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.

中文翻译：

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分解空间上 $\Gamma_{k}$ 的各向异性子群的不动点

我们研究了有限维复向量空间正交分解的偏序集的等变同伦类型。假设 n 是素数 p 的幂，并且 D 是 U(n) 的基本阿贝尔 p 子群，通过正则表示作用于复数 n 空间。我们证明了 D 的不动点空间作用于复数 n 空间的分解偏序集，它包含 GL(k) 的 Tits 构建的未归约悬浮作为缩回，它是 (k-1) 维球体的楔形。令 Gamma 是 U(n) 的射影基本阿贝尔子群，它包含 U(n) 的中心并在复 n 空间上不可约地作用。我们证明了 Gamma 的不动点空间作用于复 n 空间的适当正交分解的空间与辛 Tits 建筑物同胚，它也是 (k-1) 维球体的楔形。作为这些结果的结果，我们发现 Gamma 的任何各向同性子群的不动点空间都包含（k-1）维球体的楔形作为缩回。我们基于更一般的分支猜想对 D 的不动点空间的完全同伦类型进行了猜想，并证明该猜想与我们的结果一致。