Let $G = \operatorname{GSp}_{2n}$ be the group of symplectic similitudes. I am trying to work out certain cocharacters and have a question on the root system of this group:
A maximal torus $T$ in $G$ is given by the diagonal matrices $$ t = \operatorname{diag} (t_1, ..., t_n, t_n', ..., t_1') \quad \text{such that} \quad t_i t_i' = c$$ for some $c$ and all $i$. This gives $n+1$ characters: $e_i\colon t \mapsto t_i$ and $e_0 \colon t \mapsto c$ and the positive roots then have the form $e_i - e_j, e_i + e_j - e_0$ and $2e_i - e_0$ (see for example https://www.math.purdue.edu/~fshahidi/articles/Asgari%20&%20Shahidi%20%5B2006,%2054pp%5D---Generic%20transfer%20for%20general%20spin%20groups.pdf, p. 149).
However, the Dynkin diagram of $G$ is supposed to be of type $C_n$ (even in the paper quoted above). How can this identification be made? As I see it, $X^\ast(T)$ does not even have the right rank, which is $n+1$ but would have to be $n$. What am I missing?
And something else that confuses me: I am actually looking for minuscule cocharacters $\mu$, i.e. such that $|\langle \mu, \alpha\rangle| \leq 1$ for all positive roots $\alpha$. If I used the roots computed above, then for example $(1, ..., 1)$ (with $n+1$ entries) satisfies this. However, using the general notation (e.g. Bourbaki, Lie algebras 4-6, Tables) for root systems of type $C_n$, no non-zero such element exists, since the elements $2e_i$ are positive roots, which then force $\mu = 0$.