Quasi-positive curvature on a biquotient of $Sp(3)$

Abstract

Suppose $\phi_3:Sp(1)\rightarrow Sp(2)$ denotes the unique irreducible $4$-dimensional representation of $Sp(1) = SU(2)$ and consider the two subgroups $H_1, H_2\subseteq Sp(3)$ with $H_1 = \{\operatorname{diag}(\phi_3(q_1), q_1): q_1 \in Sp(1)\}$ and $H_2 = \{\operatorname{diag}(\phi_3(q_2),1):q_2\in Sp(1)\}$. We show that the biquotient $H_1\backslash Sp(3)/H_2$ admits a quasi-positively curved Riemannian metric.