Unipotent radical

Consider connected closed subgroups H of G which are normal and solvable. If Ht and H2 are such, so is (the closure of) H H, since the dimensions cannot increase forever, there is actually a largest such subgroup. We denote it by R and call it the radical of G. By (10.3), the unipotent elements in R form a normal subgroup U, the unipotent radical. We call G semisimple if R is trivial, reductive if U is trivial. The theorem then (for char(fc) = 0) is that all representations are sums of irreducibles iff G is reductive. It is not hard to see this condition implies G reductive (cf. Ex. 20) the converse is the hard part. We of course know the result for R, since by (10.3) it is a torus we also know that this R is central (7.7), which implies that the R-eigenspaces in a representation are G-invariant. The heart of the result then is the semisimple case. This can for instance be deduced from the corresponding result on Lie algebras. 

Let k be algebraically closed, G an algebraic matrix group inside GL (fc). Assume k is G-irreducible. Prove G is reductive. [The unipotent radical is normal and fixes a nontrivial subspace of vectors.]... 

The construction of A, commutes with base extension, since n0(CD L) = n0(CD) L. Hence to prove G, is of multiplicative type and G unipotent we may assume k = k. Then each CD/Rad CD is a product of copies of k, and the homomorphisms to k are group-like elements spanning C,. Thus A, is spanned by group-likes, and G, is diagonalizable. Also, any group-like b in C defines a homomorphism CD - k such a homomorphism vanishes on the radical, so b is in C,. Thus the other tensor factor of A, representing G , has no nontrivial group-likes. Hence by the previous corollary G is unipotent. ... 

G, resp. la dimension des sous-groupes de Cartan de G, resp. la dimension du quotient de G (ou encore de V), par son radical, (cf BIBLE pour toutss ces notions). Introduisons aussi le rang unipotent / u( ) = =... 

L exemple (XVII 6.4 a) ) fournit un exemple de groupe lisse sur un corps k, dont le radical unipotent n est pas ddfini sur k. La proposition suivante donne une mAthode generale pour obtenir de tels groupes ... 

Montrons d abord comment le corollaire rdsulte de la proposition, i) Soit U un radical unipotent de G. Alors est le radical unipotent de G, done majore R, puisque R est lisse unipotent connexe d apres i), done U = G d aprds ii). Or admet H comme quotient, et H n est pas unipotent par hypothdse, done G n est pas unipotent, d ofi une contradiction. 

---