Reductive group

2020 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]

A reductive group is a linear algebraic group $G$ (over an algebraically closed field $K$) that satisfies one of the following equivalent conditions:

1) the radical of the connected component $G^0$ of the unit element of $G$ is an algebraic torus;

2) the unipotent radical of the group $G^0$ is trivial; or

3) the group $G^0$ is a product of closed normal subgroups $S$ and $T$ that are a semi-simple algebraic group and an algebraic torus, respectively.

In this case $S$ is the commutator subgroup of $G^0$ and $T$ coincides with the radical of $G^0$ as well as with the connected component of the unit element of its centre; $S\cap T$ is finite, and any semi-simple or unipotent subgroup of the group $G^0$ is contained in $S$.

A linear algebraic group $G$ is called linearly reductive if either of the two following equivalent conditions is fulfilled:

a) each rational linear representation of $G$ is completely reducible (cf. Reducible representation); or

b) for each rational linear representation $\rho: G\to \def\GL{ {\rm GL}}\GL(W)$ and any $\rho(G)$-invariant vector $w\in W\setminus\{0\}$ there is a $\rho(G)$-invariant linear function $f$ on $W$ such that $f(w)\ne 0$.

Any linearly reductive group is reductive. If the characteristic of the field $K$ is 0, the converse is true. This is not the case when $\mathop{ {\rm char}} K > 0$: A connected linearly reductive group is an algebraic torus. However, even in the general case, a reductive group can be described in terms of its representation theory. A linear algebraic group $G$ is called geometrically reductive (or semi-reductive) if for each rational linear representation $\rho: G\to \GL(W)$ and any $\rho(G)$-invariant vector $w\in W\setminus\{0\}$ there is a non-constant $\rho(G)$-invariant polynomial function $f$ on $W$ such that $f(w)\ne 0$. A linear algebraic group is reductive if and only if it is geometrically reductive (see Mumford hypothesis).

The generalized Hilbert theorem on invariants is true for reductive groups. The converse is also true: If $G$ is a linear algebraic group over an algebraically closed field $K$ and if for any locally finite-dimensional rational representation by automorphisms of a finitely-generated associative commutative $K$-algebra $A$ with identity the algebra of invariants $A^G$ is finitely generated, then $G$ is reductive (see [Po]).

Any finite linear group is reductive and if its order is not divisible by $\char K$, then it is also linearly reductive. Connected reductive groups have a structure theory that is largely similar to the structure theory of reductive Lie algebras (root system; Weyl group, etc., see [Hu]). This theory extends to groups $G_k$ where $G$ is a connected reductive group defined over a subfield $k\subset K$ and $G_k$ is the group of its $k$-rational points (see [BoTi]). In this case the role of Borel subgroups (cf. Borel subgroup), maximal tori (cf. Maximal torus) and Weyl groups is played by minimal parabolic subgroups (cf. Parabolic subgroup) defined over $k$, maximal tori split over $k$, and relative Weyl groups (see Weyl group), respectively. Any two minimal parabolic subgroups of $G$ that are defined over $k$ are conjugate by an element of $G_k$; this is also true for any two maximal $k$-split tori of $G$.

If $G$ is a connected reductive group defined over a field $k$, then $G$ is a split group over a separable extension of finite degree of $k$; if, in addition, $k$ is an infinite field, then $G_k$ is dense in $G$ in the Zariski topology. If $G$ is a reductive group and $H$ is a closed subgroup of it, then the quotient space $G/H$ is affine if and only if $H$ is reductive. A linear algebraic group over a field of characteristic 0 is reductive if and only if its Lie algebra is a reductive Lie algebra (cf. Lie algebra, reductive). If $K=\mathbf C$, this is also equivalent to $G$ being the complexification of a compact Lie group (see Complexification of a Lie group).

References

[BoTi]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[Hu]	J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Zbl 0325.20039
[Po]	V.L. Popov, "Hilbert's theorem on invariants" Soviet Math. Dokl., 20 : 6 (1979) pp. 1318–1322 Dokl. Akad. Nauk SSSR, 249 : 3 (1979) pp. 551–555
[Sp]	T.A. Springer, "Invariant theory", Lect. notes in math., 585, Springer (1977) MR0447428 Zbl 0346.20020