4

1. SOME THEORY OF LINEAR ALGEBRAIC GROUPS

degree of F(K) over F. If K is not connected, then dim(if) is defined to be

dim(if°).

PROPOSITION

1.1.6. Let K be an algebraic group. Then

(a) dim(K) oo;

(b) dim(if) = 0 if and only if K is finite;

(c) If H is a closed subgroup of K, then dim(H) dim(K), with equality holding

if and only if K H;

(d) The set of connected subgroups of K satisfies the maximum condition;

(e) For any morphism \ : K — H is of algebraic groups, we have dim(if) =

dim(ker(/)) -f dim(/(if)); and

(f) dim(ifi x if2) = dim(ifi) + dim(if2).

EXAMPLES

1.1.7. For any integer n 1, the following subgroups of GLn(F)

are closed: SLn(F); T

n

, the group of all upper triangular nonsingular matrices; D

n

,

the group of all diagonal nonsingular matrices; t/

n

, the subgroup of Tn consisting

of all upper triangular matrices all of whose diagonal entries equal 1.

For any n-dimensional vector space V over JP, the algebraic group structure on

GLn(F) can be transported to one on GL(V) by choice of a basis of V, and the

resulting algebraic group structure on GL(V) is independent of the choice of basis.

Stabilizers in GL(V) of subspaces, multilinear forms and quadratic forms are all

closed subgroups. Finally, the natural action of SL(V) on V*£)V yields a morphism

SL(V) - 5L(y* ® V) with kernel Z(SL(V)) and one may define PSL(V) to be

the image of this morphism.

1.2. Jordan Decomposition

DEFINITION 1.2.1. Let K be an algebraic group. A mapping tp : K — K is

rational if and only if for every polynomial function / € F[K], the composite foi/j

again lies in F[K]. In this case t/* : F[K] — F[K] is the F-algebra homomorphism

defined by ^*(/) = / o ip.

Let g G K and let pg : K — K be right translation by g (so that pg(x) = xg).

Then pg is rational, and g is called semisimple (resp. unipotent) if and only if the

jP-linear transformation p* of F[K] is semisimple (resp. locally unipotent). This

condition means that F[K] is the sum of finite-dimensional p*-invariant subspaces

on each of which the restriction of p* is diagonalizable (resp. p* — 1 is nilpotent).

Finally, let ^ be an automorphism of the algebraic group K. Then ^ is

semisimple if and only if the linear transformation ip* is semisimple.

We remark that one strictly should say "g is semisimple with respect to if,"

etc., as the definition depends a priori on if. But this is unnecessary in view of

1.2.2b below.

PROPOSITION 1.2.2. Let K be an algebraic group.

(a) If (j) : if — H is a morphism of algebraic groups and g G if is semisimple

{resp. unipotent), then 4(g) is semisimple (resp. unipotent);

(b) If H is a closed subgroup of K and g G H, then g is semisimple (resp.

unipotent) with respect to H if and only if it is semisimple (resp. unipotent)

with respect to K;

(c)

If 9 € K

is

semisimple, then the inner automorphism ig : x i—

xg

of K is a

semisimple automorphism of K;