Suppose $T$ is a maximal torus in a linear algebraic group, for example, the diagonal matrices in $GL_n$.

In this context, what does it most commonly mean when authors refer to the irreducible characters $\operatorname{Irr}(T)$ of $T$? Is this meant in the sense of an abstract group? I'm confused because the set $X(T)$ of algebraic group homomorphisms $T\to\mathbb{G}_m$ is also referred to as the group of characters.

For example, if $T$ is the usual maximal torus in $GL_2$, then $X(T)=\mathbb{Z}\chi_1\oplus\mathbb{Z}\chi_2$ is a free abelian group, where $\chi_i(\operatorname{diag}(x_1,x_2))=x_i$. Is there a convention I'm not aware of where $\operatorname{Irr}(T)$ would refer to the $\mathbb{Z}$-basis $\{\chi_1,\chi_2\}$ of the character group? Or does it refer to something else?


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