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Let $G$ be a reductive group with maximal torus $T$. One knows that the equivariant cohomology ring of a point with rational coefficients is $\mathbb{Q}[X^*(T)]^W$, and also there is an equivariant Kunneth formula

$$ H^*_G(X \times Y) = H_G^*(X) \otimes_{H_G^*(pt)} H_G^*(Y) $$

Now suppose we instead take coefficients in a field $k$ with positive characteristic $\ell$. Under what condition on $\ell$ does the above story still go through? (I expect $\ell \nmid W$ is good enough, but can one do better?).

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  • $\begingroup$ Here is a somewhat related MO post: mathoverflow.net/questions/324826/… $\endgroup$
    – Jason Starr
    Commented Aug 5, 2023 at 19:13
  • $\begingroup$ Actually, a more relevant answer is here: mathoverflow.net/questions/440691/… You can use the computation of cohomology of $G/T$ (which is homotopic to $G/B$) with coefficients of arbitrary characteristic and the computation of the cohomology of $BT$ to write down a spectral sequence that converges to the cohomology of $BG$. $\endgroup$
    – Jason Starr
    Commented Aug 7, 2023 at 0:49
  • $\begingroup$ I see that $H^*(G/T)$ and $H^*(BT)$ are even. But in the spectral sequence $H^*(BG, H^*(G/T)) \implies H^*(BT)$, but I guess this doesn't immediately rule out $H^*(BG)$ having odd terms that cancel. $\endgroup$
    – user333154
    Commented Aug 7, 2023 at 5:08
  • $\begingroup$ Yes indeed. You have to prove a special property of the transgressions. $\endgroup$
    – Jason Starr
    Commented Aug 8, 2023 at 9:47

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