Equivariant cohomology and complex non-degenerate bilinear forms Let $M = GL(n,\mathbb{C})$ be the set of non-degenerate bilinear forms on $\mathbb{C}^n$ (not necessarily symmetric). The general linear group $G = GL(n,\mathbb{C})$ acts on $M$ in the usual way
$$G\times M \rightarrow M,~~~~~~(A,X) \mapsto AXA^T.$$
What is the equivariant cohomology $H^*_G(M)$?
 A: I'll assume that you mean the Borel cohomology $H^*(EG\times_GM)$.  If so, the answer is
$$ \mathbb{Z}[\![c_1,\dotsc,c_n]\!]\otimes\Lambda^*(a_1,\dotsc,a_n), $$
where $|c_k|=2k$ and $|a_k|=2k-1$.  To see this, first note that you can replace everything by maximal compact subgroups without changing the homotopy type.  This means that the relevant space is $EU(n)\times_{U(n)}U(n)^{\text{ad}}$, where $U(n)^{\text{ad}}$ refers to the space $U(n)$ with $U(n)$ acting on it by conjugation.  This can be described in a different way as follows.  For any complex vector bundle $V$ (with Hermitian inner product) over a space $X$, we can consider the fibre bundle $U(V)$ whose points are pairs $(x,g)$, where $x\in X$ and $g$ is a unitary automorphism of $V_x$.  If we take $V$ to be the tautological bundle over $BU(n)$ then $U(V)$ is easily identified with $EU(n)\times_{U(n)}U(n)^{\text{ad}}$.  For any $V$ and $X$ it is known that 
$$ H^*(U(V)) = H^*(X) \otimes\Lambda^*(a_1,\dotsc,a_n), $$
and my claim above is a special case of this.  The key ingredient in the proof is Miller's stable splitiing theorem for $U(n)$, in the equivariant form proved by Nitu Kitchloo; some additional details are in my paper "Common subbundles and intersections of divisors".
A: Update: I managed to partially answer this question after posing it. For coefficient fields relatively prime to $n$!, there is an isomorphism  $ H_G^*(M) \cong H^*(BLO(n))$ where $BLO(n)$ is the classifying space of the continuous loop group $LO(n)$, while in characteristic 2 there is an isomorphism $ H_G^*(M) \cong H^*(BLU(n))$.
Details are found in Corollary 6.4 of https://arxiv.org/abs/1312.7450.
