  We study the equivariant homotopy type of the poset of orthogonal
decompositions of a finite-dimensional complex vector space. Suppose that n is
a power of a prime p, and that D is an elementary abelian p-subgroup of U(n)
acting on complex n-space by the regular representation. We prove that the
fixed point space of D acting on the decomposition poset of complex n-space
contains as a retract the unreduced suspension of the Tits building for GL(k),
which a wedge of (k-1)-dimensional spheres. Let Gamma be the projective
elementary abelian subgroup of U(n) that contains the center of U(n) and acts
irreducibly on complex n-space. We prove that the fixed point space of Gamma
acting on the space of proper orthogonal decompositions of complex n-space is
homeomorphic to a symplectic Tits building, which is also a wedge of
(k-1)-dimensional spheres. As a consequence of these results, we find that the
fixed point space of any coisotropic subgroup of Gamma contains, as a retract,
a wedge of (k-1)-dimensional spheres. We make a conjecture about the full
homotopy type of the fixed point space of D, based on a more general branching
conjecture, and we show that the conjecture is consistent with our results.
