I am trying to understand how the Weyl character formula works for GL_n.
In Milne's Algebraic groups, he gives the character formula in the following setting : Let $G$ be a connected split reductive group over a field, let $T$ be a maximal split torus and $B$ a Borel containing it. Denote $\rho$ to be the half-sum of the corresponding positive roots. Just before theorem 22.50, Milne quotes Iversen's "the geometry of algebraic groups" (1976) to say that because $\mathrm{Pic}(G)$ is trivial, $\rho$ is actually a character of $T$, which then allows him to state and prove the character formula.
In Iversen, the same thing is said.
But for $G=GL_n$, according to this question the Picard group vanishes, however the half-sum is only a character if $n$ is odd, since it equals $\left(\frac{n-1}{2}, \frac{n-3}{2},...,\frac{1-n}{2}\right)$ in the standard basis of $X(T)$, for the standard torus and standard Borel.
I can't figure out which part of this is wrong. Is there an extra hypothesis I did not see for the result ? Is the Picard group of $GL_n$ not trivial ? Or am I making a stupid mistake computing the half-sum of positive roots ?
Thank you for answering.