The splitting-rank derived Satake equivalence

Abstract

In this thesis, we examine the geometry of symmetric spaces, using the lens of the Langlandsprogram. Symmetric spaces are a useful class of topological manifolds including such familiar spaces as the spheres, real and complex projective spaces, and Grassmannians. These are a class of homogeneous spaces–quotients of compact Lie groups Gc by subgroups Kc. It is known from the time of Bott and Mitchell [BS58; Mit88] that the combinatorial data attached to Gc and Kc controls the geometry of the symmetric space. We elaborate on this relationship. Ben-Zvi, Sakellaridis, and Venkatesh postulate a much stronger result–the derived algebraic geometry of a symmetric space is completely described by the representation theory of an auxiliary ”dual” group G∨ X attached to the space [BZSV]. (In fact, they conjecture much more–the result should hold for many spherical varieties as well.) These Relative Langlands Conjectures directly generalize the Langlands program and its geometric counterpart from compact Lie groups to other spaces. We prove the conjectures of [BZSV] for the case of splitting-rank symmetric spaces.