Endomorphisms of Linear Algebraic Groups: Number 80 by Robert Steinberg

By Robert Steinberg

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Sl 1T' , � n - dim t Iwal 8 -1 Wa. ,. I • 2k , both sums ENOO )l)RPlnSMS OP LINEAR AlDEBRAIC GROUPS " over those tor yh1 ch ( a ) holds . Assume, for example, that 2 of order d1agrams . DS ' and It �. n cr The groups W " of 3 . he equat10ns read 1s ot type W W and the correspond1ng groups B4 • and 35 Cn , c01nc1de and express ot types F4 , C3 W1I(7 36 + 27 2 • 6 and 1 + X Al 1 2 + 3 - �. on the other hand , then the equat10ns � as a sum of b1nomial coett1 c1ents 1n the usual way . For the proot ot nec essar11y 11near.

Assume ra and W as above , that L - La , and that E in 3 . 6 is identitled with a subset ot X in the natural way. It (f is an automorphism of ra which flxes E and some t a £ ra such that Cl (ta ) + 1 tor every Cl £ E , then c:r tlxes a fundamental chamber tor W. 4. ) Structure ot algebraic groups . In this section we recall the main propertles ot 11near algebraiC groups and introduce some notatlons which will be used henceforth. The basic source of thls material is [19 ] . K will be an algebraically clos ed tield , over which all algebraic groups Will be taken.

12 £o llows . Our purpos e 1 s to The trans 1t1on to algeb ra1 c tor1 . c arry over the results of the prec ed1ng s ect10n of algebra1 c tor1 . K b e an alg eb ra1cally c los ed fi eld and Let charact er1st 1 c exponent (see §6 ) . p 1ts An algeb ra1 c torus shall mean an algebra1c group 1somorphi c to the produ ct of a finite numb er K* . 2 X (a ) E: t £ T X T T a V the torus 1 . e . homomorphi sms Z - duality Wlth X Obs erve that (algebra1cally) and 5 . 1. that 1f T. l , 1nto to a real vecto r spa ce , and [19 , p .

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