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

We consider Iwahori-Coulomb branches ${\mathcal{A}}_{G,\mathbf{N},\mathbf{V}}^{\mathrm{F}\mathrm{l}}$, which are the affine flag analogs of the original Coulomb branches ${\mathcal{A}}_{G,\mathbf{N}}^{\mathrm{G}\mathrm{r}}$ defined by Braverman, Finkelberg, and Nakajima. For any conical symplectic resolution $X$, we prove that the ${\mathcal{A}}_{G,\mathbf{N},\mathbf{V}}^{\mathrm{F}\mathrm{l}}$-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 $X=T^{\ast }(G\mathrm{/}P)$, the cotangent bundle of a flag variety, for which the Iwahori-Coulomb branch is isomorphic to the trigonometric double affine Hecke algebra ${\mathcal{H}}_{G,\mathrm{\hslash },k}$. 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 $G\mathrm{/}P$ 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 $T^{\ast }(G\mathrm{/}P)$ 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, ${\mathcal{A}}_{G,{\mathfrak{g}}^{\ast }}^{\mathrm{G}\mathrm{r}}$ is isomorphic to the spherical subalgebra of ${\mathcal{H}}_{G,\mathrm{\hslash },k}$.