# Fixed points of coisotropic subgroups of $\Gamma_{k}$ on decomposition spaces

## TL;DR

This paper investigates the homotopy types of fixed point spaces of certain subgroups acting on decomposition spaces of complex vector spaces, revealing they contain wedges of spheres and relating to Tits buildings.

## Contribution

It establishes the homotopy equivalence of fixed point spaces to wedges of spheres for specific subgroups, connecting to Tits buildings and proposing a conjecture on the full homotopy type.

## Key findings

- Fixed point spaces contain wedges of (k-1)-dimensional spheres.
- Fixed points of Gamma relate to symplectic Tits buildings.
- Conjecture on the full homotopy type aligns with results.

## Abstract

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.