Extended Verma-trace correspondence at integral parameters

Prove that the normalized graded trace of the Verma module indexed by a crystal element equals the q=2\hbar-specialized vertex function computed in the equivariant cohomology of the corresponding cocharacter-fixed component.

Background

For an integral parameter determined by a cocharacter, connected components of the corresponding Higgs-branch fixed locus carry a monomial-crystal structure, while Verma modules in the Coulomb-branch category are indexed by the same crystal. The conjecture extends the isolated-fixed-point correspondence to positive-dimensional fixed components and singular parameters.

The statement is designed to compare traces of Verma modules with vertex functions evaluated using the cocharacter-equivariant theory, where the order of specialization matters.

References

It is natural to formulate the following refinement of Conjecture \ref{conj: Verma traces as vertex functions}. $\widetilde{\operatorname{tr}{\Delta(b)}(\tau) = V{(\tau),\tilde{\gamma}(\mathbb{C}\times)}{q=2\hbar,p}.$

— The quantum Hikita conjecture via quasimaps  (2608.16746 - Dinkins et al., 17 Aug 2026) in Conjecture 4.14, Section 4.3

It is an interesting question whether there is still a “reasonable” way to specialize $V{(-)}$ at $q=2\hbar$.

— The quantum Hikita conjecture via quasimaps  (2608.16746 - Dinkins et al., 17 Aug 2026) in Conjecture-free discussion immediately after Conjecture 3.19, Section 3.4

It could still be true that for appropriately chosen classes $a \in H_*(\mathfrak{L}{\tilde{\gamma}(\mathbb{C}\times)})(q)[[{\boldsymbol{z}]]$ the integrals $\int_{\widetilde{\mathcal M}_H{\tilde{\gamma}(\mathbb{C}\times)} V{(\tau)}\cap a$ have $2\hbar=q=2$ specializations and every graded trace can be obtained this way.

— The quantum Hikita conjecture via quasimaps  (2608.16746 - Dinkins et al., 17 Aug 2026) in Section 4.4, final paragraph of subsection “An application to graded traces”