Geometric Langlands and mathematical physics

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The proposal has 4 parts: 1. Hecke eigen-functions over non-archimedian local fields (with D.Kazhdan). Let X be a smooth projective curve curve over a finite field k and let G be a split reductive group. The unramified part of the classical theory of autormorphic forms for the group G and the global field k(X) of rational functions on X studies eigen-functions of the so called Hecke operators which act on functions on k-points of the moduli space BunG_G(X) of G-bundles on X. The (now proved in this case) Langlands conjecture describes the eigen-values in terms of l-adic local systems on X with structure group G^ - the Langlands dual group of G. It was shown in an old work of myself and D.Kazhdan that if one replaces k by a local field, and if one works with half-forms on K-points on Bun_G(X), one can again define Hecke operators. For archimedian K Etingof, Frenkel and Kazhdan conjectured that these operators act on the space of L^2 half-forms and have discrete common spectrum. They conjecturally describe the spectrum in terms of G^ (similar conjectures were produced by D.Gaiotto using quantum field theory). We propose to study similar problem for non-archimedian K. We have already obtained some partial results for G=SL(2). 2. Some category equivalences arising from 3d and 4d quantum field theories (with M.Finkelberg and R.Travkin). We would like to prove a conjecture of D.Gaiotto (motivated by physics arguments) describing categories of representations of various quantum super-groups (e.g. quantized GL(N|M)) in terms of certain categories of sheaves on the affine Grassmannian of some algebraic group. We also explain potential applications of this problem to local geometric Langlands correspondence. We discuss some other equivalences of categories also motivated by physical considerations. 3. Categorification of asymptotic Hecke algebra (with D.Kazhdan and S.Dawydiak) The affine Hecke algebra H attached to a reductive group G is one of the central objects of geometric representation theory. It is an algebra over the ring of Laurent polynomials in some variable q (let us call it A). Following a famous work of Kazhdan and Lusztig, Bezrukavnikov defined two categorifications of H (i.e. monoidal categories whose K-ring is isomorphic to H) and proved their equivalence. One categorification is in terms of certan constructible sheaves on the affine flag variety of G. The other is in terms of coherent sheaves on the so called derived Steinberg variety of G^. On the other hand, Lusztig has defined certain algebra J over the integers which has some very intriguing relation with H. We propose to define similar categorifications of J. This should help understand its relation with H.