18.745 Introduction to Lie Algebras Fall 2004

Lecture 24

Prof. Victor Kač Scribes: Maksim Lipyanskiy and Yakov Shapiro

For the last two lectures 𝔤 will be a finite-dimensional semisimple Lie algebra over an algebraically closed field 𝔽 of characteristic 0.

Choose a Cartan subalgebra 𝔥 in 𝔤. Let Δ ⊂ 𝔥^{∗} be the set of roots, Δ_{+} the subset of positive roots. Let 𝔤 = 𝔥 ⊕ (⨁_{α∈Δ} 𝔤_{α}) be the root space decomposition of 𝔤.

Recall that 𝔤_{α} = 𝔽E_{α}. Let 𝔫_{+} = ⨁_{α∈Δ_{+}} 𝔤_{α}, and 𝔫_{−} = ⨁_{α∈Δ_{−}} 𝔤_{α}. We then have the triangular decomposition 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}. Let 𝔟 = 𝔥 ⊕ 𝔫_{+}; 𝔟 is called a Borel subalgebra of 𝔤. Note that [𝔟, 𝔟] = 𝔫_{+}. Since 𝔫_{+} is a nilpotent subalgebra of 𝔤, 𝔟 is a solvable subalgebra.

If H_{1}, H_{2}, … H_{r} is a basis of 𝔥, then {E_{−β} (β ∈ Δ_{+}), H_{i} (i = 1, 2, … r), E_{β} (β ∈ Δ_{+})} form an ordered basis of 𝔤 if we choose an ordering on the positive roots β_{1}, β_{2}, … β_{N} (where r + 2N = dim 𝔤). Then, by the Poincaré-Birkhoff-Witt theorem, the elements

E_{−β_{1}}^{m_{1}}E_{−β_{2}}^{m_{2}} … E_{−β_{N}}^{m_{n}} h_{1}^{s_{1}}h_{2}^{s_{2}} … h_{r}^{s_{r}}E_{β_{1}}^{n_{1}}E_{β_{2}}^{n_{2}} … E_{β_{r}}^{n_{r}} (m_{i}, s_{i}, n_{i} ∈ ℤ_{+})

form a basis of the universal enveloping algebra U(𝔤) of 𝔤. (When all m_{i}, s_{i}, and n_{i} are zero, the product is 1).

A highest weight 𝔤-module with highest weight Λ ∈ 𝔥^{∗} is defined by the property that it contains a non-zero vector v_{Λ} such that

(1) hv_{Λ} = Λ(h)v_{Λ} for all h ∈ 𝔥.

(2) 𝔫_{+}v_{Λ} = 0

(3) U(𝔤)v_{λ} = V

By the above description of the basis of U(𝔤), properties (1) and (2) imply that (3) is equivalent to the following:

(3’) U(𝔫_{−})v_{Λ} = V

For μ ∈ 𝔥^{∗}, the weight subspace of a 𝔤-module V is given by V_{μ} = {v ∈ V : hv = μ(h)v for all h ∈ 𝔥}. If V_{μ} is non-zero, μ is called a weight of the 𝔤-module V.

A non-zero vector v ∈ V_{μ} is called singular if 𝔫_{+}v = 0. If one exists, μ is called a singular weight.

Example. Any Λ ∈ 𝔥^{∗} is a singular weight of a highest weight 𝔤-module with highest weight Λ.

Notation. Given Λ ∈ 𝔥^{∗}, let D(Λ) = {Λ − ∑_{i=1}^{r} k_{i}α_{i} : k_{i} ∈ ℤ_{+}} ⊂ 𝔥^{∗}, where Π = {α_{1}, α_{2}, … α_{r}} is the set of simple roots of 𝔤.

Proposition 1 Let V be a highest weight 𝔤-module with highest weight Λ. Then (a) V = ⨁_{λ∈D(Λ)} V_{λ}

(b) V_{Λ} = 𝔽v_{Λ} and dim V_{λ} < ∞

(c) V is an irreducible 𝔤-module if and only if Λ is its only singular weight.

(d) V contains a unique proper maximal submodule.

(e) If v is a singular vector with weight λ, then Ω(v) = (λ + 2ρ, λ)v, where (· , ·) is a nondegenerate symmetric invariant bilinear form on 𝔤, Ω is the corresponding Casimir operator, and 2ρ = ∑_{α∈Δ_{+}} α.

(f) Ω|_{V} = (Λ + 2ρ, Λ)Id_{V}

(g) If λ is a singular weight, then (λ + ρ, λ + ρ) = (Λ + ρ, Λ + ρ).

(h) If Λ ∈ 𝔥_{ℝ}^{∗} = ℝΔ, then V has finitely many singular weights.

Proof. (a) and (b) follow from the property (3’), which shows that any vector in V is a linear combination of elements of the form E_{−β_{i_{1}}} E_{−β_{i_{2}}} … E_{−β_{i_{k}}} v_{Λ}, and each such element has weight Λ − β_{i_{1}} − β_{i_{2}} − … − β_{i_{k}}.

(c) By a lemma from lecture 19, for any 𝔤-submodule U of V

U = ⨁_{λ∈D(Λ)}(U ∩ V_{λ})  (1)

Now let λ = Λ − ∑_{i} k_{i}α_{i}, where all k_{i}-s are in ℤ_{+}. Choose λ to be an element of D(Λ) of minimal height for which U ∩ V_{λ} ≠ 0. Let v ∈ (U ∩ V_{λ}) be non-zero. Then for any α ∈ Δ_{+} we will E_{α}v ∈ V_{λ+α}, and E_{α}v has a weight smaller than the weight of v. Therefore, by our choice of λ, E_{α}v = 0 for all α ∈ Δ_{+}, so v must be a singular vector.

Conversely, if v is a singular vector of weight λ, then U(𝔤)v = U(𝔫_{−})v, and this is a proper submodule of V unless λ = Λ.

(d) also follows from (1), because the sum of all proper submodules of V satisfies (1) and does not contain v_{λ}, so it is the only maximal proper submodule.

(e) Recall that Ω = ∑_{i} u_{i}v_{i}, where (u_{i}, v_{j}) = δ_{ij}, {u_{i}} is a basis of 𝔤 and {v_{i}} is the dual basis. Since Ω is independent on the choice of basis, we can make any selection we want. We will use {E_{α} (α ∈ Δ_{+}), E_{−α} (α ∈ Δ_{+}), H_{i} (i = 1, 2, … n)} as a basis {u_{i}} for 𝔤. The dual basis is {E_{−α} (α ∈ Δ_{+}), E_{α} (α ∈ Δ_{+}), H^{i} (i = 1, 2, … n)}, where (E_{α}, E_{−α}) = 1 for all α ∈ Δ_{+} and (H_{i}, H^{j}) = δ_{ij}. We now have

Ω = ∑_{α∈Δ_{+}}(E_{α}E_{−α} + E_{−α}E_{α}) + ∑_{i} H_{i}H^{i} = 2 ∑_{α∈Δ_{+}} E_{−α}E_{α} + 2ν^{−1}(ρ) + ∑_{i} H_{i}H^{i}

because [E_{−α}, E_{α}] = (E_{−α}, E_{α})ν^{−1}(α) = ν^{−1}(α).

Therefore, if v is a singular vector with weight λ, then Ω(v) = 2λ(ν^{−1}(ρ)) + ∑_{i} λ(H_{i})λ(H^{i}) = 2(λ, ρ) + (λ, λ). This proves (e).

(f) By (e), if v_{Λ} is the highest weight vector in V_{Λ}, then Ωv_{Λ} = (Λ + 2ρ, Λ)v_{Λ}. But any other vector in V has the form gv_{Λ}, where g ∈ U(𝔤). However, Ω commutes with 𝔤 and hence with U(𝔤). So Ω(gv) = gΩ(v). Therefore, Ω(gv_{Λ}) = gΩ(v_{Λ}) = (Λ + 2ρ, Λ)gv_{Λ}. Thus (f) holds.

(g) follows from (e) and (f): together they imply that (Λ + 2ρ, Λ) = (λ + 2ρ, λ).

(h) All weights of V lie in D_{Λ}, hence they also lie in 𝔥_{ℝ}^{∗}. So the set of singular weights must lie in the intersection of a discrete subset D(Λ) of the Euclidean space with the compact subset given by {λ||λ + ρ|^{2} = |Λ + ρ|^{2}}. Hence the set of singular weights is contained in a finite set. □

A Verma module with highest weight Λ ∈ 𝔥^{∗}, denoted by M(Λ), is a highest weight module with highest weight Λ such that any other highest weight module with highest weight Λ is a quotient of M(Λ).

Proposition 2 (a) For any Λ ∈ 𝔥^{∗}, M(Λ) exists and is unique up to isomorphism.

(b) The vectors E_{−β_{1}}^{k_{1}}E_{−β_{2}}^{k_{2}} … E_{−β_{n}}^{k_{n}}, where k_{i} ∈ ℤ_{+} and Δ_{+} = {β_{1}, β_{2}, … β_{n}}, form a basis of M(Λ).

(c) M(Λ) has a unique irreducible quotient, L(λ) = M(λ)/N(λ), where N(λ) is the unique maximum submodule of M(λ).

(d) M(Λ) ≅ M(Λ′) (respectively L(Λ) ≅ L(Λ′)) if and only if Λ = Λ′.

Proof. (a) We construct M(Λ) as U(𝔤)/U(𝔤) < 𝔫_{+}, h − Λ(h) (h ∈ 𝔥) >, where 𝔤 acts by multiplication on the left, and the highest weight vector v_{Λ} is the image of 1 in U(𝔤). Uniqueness is proved by universality.

(b) is clear from the description of the basis of U(𝔤), which shows that no non-zero linear combination of the vectors from (b) can belong to the left ideal described above.

(c) follows from part (d) of the previous proposition.

(d) follows from (c), since the set of roots of M(Λ) is D(Λ), but D(Λ) = D(Λ′) iff Λ = Λ′. The second part of (d) also follows from (c), because L(Λ) determines its highest weight Λ uniquely. □

Example. For sl_{2} =< E, F, H >, 𝔥 = ℂH and each Λ ∈ 𝔥^{∗} is just given by a number Λ = Λ(H). Also, we have ρ(H) = 1. Then M(Λ) is the span of v_{Λ}, F v_{Λ}, F^{2}v_{Λ}, . . .. F acts on it in the obvious way, and the action of E and H is determined by the key lemma: E(F^{k}v_{Λ}) = k(Λ − k + 1)F^{k−1}v_{Λ} and H(F^{k}v) = (Λ − 2k)v_{Λ}. By the key lemma, M(Λ) is irreducible unless Λ ∈ ℤ_{+}. If Λ ∈ ℤ_{+}, then F^{k+1}v_{Λ} is a singular vector, and F(Λ) =< F^{Λ+i}v_{Λ} : i = 1, 2, … > is a submodule of M(Λ), isomorphic to the Verma module with highest weight Λ − 2(Λ + 1) = −Λ − 2 .