Lectures on Chevalley Groups

( b ) H is maximal n i l p o t e n t and every subgroup of f i n i t e l a d e x . . i s of f i n i t e index i n i t s normalizer. Proof of Theorem 6: ( a ) Map Ga -> #, by x,: t -> x,(t) . This is a r a t i o n a l homomorphism. So s i n c e Ga i s a connected , ~ g e b r a i c group s o i s Xa . Hence G is a lgebra ic and connected.. Let R = r a d G . Since R is so lvable and normal it i s f i n i t e by t h e Corol lary t o Theorem 5. Since R is a l s o connected R s 1 , and hence G i s semisimple. 4, (b and c ) H is t h e image of Gm under (t1,-*-,tC) -> 4 fl h i ( t i ) and hence i s a l g e b r a i c and connected; so B = U H is i-1 connected, a l g e b r a i c , and so lvable . Let G1 3 B . Then # G1 3 B wa B (some simple r o o t a) , s o G1 > , and hence by Corol la ry 6 of Theorem 4' G1 is not so lvab le and hence ( b ) holds. H is a maximal connected diagonal izable subgroup of B ( f o r any l a r g e r subgroup must i n t e r s e c t U n o n t r i v i a l l y ) . Hence H is a maximal connected diagonal izable subgroup of G (by a theorem i n Chevalleyts jgminaix-e); 3 0 ( c ) holds . 1 ( d ) is c l e a r . To prove ( e ) it s u f f i c e s by (iii) t o prove: Lemma 34: Let %, = [ x a ( t ) l t E kl and ha = [ h a ( t ) l t E k"] Then: ( a ) Fa is defined over ko and xa: G -> $a is a an isomorphism over ko . ( b ) 3, i s defined over ko and ha: Gm ->h is a a homomorphism over ko . Proof: Let {vi] be a b a s i s of M formed of weight vec to r s . Choose vi s o t h a t Xavi # 0 , then wr i t e Xavi = 2 c . .V , and 15 j choose v s o t h a t c i j # 0 . If vi is of weight p , t hen j 2 2 v is of weight p + a . Since x,(t) = 1 + t X a + t xa/2 + * * = j it follows t h a t if a i j is t h e (i , j ) matr ic coordinate (i # j ) func t ion t h e n a i j ( % ( t ) ) = c .t . A l l o t h e r c o e f f i c i e n t s of i~ x,(t) a r e polynomials over k i n t , hence a l s o i n 0 a i j This s e t of polynomial r e l a t i o n s de f ines ) (a a s a group over 1 k, . Now b-a : x a ( t ) -> t is an i n v e r s e of ia , s o t h e i j i j map x, i s an isoinorphism over ko . The proof of ( b ) i s l e f t a s an e x c e r c i s e . We can r ecove r t h e l a t t i c e s =o and L from t h e group G A a s fo l l ows . Le t ;i e L . 3 e f i n e : H --> G by $(wi(ti) ) = m ~ t ~ ~ ( ~ i ) . This i s a c h a r a c t e r def ined over ko . $1 g e n e r a t e s A a l a t t i c e L , t h e c h a r a c t e r group of H . The X,'s a r e de te r . mined by H a s t h e unique minimal un ipo t en t subgroups normalized A by H . If h = m h i ( t i ) t h e n h x,(t)hel = x a ( e ( h ) t ) where A A A a ( h ) = ntia(Hi) . a is c a l l e d a g l o b a l . Define Lo A A h t h e l a t t i c e gene ra t ed by a l l a . Then Lo C L . A S x e r c i s e : There e x i s t s a W-iscmorphism: L -> L such t h a t h A h A Lo -> L !J ----> ; c , and a -> a . (The a c t i o n of W on L 0 ' is g iven by t h e a c t i o n of N/H on t h e c h a r a c t e r g r o u p ) . We summarize our r e s u l t s i n : Ex i s t ence Theorem: Given a r o o t system C , a l a t t i c e L wi th -.-Lo C L C L1 (where Lo and L1 a r e t h e r o o t and weight l a t t i c e s , r e s p e c t i v e l y ) , and an a l g e b r a i c a l l y c lo sed f i e l d lc , t h e n t h e r e e x i s t s a semisiinple a l g e b r a i c group G def ined over k such t h a t L and L a r e r e a l i z e d a s t h e l a t t i c e s of g l o b a l r o o t s and char0 a c t e r s , r e s p e c t i v e l y , r e l a t i v e t o a maximal t o r u s . Furthermore 61 G , ga,. .. can be t aken over t h e prime f i e l d . The c l a s s i f i c a t i o n theorem, t h a t up t o k-isoixorphism every '% !!semisimple a l g e b r a i c group over k has been ob ta ined above, i s .& ... , H g iven by " I-'1 1 ( tl, >t4) -> ) is an isomorphisin over k of a lgej= 0 b r a i c groups. T t Proof: Wri te Hi = X n . .H n E . Given i t j ] we can f i n d 1 i j I1 . L { t i ] such t h a t tT. = Tti iJ ( f o r d e t ( n . . ) # 0 and kq. i s J -! 1 J I t 1 d i v i s i b l e ) . Then n h . (t . ) a c t s on V as m u l t i p l i c a t i o n by i J J P J $(Hi) = pi , i . e . a s n h i ( t i ) . This shows t h a t cp J 1 maps G~ onto i! . Clea r ly rp i s a r a t i o n a l ii1agi2ing def ined m over k . Let {pi] be t h e b a s i s of L dual t o { H i ] l ( i . e . 0 t , P ( H j ) np ;;. (H . ) = 6 . . ) . Wri te p i = .Z n p . Then n(n"t 1 J 1 3 , L E L P ., , j j 1 I -. 1 = ti 2 S O CP e x i s t s and is de f ined over lc . 0 Theorem 7: Let k be an a l g e b r a i c a l l y c losed f i e l d and ko t h e prime s u b f i e l d . Le t G be a Chevalley group p a r a n e t r i z e d by k and viewed a s an a l g e b r a i c group def ined over ko a s above. Then; 62 ( a ) U-HU is an open s u b v a r i e t y of G def ined over ko . ( b ) If n i s t h e number of p o s i t i v e r o o t s , t h e n t h e map kn k*.L CP: x kn -> U-HU def ined by v' x a ( t a ) n h i ( t i ) YO xa( ta) is an isomorphism of a 0 a of v a r i e k i e s over k . 0 Proof : ( a ) We cons ider t h e n a t u r a l a c t i o n of G on An r e l a t i v e t o a b a s i s (Yl , Y 2 , . . . , Y r ) over k made up of products 0 t 1 of His and Xas such t h a t Y1 = Ax,(u > 0 ) . For x E G w e s e t xYi = Z, a . . (x)Y and t h e n d = afl 1~ j , a f u n c t i o n on G over ko . We c la im t h a t x E U-HU = U-B i f and on ly i f d ( x ) # 0 . Assume x E U-B . Since B f i x e s Y1 up t o a nonzero m u l t i p l e and if u E Ut h e n uX, E Xa + % + x kXg , it fo l lows h t ( p ) < h t ( a ) t h a t d ( x ) # 0 . If x E Uw B wi th w E W , w # 1 , t h e same cons ide ra t ions show t h a t d ( x ) = 0 . I f wo E E makes a l l posi t i v e r o o t s n e g a t i v e t h e n by t h e equa t ion woU'w B = B wow B and 1 Theorem 4 t h e two cases above a r e exc lus ive and exhaus t ive , whence ( a ) . ( b j The map cp i s composed of t h e two maps Y'= (y1y\Y2, Y3) : ( t a l a > 0 x (ti) x ( t a l a > 0 -> Ux H x u , and 8: Ux H x U -> U-HU . We w i l l show t h a t t h e s e a r e i somorphisms over ko . For y2 t h i s fo l lows from Lemma 35. Cons i d e r y3 . Le t [vi ] be a b a s i s f o r V , t h e under ly ing vec tor space , made up of weight v e c t o r s i n t h e l a t t i c e i , and f i j t h e ?. norresponding coo rd ina t e f u n c t i o n s on End V , For each r o o t a -qm choose i = i ( a ) , j = j(a) ni = n ( a ) a s i n t h e proof of Lemma 34. s e t x = 1.1.x g ( t g ) . Choosing an o rde r ing of t h e p o s i t i v e 8 > 0 r o o t s c o n s i s t e n t wi th a d d i t i o n , we s e e a t once t l i a ' ~ f i ( a ) , j ( a ) ( X I = n ( a ) t a + an i n t e ~ r a l polynomial in t h e e a r l i e r t t s and t h a t f i j ( x ) is an i n t e g r a l polynomial i n t h e t f s f o r a l l i , j . Thus v3 is an isomorphism over k , and s i n i l a r l y f o r Yl . 0 To prove 0 i s an isomorphism we order t h e vi so t h a t U-,H;U c o n s i s t r e s p e c t i v e l y of subdiagonal un ipo t en t , d iagona l , super:, .<.,<...., . .. . , .:._ -c d iagonal un ipo ten t ma t r i ce s ( s e e Lemma 18, Cor. 3 ) , and t h e n w e . .. , . . . may assume t h a t t h e y c o n s i s t 'of a l l of t h e i n v e r t i b l e mat r ices of t h e s e t y p e s . Le t x = u-hu be i n U-HU and l e t t h e subdiagonal e n t r i e s of u , t l ie d iagona l e n t r i e s of h , t h e s u ~ e r d i a a o n a l e n t r i e s of u be l a b e l l e d t i wi th i > j i = j i < j r e I s p e c t i v e l y . We order t h e i n d i c e s s o t h a t i j precedes kg i n I case i 5 k, j 5 * and i j # kt . Then i n t h e t h r e e cases above I f . .(x) = t . i t . . , r e s p . t i j ~ t . . t . . , i n c r e a s e d by a n 1 J 1 J J J 11 1J i n t e g r a l polynomial i n t ' s preceed ing t i j . We may now induct i v e l y s o l v e f o r t l ie t f s a s r a t i o n a l forms over i n t h e f l s , t h e d i v i s i o n by t h e forms r e p r e s e n t i n g t l ie t . . I s be ing J J j u s t i f i e d by t h e f a c t t h a t t h e y a r e nonzero oil UoHU . Thus 8 i s an isomorphism over ko and ( b ) fo l lows . minors Call 1 , /::: 21 c o n s i s t s of a l l ( a i j ) such t h a t t h e a r e n ~ n s i n g ~ l a r . I 64 k Remark : It e a s i l y fo l lows t h a t t h e L i e a lgeb ra of G i s d We can now e a s i l y prove t h e fo l lowing important f a c t ( b u t w i l l r e f e r t h e r e a d e r t o ~ g m i n a i r e Bourbaki , Exp 219 i n s t e a d ) . Let G be a Chevalley group over Q , viewed a s above a s an a l g e b r a i c maLric group over $ , t h e prime f i e l d , and I t h e corresponding i d e a l over 22 ( c o n s i s t i n g of a l l polynomials over which van ish oil G ) . Then t h e s e t of ze ros of I i n any a lge b r a i c a l l y c l o s e d f i e l d k i s j u s t t h e Chevalley group over k of t h e same t y p e (same r o o t sys tem and same weight l a t t i c e ) a s G . Thus we have a f u n c t o r i a l d e f i n i t i o n i n t e rms of equa t ions of a l l of t h e semisimple a l g e b r a i c groups of any g iven t y p e . 1 Co ro l l a ry 1: Let k,ko , G ,V be as above. Let G be a Chevalley -. group construcZzd u s i n g V ' i n s t e a d of V b u t wi th t h e same . I Assume t h a t LV > LVT . Then t h e homomorphir;ia ~ 3 : G -> G I t a k i n g x,(t) ---3 x,(t) f o r a l l a and t i s a homomorphism I of a l g e b r a i c groups over ko . 1 Proof: Consider f i r s t cp I U-HU . By Theorem 7 we need on ly show t h a t cp [ H i s r a t i o n a l over ko The nonzero coo rd ina t e s of 1 T P ( H i ) 1 h i ( t i ) a r e TT -L ( E LV1) . The nonzero coo rd ina t e s i A H i ) of h . . a r e t i ( i i E LV) . Each of t h e former i s a 1 1 1 monomial i n t h e l a t t e r (because LVl C LV) , and hence i s r a t i o n a l 1 over k . Now f o r w E W, uW ( r e s p . w , ) can be chosen wi th 0 c o e f f i c i e n t s i n ko ( f o r w a ( l ) = ( 1 ) ( 1 ) a(1)) , s o t h a t 1 cp J(: U-B i s r a t i o n a l over ko . Since B.LB C w w -'u-B , we conc lude t h a t i s r a t i o n a l over ko 65 Coro l la ry . . 2: The homomorphism c p .: SL -a 2 > < K a , ) ( a > (of 1 Corol la ry 6 t o Theorem 4 ) i s a homomorphism of a l g e b r a i c groups over lco . Proof : This i s a s p e c i a l case of Coro l la ry 1. 7 C o r o l l a r y L : Assume Z,V, and 4 a r e f i x e d , t h a t V i s univ e r s a l , k C 1; a r e f i e l d s and Gk and GK a r e t h e corresponding Chevalley groups. Then Gk = GK n GLM, . Proof : C l e a r l y G C GK n GLM, . Suppose x E G K n G L M j k . 1c I Then x = uhw,v ( s e e Theorem 4 ) with W w d e f i n e d over t h e prime f i e l d . We must show t h a t xu,' E G k , i . e . uhuE G k -..1 where u = U W v w W . Write uhu= ('17 h i ( t i I a F x,(%) a > oAa 1 0 with ta, ti E K . Applying cp-' of Theorem 7 , we g e t > 0 X ( t i) X ( t a l a < 0 Since uhui s de f ined over k and cp-' i s def ined over ko , a l l t,, ti E k . Hence uhuE Gg . Remark: Suppose k = (t and G is a Chevalley group over k . Then G has t h e s t r u c t u r e of a complex L ie group, and a l l t h e preceding s t a t emen t s have obvious mod i f i ca t ions i n t h e language of L i e g roups , a l l of which a r e t r u e . For example, a l l complex semisimple L ie grougs a r e inc luded i n t h e c o n s t r u c t i o n , and i n Theorem 7 i s an isomorphism of complex a n a l y t i c manifolds .