Chapter VI. Representation Theory
FC14 — Humphreys, Introduction to Lie Algebras and Representation Theory.
This chapter contains 21 definitions, 15 theorems, 8 propositions, 14 lemmas, 9 corollaries, 2 examples, 6 constructions, 1 convention, 1 notation and 2 formulas.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: representation-theory setting
L is semisimple over algebraically closed F of characteristic zero; H is a fixed CSA, Φ its root system, Δ a base, and W the Weyl group. Finite-dimensional modules are the main object, though selected infinite-dimensional modules are admitted.
Definition: weight and weight space of an L-module
For an L-module V and λ∈H*, V_λ={v | h·v=λ(h)v for all h∈H}. If V_λ≠0 it is a weight space and λ is a weight. For finite-dimensional V, V is the direct sum of its weight spaces.
Lemma: root spaces shift weights
For any L-module V, L_α maps V_λ into V_{λ+α}; the sum V′ of all weight spaces is direct and is an L-submodule; if V is finite-dimensional, V=V′.
Definition: maximal vector
A maximal vector of weight λ is a nonzero v∈V_λ annihilated by all positive-root spaces L_α; it is enough to require annihilation by the simple-root spaces.
Definition: standard cyclic module and highest weight
If V=U(L)v^+ for a maximal vector v^+ of weight λ, V is a standard cyclic module of weight λ and λ is its highest weight.
Theorem: structure of a standard cyclic module
For standard cyclic V of highest weight λ: V is spanned by PBW monomials in negative-root vectors applied to v^+; all weights are ≤λ; every weight space is finite-dimensional and dim V_λ=1; every submodule is a sum of weight spaces; V is indecomposable with a unique maximal proper submodule and unique irreducible quotient; every nonzero quotient is standard cyclic of highest weight λ.
Corollary: uniqueness of maximal vector in an irreducible highest-weight module
If a standard cyclic module is irreducible, its maximal vector is unique up to nonzero scalar.
Theorem: uniqueness of irreducible standard cyclic modules
Any two irreducible standard cyclic L-modules with the same highest weight λ are isomorphic.
Construction: one-dimensional Borel module Dλ
For the standard Borel B=H⊕N^+, define the one-dimensional B-module D_λ=Fv^+ by (h+n)v+=λ(h)v+.
Definition/construction: Verma module Z(λ)
The induced module Z(λ)=U(L)⊗_{U(B)}D_λ (the Verma module) is a standard cyclic module of highest weight λ. As a left U(N^−)-module it is canonically U(N^−).
Construction: Verma module by generators and relations
Let I(λ) be the left ideal of U(L) generated by positive-root vectors and h−λ(h)1. Then Z(λ)≅U(L)/I(λ).
Theorem: existence of irreducible highest-weight module V(λ)
For every λ∈H*, the Verma module Z(λ) has a unique maximal submodule Y(λ), and V(λ)=Z(λ)/Y(λ) is the irreducible standard cyclic module of highest weight λ.
Theorem/recorded result: Verma-module Hom-space dimension
The source records Verma’s result that Hom_L(Z(μ),Z(λ)) is either zero or one-dimensional. Verma proved a sufficient condition for the one-dimensional case, later proved necessary by Bernstein–Gelʹfand–Gelʹfand; the condition itself is not restated here.
Theorem: necessary integrality condition for finite dimension
If a finite-dimensional irreducible L-module has highest weight λ, then λ(h_i)=⟨λ,α_i⟩ is a nonnegative integer for every simple root α_i.
Definition: integral and dominant integral weights
A linear function λ∈H* is integral if λ(h_α)∈Z for all roots (equivalently simple roots), and dominant integral if the simple values are nonnegative. The integral weights form the weight lattice Λ and dominant integral weights form Λ^+.
Notation: weight set Π(V) and Π(λ)
Π(V) denotes the set of weights of an L-module V; Π(λ) abbreviates Π(V(λ)).
Theorem: dominant integral highest weights give finite-dimensional modules
If λ is dominant integral, V(λ) is finite-dimensional. W permutes Π(λ), and dim V_μ=dim V_{wμ} for every w∈W.
Corollary: classification of finite-dimensional irreducible modules
λ↦V(λ) gives a bijection from dominant integral weights Λ^+ to isomorphism classes of finite-dimensional irreducible L-modules.
Lemma: commutator formulas in U(L) for simple sl2-triples
For standard generators and k≥0: [x_i,y_j^{k+1}]=0 for i≠j; [h_i,y_j^{k+1}]=−(k+1)c_{ji}y_j^{k+1}; and [x_i,y_i^{k+1}]=−(k+1)y_i^k(k−h_i).
Proposition: weight strings in finite-dimensional highest-weight modules
For λ∈Λ^+, μ∈Π(λ), α∈Φ, the α-string through μ is connected, is reversed by σ_α, and if it runs μ−rα,…,μ+qα then r−q=⟨μ,α⟩.
Proposition: weight set is saturated
For λ∈Λ^+, Π(λ) is saturated. Equivalently μ∈Λ belongs to Π(λ) iff μ and every W-conjugate of μ are ≤λ.
Definition: annihilator ideal J(λ)
For λ∈Λ^+, J(λ) is the left ideal of U(L) annihilating a maximal vector of V(λ). The canonical quotient U(L)/I(λ)=Z(λ)→V(λ) is induced by I(λ)⊆J(λ).
Theorem: generators and relations for V(λ)
If m_i=⟨λ,α_i⟩, then J(λ) is generated by I(λ) together with y_i^{m_i+1} for the simple negative-root generators y_i.
Lemma: commutator with a power in an associative algebra
For elements y,z of an associative algebra, [y^k,z]=Σ_{i=0}^{k−1} yi[y,z]y{k−1−i}.
Definition: weight multiplicity
For λ∈Λ^+ and integral μ, m_λ(μ)=dim V(λ)_μ, taken as zero when μ is not a weight. For fixed λ the source writes m(μ).
Definition/construction: universal Casimir element
Choose Killing-dual bases h_i,k_i of H and root vectors x_α,z_α with κ(x_α,z_α)=1. The element c_L=Σh_ik_i+Σ_{α∈Φ}x_αz_α∈U(L) is the universal Casimir element. Its image under any representation commutes with L.
Lemma: uniqueness of invariant nondegenerate forms on a simple Lie algebra
On a simple Lie algebra, any two nondegenerate symmetric associative bilinear forms are nonzero scalar multiples.
Proposition: universal and representation Casimir elements
For a nontrivial representation of simple L, the image of c_L is a nonzero scalar multiple of its ordinary Casimir operator; for semisimple L, c_L is the sum of the universal Casimirs of the simple ideals and its image still centralizes the representation.
Formula: trace of xαzα on a weight space
For μ∈Π(λ) and α∈Φ, the source derives Tr_{V_μ}(x_αz_α)=Σ_{i≥0}m(μ+iα)(μ+iα,α), with the finite sum determined by the α-string through μ.
Theorem: Freudenthal multiplicity formula
For irreducible V(λ), λ∈Λ^+, and μ∈Λ, multiplicities satisfy [(λ+δ,λ+δ)−(μ+δ,μ+δ)]m(μ)=2Σ_{α>0}Σ_{i≥1}m(μ+iα)(μ+iα,α), where δ is the half-sum of positive roots and m(λ)=1.
Examples: Freudenthal multiplicities for A2 and G2
The source works explicit multiplicity tables for an A_2 highest weight and for a range of G_2 highest weights, illustrating recursive calculation by levels.
Definition: integral weight group ring Z[Λ]
Z[Λ] is the free Z-module on basis e(λ), λ∈Λ, with multiplication e(λ)e(μ)=e(λ+μ). W acts by w(e(λ))=e(wλ).
Definition: formal character
For finite-dimensional V, ch V=Σ_μ(dim V_μ)e(μ)∈Z[Λ]. In particular ch_λ denotes the formal character of V(λ); it is W-invariant.
Proposition: W-invariants have a unique character expansion
Every W-invariant element of Z[Λ] is uniquely a Z-linear combination of the irreducible characters ch_λ, λ∈Λ^+.
Proposition: characters multiply under tensor products
For finite-dimensional L-modules V,W, ch(V⊗W)=ch(V)ch(W).
Definition: polynomial functions and invariant polynomial algebras
For finite-dimensional V, P(V)=S(V*) is the algebra of polynomial functions. P(H)^W denotes Weyl-invariant polynomials on H; with G=Int L, P(L)^G denotes invariant polynomial functions on L.
Construction: trace polynomial
For a finite-dimensional representation φ and k≥0, x↦Tr(φ(x)^k) is a G-invariant polynomial on L. Such functions are called trace polynomials.
Construction: Chevalley restriction map
Restriction from L to H gives an algebra homomorphism θ:P(L)G→P(H)W.
Theorem: Chevalley restriction surjectivity
The restriction map θ:P(L)G→P(H)W is surjective.
Corollary: trace polynomials generate invariant polynomials after restriction
The subalgebra generated by trace polynomials maps isomorphically onto P(H)^W; the source records the full injectivity of Chevalley restriction via the appendix density argument.
Definition: center of U(L)
Z(U(L)) is the set of enveloping-algebra elements commuting with all of U(L), equivalently with all x∈L.
Lemma: center equals inner-automorphism invariants
Z(U(L)) is precisely the subspace of U(L) fixed by G=Int L.
Definition: central character χλ
For z∈Z(U(L)), z acts on a Verma module Z(λ) by a scalar χ_λ(z); χ_λ:Z(U(L))→F is an F-algebra homomorphism, the character determined by λ. The same scalar acts on submodules and quotients.
Definition: linkage of weights
Weights λ,μ∈H* are linked, λ∼μ, if λ+δ and μ+δ are W-conjugate. This is an equivalence relation.
Proposition: maximal vector inside an integral Verma module
If λ is integral and m=⟨λ,α⟩≥0, then the coset of y_α^{m+1} in Z(λ) is a maximal vector of weight λ−(m+1)α.
Corollary: central character equality across a simple affine reflection
Let λ be integral and α a root, and set μ=σ_α(λ+δ)−δ. Then χ_λ=χ_μ.
Corollary: linked integral weights have equal central characters
If integral λ,μ are linked then χ_λ=χ_μ.
Construction: PBW projection and Harish–Chandra homomorphism
Using a PBW ordering N−,H,N+, define ξ:U(L)→U(H) by retaining only PBW monomials in H. Its restriction to the center is multiplicative. Composing with the shift h_i↦h_i−1 gives the Harish–Chandra map ψ:Z(U(L))→S(H)^W, satisfying χ_λ(z)=(λ+δ)(ψ(z)).
Theorem: Harish–Chandra character theorem
For λ,μ∈H*, χ_λ=χ_μ iff λ and μ are linked, equivalently λ+δ and μ+δ lie in the same W-orbit.
Lemma: W-orbits separated by invariant polynomials
Distinct W-orbits in H* are separated by some element of S(H)^W.
Theorem: surjectivity of the Harish–Chandra map
The Harish–Chandra homomorphism ψ maps Z(U(L)) onto S(H)^W.
Definition: Zariski topology and polynomial functions on affine space
On affine n-space F^n, closed sets are common zero sets of polynomial ideals; this defines the Zariski topology. Polynomial functions are evaluations of polynomials.
Definition: irreducible topological subset
A subset is irreducible if it cannot be covered by two proper closed subsets.
Lemma: affine space is irreducible
Affine n-space over the infinite field F is irreducible.
Corollary: nonempty Zariski opens are dense
Every nonempty Zariski-open subset of affine n-space is dense.
Definition/result: p-regular elements
Using the characteristic polynomial of ad x, p-rank is the smallest possible multiplicity of zero as an ad-eigenvalue; x is p-regular when it attains it. The p-regular set is nonempty open and dense, and equals the set of regular semisimple elements; p-rank equals rank L.
Corollary: injectivity of Chevalley restriction
A G-invariant polynomial on L vanishing on H vanishes on the dense regular-semisimple locus and hence identically. Thus θ:P(L)G→P(H)W is injective.
Theorem/recorded result: Chevalley polynomiality theorem for Weyl invariants
The source records Chevalley’s theorem that the invariant algebra S(H)^W is a polynomial algebra.
Definition: convolution algebra X of formal weight functions
X consists of F-valued functions on H* whose support is contained in finitely many downward positive-root cones; convolution makes X a commutative algebra containing finite and Verma-module formal characters.
Definition: Kostant partition function
p(λ) counts representations of −λ as a nonnegative integral sum of positive roots; viewed as an element of X it is the formal character of a Verma module after the appropriate shift.
Definition: Weyl function
The Weyl function is q=∏{α>0}(e{α/2}−e_{−α/2}) in the convolution algebra X.
Lemma: Lemma 24.1A — product formulas for p and q
If f_α is the geometric-series function supported on nonpositive multiples of a positive root α, then p=∏{α>0}f_α, (e_0−e{−α})*f_α=e_0, and q=e_δ∏{α>0}(e_0−e{−α}).
Lemma: Lemma 24.1B — Weyl anti-invariance of q
For w∈W, w(q)=sn(w)q, where sn(w)=(−1)^{ℓ(w)}.
Lemma: Lemma 24.1C — inverse identity
The convolution identity qpe_{−δ}=e_0 holds.
Lemma: Lemma 24.1D — Verma-character coefficient formula
For a Verma module Z(λ), ch Z(λ)(μ)=p(μ−λ)=(p*e_λ)(μ).
Lemma: Lemma 24.1E — q times a Verma character
For every λ∈H, qch Z(λ)=e_λ.
Definition: category Mλ used in the multiplicity proof
M_λ consists of weight modules whose center acts via χ_λ and whose formal character lies in X; it is closed under submodules, quotients, and finite direct sums. By Harish–Chandra, M_λ=M_μ exactly for linked λ,μ.
Lemma: nonzero modules in Mλ contain maximal vectors
Every nonzero module in M_λ has a maximal vector.
Proposition: composition factors of a Verma module
Z(λ) has a finite composition series; every factor is V(μ) for μ≤λ linked to λ; V(λ) occurs with multiplicity one.
Corollary: irreducible character as integer combination of Verma characters
For λ∈H*, ch V(λ) is a Z-linear combination of ch Z(μ) over μ≤λ linked to λ, with coefficient one at μ=λ.
Theorem: Kostant multiplicity formula
For λ∈Λ^+, m_λ(μ)=Σ_{w∈W}sn(w) p(μ+δ−w(λ+δ)).
Lemma: Weyl denominator formula
q=Σ_{w∈W}sn(w)e_{wδ}.
Theorem: Weyl character formula
For λ∈Λ^+, (Σ_{w∈W}sn(w)e_{wδ}) ch V(λ)=Σ_{w∈W}sn(w)e_{w(λ+δ)}; equivalently ch V(λ) is the quotient of these alternating sums.
Corollary: Weyl dimension formula
Writing deg(λ)=dim V(λ), the Weyl dimension formula is deg(λ)=∏{α>0}(λ+δ,α)/(δ,α)=∏{α>0}⟨λ+δ,α⟩/⟨δ,α⟩.
Examples: Weyl dimensions in rank one and two
The source specializes the dimension formula to A_1,A_2,B_2,G_2 and recovers dim of the adjoint and seven-dimensional G_2 modules.
Theorem: Steinberg tensor-product multiplicity formula
For dominant λ′,λ″, the multiplicity of V(λ) in V(λ′)⊗V(λ″) is Σ_{σ,τ∈W}sn(στ) p(λ+2δ−σ(λ′+δ)−τ(λ″+δ)).
Formula: Appendix Casimir eigenvalue on a standard cyclic module
On a standard cyclic module of highest weight λ, the universal Casimir element acts by the scalar (λ+δ,λ+δ)−(δ,δ).
Proposition: Appendix integral-weight composition-series criterion
If Z is a standard cyclic module of integral highest weight λ, then Z has a finite composition series whose factors are V(μ) with μ≤λ and (μ+δ,μ+δ)=(λ+δ,λ+δ).
Construction/comparison: Appendix triangular character inversion and Casimir-only proof route
For integral λ, the Appendix orders the finitely many μ≤λ with the same shifted norm, inverts the resulting upper-unitriangular Verma-character matrix over Z, and derives Weyl’s and Kostant’s formulas using W-anti-invariance of q and Lemma 13.4C, avoiding the full Harish–Chandra theorem.