23 June 2021

Hankyung Ko - Monoidal actions of the Hecke category and Kostant's problem

Let g be a semisimple complex Lie algebra and let M be a g-module. Consider A(M), the space of linear endomorphisms on M on which the adjoint action of g is finite. A classical question of Kostant is: for which simple module M is the canonical map from U(g) to A(M) surjective? I will reformulate this problem using the language of monoidal (or 2-)categories. If M belongs to the BGG category O, then the answer to the problem is equivalent to equivalence of certain categorical actions of the Hecke category, and is determined by decomposing the action of translation functors on M. This leads to a conjectural answer to Kostant's problem in terms of the Kazhdan-Lusztig basis for the Hecke algebra. This is a joint work with Walter Mazorchuk and Rafael Mrden.