Levi subgroup - Knowlpedia

Definition

Let $G$ be a connected reductive group and let $P\subseteq G$ be a parabolic subgroup. A Levi subgroup of $P$ is a reductive closed subgroup $M\subseteq P$ for which multiplication induces an isomorphism

M\ltimes R_u(P)\xrightarrow{\sim}P,

where $\ltimes$ denotes a semidirect product and $R_u(P)$ is the unipotent radical.

Existence and nonuniqueness

Parabolic subgroups of connected reductive groups over a field admit Levi subgroups over that field. Any two Levi subgroups of a fixed $P$ are conjugate by an element of $R_u(P)$ . Thus “the Levi of $P$ ” usually means a choice, while its conjugacy class inside $P$ is canonical.

If $P=P_G(\lambda)$ is defined by a cocharacter $\lambda:\mathbb G_m\to G$ , then the centralizer $Z_G(\lambda)$ is a Levi subgroup and $R_u(P)=U_G(\lambda)$ .

Representation-theoretic role

Over a nonarchimedean local field, parabolic induction starts with a representation of $M(F)$ , inflates it to $P(F)$ , and induces it to $G(F)$ . The adjoint construction is the Jacquet module. Levi subgroups also index the terms in the trace formula and the blocks in the Bernstein decomposition.

References

Brian Conrad, Reductive Group Schemes, §5.4. Author notes.
Armand Borel and Jacques Tits, “Groupes réductifs,” Publications Mathématiques de l'IHÉS 27 (1965), 55–150. Numdam.