  For manifolds with geodesic flow that is ergodic on the unit tangent bundle,
the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions
become equidistributed as the eigenvalue goes to infinity. For a locally
symmetric space with a universal cover that is a product of several upper half
planes, the geodesic flow has constants of motion so it can not be ergodic. It
is, however, ergodic when restricted to the submanifolds defined by these
constants. In accordance, we show that almost all eigenfunctions become
equidistributed on these submanifolds.
