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# Homology, Homotopy and Applications

## Volume 22 (2020)

### Number 1

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

Pages: 77 – 96

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a6

#### Authors

#### Abstract

We study the equivariant homotopy type of the poset ${\mathcal{L}}_{p^k}$ of orthogonal decompositions of ${\mathbb{C}}^{p^k}$. The fixed point space of the $p$-radical subgroup $\Gamma_{k}\subset U(p^k)$ acting on ${\mathcal{L}}_{p^k}$ is shown to be homeomorphic to a symplectic Tits building, a wedge of $(k-1)$-dimensional spheres. Our second result concerns $\Delta_{k}=({\mathbb{Z}}/p)^{k}\subset U(p^k)$ acting on ${\mathbb{C}}^{p^k}$ by the regular representation. We identify a retract of the fixed point space of $\Delta_{k}$ acting on ${\mathcal{L}}_{p^k}$. This retract has the homotopy type of the unreduced suspension of the Tits building for $\operatorname{GL}_{k}\!\left({\mathbb{F}}_{p}\right)$, 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 $\Gamma_{k}$ 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 $\Delta_{k}$ acting on ${\mathcal{L}}_{p^k}$, based on a more general branching conjecture, and we show that the conjecture is consistent with our results.

#### Keywords

Tits building, decomposition space, fixed points, unitary group

#### 2010 Mathematics Subject Classification

55N91, 55P65, 55R45

The first author was supported in part by the Swedish Research Council, grant number 2016-05440.

Received 25 March 2017

Received revised 24 November 2018

Published 30 October 2019