Simple Singularities and Simple Algebraic Groups
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815 Peter Slodowy
Simple Singularities and Simple Algebraic Groups II
III
Springer-Verlag Berlin Heidelberg New York 1980
Author Peter Slodowy Mathematisches Institut der Universit~t Bonn Wegelerstr. 10 5300 Bonn Federal Republic of Germany
AMS Subject Classifications (1980): 14 B05, 14 D15, 17 B20, 20G15 ISBN 3-540-10026-1 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10026-1 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Introduction B y a r a t i o n a l double p o i n t or a simple s i n g u l a r i t y
(in the introduction,
C , for simplicity) we u n d e r s t a n d the s i n g u l a r i t y of the q u o t i e n t of a c t i o n of a finite s u b g r o u p o f
SL2(C)
say over
C2
b y the
° In the m i n i m a l r e s o l u t i o n of such a singu-
larity an i n t e r s e c t i o n c o n f i g u r a t i o n of the c o m p o n e n t s o f the e x c e p t i o n a l d i v i s o r appears w h i c h can be d e s c r i b e d in a simple way by a Dynkin d i a g r a m of type E6 , E7
or
A r , Dr ,
E 8 . U p to a n a l y t i c isomorphism, t h e s e d i a g r a m s c l a s s i f y the c o r r e -
sponding singularities
(for d e t a i l s see 6.1). Moreover, these d i a g r a m s also c l a s s i f y
just those simple Lie algebras a n d Lie groups w h i c h have root systems w i t h o n l y roots of equal length. Besides the c o n n e c t i o n b e t w e e n the rational double points and Dynkin d i a g r a m s m e n t i o n e d above, w h i c h h a s
been known since the work of Du Val
(cf.
[DV]) and goes
b a c k e s s e n t i a l l y to the c l a i m that the integral i n t e r s e c t i o n form for the c o m p o n e n t s of the e x c e p t i o n a l d i v i s o r is negative d e f i n i t e were d i s c o v e r e d by B r i e s k o r n in the works
(ef.
[Brl] and
[MI],
[Arl]), further c o n n e c t i o n s
[Br3] in i n v e s t i g a t i n g the simul-
taneous r e s o l u t i o n of h o l o m o r p h i c maps w i t h simple singularities,
c o n n e c t i o n s with
v a r i o u s o t h e r structures linked to Dynkin d i a g r a m s such as the Weyl groups, Weyl chambers, the Coxeter numbers and the Coxeter transformations. W i t h k n o w l e d g e of these results and those of K o s t a n t and S t e i n b e r g on the q u o t i e n t s group
G
y
: g + h/w
and
X : G + T/W
(ef.
[Ko2],
of a simple Lie algebra
g
[Stl])
and Lie
b y the o p e r a t i o n of the adjoint
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