Shifted twisted Yangians and affine Grassmannian islices
Kang Lu, Weiqiang Wang, Alex Weekes
TL;DR
The paper develops a comprehensive framework linking shifted iYangians to geometric objects arising from affine Grassmannians and their fixed points under Poisson involutions. Through iGKLO representations, it identifies truncated shifted iYangians with quantizations of top-dimensional components of affine Grassmannian islices and proves that the associated graded algebras recover coordinate rings of fixed-point varieties ${}^{\imath}\mathcal{W}_{\mu}$. In type AI it further identifies affine Grassmannian islices with nilpotent Slodowy slices of type BCD, and extends these insights to an $\,\imath$-fication of Coulomb branches, proposing a framework where iCoulomb branches normalize fixed-point slices and whose deformations correspond to GKLO-type quantizations. The work thus connects quantum groups, Dirac reductions, and Coulomb-branch constructions to a unified picture of fixed-point geometry in the affine Grassmannian, with substantial implications for representation theory and mathematical physics. These results pave the way for broader q-deformations and general Satake-diagram extensions, enriching the interplay between quantum algebras and classical Poisson varieties.
Abstract
In a prequel we introduced the shifted iYangians ${}^\imath Y_μ$ associated to quasi-split Satake diagrams of type ADE and even spherical coweights $μ$, and constructed the iGKLO representations of ${}^\imath Y_μ$, which factor through truncated shifted iYangians ${}^\imath Y_μ^λ$. In this paper, we show that ${}^\imath Y_μ$ quantizes the involutive fixed point locus ${}^\imath W_μ$ arising from affine Grassmannians of type ADE, and supply strong evidence toward the expectation that ${}^\imath Y_μ^λ$ quantizes a top-dimensional component of the affine Grassmannian islice ${}^\imath\overline{W}_μ^λ$. We identify the islices ${}^\imath\overline{W}_μ^λ$ in type AI with suitable nilpotent Slodowy slices of type BCD, building on the work of Lusztig and Mirković-Vybornov in type A. We propose a framework for producing ortho-symplectic (and hybrid) Coulomb branches from split (and nonsplit) Satake framed double quivers, which are conjectured to relate closely to the islices ${}^\imath\overline{W}_μ^λ$ and the algebras ${}^\imath Y_μ^λ$.