Special Orthogonal Group Scheme SO_n

Setup: the quadratic form

Fix the standard split quadratic form of rank \(n\). Over \(\mathbb{Z}\):

• For \(n = 2m+1\) (odd): \[ Q = x_0^2 + x_1 x_{n-1} + x_2 x_{n-2} + \cdots + x_m x_{m+1} \] with Gram matrix \(I_1 \oplus \begin{pmatrix}0&1\\1&0\end{pmatrix}^m\).

• For \(n = 2m\) (even): \[ Q = x_1 x_n + x_2 x_{n-1} + \cdots + x_m x_{m+1} \] with Gram matrix \(\begin{pmatrix}0&1\\1&0\end{pmatrix}^m\).

Write the associated bilinear form as \(B(v,w) = Q(v+w) - Q(v) - Q(w)\), and its matrix as \(M\) (\(M = 2\,\mathrm{Gram}(Q)\) when \(2\) is invertible).

Definition

Orthogonal group \(\mathrm{O}(Q)\)

\[ \mathrm{O}(Q)(R) = \{A \in \mathrm{GL}_n(R) : Q(Av) = Q(v) \text{ for all } v \in R^n\}. \] In terms of matrices (when \(2\) is a unit in \(R\)): \[ \mathrm{O}(Q)(R) = \{A \in \mathrm{GL}_n(R) : A^T M A = M\}. \] This is a closed subgroup scheme of \(\mathrm{GL}_n\).

Special orthogonal group \(\mathrm{SO}(Q)\)

\[ \mathrm{SO}(Q) = \ker\bigl(\det : \mathrm{O}(Q) \to \mathbb{G}_m\bigr). \] Concretely, \[ \mathrm{SO}(Q)(R) = \{A \in \mathrm{O}(Q)(R) : \det A = 1\}. \]

Coordinate Hopf algebra

\[ \mathcal{O}(\mathrm{SO}_n) = \mathbb{Z}[x_{ij}] / (A^T M A - M,\; \det(x_{ij}) - 1). \] The Hopf algebra structure (comultiplication \(\Delta(x_{ij}) = \sum_k x_{ik} \otimes x_{kj}\), counit \(\varepsilon(x_{ij}) = \delta_{ij}\), antipode via the inverse matrix) is inherited from \(\mathrm{GL}_n\).

Smoothness and structure

In characteristic \(\ne 2\):

• \(\mathrm{O}_n\) and \(\mathrm{SO}_n\) are smooth over \(\mathbb{Z}[1/2]\).
• \(\mathrm{O}_n\) has two connected components: the identity component \(\mathrm{SO}_n\) (det \(= 1\)) and the component of reflections (det \(= -1\)).
• \(\dim \mathrm{SO}_n = \tfrac{n(n-1)}{2}\).

In characteristic \(2\), the correct definition uses quadratic forms directly (not the bilinear form), and \(\mathrm{SO}_n\) is defined differently to remain smooth.

Lie algebra

\[ \operatorname{Lie}(\mathrm{SO}_n) = \mathfrak{so}_n = \{X \in M_n : X^T M + M X = 0\} = \{X \in M_n : X^T = -X\} \quad (\text{for }M = I_n). \] Skew-symmetric matrices; dimension \(\frac{n(n-1)}{2}\).

Root data and types

Type \(B_n\): \(\mathrm{SO}_{2n+1}\)

Maximal split torus \(T = \mathrm{diag}(t_1, \dots, t_n, 1, t_n^{-1}, \dots, t_1^{-1})\). \[ \Phi = \{\pm\varepsilon_i \pm \varepsilon_j : i<j\} \cup \{\pm\varepsilon_i\}, \quad W = (\mathbb{Z}/2)^n \rtimes S_n. \] Simple roots: \(\varepsilon_i - \varepsilon_{i+1}\) for \(i < n\), and \(\varepsilon_n\) (the unique short simple root).

Type \(D_n\): \(\mathrm{SO}_{2n}\)

Maximal split torus \(T = \mathrm{diag}(t_1,\dots,t_n,t_n^{-1},\dots,t_1^{-1})\). \[ \Phi = \{\pm\varepsilon_i \pm \varepsilon_j : i < j\},\quad W = (\mathbb{Z}/2)^{n-1} \rtimes S_n. \] Simple roots: \(\varepsilon_i - \varepsilon_{i+1}\) for \(i < n\), and \(\varepsilon_{n-1} + \varepsilon_n\).

Centre and (co)spin covers

Langlands duals