 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.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)