Index

P –frame bundle, 274, 384, 421, 605

normal, 277

W

p

, 325

Wp , 325

Σ–height, 292

α–string through β, 167

r–determined, 562

|1|–grading, 296

|k|–grading, 238

absolute derivative, 38

action, 24

effective, 25

free, 25

transitive, 25

adjoint action, 20

admissible

family of curves, 110

aﬃne extension, 47

AHS structure, 137

algebraic bracket

of adjoint tractors, 85

ambient connection

for contact projective structures, 582

for projective structures, 528

ambient metric, 531

associated bundle, 28

associated graded

vector space, 235

atlas, 15

automorphism, 163

infinitesimal, 97

inner, 163

Bianchi identity

general, 88

reductive, 90

biholomorphism, 32

Borel fixed point theorem, 307

Borel subalgebra, 291

standard, 184, 291

Borel–Moore homology, 337

Borel–Weil theorem, 185

Bott–Borel–Weil theorem, 357

Kostant’s version, 351

Bruhat decomposition, 333

Bruhat order, 324

bundle of scales, 503

conformal, 124

canonical curves

in Cartan geometries, 110

on homogeneous spaces, 69

preferred parametrization, 568

Carnot group, 267

Cartan connection, 71

conformal, 128

Cartan decomposition, 202

global, 206

Cartan geometry, 71

Cartan involution, 202

global, 206

Cartan matrix, 169

Cartan product, 186

Cartan subalgebra

θ–stable, 213

complex, 162

maximally compact, 213

maximally noncompact, 213

real, 200, 213

Cartan’s criteria, 150

Casimir element, 190

Casimir operator

of a representation, 152

center

of a Lie algebra, 144

of the universal enveloping algebra, 192

central character, 192

centralizer, 163

chains, 574, 584, 591

character, 194

chart, 15

Christoffel symbols, 39

cocycle

of transition functions, 27

commutator ideal, 142

complete reducibility, 152

complexification

623