Index of terminology

The reference numbers indicate respectively the chapter, paragraph and number.

absolutely simple (Lie algebra): VIII.3.2

adjoint (representation of $\mathbf{S}\mathbf{L}(2, k)$)$:$ VIII.1.4

admissible (lattice): VIII.12.8

alcove: IX.5.2

anti-hermitian (endomorphism): IX.1.1

anti-invariant: IX.7.4

biorder (of a Lie algebra): VIII.12.1

Borel (subalgebra): VIII.3.3, VIII.3.5

canonical (Cartan subalgebra, root system, basis): VIII.5.3

canonical (involution of $\mathfrak{s}\mathfrak{l}(2, k)$)$:$ VIII.1.1

Cartan (subalgebra): VII.2.1

Casimir element: IX.7.6

central (character of a $\mathfrak{g}$-module): VIII.6.1

central (function): IX.8.3

chamber: IX.5.2

character (of a $\mathfrak{g}$-module): VIII.7.7

Chevalley (order): VIII.12.7

Chevalley (system): VIII.2.4

classical (splittable Lie algebra): VIII.3.2

clean (subgroup): IX.4, Exerc. 15

Clebsch-Gordan (formula): VIII.9.4

compact (Lie algebra): IX.1.3

compact (real form of a complex Lie algebra): IX.3.1

compatible (representations): VII.3.1, VIII.1.4

condition (AC): VII.1.1

conjugation: IX.3.1

contravariant diagram: IX.4.9

covariant diagram: IX.4.9

Coxeter element: IX.4, Exerc. 14

dominant: IX.7.1

eigenvalue: VII.1.1

eigenvector: VII.1.1 elementary (automorphism): VII.3.1

embedding (of a manifold in the neighbourhood of a subset): IX.9.1 exceptional (splittable simple Lie algebra): VIII.3.2

exponential: VIII.8.1

facet (associated to a parabolic subset): VIII.3.4

Fitting (decomposition of a module): VII.1.1

flag: VIII.13.1

Fourier cotransform: IX.8.1

Fourier transform: IX.8.1

framing: IX.4.10

framing (of a split semi-simple Lie algebra): VIII.4.1

framed (semi-simple Lie algebra): VIII.4.1

fundamental (dominant weight): IX.7.1

fundamental (representations, simple $\mathfrak{g}$-modules): VIII.7.2

generating family (defined by a framing): VIII.4.1

Harish-Chandra (homomorphism): VIII.6.4

Hermann Weyl (formula of –): VIII.9.1

homogeneous symmetric space: IX.1, Exerc. 8

invariant (polynomial function): VIII.8.3

involutive (Lie algebra): IX.1, Exerc. 7

irreducible (involutive Lie algebra): IX.1, Exerc. 7

irreducible (homogeneous symmetric space): IX.1, Exerc. 8

isotropic (flag): VIII.13.2, VIII.13.3

isotypical (component of highest weight $\lambda$ )$:$ VIII.7.2

Jacobson-Morozov (theorem): VIII.11.2

lattice (in a $\mathbf{Q}$-vector space): VIII.12.1

linear tube: IX.9.3

maximal torus: IX.2.2

minuscule (weight): VIII.7.3

moderately increasing: IX.8.2

nilpotent (component of an element): VII.1.3

nilspace: VII.1.1

nodal group (of a torus): IX.4.2

nodal vector: IX.4.5

order (of a $\mathbf{Q}$-algebra): VIII.12.1

orthogonal (Lie algebra): VIII.13.2

orthogonal (irreducible representation): VIII.7.5

parabolic (subalgebra): VIII.3.4, VIII.3.5

partition (into positive roots): VIII.9.1

permissible (lattice): VIII.12.6

positive root: IX.7.1

primary (subspace): VII.1.1

primitive (element of a module): VIII.1.2, VIII,6.1

principal (nilpotent element, simple element, $\mathfrak{s}\mathfrak{l}_2$-triplet): VIII.11.4 principal (orbit): IX.9.4

principal (subgroup): IX.4, Exerc. 18

proper (subspace): VII.1.1

quasi-maximal (isotropic flag): VIII.13.4

R-extremal (element): VIII.7.2

R-saturated (subset): VIII.7.2

radical (subgroup): IX.4.7

rank (of a Lie algebra): VII.2.2

rank (of a Lie group): IX.2.4

rapidly decreasing: IX.8.2

real form (of a complex Lie algebra): IX.3.1

reduced (root diagram): IX.4.8

regular (element for a linear representation): VII.4.1

regular (element of a Lie algebra): VII.2.2

regular (element of a Lie group): VII.4.2

representation ring (of a Lie algebra): VIII.7.6

root: IX.4.4

root (of $(\mathfrak{g},\mathfrak{h})$)$:$ VIII.2.2

root diagram: IX.4.8

semi-simple (component of an element): VII.1.3

semi-spinorial (representation): VIII.13.4

simple (element): VIII.11.3

simple (root): IX.7.1

singular hyperplane: IX.5.2

$\mathfrak{s}\mathfrak{l}_2$-triplet: VIII.11.1

spinorial (representation): VIII.13.2, VIII.13.4

split (reductive Lie algebra): VIII.2.1

splittable (Lie subalgebra): VII.5.1, VIII.10

splittable (Cartan subalgebra, reductive Lie algebra): VIII.2.1 splittable envelope (of a subalgebra of $\mathfrak{g}\mathfrak{l}(V)$)$:$ VII.5.2

subgroup (Cartan –): IX.2.2

subgroup (of maximum rank): IX.2.4

subgroup (principal –): IX.4, Exerc. 18

symplectic (Lie algebra): VIII.13.3

symplectic (irreducible representation): VIII.7.5

tangent (linear map of a polynomial map): VII.App.I.2

topology of compact $C^r$-convergence: IX.6.4

torsion prime number: IX.5, Exerc. 9 to 11

torus: IX.1.2

transversal: IX.9.3 and IX.9, Exerc. 16

type $(A_n,B_n, . . .):$ IX.3.3

type (of a representation): IX.App.II.1, IX.App.II.2

vector (group): IX.1.2

very regular (element): IX.5.1 weights (of a $\mathfrak{g}$-module): VII.1.1, VIII.1.2, VIII.6.1 weights (of a representation): IX.4.3

weights (group of – of a split Lie algebra): VIII.2.2 Weyl (group): IX.2.5

Weyl (group of a split algebra): VIII.2.2

Witt (basis): VIII.13.2, VIII.13.3, VIII.13.4

Zariski (topology): VII.App.I.1