Definition Symplectic Group Scheme Sp_{2n}
definition admitted

Symplectic Group Scheme Sp_{2n}

Setup

Fix the standard symplectic form on \(\mathbb{Z}^{2n}\): let \[ J = J_n = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} \in M_{2n}(\mathbb{Z}). \] This is a non-degenerate alternating bilinear form: \(J^T = -J\), \(\det J = 1\).

Definition (functor of points)

\[ \mathrm{Sp}_{2n}(R) = \{A \in \mathrm{GL}_{2n}(R) : A^T J A = J\}. \] Equivalently, \(A\) preserves the alternating form \(\omega(v,w) = v^T J w\): \(\omega(Av, Aw) = \omega(v, w)\) for all \(v, w \in R^{2n}\).

Coordinate Hopf algebra

The coordinate ring is \[ \mathcal{O}(\mathrm{Sp}_{2n}) = \mathbb{Z}[x_{ij}]_{1 \le i,j \le 2n} / I \] where \(I\) is generated by the \(4n^2\) polynomial entries of the matrix equation \(X^T J X - J = 0\). This is a quotient of \(\mathcal{O}(\mathrm{GL}_{2n})\) via the closed embedding \(\mathrm{Sp}_{2n} \hookrightarrow \mathrm{GL}_{2n}\).

The Hopf algebra structure (comultiplication, counit, antipode) is inherited from \(\mathrm{GL}_{2n}\).

Key properties

  • Dimension: \(n(2n+1)\). (The symplectic Lie algebra has \(n^2\) diagonal blocks and \(\frac{n(n+1)}{2}\cdot 2\) off-diagonal.)
  • Smoothness: \(\mathrm{Sp}_{2n}\) is smooth over \(\mathbb{Z}\) (the linearisation of \(A^T J A = J\) at the identity is \(X^T J + J X = 0\), which has the expected rank \(n(2n+1)\) as the dimension of the symplectic Lie algebra).
  • Simply connected: \(\mathrm{Sp}_{2n}\) is the simply connected form of type \(C_n\). It has trivial centre only for \(n = 0\); for \(n \ge 1\), \(Z(\mathrm{Sp}_{2n}) = \mu_2 = \{\pm I_{2n}\}\).
  • Determinant: Every symplectic matrix has \(\det A = 1\). Thus \(\mathrm{Sp}_{2n} \subseteq \mathrm{SL}_{2n}\) as group schemes.
  • No non-trivial inner forms: Over any field, there is a unique inner form of \(\mathrm{Sp}_{2n}\) up to isomorphism (symplectic groups are their own adjoint forms modulo centre, and all alternating forms of a given dimension over a field are isomorphic).

Lie algebra

\[ \operatorname{Lie}(\mathrm{Sp}_{2n}) = \mathfrak{sp}_{2n} = \{X \in M_{2n} : X^T J + J X = 0\}. \] Equivalently, \(X = \begin{pmatrix} A & B \\ C & -A^T \end{pmatrix}\) with \(B = B^T\), \(C = C^T\). Dimension: \(n(2n+1) = 2n^2 + n\).

Maximal torus and root datum

The standard maximal torus is \[ T = \{\mathrm{diag}(t_1,\dots,t_n,t_n^{-1},\dots,t_1^{-1}) : t_i \in \mathbb{G}_m\}. \] With \(\varepsilon_i : T \to \mathbb{G}_m\) picking the \(i\)-th coordinate:

\[ X^*(T) = \mathbb{Z}\varepsilon_1 \oplus \cdots \oplus \mathbb{Z}\varepsilon_n, \] \[ \Phi = \{\pm\varepsilon_i \pm \varepsilon_j : i < j\} \cup \{\pm 2\varepsilon_i : 1 \le i \le n\}. \]

This is the root system of type \(C_n\). The simple roots are \[ \alpha_i = \varepsilon_i - \varepsilon_{i+1}\;(i < n),\quad \alpha_n = 2\varepsilon_n. \] The long roots \(\pm 2\varepsilon_i\) are the symplectic roots.

Langlands dual

\(\widehat{\mathrm{Sp}_{2n}} = \mathrm{SO}_{2n+1}\) (type \(B_n\)). The dual root datum swaps \(C_n \leftrightarrow B_n\).

Also in