Equivariant $K$-theory of quaternionic flag manifolds

Abstract

We consider the manifold $F{l}_n(\mathbb{H})=Sp(n)/Sp(1{)}^n$ 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 $K_T(F{l}_n(\mathbb{H}))$ as a subring of $K_T(Sp(n)/T)$. This ring is well known, since $Sp(n)/T$ is a complex flag variety.

Final version: 18 pages