Index — Tome III

A combined index of notation and terminology for Tome III. Page numbers refer to the 2011 SMF re-edition.

Entry	Pages
$a d (G)$	127
Adjoint (canonical decomposition)	243
Adjoint $S$ -reductive group	127
Adjoint root datum	91
$a d (R)$	95, 98
$α *$	110
Paired (sections of $g_{α}$ and $g_{- α}$ )	110
$A_{S} (R)$	223
$Aut_{S - g r .} (G)$	217
Automorphisms of a reductive group	216
Automorphisms of a root datum	98
Automorphisms of Borel subgroups of reductive groups	248
$Aut (R)$	98
$Aut_{s} (R)$	99
$B^{a d}$	224
Borel (subgroups of)	131
$B or (G)$	156
Bruhat (decomposition of)	149, 151
Cartan (subgroups of)	128
Center of an $S$ -reductive group	119
Weyl chambers	83
Chevalley (group scheme of)	209
$C_{ℓ}^{cr i t}$	292
Closed (set of roots)	71
Coroot of a reductive group scheme	110, 111
Coroot of a root datum	64
Infinitesimal coroot $H_{α}$	114
$cor a d (G)$	171
Coradical (torus)	171
Coradical of a root datum	92
$cor a d (R)$	92
Killing couple	135
$C r i t (G)$	292
Critical (subgroup)	292
$C$ -critical (torus)	292
$C_{i} (T)$	262
`CT`	293
Splitting of a reductive group	113
Split (reductive group)	113
Split (torus)	10
Splittable (reductive group)	113
$d \overset{e}{ˊ} r (R)$	95
Dynkin diagram of a root datum	105
Root datum	63
Root datum (adjoint)	91
Root datum (dual)	64
Root datum (induced or coinduced)	94
Root datum (irreducible)	102
Root datum (reduced)	68
Root datum (relative)	325
Root datum (semisimple)	64
Root datum (simply connected)	91
Root datum (trivial)	64
Root datum (twisted)	112
Pinned root data	179
$Dy n (G)$	228
$Dy n_{0} (G)$	244
$Dy n_{0, t} (G)$	244
$E_{Δ} (R)$	98
$E_{Δ}^{s} (R)$	99
Generation of $G (S)$ by the $U_{α} (S)$ ( $S$ local)	151
Pinnings	177
Pinned ( $S$ -reductive group)	178
Essentially free ( $S$ -scheme)	132, 152
`exp`	29
$exp_{α}$	109
Exponential (map)	29
(F) (formula)	35
$F^{α}$	12
$F ib (S, G)$	215
Form of $G$ over $S$	222
$g_{α}$	5
$Γ_{0} (R)$	64
Generators and relations for a pinned group	182
$G e n (G)$	299
$G e n (P / Q)$	299
$G e n (/ Q)$	299
$G^{q - \overset{e}{ˊ} p .}$	233
Big cell	118, 271
Pinned group	216
Group of type (RA)	130
Group of type (RR)	128
Derived group of an $S$ -reductive group	171
Quasi-pinned group	230
Weyl group (extended)	212
Weyl group of a torus	6
Weyl group of a root datum	65
Weyl group of a root datum (generators and relations)	86
Weyl group of $(G, T) / S$	116
Weyl group in semisimple rank 1	44
Reductive group over an algebraically closed field	4
Reductive group over an arbitrary base	12
Semisimple group associated to an $S$ -reductive group	127
Semisimple group over an algebraically closed field	5
Semisimple group over an arbitrary base	12
$G_{S}^{\overset{ˊ}{E} p} (R)$	209
Indivisible (root)	68
Central isogeny of $S$ -reductive groups	124
Isogeny of $S$ -reductive groups	124
Isogeny of root data	90
$I so m^{e x t} (G, G^{'})$	220
$I so m^{in t, u} (G, G^{'})$	226
Isotrivial (locally)	237
Isotrivial (semi-locally)	237
Isotypic (component)	243
$K i l (G)$	156
$Λ (R)$	97
$L e v (P)$	282
Cartan matrix of a root datum	104
Morphisms of root data	90
Morphisms of pinned groups	178
Morphisms of split groups	122
$Opp (B)$	158
$Opp (G)$	305
$Opp (/ P)$	305
Opposite (parabolic subgroups)	303
$or d_{Δ} (α)$	76
$O f (E)$	289
$p_{α}$	27, 109
$P a r (G)$	289
`PL`	293
`PLT`	293
$p$ -morphisms of pinned root data	179
$p$ -morphisms of reduced root data	100
Weights of a root datum	97
Fundamental weights	97
General position (Borel subgroups in)	158
Relative position of two parabolic subgroups	296
Standard position of two parabolic subgroups	308
Transversal position of two parabolic subgroups	297
`PT`	293
Quasi-splittable ( $S$ -reductive group)	230
`Qép`	232
Quasi-pinnings	230
Central quotients of reductive groups	126
Chevalley's rule	211
Root of a reductive group scheme	13, 109
Root of a root datum	64
Infinitesimal root $α$	114
$r a d_{u} (G)$	2
$r a d (G)$	127
Radical of an $S$ -reductive group	127
Radical of an algebraic group	2
Radical of a parabolic subgroup	284
Radical of a root datum	92
Unipotent radical	2
Unipotent radical of a type (RC) subgroup	166, 281
Unipotent radical of a parabolic subgroup	281, 283
$r a d (R)$	92
$r a d_{u} (P)$	281, 283
Reductive rank of a smooth affine $k$ -group	4
Reductive rank of a root datum	64
Semisimple rank of a smooth affine $k$ -group	5
Semisimple rank of a root datum	64
$r g^{re d} (G)$	4
$r g^{re d} (R)$	64
$r g^{ss} (G)$	5
$r g^{ss} (R)$	64
$R$ , $R *$ (root data)	64
`Redext`	232
`Rev`	232
$R (G)$	179
$R$ (scheme of roots)	15
$s_{α} (t)$	110
Dynkin scheme of a reductive group	228
Scheme of coroots of an $S$ -reductive group	111
Scheme of roots of an $S$ -reductive group	15
Scheme of parabolic subgroups of a reductive group	289
Scheme of types of parabolic subgroups of a reductive group	291
Reductive group scheme	8
Henselian local scheme	220, 223, 261, 330
Semi-local scheme	162, 230, 239, 241, 287, 288, 311, 318, 322
$sc (R)$	95
$ss (R)$	95
$s_{G}$	236
Simply connected $S$ -reductive group	127
Simply connected (canonical decomposition)	243
Simply connected root datum	91
Subgroups with commutative quotients	174
Cartan subgroups	128
Minimal parabolic subgroups	316
Borel subgroups	131
Borel subgroups of a split reductive group	139
Subgroups of type (R)	131
Type (R) subgroups with reductive fibers	162
Type (R) subgroups of a split reductive group	137
Subgroups of type (RC)	165
Parabolic subgroups	131, 279
Critical reductive subgroups	163
$St an d (G)$	309
Root system	64
Root system of a reductive group scheme	14
Positive root system	73
Simple root system	71
Chevalley system	269
Elementary system	28
Elementary system (generators and relations)	58
Chevalley systems	209
$t_{α β} = (w_{α} w_{β})^{n_{α β}}$	180
$T^{a d}$	224
$T_{α}$	7
Uniqueness theorem	207
Theorem 90	215
Existence theorem	267
Bruhat theorem	149, 151
Conjugacy theorem	311
Fundamental theorem	202
Coradical torus of an $S$ -reductive group	171
Critical torus	163
Split torus	10
Maximal torus of an $S$ -group $G$	10, 23
Trivialized torus	10
$T or (G)$	156
Strict transporter of two type (R) subgroups	134
Strict transporter of parabolic subgroups	280
Braid relations	212
Type of a reductive group	115
Type at a point $s$	115
$t$	289
$t (P)$	289
$U_{α}$	33, 110
$V^{\times}$	16
$w_{α}$	7
$w_{α} (X)$	111
$W (T)$	6
$W (F)_{α}$	12
$W (R)$	65, 98
$W * (R) = W (R *)$	65
$W *$	212
$X_{α}$	109
$Z_{α}$	7
$Z (R)$	98
$Z^{1} (S^{'} / S, G)$	262

Séminaire de Géométrie Algébrique du Bois-Marie — English

Index — Tome III