Equivariant Chern character operators and Okounkov's conjecture
Mazen M. Alhwaimel, Zhenbo Qin
TL;DR
This work advances the understanding of equivariant Chern character operators on the torus-equivariant cohomology of Hilbert schemes ${(\,\mathbb{C}^2\,)}^{[n]}$ by connecting geometry to symmetric-function theory through deformed vertex operators and transformed Macdonald/Jack bases. It provides explicit expressions for the equivariant Chern character operators in low degree ($k=0,1,2$) and reveals their vertex-operator structure, while framing higher-degree terms via leading-term conjectures tied to generalized partitions. Using Ext-vertex operators, it partially verifies Okounkov's Conjecture in the equivariant setting: the reduced generating series for single insertions lies in $\mathbf{qMZV}[t_1,t_2]$ with weight bounds, and multi-insertion cases with $k_i\in\{0,1,2\}$ satisfy polynomial weight/degre constraints in a bi-bracket algebra. The paper also develops a theory of equivariant higher-order derivatives of Heisenberg and Chern character operators, delivering a closed form for $\mathfrak a_{-n}^{(k)}$ and establishing a universal leading-term pattern indexed by generalized partitions, with a key inductive result $f^{(k)}_{0,\lambda}=n^k k!$.
Abstract
In this paper, we study the Chern character operators on the equivariant cohomology of the Hilbert schemes of points in the complex affine plane $C^2$ with the action of the torus $(C^*)^2$, and partially verify Okounkov's Conjecture [Oko, Conjecture 2] in this setting. Our main idea is to apply the connection between the equivariant cohomology of these Hilbert schemes and the ring of symmetric functions, via the deformed vertex operators of Cheng and Wang [CW], (the integral form of) the Jack symmetric functions and the transformed Macdonald symmetric functions of Garsia and Haiman [GH, Hai].