Rational Representations of Algebraic Groups Tensor Products and Fil
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1140 Stephen Donkin
Rational Representations of Algebraic Groups: Tensor Products and Filtrations
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Stephen Donkin School of Mathematical Sciences, Queen Mary College Mile End Road London E1 4NS, England
Mathematics Subject Classification (1980): 20G ISBN 3-540-15668-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15668-2 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translating, 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
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Table of Contents
Introduction
1
Homological algebra
7
1.1
Induction
7
Chapter 1.
1.2
Injective modules for soluble groups
12
1.3
Reductive groups
14
1.4
B cohomology
15
1.5
Induced modules
16
1.6
Kempf's Vanishing Theorem
18
1.7
Parabolic subgroups
18
More homological algebra
21
Chapter 2. 2.1
Applications of the Vanishing Theorem
21
2.2
Euler characteristics
22
2.3
Some useful results
26
Reductions
31
Good filtrations
31
Chapter 3. 3.1 3.2
Good filtrations for reductive groups
35
3.3
Canonical filtration
38
3.4
Good filtrations for semisimple groups
40
3.5
Good filtrations for sernisimple, simply connected groups
46
3.6
The canonical filtration revisited
49
3.7
A useful lemma
50
The classical groups
52
Exterior powers
52
Chapter 4. 4.1 4.2
Miniscule weights
53
4.3
Exceptional weights
53
4.4
The first fundamental dominant weight
54
VI
4.5
Some module homomorphisms
56
4.6
Character formulas
57
4.7
The canonical filtration again
59
4.8
The restriction of Y(A to P Q r) The restriction of Y(A ) to the maximal parabolic subgroups r
60
Homological algebra revisited
76
4.9
Chapter 5.
70
Chapter 6.
86
Chapter 7.
89
Introduction
89
7.2
Y (AI)
89
7.3
Y (A
7.4
Odd characteristic
7.5
Exact sequences
7.6 7.7
The restriction of Y(A ) to P Q 2 M, N/E and Y /S
7.8
l:
,',\ {a 3}
110
7.9
l:
,',\{a }
112
7.10
l:
117
7.11
,',\{a } l
l: arbitrary
124
7.12
Y(A
124
7.1
92
4)
98
99
2
7.13
Y(A
2)
and
2)
109
128
are true
Chapter 8.
Chapter 9.
101
133
E
140
7
9.1
Y (AI)
140
9.2
Y (A
142
9.3
6) p '" 2 or 7
9.4
The restriction of Y(A
9.5 9.6 9.7 9.8
.
P"
143
7)
to P
n
147
(D)
155
Character formulas
157
L.
.
P" t..
Y (A
7
)
(E)
160 168
VII
Chapter 10. 10.1
E
169
8
169
Y (AI)
10.2
p
10.3
The restriction of Y (1.. ) to P 7 n · PL (D)
178
Orbits and characters · PL (E)
187
10.4 10.5 10.6 10.7
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