Stable Bases of the Springer Resolution and Representation Theory

Abstract

In this expository note, we review the recent developments about Maulik and Okounkov’s stable bases for the Springer resolution. In the cohomology case, we compute the action of the graded affine Hecke algebra on the stable basis, which is used to obtain the localization formulae. We further identify the stable bases with the Chern–Schwartz–MacPherson classes of the Schubert cells. This relation is used to prove the positivity conjecture of Aluffi and Mihalcea. For the K theory stable basis, we first compute the action of the affine Hecke algebra on it, which is used to deduce the localization formulae via root polynomial method. Similar as the cohomology case, they are also identified with the motivic Chern classes of the Schubert cells. This identification is used to prove the Bump–Nakasuji–Naruse conjecture about the unramified principal series of the Langlands dual group over non-Archimedean local fields. In the end, we study the wall R-matrices, which relate stable bases for different alcoves. As an application, we give a categorification of the stable bases via the localization of Lie algebras over positive characteristic fields.