New PDF release: Endomorphisms of Linear Algebraic Groups: Number 80

By Robert Steinberg

ISBN-10: 0821812807

ISBN-13: 9780821812808

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Extra info for Endomorphisms of Linear Algebraic Groups: Number 80

Sample text

Po i s a reflectlon 1 1 - w - group by - + w' 4 . ,0 wit ) . , ls chosen ln henc e ln to If and w l s just ls a reflectlon ln V above t Wt , then 10 by the defiD1tlon of 10 , e riPo • From th1s (a ) f ollows . ,p fl C . 6. grOUp of flxed polnts of C. L . 10 . Assume as ln 4 . 2!! T, � so that � C o C denot e th e any subgroup of ENOOMORPHISMS OF LINEAR ALGEBRAIC GRO UPS W} { Cl - wl t I w E: For c E: C an integer for every a C • Co V c E: P-1c · choose The condIt10n 4 . 7. II E: c E: t C 39 • As is well known : is equivalent to : (vc, a ) � (� 3.

T/ (l - a ) T l s ln the kernel and The kernel , 1 s Just the orthogonal proJect10n ot L on (1" generat e a the restrl ct10n Va n ( l - O") V Vrr ' + L) , hence 1s a latt1ce g enerated by the pro Ject10ns ot the basic trans lat10ns � (u u c 11r ), 4 . 10 . w c so that (c t . 30) 4. 9 holds 1n th1s cas e als o . Let eVeryth1ng be as 1n 4 . m. It becomes we - proje ct - wt . U t c T. 'ben those torm a retlection sub­ - rr) T , the condit10n so that 4 . 10 tollows tram 4. 2 and 4. 9. In the same way we can extend 4 .

M i s a conne ct ed reduct1ve group. Because of 7 . 5 this is a consequenc e of tbe following result , whic h is s lightly more general becaus e � need not b e � an auto­ s em1simple . 8. 2 . Theorem. morphi sm of � T £! Let G b e as i n 8 . 1 � which fixes a Borel subgroup G B. h!!! G � B and a maximal is a connect ed reductive group. Sinc e the s imple components of G are all s1mply connected ( s e e , e . g . , 6 . 4 and the d1s cus sion at the start o f §6) and � ENOO)l)RPHISMS OF we permutes them , LI NEAR ALGEBRAIC GROUPS G may as sume that the notat10ns o f §6 a s soc1ated with 53 1ts elf 1 s s1mple .

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Endomorphisms of Linear Algebraic Groups: Number 80 by Robert Steinberg

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