TITLE: Where can I find basic "computations" of equivariant stable homotopy groups?
QUESTION [6 upvotes]: I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (should it actually be a compact Lie group?) in my question. Also, for a group $G$ the corresponding stable homotopy category $SH_G$ (should I use another notation?) also depends on the choice of a universe; does one get a canonical category taking a complete universe? 
As far as I understand equivariant stable homotopy theory, in $SH_G$ one should take spectra $\Sigma_G (G/H)_+$ corresponding to all homogenious spaces $G/H$ to obtain a family of compact generators. Now, I want to know something on morphisms between these generators. In particular, is the group $Mor_{SH_G} (\Sigma_G (G/H_1)_+, \Sigma_G (G/H_2)_+[i])$ zero for any $i>0$ and all subgroups of $G$? This question seems to be related to A heart for stable equivariant homotopy theory
Is there any text where I can find results of this sort? Possibly, one can say something about these morphism groups (say) using duality and apply some well-known results after that? 
Any hints would be very welcome!

REPLY [2 votes]: Since Denis gave the right reference, namely http://www.math.uchicago.edu/~may/BOOKS/equi.pdf, I did not follow up and answer this question.  We can work with any compact Lie group and any complete universe.
Working in the equivariant stable category always, with $S = S^0$,
$S^{-n}$ a negative sphere $G$-spectrum and $S^0$ and $S^n$
sphere spaces with trivial $G$-action, 
$$ \pi_{-n}(S) = [S^{-n},S]_G = [S,\Sigma^{\infty} S^n]_G = [S^0, QS^n]_G = 0 $$
where $Q^G$ is the functor $(colim_V \Omega^V \Sigma^V(-))^G$, with $V$ running through
representations of $G$; $Q^G(X)$ is an $(n-1)$-connected $G$-space if $X$ is so.  Replacing $S$ 
by $\Sigma^{\infty} G/H_+$, the same  argument works, using change of groups so that one is now looking at $[-,-]_H$. More generally, the negative homotopy groups of the suspension $G$-spectrum of any $G$-space are zero, and the non-negative groups are calculated by the tom Dieck splitting theorem.