Shifted twisted Yangians of quasi-split ADE types

TL;DR

This paper introduces shifted twisted Yangians for quasi-split ADE types, constructs their PBW bases and representations, and links them to finite W-algebras and affine Grassmannian slices, expanding the algebraic framework in this area.

Contribution

It defines new classes of shifted iYangians for quasi-split ADE types, establishes their PBW bases, and connects them to finite W-algebras and geometric structures.

Findings

Established PBW bases for shifted iYangians.

Constructed iGKLO representations factoring through quotients.

Identified connections to finite W-algebras of type BCD.

Abstract

Associated to all quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, we introduce the shifted iYangians ${}^\imath Y_\mu$ and establish their PBW bases. We construct the iGKLO representations of ${}^\imath Y_\mu$, which factor through quotients called truncated shifted iYangians ${}^\imath Y_\mu^\lambda$. In type AI with $\mu$ dominant, a variant of ${}^\imath Y_\mu^{N\varpi_1^\vee}$ is identified with the truncated shifted iYangians in another definition, which are isomorphic to finite W-algebras of type BCD. These new family of algebras has connections and applications to fixed point loci of affine Grassmannian slices which will be developed in a sequel.