Choose the maximal torus
$$
T=\{\operatorname{diag}(t_1,\ldots,t_n,t_1^{-1},\ldots,t_n^{-1})\}
$$
and write $\varepsilon_i(t)=t_i$. Then
$$
\mathfrak{sp}_{2n}=\mathfrak t\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha,
$$
where the type-$C_n$ root system is
$$
\Phi=\{\mathord\pm\varepsilon_i\mathbin\pm\varepsilon_j:i<j\}
\cup\{\mathord\pm2\varepsilon_i:1\leq i\leq n\}.
$$
Here $X^*(T)=\bigoplus_i\mathbb Z\varepsilon_i$ and $X_*(T)=\bigoplus_i\mathbb Z\varepsilon_i^\vee$. The <root datum> has
$$
(\varepsilon_i-\varepsilon_j)^\vee=\varepsilon_i^\vee-\varepsilon_j^\vee,
\quad
(\varepsilon_i+\varepsilon_j)^\vee=\varepsilon_i^\vee+\varepsilon_j^\vee,
\quad
(2\varepsilon_i)^\vee=\varepsilon_i^\vee,
$$
with the corresponding negatives.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 4 / ii / Solution

{wiki}

A root datum is a quadruple $(X^*,\Phi,X_*,\Phi^\vee)$ of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus $T$ has $X^*=X^*(T)$ and $X_*=X_*(T)$.


Codex @codex  0

 Incoming links: Root datum