Let $G$ be an elementary abelian group, so that $G = (\mathbb{Z}/p)^k$ for some $k$. We can then compute the Morava $K$-theory of $BG = (BZ/p)^k$ pretty easily: $K(n)^\*(BG) = K(n)^\*[[x]]/[p](x) (x)\_{K(n)^\*} ... (x)\_{K(n)^\*} K(n)^\*[[x]]/[p](x)$, the tensor being taken $k$ times. We know that $[p](x) = x^{p^n}$. What is this cohomology $K(n)^\*(BG)$ as a representation of $GL\_k(\mathbb{F}$\_p)$?
Maybe we can compute the topological cyclic homology via a slice spectral sequence computation.



Computation of cohomology of Morava $K$-Theory
Bjorn Schuster and I looked at this problem in our article `On the $GL(V)$-module structure of $K(n)^\*(BV)$' Math Proc Cambridge Phil Soc vol 122 (1997) pp73--89. In particular, we show that in at least some cases it is not a permutation module. If you don't have access to the journal, there is a preprint that started off on Dave Benson's preprint server and was put on ArXiv in 2007: <https://doi.org/10.48550/arXiv.0711.5016>


Before our article, Nick Kuhn had shown that $K(n)^\*(BV)$ always has the same Brauer character as a permutation module for $GL(V)$ in `Morava $K$-theories of some classifying spaces' Trans Amer Math Soc vol 304 (1987) 193--205, and this motivated our work.


