The Hodge theory of the Hecke category

TL;DR

This paper surveys joint work revealing Hodge theoretic structures within the Hecke category, connecting ideas from Hodge theory to representation theory and comparing with other combinatorial Hodge theories.

Contribution

It uncovers Hodge theoretic structures in the Hecke category, advancing understanding of its combinatorial and geometric properties.

Findings

Hodge structures are present in the Hecke category

Connections between Hodge theory and representation theory are established

Comparisons with other combinatorial Hodge theories are outlined

Abstract

Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category ("Soergel bimodules"). We also outline similarities and differences to other combinatorial Hodge theories.