Hessenberg varieties are subvarieties of flag varieties whose cohomology comes with a natural Weyl group action. In type A, these representations are closely connected to positivity questions in the Stanley-Stembridge and Haiman conjectures. This raises the question of whether such positivity phenomena extend naturally to other Lie types. A first step is to compute these Weyl group representations.

After briefly reviewing Hessenberg varieties and our recent results relating them to Lusztig varieties, I will present our construction of the Hessenberg functor on the Hecke category. This functor gives rise to a representation-valued trace on the Hecke algebra, called the Hessenberg character, which recovers the graded cohomology representations of a natural class of Hessenberg varieties. We then identify the Hessenberg functor with a microlocalization of the functor underlying Lusztig’s character, allowing us to use his theory of character sheaves and nonabelian Fourier transform to obtain an explicit character formula.

Based on joint work with Patrick Brosnan and Jaehyun Hong.