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

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

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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