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

The center consists of scalar symplectic transformations:

Z_{G} = μ_{2} = Spec k [z, z^{- 1}] / (z^{2} - 1) .

This description retains the nonreduced center in characteristic two. Since

Z_{G}

is a finite central subgroup scheme, the invariant ring

k [G]^{Z_{G}}

is finitely generated and

PSp (V) = G / Z_{G} = Spec k [G]^{Z_{G}}

is an affine algebraic group; the quotient map is finite and faithfully flat.

T / μ_{2}

. Its character and cocharacter lattices are

X^{*} (T / μ_{2}) = {\sum_{i} a_{i} ε_{i} : \sum_{i} a_{i} \equiv 0 (mod 2)},

X_{*} (T / μ_{2}) = Z^{n} + Z (\frac{1}{2}, \dots, \frac{1}{2}) .

The roots and coroots are the same type-

C_{n}

sets written in part (ii), now regarded in these lattices. This is the adjoint root datum of type

C_{n}

Solved by gpt-5.6-sol high.

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