A geometric realization of Catalan functions

Abstract

We construct a smooth projective variety ${\mathcal{X}}_{\hat{I}}¨$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{GL}(n)$, determined by a root ideal $\hat{I}¨$. A natural family of line bundles on ${\mathcal{X}}_{\hat{I}}¨$ yields the Catalan functions -- symmetric functions introduced by Chen--Haiman and studied further by Blasiak--Morse--Pun--Summers. By analyzing the geometry of ${\mathcal{X}}_{\hat{I}}¨$, we prove the vanishing conjecture of Chen--Haiman, confirm the tame case of the vanishing conjecture of Blasiak--Morse--Pun, and establish the monotonicity conjectures of Shimozono--Weyman.

v9, 43 pages. Minor revision to v8. Lemma 3.8 (previously implicit in the proof of Theorem 3.9) is recorded and Section 4.2 is replaced with a shorter account. Few typos corrected