  We study the rational homotopy of the moduli space ${\mathcal N}_X$ of stable
vector bundles of rank two and fixed determinant of odd degree over a compact
connected Riemann surface $X$ of genus $g\geq 2$. The symplectic group
$Aut(H_1(X,{\mathbb Z}))=Sp(2g,{\mathbb Z})$ has a natural action on the
rational homotopy groups $\pi_n({\mathcal N}_X) \otimes {\mathbb Q}$. We prove
that this action extends to an action of $Sp(2g,{\mathbb C})$ on
$\pi_n({\mathcal N}_X) \otimes {\mathbb C}$. We also show that $\pi_n({\mathcal
N}_X) \otimes {\mathbb C}$ is a non-trivial $Sp(2g,{\mathbb C})$-representation
for each $n\geq 2g-1$. In particular, ${\mathcal N}_X$ is a rationally
hyperbolic space. In the special case where $g=2$, we compute the leading
$Sp(2g,{\mathbb C})$-representation occurring in $\pi_n({\mathcal N}_X) \otimes
{\mathbb C}$, for each $n$.
