Equivariant $K$-theory of quaternionic flag manifolds

TL;DR

This paper computes the equivariant $K$-theory rings of quaternionic flag manifolds under specific group actions, providing explicit descriptions and connections to complex flag varieties.

Contribution

It offers a Goresky-Kottwitz-MacPherson type description for the $Sp(1)^n$ action and characterizes the $T$-equivariant $K$-theory ring as a subring of a known complex flag variety.

Findings

GKM description for $Sp(1)^n$ action

Explicit description of $K_T(Fl_n(\mathbb{H}))$

Connection to complex flag varieties

Abstract

We consider the manifold $Fl_n(\mathbb{H})=Sp(n)/Sp(1)^n$ of all complete flags in $\mathbb{H}^n$, where $\mathbb{H}$ is the skew-field of quaternions. We study its equivariant $K$-theory rings with respect to the action of two groups: $Sp(1)^n$ and a certain canonical subgroup $T:=(S^1)^n\subset Sp(1)^n$ (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring $K_T(Fl_n(\mathbb{H}))$ as a subring of $K_T(Sp(n)/T)$. This ring is well known, since $Sp(n)/T$ is a complex flag variety.