Abstract

Recent years have seen a flurry of developments relating homotopy theoretic constructions with results in arithmetic geometry. In this talk, I will discuss another place where homotopy theory plays an important organizational role, namely geometric representation theory.  I will explain some conjectures surrounding the derived geometric Satake equivalence, which relates a category of constructible sheaves of k-modules on the affine Grassmannian of a complex Lie group G to the algebraic geometry of its Langlands dual group G^. When k is the sphere spectrum and G is the trivial group, this relationship is just the Adams-Novikov spectral sequence; and when k is an ordinary commutative ring (and G can be nontrivial), this reduces to Bezrukavnikov-Finkelberg’s derived Satake theorem.  Time permitting, I will explain how these ideas allow one to construct an associative algebra U_F(G) associated to any 1-dimensional formal group law F such that when F is the additive formal group law, U_F(G) is the usual enveloping algebra U(g), and when F is the multiplicative formal group law, U_F(G) is the quantum group U_q(G) (closely related to q-de Rham cohomology, etc.). There are many questions in this area that I do not know how to address, and I hope to share some of them.