Macdonald polynomials are symmetric polynomials whose coefficients involve parameters $q,t$. Together with their specializations, Hall-Littlewood polynomials and Schur functions, they are quite ubiquitous in representation theory. For type A, a modified version of Macdonald polynomials exhibits positivity, conjectured by Macdonald in the late 80s and proved by Haiman in 2002. In the meantime, LaPointe-Lascoux-Morse (2003) proposed a conjectural refinement of the Macdonald positivity by introducing the notion of k-Schur functions.

In this talk, we first briefly recall some background on the theory of symmetric functions and Macdonald polynomials. Then, we present the Garsia-Procesi/Garsia-Haiman modules and their characterizations due to Haiman. Using them, we explain a conjectural interpretation of k-Schur functions and the above refined Macdonald positivity proposed by Chen-Haiman (2009).

Then, we introduce a family of algebraic varieties whose Borel-Weil theory represents Catalan symmetric functions, which contain Hall-Littlewood/k-Schur functions as a proper subclass. Using the affine version of the Schur-Weyl duality, we transplant our realization of k-Schur functions to establish one of the above-mentioned conjectures by Chen-Haiman. If time permits, I will explain 1) how the above varieties resolve the vanishing conjectures by Shimozono-Weyman and Blasiak-Morse-Pun, and 2) our approach to the rest of the conjectures by Chen-Haiman.

This talk is based on arXiv:1111.4640, arXiv:2301.00862, and arXiv:2505.23202.