Quasipositive curvature on a biquotient of Sp$(3)$

Abstract

Suppose ${\varphi }_3:Sp\left(1\right)\to Sp\left(2\right)$ denotes the unique irreducible complex $4$-dimensional representation of $Sp\left(1\right)=SU\left(2\right)$, and consider the two subgroups $H_i\subseteq Sp\left(3\right)$ with $H_1=\left\{diag\left({\varphi }_3\left(q_1\right),q_1\right):q_1\in Sp\left(1\right)\right\}$ and $H_2=\left\{diag\left({\varphi }_3\left(q_2\right),1\right):q_2\in Sp\left(1\right)\right\}$. We show that the biquotient $H_1\setminus Sp\left(3\right)∕H_2$ admits a quasipositively curved Riemannian metric.