In this paper, we will study Mordell-Weil groups of Kuga fiber spaces of abelian varieties associated to the standard sympiectic representation classified by Satake. We will show the finiteness theorem for them with a few exceptions by using the Hodge theory and Borel-Wallach's vanishing theorem.
The shuffling problem is discussed as the asymptotic behavior of random walks on finite groups. We give a new characterization for asymptotic equidistribution of such random walks in terms of representations of the group. As applications, we characterize perfect groups and consider random walks on classical Weyl groups.