I am working through the paper "On the Algebraic $K$-Theory of Some Homogeneous Varieties" by Alexey Ananyevskiy and got stuck at the beginning of the second section.
The set up is the following: Let $G$ be connected reductive group and $T\subseteq G$ a maximal torus, i.e. a maximal connected abelian subgroup. It is well-known that $$X(T)=X^*(T)=\mathrm{Hom}(T,G_m) \cong \mathbb{Z}^{\mathrm{rank}(T)},$$ where $G_m$ denote the multiplicative one-dimensional irreducible affine linear algebraic group (or simply $U(1)$). The Weyl-group of $G$ w.r.t $T$ is given by $$W(G,T):= N_G(T)/Z_G(T)=N_G(T)/T=\{g \in G \ \vert \ gTg^{-1}=T\}/T.$$ The action of $N$ on $T$ by $n.t=ntn^{-1}$ induces an action of $W$ on $T$ by $[n].t = ntn^{-1}$ and this extends to an action on $X^*(T)$ by $$([n].\chi)(t) := \chi([n].t) \qquad (t \in T)$$ for any $(\chi: T \longrightarrow G_m) \in X^*(T)$. Now let $$\mathrm{Ch}= \mathrm{Hom}(Z(G), G_m),$$ where $Z(G)$ is the center of the group $G$. Ananyevskiy claims, that there is a Well-equivariant $\mathrm{Ch}$-grading on $X^*(T)$, which for me means, that we may write $$X^*(T)= \bigoplus_{\theta \in \mathrm{Ch}} X_{\theta}.$$ My question is: What is this grading and where does it come from? Unfortunately I could not find any references for it. It might also be trivial.
Let $T \cong (\mathbf{C}^\times)^n$ be a complex torus and let $S \subseteq T$ be a subtorus. Given $\lambda: S \to \mathbf{C}^\times$ a character of $S$, we define $$X(T)_\lambda=\{\phi \in X(T) \ | \ \phi|_S=\lambda \}.$$ These are not subgroups but rather cosets, and $X(T)$ is not their direct sum but rather their disjoint union, so this doesn't give a grading in the usual sense (and of course, if the rank of $S$ is at least one, the equation $$X(T)=\bigoplus_{\lambda \in X(S)} X(T)_\lambda$$ with $X(T)_\lambda$ a subgroup would imply that $X(T)$ is of infinite rank and is hence impossible).
This does, however, produce a $X(S)$-grading on the group algebra $\mathbf{Z}[X(T)]$ in the usual sense, and presumably this is what the author means to consider. Of course, if you have a finite subgroup $W$ of the automorphism group of $T$ that fixes the elements of $S$, this grading will be $W$-invariant.