  We study rotation of invariant vectors in tensor products of minuscule
representations. We define a combinatorial notion of rotation of minuscule
Littelmann paths. Using affine Grassmannians, we show that this rotation action
is realized geometrically as rotation of components of the Satake fibre. As a
consequence, we have a basis for invariant spaces which is permuted by rotation
(up to global sign). Finally, we diagonalize the rotation operator by showing
that its eigenspaces are given by intersection homology of quiver varieties. As
a consequence, we generalize Rhoades' work on the cyclic sieving phenomenon.
