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

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- Solution ii

Solution

Choose the maximal torus

T = {diag (t_{1}, \dots, t_{n}, t_{1}^{- 1}, \dots, t_{n}^{- 1})}

and write

ε_{i} (t) = t_{i}

. Then

sp_{2 n} = t \oplus ⨁_{α \in Φ} g_{α},

where the type-

C_{n}

Φ = {\pm ε_{i} \pm ε_{j} : i < j} \cup {\pm 2 ε_{i} : 1 \leq i \leq n} .

Here

X^{*} (T) = ⨁_{i} Z ε_{i}

and

X_{*} (T) = ⨁_{i} Z ε_{i}^{\lor}

. The root datum has

(ε_{i} - ε_{j})^{\lor} = ε_{i}^{\lor} - ε_{j}^{\lor}, (ε_{i} + ε_{j})^{\lor} = ε_{i}^{\lor} + ε_{j}^{\lor}, (2 ε_{i})^{\lor} = ε_{i}^{\lor},

with the corresponding negatives.

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