Reductive group

Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field.

over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of

over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of

For a perfect field k, that can be avoided: a linear algebraic group G over k is reductive if and only if every smooth connected unipotent normal k-subgroup of G is trivial.

For a general field k, different quadratic forms of dimension n can yield non-isomorphic simple groups SO(q) over k, although they all have the same base change to the algebraic closure

In more detail: an affine group scheme G of finite type over a field k is called linearly reductive if its finite-dimensional representations are completely reducible.

[7] For k of characteristic p>0, however, Masayoshi Nagata showed that G is linearly reductive if and only if Go is of multiplicative type and G/Go has order prime to p.[8] The classification of reductive algebraic groups is in terms of the associated root system, as in the theories of complex semisimple Lie algebras or compact Lie groups.

together with 1-dimensional subspaces indexed by the set Φ of roots: For example, when G is the group GL(n), its Lie algebra

There are finitely many Borel subgroups containing a given maximal torus, and they are permuted simply transitively by the Weyl group (acting by conjugation).

Root systems are classified by the corresponding Dynkin diagram, which is a finite graph (with some edges directed or multiple).

In short, the Dynkin diagram describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant inner product on the weight lattice.

For a split reductive group G over a field k, an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of G, but also a copy of the additive group Ga in G with the given Lie algebra, called a root subgroup Uα.

For a split reductive group G over a field k, the smooth connected subgroups of G that contain a given Borel subgroup B of G are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram).

These are classified by their Satake diagram; or one can just refer to the list of simple Lie groups (up to finite coverings).

For a connected real reductive group G, the quotient manifold G/K of G by a maximal compact subgroup K is a symmetric space of non-compact type.

For a reductive group G over a field k that is complete with respect to a discrete valuation (such as the p-adic numbers Qp), the affine building X of G plays the role of the symmetric space.

For a split reductive group G over a field k of positive characteristic, the situation is far more subtle, because representations of G are typically not direct sums of irreducibles.

One important result is that the dimension and character of L(λ) are known when the characteristic p of k is much bigger than the Coxeter number of G, by Henning Andersen, Jens Jantzen, and Wolfgang Soergel (proving Lusztig's conjecture in that case).

[21] For any prime p, Simon Riche and Geordie Williamson conjectured the irreducible characters of a reductive group in terms of the p-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.

By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field.

A reductive group over a field k is called isotropic if it has k-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise anisotropic.

For example, let D be a division algebra with center a p-adic field k. Suppose that the dimension of D over k is finite and greater than 1.

For a Lie group G, a lattice in G means a discrete subgroup Γ of G such that the manifold G/Γ has finite volume (with respect to a G-invariant measure).

These problems motivate the systematic study of G-torsors, especially for reductive groups G. When possible, one hopes to classify G-torsors using cohomological invariants, which are invariants taking values in Galois cohomology with abelian coefficient groups M, Ha(k,M).

In this direction, Steinberg proved Serre's "Conjecture I": for a connected linear algebraic group G over a perfect field of cohomological dimension at most 1, H1(k,G) = 1.

Serre's Conjecture II predicts that for a simply connected semisimple group G over a field of cohomological dimension at most 2, H1(k,G) = 1.

More generally, for any number field k, Martin Kneser, Günter Harder and Vladimir Chernousov (1989) proved the Hasse principle: for a simply connected semisimple group G over k, the map is bijective.

The analogous result for a global field k of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group G over k, H1(k,G) is trivial (since k has no real places).

[36] In the slightly different case of an adjoint group G over a number field k, the Hasse principle holds in a weaker form: the natural map is injective.

[37] For G = PGL(n), this amounts to the Albert–Brauer–Hasse–Noether theorem, saying that a central simple algebra over a number field is determined by its local invariants.