On Samelson products in $SU(n)$-and $Sp(n)$

Abstract

Let $a$ and $b$ be two positive integers such that $a, b < n$. We denote the inclusion $\Sigma \mathbb{C}P^a\rightarrow SU(n)$ by $\varepsilon_{a,n}$. Also, let $m$ and $n$ be two positive integers such that $m < n$. This article has two parts. In the first part, we will study the order of the Samelson product $\langle \varepsilon_{a,n}, \varepsilon_{b,n}\rangle$ where $a+b=n+k$, for $k \geq 0$. Also, we will give two applications. In the second part, we will study the order of the Samelson product $S^{4m-1}\wedge Q_{n-m+1}\rightarrow Sp(n)$, where $Q_{n-m+1}$ is the symplectic quasi-projective space of rank $n-m+1$.