So equivariant cohomology of a topological space $X$ with a $G$-action is given by computing cohomology of the homotopy quotient:
$H^{*}_{G}(X) = H^{*}(X \times_G EG)$, where $EG$ is a contractible space with a free $G$-action.
But then something that's been puzzling me, and I'm sure the answer is probably easy and I'm just being dumb, but I'm interested in a subgroup of $G$, say $H \leq G$, and according to a paper I'm reading, there's supposed to be a "forgetful" map that looks like $H^{*}_G(X) \rightarrow H^{*}_H(X)$ induced by the inclusion of groups.
The case I'm studying is that $X \cong pt$ and that $G = (C^{*})^n$ (the $n$-dimensional algebraic torus), $H = S^1$ a rank-$1$ torus in $G$ given by:
$G = \{ diag(t_1,...,t_n) : t_i \in \mathbb{C} \} $ and $H = \{ diag(t^n,t^{n-1},...,t) : \|t\| = 1, t \in \mathbb{C}\}$
where $diag(a_1,...,a_n)$ is a diagonal matrix with entries $a_1,...,a_n$.
I know that $H^{*}_G(pt) \cong \mathbb{C}[t_1,...,t_n]$ by computing it explicitly, and I know that $H^{*}_H(pt) \cong \mathbb{C}[t]$ by virtually the same computation (for the first, the homotopy quotient is $(\mathbb{C}P^{\infty})^n$, and just $\mathbb{C}P^{\infty}$ for the second). But I cannot for the life of me figure out what the map $\pi^{*}: \mathbb{C}[t_1,...,t_n] \rightarrow \mathbb{C}[t]$ induced from the inclusion of groups $H \leq G$ is. Can anyone help me out or point me to a reference?
Thanks in advance!
(Context: $H$ preserves the Springer fiber and $G$ is the torus that acts on the flag variety)
EDIT: Just in case anyone was wondering and doesn't care about how we arrived at the answer, the generator $t^i$ is mapped to $(i-1)t$.