Iwahori-Coulomb branches, stable envelopes, and quantum cohomology of cotangent bundles of flag varieties

TL;DR

This work develops the flavor-deformed Iwahori-Coulomb branch algebras $\mathcal{A}^{\mathrm{Fl}}_{G,\mathbf{N},\mathbf{V}}$ and proves a polynomiality property for their action on localized torus-equivariant quantum cohomology via shift operators and stable envelopes. Specializing to $X=T^{*}(G/P)$, the authors identify the Iwahori-Coulomb branch with the trigonometric double affine Hecke algebra and give explicit action formulas in terms of Demazure–Lusztig elements, stable envelopes, and the Schubert basis; they further derive a confluent limit yielding Peterson’s quantum equals affine theorem and construct Namikawa-Weyl-group actions preserving the quantum product. A central outcome is the geometric realization of the tDAHA action on quantum cohomology and the demonstration that the spherical subalgebra of the tDAHA corresponds to the Gr-Coulomb branch after a parameter shift, linking Coulomb branch algebras to established Hecke-algebra frameworks. The results provide new tools for computing Coulomb-branch actions, enable explicit algebraic descriptions via stable envelopes, and reveal deep connections between Coulomb branches, DAHA theory, and quantum cohomology with applications to Namikawa-Weyl symmetries and quantum Peterson-type correspondences.

Abstract

We consider Iwahori-Coulomb branches $\mathcal{A}_{G,\mathbf{N},\mathbf{V}}^{\mathrm{Fl}}$, which are the affine flag analogs of the original Coulomb branches $\mathcal{A}_{G,\mathbf{N}}^{\mathrm{Gr}}$ defined by Braverman, Finkelberg, and Nakajima. For any conical symplectic resolution $X$, we prove that the $\mathcal{A}_{G,\mathbf{N},\mathbf{V}}^{\mathrm{Fl}}$-action on the localized equivariant quantum cohomology of $X$, induced by shift operators, satisfies a polynomiality property in terms of stable envelopes. We then study the case $X = T^*(G/P)$, the cotangent bundle of a flag variety, for which the Iwahori-Coulomb branch is isomorphic to the trigonometric double affine Hecke algebra $\mathcal{H}_{G,\hbar,k}$. The polynomiality property enables us to compute explicitly the above action in terms of the Demazure-Lusztig elements and stable envelopes. Applications include: (1) Computation of the Iwarhori-Coulomb branch action for $G/P$ by taking the confluent limit, recovering Peterson-Lam-Shimozono's theorem. (2) Construction of an explicit Namikawa-Weyl group action on the equivariant quantum cohomology of $T^*(G/P)$ that preserves the quantum product, extending a result of Li-Su-Xiong. (3) Proof of a conjecture of Braverman-Finkelberg-Nakajima stating that, up to a shift of the dilation parameter, $\mathcal{A}_{G,\mathfrak{g}^*}^{\mathrm{Gr}}$ is isomorphic to the spherical subalgebra of $\mathcal{H}_{G,\hbar,k}$.

Theorems & Definitions (117)

• Theorem A
• Corollary 1
• Proposition 2
• Corollary 3
• Theorem B
• Theorem C
• Corollary 4
• Proposition 5
• Corollary 6
• Corollary 7