Curved Hecke categories

The Hecke algebra admits an involution which preserves the standard basis and exchanges the canonical basis with its dual. This involution is categorified by "monoidal Koszul duality" for Hecke categories, studied in positive characteristic in my previous joint work with Achar, Riche, and Williamson. In this talk I will explain the following rough statement: "The Koszul dual of the Hecke category is equivalent to the derived category of bimodules for a particular Koszul complex in the Hecke category of the Langlands dual." This is motivated by the curved Koszul duality of Positselski and Burke. If time permits, I will also discuss the connection of this duality to a conjectural symmetry in link homology. Based on joint work with Matt Hogancamp.