Let $G$ be a finite group and $E$ be a genuine $H$-spectrum for $H\leq G.$ Then for any subgroup $K$ of $G$, consider the $K$-spectrum $X=Res^G_K Ind^G_H(E).$

Is there any reference for computing the homotopy groups of the geometric fixed point $\Phi^K(X): = (\widetilde{E\mathcal{P}}\wedge X)^K?$ Here $\widetilde{E\mathcal{P}}$ is a $K$-space given the following fixed point data:

$$\widetilde{E\mathcal{P}}^L= \begin{cases} S^0, & \text{ if }L=K\\ \ast, & \text { if } L \text{ is any proper subgroup of } K.\end{cases} $$ Maybe the double coset formula could be useful.

Thank you so much in advance. Any help will be appreciated.

  • 2
    $\begingroup$ back of the envelope guess: if $K$ is not subconjugate to $H$ then you should get 0, if $K$ is subconjugate to $H$ then you should end up with $|G/H|$-copies of $\Phi^K(E)$, I think. Does that sound right? $\endgroup$ Sep 14, 2020 at 12:11
  • $\begingroup$ @DylanWilson: Yes, it sounds interesting. Can you please write down the key arguments for it? $\endgroup$ Sep 14, 2020 at 12:24
  • $\begingroup$ As you said, I think it's the double coset formula. $\Phi^K$ annihilates anything induced from a proper subgroup of $K$, so you just have to ask- when is one of the terms in the double coset formula not induced from a proper subgroup? That happens only when $K \cap g^{-1}Hg=K$ for some $g$, i.e. when $K$ is subconjugate to $H$. $\endgroup$ Sep 14, 2020 at 12:43


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy