Topic Levi Subgroups and Unipotent Radicals
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Levi Subgroups and Unipotent Radicals

For each parabolic subgroup \(P\) of a reductive group \(G\), there is a canonical decomposition \(P = L \ltimes R_u(P)\) into a Levi subgroup \(L\) (reductive) and the unipotent radical \(R_u(P)\) (unipotent). This decomposition is the algebraic-geometric backbone of parabolic induction and the structure theory of \(G\).

The unipotent radical

For any connected linear algebraic group \(H\) over a field \(k\), the unipotent radical \(R_u(H)\) is the maximal connected unipotent normal subgroup of \(H_{\bar k}\) (i.e.\ the geometric unipotent radical, defined over \(\bar k\) and descending under suitable hypotheses).

For a parabolic \(P\) inside a connected reductive \(G\):

  • \(R_u(P)\) is the unipotent radical of \(P\);
  • \(R_u(P)\) is a (split, geometrically) unipotent group;
  • the quotient \(P / R_u(P)\) is reductive.

Levi subgroups

A Levi subgroup of \(P\) is a closed reductive subgroup \(L \subseteq P\) such that \[ P \;=\; L \ltimes R_u(P). \]

Theorem (Levi decomposition, char \(0\); analogue in char \(p\)). Every parabolic \(P\) admits a Levi subgroup. Levi subgroups are conjugate under \(R_u(P)(k)\); in particular, there is a unique Levi up to \(R_u(P)\)-conjugacy.

In good characteristic the existence is straightforward via the exponential of a graded nilpotent Lie algebra; in general it requires more care (see Conrad–Gabber–Prasad for the pseudo-reductive generalisation).

Standard Levis from root data

Fix a maximal torus \(T \subseteq G\) and simple roots \(\Delta\). For \(I \subseteq \Delta\), the standard parabolic \(P_I = B W_I B\) has:

  • Levi \(L_I\): the subgroup generated by \(T\) and the root subgroups \(U_{\pm \alpha}\) for \(\alpha \in \mathbb{Z} I \cap \Phi\). The root system of \(L_I\) is \(\Phi_I := \mathbb{Z} I \cap \Phi\).
  • Unipotent radical \(R_u(P_I) = U_I\): generated by the root subgroups \(U_\alpha\) for \(\alpha \in \Phi^+ \setminus \Phi_I\).

The Bruhat decomposition of \(L_I\) is the restriction of that of \(G\) to the parabolic Weyl subgroup \(W_I\).

Examples

\(P \subseteq \mathrm{GL}_n\) for the composition \(n = n_1 + \cdots + n_k\)

  • \(P\) = block upper-triangular matrices with diagonal block sizes \(n_1, \dots, n_k\).
  • \(L = \mathrm{GL}_{n_1} \times \cdots \times \mathrm{GL}_{n_k}\).
  • \(R_u(P)\) = block strict-upper-triangular matrices (identity on diagonal blocks).

Minimal parabolics

For \(P = B\) a Borel: \(L = T\), \(R_u(B) = U\) (the maximal unipotent subgroup). This is the "smallest" Levi decomposition.

\(P = G\)

\(P = G\) is itself parabolic (trivially containing every Borel). The Levi is \(L = G\); the unipotent radical is trivial.

Maximal parabolics in \(\mathrm{Sp}_{2n}\)

Stabilisers of isotropic flags; the Levi has the form \(\mathrm{GL}_a \times \mathrm{Sp}_{2(n-a)}\) for some \(a \le n\).

Opposite parabolics

For each parabolic \(P\) with Levi \(L\), there is an opposite parabolic \(P^- = L \ltimes R_u(P^-)\) with the same Levi \(L\) and "opposite" unipotent radical (generated by the negative roots \(-\alpha\) for \(\alpha \in \Phi^+ \setminus \Phi_I\)).

Then \(P \cap P^- = L\), and \(R_u(P) \cdot L \cdot R_u(P^-)\) is a big cell \(\Omega \subseteq G\): open, dense, isomorphic as a variety to \(R_u(P) \times L \times R_u(P^-)\).

Parabolic induction

The Levi decomposition is the input to parabolic induction:

  • For representations of \(G(\mathbb{F}_q)\) (finite reductive groups): the Harish-Chandra functor \(R_L^G : \mathrm{Rep}(L) \to \mathrm{Rep}(G)\) inflates from \(L\) to \(P\) and induces to \(G\).
  • For representations of \(G(F)\) (\(F\) a local field): the normalised parabolic induction \(i_P^G\) is the foundation of the Langlands–Bernstein–Zelevinsky classification.
  • For automorphic representations: Eisenstein series, residues, and the constant-term map all involve the Levi decomposition.

Role in Langlands

  • Cuspidal supports. Every irreducible representation of \(G(F)\) has a unique (up to conjugacy) cuspidal support \((L, \sigma)\) — a Levi and a supercuspidal \(\sigma\) of \(L\) — such that the representation appears in \(i_P^G(\sigma)\).
  • Bernstein decomposition. \(\mathrm{Rep}(G(F))\) decomposes into blocks indexed by cuspidal-support orbits.
  • Endoscopy. Endoscopic groups arise as Levis of inner forms in a systematic way, and Langlands–Shelstad transfer is compatible with parabolic induction.
  • Geometric Langlands. Levi subgroups define constant-term functors on \(\mathrm{Bun}_G\), the foundation of Eisenstein series in the geometric setting.

References

  • Borel, Linear Algebraic Groups (1991), §11, §14.
  • Springer, Linear Algebraic Groups (1998), §8.
  • Conrad–Gabber–Prasad, Pseudo-reductive groups (2010, 2015).
  • Renard, Représentations des groupes réductifs p-adiques (2010).

Prerequisite Topics

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