Syu Kato part1 "A geometric realization of Catalan functions"
Channel: Schubert Seminar
Date: 2023-02-06

Abstract: In the late 1980s, R.Brylinski observed that (a version of) the Hall-Littlewood polynomials of a simple algebraic group $G$ are the Euler-Poincar\'e characteristics of line bundles on the total space of the cotangent bundle of the flag manifold of $G$. This observation gained a geometric meaning when Broer proved the higher cohomology vanishing of these line bundles. These ideas are polished by the works of Shimozono-Weyman and Chen’s Ph.D thesis for $G = \mathop{GL} ( n )$ (instead of simple algebraic groups), and the counterpart of Brylinski’s observation was subsequently proved by Blasiak-Morse-Pun(-Summers) as the theory of Catalan functions. In this talk, we briefly discuss some of these ideas, and then exhibit a smooth algebraic variety whose positive line bundles realize the Catalan polynomials as the graded characters of their global sections. This particularly implies the counterpart of Broer’s result in the theory of Catalan functions.