In-Person Talk 

I will explain how ideas from the (geometric) Langlands program help solve the following purely algebraic problem: describe the ring of conjugation-invariant functions on the scheme of commuting pairs in a complex reductive group. The answer was known up to nilpotents, and we show that this ring is indeed reduced. We also describe the ring of invariant functions on the derived version of the commuting scheme. The proof brings in seemingly unrelated objects such as the affine Hecke category and character sheaves (of the Langlands dual group).

This is joint work with Penghui Li and David Nadler.