A geometric realization of Catalan functions

Abstract

We construct smooth projective varieties $\sX_Ψ$, each of which compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\GL(n)$. Each variety $\sX_Ψ$ carries a natural family of line bundles whose spaces of global sections realize, as graded characters, the Catalan functions introduced in Chen's thesis and further studied by Blasiak, Morse, Pun, and Summers. Using the geometry of $\sX_Ψ$, we prove the Chen--Haiman vanishing conjecture and confirm the tame case of the Blasiak--Morse--Pun vanishing conjecture. We further establish the Shimozono--Weyman monotonicity conjectures.

v10, 48 pages. Revised exposition, with no changes to the mathematical content except for weaker assumptions in Corollary 5.17. References updated