I'm having the following problem: let $T \subset G := SO(2k)$ be the maximal torus acting on $V := \mathbb{R}^{2k}$ by linear transformations on each $2$-dimensional component. Denote by $V_T := (V \times ET) / T$ the homotopy quotient, where $ET \to BT$ is the universal principal bundle associated with $T$. I would like to compute the Euler class of the (oriented) vector bundle (with fiber $V$) \begin{equation} \pi : V_T \to BT, \end{equation} in terms of the cohomology groups $H^*(BT)$.

So far, I have the following, mainly coming from Chern-Weil theory: we first consider the action of the whole group $G$ (we still denote by $\pi$ the vector bundle above for this action). The Chern-Weil homomorphism (which is an isomorphism here since $G$ is compact) provides an algebra isomorphism \begin{equation} \xi : S(g^*)^G \simeq H^*(BG), \end{equation} where $S(g^*)^G$ is the algebra of (say complex-valued if we look at cohomology with complex coefficients) $G$-invariant polynomial functions on the Lie algebra $g = so(2k)$. This isomorphism is given by applying polynomials on $g$ to the curvature $\Omega$ of any given connection form on the universal principal vector bundle $EG \to BG$, and one can show that the classes obtained this way are independent on the choices made. A particular element in $S(g^*)^G$ is called the Pfaffian $Pf$, and its image $Pf(\Omega)$ through $\xi$ is called an Euler form for $EG \to BG$. Now, choose a metric on the bundle $\pi$, and consider the associated principal bundle, that is the bundle $$\pi_F : F(V_G) := (F(V) \times EG) \to BG,$$ where $F(V)$ is the set of oriented orthonormal frames on V. Of course, $G$ acts freely on the total space $F(V_G)$, and one can see that it is also contractible, since it is homeomorphic to $(G \times EG) / G$. Therefore $\pi_F$ can be seen as the universal principal bundle of $G$, and the Euler form $Pf(\Omega)$ as an Euler form for $\pi_F$.

My questions are the following:

- How can I prove that this Euler form is indeed a representative of the Euler class of the vector bundle $\pi$ ?
- Is there a preferred curvature form for which the computation of $Pf(\Omega)$ would be simple ?
- Is there a more direct way (without passing through the Chern-Weil homomorphism) of computing the Euler class of $\pi$ ?
- Going back to the action of the torus $T$, how can I compute the Euler class of the vector bundle $\pi : V_T \to BT$ in terms of the cohomology groups $H^*(BT) \simeq \mathbb{C}[u_1,...,u_k]$ (I should find that it is equal to $u_1...u_k$, where $u_i$ are generators of degree $2$ of $H^*(BT)$) ?

My ultimate goal is to answer question 4, but it seems to me that understanding the above is necessary. Of course, if someone has a more straightforward way of treating question 4, it would be great.

I apologize in advance if I don't seem clear (I'm not a specialist of Chern-Weil theory).

Thanks a lot.