Shifted twisted Yangians and affine Grassmannian islices

Abstract

In a prequel we introduced the shifted iYangians ${}^{ı}{Y}_{\mu }$ associated to quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, and constructed the iGKLO representations of ${}^{ı}{Y}_{\mu }$, which factor through truncated shifted iYangians ${}^{ı}{Y}_{\mu }^{\lambda }$. In this paper, we show that ${}^{ı}{Y}_{\mu }$ quantizes the involutive fixed point locus ${}^{ı}{W}_{\mu }$ arising from affine Grassmannians of type ADE, and supply strong evidence toward the expectation that ${}^{ı}{Y}_{\mu }^{\lambda }$ quantizes a top-dimensional component of the affine Grassmannian islice ${}^{ı}{\overline{W}}_{\mu }^{\lambda }$. We identify the islices ${}^{ı}{\overline{W}}_{\mu }^{\lambda }$ 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 ${}^{ı}{\overline{W}}_{\mu }^{\lambda }$ and the algebras ${}^{ı}{Y}_{\mu }^{\lambda }$.

v3, 63 pages, simplified notation in Section 6 via a coweight-partition identification, other mild edits