Parabolic Subgroups

A parabolic subgroup is a closed subgroup of a reductive group that contains a Borel subgroup. Parabolics interpolate between Borel subgroups (minimal parabolics) and the full group, and they index the flag varieties \(G/P\), which are the geometric scaffolding of representation theory.

Definition

Let \(G\) be a connected reductive group over a field \(k\). A closed subgroup \(P \subseteq G\) is parabolic if \(P_{\bar k}\) contains a Borel subgroup of \(G_{\bar k}\).

Equivalently, \(P\) is parabolic iff the quotient \(G/P\) is a projective variety. (For an arbitrary closed subgroup, \(G/P\) is only quasi-projective; properness over Spec \(k\) characterises parabolic.)

Standard parabolics

Fix a Borel \(B \supseteq T\) (\(T\) a maximal torus) and a set of simple roots \(\Delta\). For each subset \(I \subseteq \Delta\), the standard parabolic \[ P_I \;=\; B \cdot W_I \cdot B \quad (\text{where } W_I = \langle s_\alpha : \alpha \in I \rangle) \] is a parabolic subgroup, and every parabolic containing \(B\) is of this form.

Moreover, every parabolic of \(G\) is \(G\)-conjugate to a unique standard parabolic.

Levi decomposition

Theorem. Every parabolic \(P \subseteq G\) admits a semidirect product decomposition \[ P \;=\; L \ltimes R_u(P), \] where \(R_u(P)\) is the unipotent radical of \(P\), and \(L\) (the Levi subgroup) is reductive. The decomposition is unique up to conjugation by \(R_u(P)\); the Levi subgroups containing \(T\) are parametrised by reflection subgroups of the Weyl group.

For the standard parabolic \(P_I\): \(L_I\) has root system \(\Phi_I = \mathbb{Z} I \cap \Phi\), and \(R_u(P_I)\) is generated by the positive roots not in \(\Phi_I\).

Examples

\(\operatorname{GL}_n\)

Parabolics of \(\operatorname{GL}_n\) are stabilisers of partial flags \(0 = V_0 \subset V_1 \subset \cdots \subset V_k = k^n\).

The standard parabolics correspond to compositions of \(n\) (ordered partitions): if \(n = n_1 + \cdots + n_k\), the standard parabolic is the block upper-triangular subgroup with blocks of sizes \(n_1, \dots, n_k\). The Levi is \(\mathrm{GL}_{n_1} \times \cdots \times \mathrm{GL}_{n_k}\); the unipotent radical is the block-strict-upper-triangular subgroup.

\(\operatorname{Sp}_{2n}\) and \(\operatorname{SO}_n\)

Parabolics correspond to isotropic flags of subspaces of the ambient vector space; the Levi has a "GL × Sp" or "GL × SO" form depending on the dimensions.

Geometry of the flag variety

The variety \(G/P\) is:

projective and smooth;
rational (in fact built from \(\mathbb{A}^1\)'s via Bruhat cells);
decomposed into Schubert cells \(BwP/P\) for \(w \in W^I\) (representatives for \(W/W_I\)), of dimension \(\ell(w)\);
the closures \(\overline{BwP/P}\) are Schubert varieties, key objects in geometric representation theory.

Parabolic induction

For a representation \(\sigma\) of \(L\) (the Levi of \(P\)), the parabolically induced representation \[ \operatorname{Ind}_P^G(\sigma) \;=\; \{f : G \to V_\sigma : f(pg) = \sigma(p) f(g)\} \] is the fundamental construction in representation theory of reductive groups:

For \(G\) over a local field: produces principal series and their generalisations.
For \(G\) over a finite field: gives Harish-Chandra induction, the starting point for Lusztig's classification.
For \(G\) over a global field: Eisenstein series and the residual spectrum.

Role in Langlands

Local Langlands for non-tempered representations is organised by Levi subgroups via parabolic induction (the Langlands classification).
Constant terms of automorphic forms are computed along parabolic subgroups; they record the cuspidal data.
Bernstein decomposition of the category of smooth representations of \(G(F)\) is indexed by pairs \((M, \sigma)\) of Levis and supercuspidal representations.
Affine Hecke algebras and Iwahori-Hecke algebras arise from the parabolic-Bruhat decomposition of \(G/B\) and parahoric structures in the affine setting.