In 1995, Stanley introduced the chromatic symmetric function for a graph and studied its expansion in terms of various symmetric functions. Shareshian-Wachs introduced its t-analogue called chromatic quasisymmetric function for a graph whose vertices are labeled by 1,2,…,n. The chromatic quasisymmetric functions for unit interval graphs are known to be related to the cohomology of regular semisimple Hessenberg varieties of type A and characters of finite Hecke algebra of type A at certain Kazhdan-Lusztig basis elements.

In this talk, we introduce a (q,t)-analogue of the chromatic symmetric functions based on a recent work of Syu Kato on a new geometric interpretation of the chromatic symmetric functions for unit interval graphs. We also introduce a (q,t)-deformation of the usual ring structure on symmetric functions so that the (q,t)-chromatic symmetric functions are multiplicative with respect to the new multiplication. We show that the expansions of (q,t)-chromatic symmetric functions in terms of a (q,t)-analogue of elementary symmetric functions essentially coincide with the the expansions of the chromatic quasisymmetric functions in terms of the usual elementary symmetric functions. By specializing q at infinity, we obtain a linear relation between the e-expansion coefficients of chromatic quasisymmetric functions, which leads to our proof of the Stanley-Stembridge conjecture.