# Weyl map for $SU(n)$

Let $$G= SU(n)$$ and let $$\mathbb{T}$$ be the maximal torus in $$G$$ given by diagonal matrices. We have $$H^*(G,\mathbb{Q}) \cong \Lambda_{\mathbb{Q}}[x_3, x_5, \dots, x_{2n-1}] \ .$$ Now consider the Weyl map $$p \colon G/\mathbb{T} \times \mathbb{T} \to G \quad , \quad ([g],z) \mapsto gzg^{-1}\ .$$ The induced map in rational cohomology $$p^* \colon H^*(G,\mathbb{Q}) \to H^*(G/\mathbb{T}, \mathbb{Q}) \otimes H^*(\mathbb{T},\mathbb{Q})$$ is injective. In fact, if we restrict the codomain to fixed points with respect to a certain action of the Weyl group $$W$$ it becomes an isomorphism (see for example Reeder, On the Cohomology of Compact Lie Groups). The cohomology ring $$H^*(\mathbb{T},\mathbb{Q})$$ is isomorphic to another exterior algebra and there are also explicit descriptions of $$H^*(G/\mathbb{T},\mathbb{Q})$$ (see Reeder's paper again for a reference).

Is there a formula describing $$p^*(x_{2i+1})$$ in terms of any set of natural generators for the codomain?

First, let me fix generators for $$H^*(SU(n))$$ and $$H^*(SU(n)/\mathbb T)$$: For the first, consider the vector bundle on $$\Sigma SU(n)$$ with clutching map $$\operatorname{id}_{SU(n)}$$, i.e. with classifying map $$f_n\colon\Sigma SU(n)\simeq \Sigma\Omega BSU(n)\to BSU(n)$$, and let $$\Sigma x_{2i-1} = f^*c_i$$. For the second, let $$\pi_k\colon \mathbb T\subset U(1)^k\to U(1)$$ be the projection to the $$k$$-th coordinate and $$L_k = SU(n)\times_{T,\pi_k} \mathbb C$$ be the associated line bundle, and set $$y_k = c_1(L_k)$$; since $$\bigoplus_{k=1}^n \pi_k$$ is the restriction of the defining representation of $$SU(n)$$ to $$\mathbb T$$, the sum $$E := \bigoplus_{k=1}^n L_k$$ is trivial, so that all symmetric polynomials in the $$y_k$$ vanish since they can be expressed via Chern classes of $$E$$. The induced map $$Q[y_1,\dots,y_n]/Q[y_1,\dots,y_n]^{S_n}\to H^*(SU(n)/\mathbb T)$$ is an isomorphism; a basis for this cokernel is given by the monomials $$y_1^{\alpha_1}\dots y_n^{\alpha_n}$$ with $$0\le \alpha_k < k$$.
Now think of $$p:G/\mathbb T\times\mathbb T\to SU(n)$$ as an automorphism $$\phi$$ of $$E\boxtimes\underline{\mathbb C}$$; it splits as $$\phi = \phi_1\oplus\dots\oplus \phi_n$$, where $$\phi_k = \operatorname{id}_{L_k}\boxtimes \pi_k$$ is the automorphism of $$L_k\boxtimes \underline{\mathbb C}$$ obtained as the external tensor product of the identity of $$L_k$$ with the projection $$\pi_k:\mathbb T\subset U(1)^k\to U(1)$$.
Consider the clutching construction $$(E\boxtimes\underline{\mathbb C})^\phi\to G/\mathbb T\times\mathbb T\times S^1$$ which is obtained by gluing the two ends of $$E\boxtimes\underline{\mathbb C}$$ together using $$\phi$$. Denote by $$z_k = \pi_k^*([U(1)]), u = [S^1]$$ the obvious cohomology classes in degree $$1$$, and itdentify them and the $$y_k$$ with their image in the product. An easy argument shows that $$c_1(\mathbb C^{\pi_k}) = uz_k\in H^2(\mathbb T\times S^1)$$. By naturality, we have \begin{align*} (E\boxtimes\underline{\mathbb C})^\phi &\cong \bigoplus_{k=1}^n(L_k\boxtimes\underline{\mathbb C})^{\phi_k}\\ c_1\big((L_k\boxtimes\underline{\mathbb C})^{\phi_k}\big) &= c_1(L_k) + c_1(\underline{\mathbb C}^{\pi_k}) = y_k + uz_k\\ c\big((E\boxtimes\underline{\mathbb C})^\phi\big) &= \prod_{k=1}^n c\big((L_k\boxtimes\underline{\mathbb C})^{\phi_k}\big)\\ &= \prod_{k=1}^n (1 + y_k + uz_k)\\ &= \underbrace{c(E)}_{= 0} + u\sum_{k=1}^n z_k \sum_{S\subset \{1,\dots,n\}\smallsetminus\{k\}}\prod_{l\in S} y_l \end{align*}
It is no surprise that this expression vanishes for $$u=0$$ since we can use the chosen trivialization of $$E\boxtimes \underline{\mathbb C}$$ to descend $$(E\boxtimes \underline{\mathbb C})^\phi$$ to $$\Sigma(G/\mathbb T\times \mathbb T)$$, and the resulting vector bundle has classifying map $$f_n\circ \Sigma p$$. Chasing through the definitions, we see that $$p^* x_{2i-1} = \sum_{k=1}^n\Big[\sum_{\substack{S\subset \{1,\dots,n\}\smallsetminus\{k\}\\|S| = i - 1}}\prod_{l\in S} y_l\Big]\otimes z_k\ .$$