A geometric realization of Catalan functions
Syu Kato
TL;DR
The paper constructs a smooth projective variety ${oldsymbol{ m X}}_{oldsymboligPsi}$ that compactifies the equivariant subbundle $T^*_{oldsymboligPsi}X$ of the cotangent bundle of the flag variety for ${G}={ m GL}(n)$, with a $(G imes{ m C}^ imes)$-action. It shows that line bundles ${oldsymbol{ m O}}_{oldsymbol{ m X}}(oldsymbol ewlambda)$ encode the Catalan functions $H(oldsymboligPsi;oldsymbol ewlambda)$ as graded characters, via $H(oldsymboligPsi;oldsymbol ewlambda)=[H^0({oldsymbol{ m X}}_{oldsymboligPsi}, {oldsymbol{ m O}}_{oldsymbol{ m X}}(oldsymbol ewlambda))]^ig)$ with $q$ inverted. The authors prove the Chen–Haiman vanishing conjecture in full generality (and, consequently, the Broer vanishing and Shimozono–Weyman conjectures) in the tame case, and deduce monotonicity of Catalan-multiplicities akin to SW00. The construction provides a geometric realization of Catalan functions compatible with Demazure theories, and yields new insights into representations of $ rak{gl}(n)[z]$ via simple-head properties and filtered Demazure structures, with corollaries for nilpotent orbit closures and topological field-theory-inspired perspectives.
Abstract
We construct a smooth projective variety $\mathscr X_Ψ$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{GL}(n)$, determined by a root ideal $Ψ$. A natural family of line bundles on $\mathscr X_Ψ$ yields the Catalan functions -- symmetric functions introduced by Chen--Haiman and studied further by Blasiak--Morse--Pun--Summers. By analyzing the geometry of $\mathscr X_Ψ$, 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.