Unipotent radical - Knowlpedia

Definition

For a smooth connected linear algebraic group $H$ over a field $k$ , the unipotent radical $R_u(H)$ is its largest smooth connected normal unipotent $k$ -subgroup.

A connected linear algebraic group is reductive precisely when its geometric unipotent radical is trivial. The word “geometric” matters over imperfect fields: formation of the largest $k$ -defined unipotent normal subgroup need not detect every unipotent subgroup after extending scalars.

Parabolic subgroups

If $P$ is a parabolic subgroup of a connected reductive group, then $R_u(P)$ is defined over the base field and the quotient $P/R_u(P)$ is reductive. Choosing a Levi subgroup $M$ splits this quotient and gives

P=M\ltimes R_u(P).

For $G=\operatorname{GL}_n$ , a standard parabolic consists of block upper triangular matrices, and its unipotent radical consists of those matrices with identity diagonal blocks.

In automorphic forms

Integration over $R_u(P)(F)\backslash R_u(P)(\mathbb A_F)$ , using the adeles of the global field $F$ , defines the constant term along $P$ . Vanishing of these terms for every proper parabolic is the cuspidality condition.

References

Brian Conrad, Reductive Group Schemes, §§5.2 and 5.4. Author notes.
Armand Borel, Linear Algebraic Groups, second edition, Graduate Texts in Mathematics 126, Springer, 1991, §§11 and 14.