Demazure operators via shifted $q=0$ affine algebras

In this article, we construct two families of idempotent operators/projection functors acting on the K-theory/derived category of coherent sheaves on $n$-step partial flag varieties. Those operators can be viewed as a generalization of the Demazure operators for the full flag variety, and the two families of idempotent operators generate respectively two algebras that are similar to the $q=0$ Hecke algebra. The main idea comes from the interpretation of the Demazure operators in terms of elements in the shifted $q=0$ affine algebra. As a main result, we show that a categorical action of the shifted $q=0$ affine algebra naturally induces actions of two variants of the $q=0$ Hecke algebra on each weight category. Finally, we calculate the action of those operators on the basis given by the Kapranov exceptional collection.Comment: Comments are welcom