Classical Groups
The classical groups are the families \(\mathrm{GL}_n\), \(\mathrm{SL}_n\), \(\mathrm{SO}_n\), \(\mathrm{Sp}_{2n}\), and \(\mathrm{U}_n\) (and their adjoint / inner-form variants). They provide the bulk of the concrete cases in the Langlands programme and a complete bestiary of root-datum types \(A_n, B_n, C_n, D_n\).
The four classical families
| Type | Group | Dynkin | Form preserved |
|---|---|---|---|
| \(A_{n-1}\) | \(\mathrm{GL}_n\), \(\mathrm{SL}_n\) | \(A_{n-1}\) | none (linear automorphisms) |
| \(B_n\) | \(\mathrm{SO}_{2n+1}\) | \(B_n\) | symmetric bilinear, dim \(2n+1\) |
| \(C_n\) | \(\mathrm{Sp}_{2n}\) | \(C_n\) | symplectic, dim \(2n\) |
| \(D_n\) | \(\mathrm{SO}_{2n}\) | \(D_n\) | symmetric bilinear, dim \(2n\) |
Each can be defined as a group preserving a non-degenerate form on a finite-dimensional vector space.
Definitions
Over a field \(k\) of characteristic \(\ne 2\):
- \(\mathrm{GL}_n = \{g \in M_n(k) : \det g \ne 0\}\).
- \(\mathrm{SL}_n = \ker(\det : \mathrm{GL}_n \to \mathbb{G}_m)\).
- \(\mathrm{O}(q) = \{g \in \mathrm{GL}(V) : q(gv) = q(v)\}\) for a non-degenerate quadratic form \(q\) on \(V\); \(\mathrm{SO}(q)\) is its identity component.
- \(\mathrm{Sp}(V, \omega) = \{g \in \mathrm{GL}(V) : \omega(gv, gw) = \omega(v, w)\}\) for a non-degenerate alternating form \(\omega\).
- \(\mathrm{U}(V, h) = \{g \in \mathrm{GL}_K(V) : h(gv, gw) = h(v, w)\}\) for a non-degenerate Hermitian form \(h\) over a quadratic extension \(K/k\).
Inner and outer forms
Over a non-algebraically-closed field \(k\), the classical groups have non-split forms parametrised by Galois cohomology:
| Group | Forms over \(k\) |
|---|---|
| \(\mathrm{GL}_n\) | Twisted by \(H^1(k, \mathrm{PGL}_n) = \mathrm{Br}(k)[n]\): inner forms are \(D^\times\) for central simple \(k\)-algebras \(D\) of degree \(n\) |
| \(\mathrm{SO}_n\) | Twisted by quadratic forms: forms are \(\mathrm{SO}(q)\) for non-isometric \(q\) of dimension \(n\) |
| \(\mathrm{Sp}_{2n}\) | No non-trivial inner forms over fields (symplectic forms are unique up to isometry) |
| \(\mathrm{U}_n\) | Forms are \(\mathrm{U}(h)\) for Hermitian forms \(h\); also depends on the quadratic extension \(K/k\) |
Outer forms involve non-trivial Galois action on the Dynkin diagram (\(A_n\) and \(D_n\) have non-trivial diagram automorphisms); \(\mathrm{U}_n\) is the outer form of \(\mathrm{GL}_n\) for the quadratic extension \(K/k\).
Root data
For the split forms:
\(\mathrm{GL}_n\) (\(A_{n-1}\))
\(X^* = \mathbb{Z}^n\), \(\Phi = \{\varepsilon_i - \varepsilon_j\}\), \(W = S_n\).
\(\mathrm{Sp}_{2n}\) (\(C_n\))
\(X^* = \mathbb{Z}^n\), \(\Phi = \{\pm \varepsilon_i \pm \varepsilon_j : i < j\} \cup \{\pm 2\varepsilon_i\}\), \(W = (\mathbb{Z}/2)^n \rtimes S_n\).
\(\mathrm{SO}_{2n+1}\) (\(B_n\))
\(\Phi = \{\pm \varepsilon_i \pm \varepsilon_j : i < j\} \cup \{\pm \varepsilon_i\}\); the dual of \(\mathrm{Sp}_{2n}\) under Langlands duality.
\(\mathrm{SO}_{2n}\) (\(D_n\))
\(\Phi = \{\pm \varepsilon_i \pm \varepsilon_j : i < j\}\); no short roots. Self-dual under Langlands duality.
Centres and fundamental groups
| \(G\) | \(Z(G)\) | \(\pi_1(G)\) |
|---|---|---|
| \(\mathrm{GL}_n\) | \(\mathbb{G}_m\) | \(\mathbb{Z}\) |
| \(\mathrm{SL}_n\) | \(\mu_n\) | \(0\) |
| \(\mathrm{PGL}_n\) | \(1\) | \(\mathbb{Z}/n\) |
| \(\mathrm{Sp}_{2n}\) | \(\mu_2\) | \(0\) |
| \(\mathrm{SO}_{2n+1}\) | \(1\) | \(\mathbb{Z}/2\) |
| \(\mathrm{SO}_{2n}\) | \(\mu_2\) | \(\mathbb{Z}/2\) (or \(\mathbb{Z}/4\) for \(n\) odd) |
| \(\mathrm{Spin}_n\) | \(\mu_2 \times \mu_2\) etc. (depends on parity) | \(0\) |
Langlands duality
| \(G\) | \(\hat G\) |
|---|---|
| \(\mathrm{GL}_n\) | \(\mathrm{GL}_n\) (self-dual) |
| \(\mathrm{SL}_n\) | \(\mathrm{PGL}_n\) |
| \(\mathrm{Sp}_{2n}\) | \(\mathrm{SO}_{2n+1}\) |
| \(\mathrm{SO}_{2n+1}\) | \(\mathrm{Sp}_{2n}\) |
| \(\mathrm{SO}_{2n}\) | \(\mathrm{SO}_{2n}\) (self-dual; \(D_n\) has non-trivial outer auto for \(n > 4\)) |
| \(\mathrm{U}_n\) | \(\mathrm{GL}_n\) (with non-trivial Galois action) |
Role in Langlands
Classical groups carry the most worked-out instances of the Langlands correspondence:
- \(\mathrm{GL}_n\): Langlands correspondence proved over local fields (Harris-Taylor, Henniart, Scholze); functional equation of \(L\)-functions; Rankin-Selberg theory.
- Symplectic and orthogonal groups: Arthur's endoscopic classification (2013) gives a complete description of the automorphic spectrum in terms of \(\mathrm{GL}_n\) data.
- Unitary groups: \(p\)-adic Langlands (Bushnell-Kutzko types); Shimura varieties (Kottwitz, Kisin); rigid inner forms (Kaletha).
References
- Weyl, The Classical Groups (1939).
- Borel, Linear Algebraic Groups (1991), §23.
- Springer, Linear Algebraic Groups (1998), §17.
- Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups (Colloquium Publications, 2013).