The invariant bilinear form on Verma modules

2.6.1. The invariant bilinear form

The study of the Verma modules rests on a g-bilinear form which connects a highest-weight Verma module with a lowest-highest-weight Verma module for the opposite highest-weight.

First, let us prove its existence and basic properties:

Proposition 2.6.1. Letg be a Z-graded Lie algebra, andλ∈h^∗.

(a) There exists a unique g-invariant bilinear form M_λ⁺×M_−λ⁻ → C satisfying v_λ⁺, v⁻_−λ

= 1 (where we denote this bilinear form by (·,·)).

(b) This form has degree 0. (This means that if we consider this bilinear form M_λ⁺ × M_−λ⁻ → C as a linear map M_λ⁺ ⊗M_−λ⁻ → C, then it is a graded map, where M_λ⁺⊗M_−λ⁻ is graded as a tensor product of graded vector spaces, and C is concentrated in degree 0.)

(c) Every g-invariant bilinear form M_λ⁺×M_−λ⁻ → C is a scalar multiple of this form (·,·).

Remark 2.6.2. Proposition 2.6.1 still holds when the ground field Cis replaced by a commutative ring k, as long as some rather weak conditions hold (for instance, it is enough that n−,n+ and h are freek-modules).

Definition 2.6.3. Let g be a Z-graded Lie algebra, and λ ∈ h^∗. According to Proposition 2.6.1(a), there exists a uniqueg-invariant bilinear formM_λ⁺×M_−λ⁻ →C satisfying v_λ⁺, v_−λ⁻

= 1 (where we denote this bilinear form by (·,·)). This form is going to be denoted by (·,·)_λ (to stress its dependency on λ). (Later we will also denote this form by (·,·)^g_λ to point out its dependency on both λ and g.)

To prove Proposition 2.6.1, we recall two facts about modules over Lie algebras:

Lemma 2.6.4. Letabe a Lie algebra, and letbbe a Lie subalgebra ofa. LetV be a b-module, andW be ana-module. Then, (Ind^a_bV)⊗W ∼= Ind^a_b(V ⊗W) asa-modules (where theW on the right hand side is to be understood as Res^a_bW). More precisely, there exists a canonicala-module isomorphism (Ind^a_bV)⊗W →Ind^a_b(V ⊗W) which maps 1⊗U(b)v

⊗w to 1⊗U(b)(v⊗w) for all v ∈V and w∈W.

Lemma 2.6.5. Letcbe a Lie algebra. Leta andb be two Lie subalgebras ofcsuch that a+b =c. Notice that a∩b is also a Lie subalgebra ofc. LetN be a b-module.

Then, Ind^a_a∩b Res^b_a∩bN∼= Res^c_a(Ind^c_bN) as a-modules.

We will give two proofs of Lemma 2.6.4: one which is direct and uses Hopf algebras;

the other which is more elementary but less direct.

First proof of Lemma 2.6.4. Remember that U(a) is a Hopf algebra (a cocommu-tative one, actually; but we won’t use this). Let us denote its antipode by S and use sumfree Sweedler notation.

Recalling that Ind^a_bV = U(a)⊗_U(b) V and Ind^a_b(V ⊗W) = U(a)⊗_U(b)(V ⊗W), we define a C-linear map φ : (Ind^a_bV)⊗W → Ind^a_b(V ⊗W) by α⊗_U(b)v

⊗w 7→

α₍₁₎⊗_U(b) v ⊗S α₍₂₎ w

. This map is easily checked to be well-defined and a-linear.

Also, we define aC-linear mapψ : Ind^a_b(V ⊗W)→(Ind^a_bV)⊗W byα⊗_U(b)(v⊗w)7→

α₍₁₎⊗_U(b)v

⊗α₍₂₎w. This map is easily checked to be well-defined. It is also easy to see that φ◦ψ = id andψ◦φ = id. Hence,φ and ψ are mutually inverse isomorphisms between the a-modules (Ind^a_bV)⊗W and Ind^a_b(V ⊗W). This proves that (Ind^a_bV)⊗ W ∼= Ind^a_b(V ⊗W) as a-modules. Moreover, the isomorphism φ : (Ind^a_bV)⊗W → Ind^a_b(V ⊗W) is canonical and maps 1⊗_U(b)v

⊗w to 1⊗_U(b)(v⊗w) for all v ∈ V and w∈W. In other words, Lemma 2.6.4 is proven.

Second proof of Lemma 2.6.4. For every a-module Y, we have Hom_a((Ind^a_bV)⊗W, Y)

=









Hom_C((Ind^a_bV)⊗W, Y)

| {z }

∼=Hom_C(^IndabV,Hom_C(W,Y))









a

∼= (Hom_C(Ind^a_bV,Hom_C(W, Y)))^a= Hom_a(Ind^a_bV,Hom_C(W, Y))

∼= Homb(V,Hom_C(W, Y)) (by Frobenius reciprocity)

=





Hom_C(V,Hom_C(W, Y))

| {z }

∼=Hom_C(V⊗W,Y)







b

∼= (Hom_C(V ⊗W, Y))^b

= Hom_b(V ⊗W, Y)∼= Homa(Ind^a_b(V ⊗W), Y) (by Frobenius reciprocity). Since this isomorphism is canonical, it gives us a natural isomorphism between the functors Hom_a((Ind^a_bV)⊗W,−) and Hom_a(Ind^a_b(V ⊗W),−). By Yoneda’s lemma, this yields that (Ind^a_bV)⊗W ∼= Ind^a_b(V ⊗W) as a-modules. It is also rather clear that thea-module isomorphism (Ind^a_bV)⊗W →Ind^a_b(V ⊗W) we have just obtained is canonical.

In order to check that this isomorphism maps 1⊗_U(b)v

⊗w to 1⊗_U(b) (v⊗w) for all v ∈ V and w ∈ W, we must retrace the proof of Yoneda’s lemma. This proof proceeds by evaluating the natural isomorphism Hom_a((Ind^a_bV)⊗W,−) → Hom_a(Ind^a_b(V ⊗W),−) at the object Ind^a_b(V ⊗W), thus obtaining an isomorphism

Hom_a((Ind^a_bV)⊗W,Ind^a_b(V ⊗W))→Hom_a(Ind^a_b(V ⊗W),Ind^a_b(V ⊗W)), and taking the preimage of id ∈ Homa(Ind^a_b(V ⊗W),Ind^a_b(V ⊗W)) under this iso-morphism. This preimage is our isomorphism (Ind^a_bV)⊗W →Ind^a_b(V ⊗W). Checking that this maps 1⊗_U(b)v

⊗w to 1⊗_U(b)(v⊗w) for allv ∈V and w∈W is a matter of routine now, and left to the reader. Lemma 2.6.4 is thus proven.

Proof of Lemma 2.6.5. Let ρ : U(a)⊗_U(a∩b)U(b) → U(c) be the C-vector space homomorphism defined by

ρ α⊗U(a∩b)β

=αβ for all α∈U(a) and β∈U(b)

(this is clearly well-defined). By Proposition 2.4.1, this map ρ is an isomorphism of left U(a)-modules and of right U(b)-modules. Hence, U(a)⊗U(a∩b)U(b) ∼= U(c) as left U(a)-modules and simultaneously right U(b)-modules. Now,

Ind^a_a∩b



Res^b_a∩b N

∼ |{z}

=U(b)⊗_U(b)N



∼= Ind^a_a∩b













Res^b_a∩b U(b)⊗_U(b)N

| {z }

=U(b)⊗_U(b)N (as aU(a∩b)-module)













= Ind^a_a∩b U(b)⊗_U(b)N

=U(a)⊗U(a∩b) U(b)⊗_U(b)N

∼= U(a)⊗_U(a∩b)U(b)

| {z }

∼=U(c)

⊗_U(b)N ∼=U(c)⊗_U(b)N

= Ind^c_bN = Res^c_a(Ind^c_bN) asa-modules.

This proves Lemma 2.6.5.

Proof of Proposition 2.6.1. We have M_λ⁺=U(g)⊗_U(h⊕n+)C^λ = Ind^g_h⊕n

(since we are suppressing the Res functors),

so that Ind^g_h⊕n−(Cλ⊗C^−λ)∼= Ind^h⊕n_h ⁺(Cλ⊗C^−λ) (as (h⊕n₊) -modules)

→ C is easily seen to map every g-invariant bilinear form (·,·) : M_λ⁺×M_−λ⁻ → C (seen as a linear map M_λ⁺⊗M_−λ⁻ → C) to the value v_λ⁺, v⁻_−λ

. Hence, there exists a uniqueg-invariant bilinear formM_λ⁺×M_−λ⁻ →C satisfying v⁺_λ, v_−λ⁻

= 1 (where we denote this bilinear form by (·,·)), and every other g-invariant bilinear form M_λ⁺×M_−λ⁻ →C must be a scalar multiple of this one. This proves Proposition 2.6.1 (a) and (c).

Now, for the proof of (b): Denote by (·,·) the unique g-invariant bilinear form M_λ⁺×M_−λ⁻ →C satisfying v⁺_λ, v_−λ⁻

= 1. Let us now prove that this bilinear form is of degree 0:

Consider the antipode S : U(g) → U(g) of the Hopf algebra U(g). This S is a graded algebra antiautomorphism satisfying S(x) = −x for every x ∈ g. It can be explicitly described by

S(x1x2...xm) = (−1)^mxmxm−1...x1 for all m∈N and x1, x2, ..., xm ∈g.

We can easily see by induction (using theg-invariance of the bilinear form (·,·)) that (v, aw) = (S(a)v, w) for all v ∈M_λ⁺ and w∈M_−λ⁻ and a ∈U(g). In particular,

av_λ⁺, bv_−λ⁻

= S(b)av⁺_λ, v_−λ⁻

for all a ∈U(g) and b∈U(g). Thus, av_λ⁺, bv_−λ⁻

= S(b)av_λ⁺, v⁻_−λ

= 0 whenever a and b are homogeneous elements of U(g) satisfying degb > −dega (this is because any two homogeneous elements a and b of U(g) satisfying degb > −dega satisfy S(b)av_λ⁺ = 0 ⁵⁰). In other words, whenever n ∈Z and m ∈ Z are integers satisfying m >−n, we have av⁺_λ, bv⁻_−λ

= 0 for every a ∈ U(g) [n] and b ∈ U(g) [m]. Since M_λ⁺[n] =

av⁺_λ | a∈U(g) [n] and M_−λ⁻ [m] =

bv_−λ⁻ | b∈U(g) [m] , this rewrites as follows: Whenever n ∈ Z and m∈Z are integers satisfyingm >−n, we have M_λ⁺[n], M_−λ⁻ [m]

= 0.

Similarly, using the formula (av, w) = (v, S(a)w) (which holds for all v ∈ M_λ⁺ and w ∈ M_−λ⁻ and a ∈ U(g)), we can show that whenever n ∈ Z and m ∈ Z are integers satisfying m <−n, we have M_λ⁺[n], M_−λ⁻ [m]

= 0.

Thus we have M_λ⁺[n], M_−λ⁻ [m]

= 0 whenever m > −n and whenever m < −n.

Hence, M_λ⁺[n], M_−λ⁻ [m]

can only be nonzero when m = −n. In other words, the form (·,·) has degree 0. This proves Proposition 2.6.1. In this proof, we have not used any properties ofC other than being a commutative ring over whichn−, n₊ and h are free modules (the latter was only used for applying consequences of Poincar´ e-Birkhoff-Witt); we thus have also verified Remark 2.6.2.

2.6.2. Generic nondegeneracy: Statement of the fact

We will later (Theorem 2.7.3) see that the bilinear form (·,·)_λ : M_λ⁺×M_−λ⁻ → C is nondegenerate if and only if the g-moduleM_λ⁺ is irreducible. This makes the question of when the form (·,·)_λ is nondegenerate an important question to study. It can, in many concrete cases, be answered by combinatorial computations. But let us first give a general result about how it is nondegenerate “ifλ is in sufficiently general position”:

Theorem 2.6.6. Assume that g is a nondegenerate Z-graded Lie algebra.

Let (·,·) be the form (·,·)_λ : M_λ⁺×M_−λ⁻ → C. (In other words, let (·,·) be the unique g-invariant bilinear form M_λ⁺×M_−λ⁻ → C satisfying v_λ⁺, v_−λ⁻

= 1. Such a form exists and is unique by Proposition 2.6.1 (a).)

In every degree, the form (·,·) is nondegenerate for generic λ. More precisely: For every n∈N, the restriction of the form (·,·) :M_λ⁺×M_−λ⁻ →CtoM_λ⁺[−n]×M_−λ⁻ [n]

is nondegenerate for generic λ.

(What “genericλ” means here may depend on the degree. Thus, we cannot claim that “for generic λ, the form (·,·) is nondegenerate in every degree”!)

The proof of this theorem will occupy the rest of Section 2.6. While the statement of Theorem 2.6.6 itself will never be used in this text, the proof involves several useful ideas and provides good examples of how to work with Verma modules computationally;

moreover, the main auxiliary result (Proposition 2.6.17) will be used later in the text.

50Proof. Let a and b be homogeneous elements of U(g) satisfying degb > −dega. Then, degb+ dega > 0, and thus the element S(b)av⁺_λ of M_λ⁺ is a homogeneous element of positive degree (since degv⁺_λ = 0), but the only homogeneous element ofM_λ⁺ of positive degree is 0 (sinceM_λ⁺ is concentrated in nonpositive degrees), so that S(b)av⁺_λ = 0.

[Note: The below proof has been written at nighttime and not been checked for mistakes. It also has not been checked for redundancies and readability.]

2.6.3. Proof of Theorem 2.6.6: Casting bilinear forms on coinvariant spaces Before we start with the proof, a general fact from representation theory:

Lemma 2.6.7. Let k be a field, and let G be a finite group. Let Λ ∈ k[G] be the element P

g∈G

g.

LetV andW be representations ofGoverk. LetB :V ×W →kbe aG-invariant bilinear form.

(a)Then, there exists one and only one bilinear formB⁰ :VG×WG →k satisfying B⁰(v, w) =B(Λv, w) = B(v,Λw) for all v ∈V and w∈W.

(Here, v denotes the projection ofv ontoV_G, andwdenotes the projection ofwonto W_G.)

(b)Assume that|G|is invertible ink (in other words, assume that charkis either 0 or coprime to |G|). If the form B is nondegenerate, then the form B⁰ constructed in Lemma 2.6.7 (a) is nondegenerate, too.

Proof of Lemma 2.6.7. Every h∈Gsatisfies hΛ =hX

g∈G

g since Λ =X

g∈G

g

!

=X

g∈G

hg=X

i∈G

i

here, we substitutedi for hg in the sum, since the map G→G, g 7→hg is a bijection

=X

g∈G

g = Λ and similarly Λh= Λ.

Also, X

g∈G

g⁻¹ =X

g∈G

g

here, we substitutedg for g⁻¹ in the sum, since the map G→G, g 7→g⁻¹ is a bijection

= Λ.

We further notice that the group G acts trivially on the G-modules k and W_G (this follows from the definitions of these modules), and thusGacts trivially on Hom (W_G, k) as well.

For every v ∈V, the map

W →k, w7→B(Λv, w)

is clearly G-equivariant (since it maps hw to

B Λ

|{z}

=hΛ

v, hw

!

=B(hΛv, hw) = B(Λv, w) (sinceB is G-invariant)

=hB(Λv, w) (since Gacts trivially on k) for every h∈G and w∈W), and thus descends to a map

W_G→k_G, w7→B(Λv, w).

Hence, we have obtained a map

V →Hom (W_G, k_G), v 7→

w7→B(Λv, w) . Since k_G=k (becauseG acts trivially on k), this rewrites as a map

V →Hom (W_G, k), v 7→(w7→B(Λv, w)). This map, too, is G-equivariant (since it mapshv to the map

W_G →k, w7→B Λh

|{z}

=Λ

v, w

!!

= (W_G →k, w7→B(Λv, w)) =h(W_G →k, w7→B(Λv, w)) (since Gacts trivially on Hom (W_G, k))

for every h∈G and v ∈V). Thus, it descends to a map

V_G→(Hom (W_G, k))_G, v 7→(w7→B(Λv, w)).

Since (Hom (WG, k))_G = Hom (WG, k) (because G acts trivially on Hom (WG, k)), this rewrites as a map

V_G →Hom (W_G, k), v 7→(w7→B(Λv, w)).

This map can be rewritten as a bilinear form V_G×W_G → k which maps (v, w) to B(Λv, w) for all v ∈V and w∈W. Since

B(Λv, w) =B X

g∈G

gv, w

!

since Λ =X

g∈G

g

!

=X

g∈G

B



gv, w

|{z}

=gg⁻¹w



=X

g∈G

B gv, gg⁻¹w

| {z }

=B(^v,g−1w)

(sinceBisG-invariant)

=X

g∈G

B v, g⁻¹w

=B











 v,X

g∈G

g⁻¹

| {z }

=Λ

w













=B(v,Λw)

for all v ∈ V and w ∈ W, we have thus proven that there exists a bilinear form B⁰ :V_G×W_G→k satisfying

B⁰(v, w) =B(Λv, w) =B(v,Λw) for all v ∈V and w∈W. The uniqueness of such a form is self-evident. This proves Lemma 2.6.7 (a).

(b) Assume that |G| is invertible in k. Assume that the form B is nondegenerate.

Consider the form B⁰ constructed in Lemma 2.6.7 (a).

Let p ∈ V_G be such that B⁰(p, W_G) = 0. Since p ∈ V_G, there exists some v ∈ V such that p = v. Consider this v. Then, every w ∈ W satisfies B(Λv, w) = 0 (since B(Λv, w) = B⁰



 v

|{z}

=p

, w

|{z}

∈WG



 ∈ B⁰(p, W_G) = 0). Hence, Λv = 0 (since B is nondegenerate).

But since the projection of V toV_G is a G-module map, we have Λv = Λv =X

g∈G

gv

|{z}

=v (sinceGacts trivially onVG)

since Λ =X

g∈G

g

!

=X

g∈G

v =|G|v.

Since|G|is invertible ink, this yieldsv = 1

|G|Λv = 0 (since Λv = 0), so thatp=v = 0.

We have thus shown that every p∈V_G such thatB⁰(p, W_G) = 0 must satisfy p= 0.

In other words, the form B⁰ is nondegenerate. Lemma 2.6.7 (b) is proven.

2.6.4. Proof of Theorem 2.6.6: The form (·,·)^◦_λ Let us formulate some standing assumptions:

Convention 2.6.8. From now on until the end of Section 2.6, we letgbe aZ-graded Lie algebra, and letλ∈h^∗. We also require thatg₀is abelian (this is condition(2)of Definition 2.5.4), but we do not require g to be nondegenerate (unless we explicitly state this).

As vector spaces, M_λ⁺ = U(n−)v_λ⁺ ∼= U(n−) (where the isomorphism maps v_λ⁺ to 1) and M_−λ⁻ = U(n₊)v⁻_−λ ∼= U(n₊) (where the isomorphism maps v⁻_−λ to 1). Thus, the bilinear form (·,·) = (·,·)_λ : M_λ⁺ ×M_−λ⁻ → C corresponds to a bilinear form U(n−)×U(n₊)→C.

For every n ∈ N, let (·,·)_λ,n denote the restriction of our form (·,·) = (·,·)_λ : M_λ⁺×M_−λ⁻ →CtoM_λ⁺[−n]×M_−λ⁻ [n]. In order to prove Theorem 2.6.6, it is enough to prove that for everyn∈N, whengis nondegenerate, this form (·,·)_λ,nis nondegenerate for generic λ.

We now introduce a C-bilinear form, which will turn out to be, in some sense, the

“highest term” of the form (·,·) with respect to λ (what this exactly means will be explained in Proposition 2.6.17).

Proposition 2.6.9. For every k ∈N, there exists one and only one C-bilinear form λk:S^k(n−)×S^k(n+)→C by

λ_k(α₁α₂...α_k, β₁β₂...β_k) = X

σ∈S_k

λ

α₁, β_σ(1) λ

α₂, β_σ(2) ...λ

α_k, β_σ(k) for all α₁, α₂, ..., α_k ∈n₋ and β₁, β₂, ..., β_k ∈n₊.

(36)

Here, we are using the following convention:

Convention 2.6.10. From now on until the end of Section 2.6, the mapλ:g₀ →C is extended to a linear map λ:g→Cby composing it with the canonical projection g→g0.

First proof of Proposition 2.6.9 (sketched). Letk ∈N. The value of X

σ∈S_k

λ

α₁, β_σ(1) λ

α₂, β_σ(2) ...λ

α_k, β_σ(k)

depends linearly on each of the α₁, α₂, ..., α_k and β₁, β₂, ..., β_k, and is invariant under any permutation of the α₁, α₂, ..., α_k and under any permutation of the β₁, β₂, ..., β_k (as is easily checked). This readily shows that we can indeed define a C-bilinear form λ_k :S^k(n−)×S^k(n₊)→C by (36). This proves Proposition 2.6.9.

Second proof of Proposition 2.6.9. Let G = S_k. Let Λ ∈ C[G] be the element P

g∈Sk

g = P

σ∈Sk

σ = P

σ∈Sk

σ⁻¹. Let V and W be the canonical representations n^⊗k₋ and n^⊗k₊ of S_k (where S_k acts by permuting the tensorands). LetB :V ×W → C be the C-bilinear form defined as the k-th tensor power of the C-bilinear formn−×n₊→C, (α, β) 7→ λ([α, β]). It is easy to see that this form is S_k-invariant (in fact, more generally, the k-th tensor power of any bilinear form is S_k-invariant). Thus, Lemma 2.6.7 (a) (applied to Cinstead of k) yields that there exists one and only one bilinear formB⁰ :V_G×W_G →Csatisfying

B⁰(v, w) = B(Λv, w) = B(v,Λw) for all v ∈V and w∈W (37) (where v denotes the projection of v onto V_G = V_Sk = S^k(n−), and w denotes the projection ofw ontoW_G =W_Sk =S^k(n₊)). Consider this form B⁰. Allα₁, α₂, ..., α_k ∈

n− and β₁, β₂, ..., β_k ∈n+ satisfy B⁰(α₁α₂...α_k, β₁β₂...β_k)

=B⁰ α1⊗α2 ⊗...⊗αk, β1⊗β2⊗...⊗βk

since α₁α₂...α_k =α₁⊗α₂⊗...⊗α_k and β₁β₂...β_k=β₁⊗β₂⊗...⊗β_k

=B(α₁⊗α₂⊗...⊗α_k,Λ (β₁⊗β₂⊗...⊗β_k))

(by (37), applied to v =α₁⊗α₂⊗...⊗α_k and w=β₁⊗β₂ ⊗...⊗β_k)

=B α₁⊗α₂⊗...⊗α_k,X

σ∈S_k

β_σ(1)⊗β_σ(2)⊗...⊗β_σ(k)

!









since Λ = P

σ∈S_k

σ⁻¹ yields Λ (β₁⊗β₂⊗...⊗β_k) = P

σ∈S_k

σ⁻¹(β₁ ⊗β₂⊗...⊗β_k)

| {z }

=βσ(1)⊗β_σ(2)⊗...⊗β_σ(k)

= P

σ∈S_k

β_σ(1)⊗β_σ(2)⊗...⊗β_σ(k)









= X

σ∈Sk

B α₁ ⊗α₂⊗...⊗α_k, β_σ(1)⊗β_σ(2)⊗...⊗β_σ(k)

| {z }

=λ([^α1,βσ(1)])^λ([^α2,βσ(2)])^...λ([^αk,βσ(k)])

(sinceBis thek-th tensor power of theC-bilinear formn−×n₊→C, (α,β)7→λ([α,β]))

= X

σ∈S_k

λ

α1, βσ(1)

λ

α2, βσ(2)

...λ

αk, βσ(k)

.

Thus, there exists aC-bilinear formλ_k:S^k(n−)×S^k(n₊)→Csatisfying (36) (namely, B⁰). On the other hand, there exists at most one C-bilinear form λ_k : S^k(n₋)× S^k(n₊) → C satisfying (36) ⁵¹. Hence, we can indeed define a C-bilinear form λ_k :S^k(n−)×S^k(n₊)→C by (36). And, moreover,

this formλ_k is the formB⁰ satisfying (37). (38) Proposition 2.6.9 is thus proven.

Definition 2.6.11. For everyk ∈N, letλ_k :S^k(n₋)×S^k(n₊)→Cbe theC-bilinear form whose existence and uniqueness is guaranteed by Proposition 2.6.9. These forms can be added together, resulting in a bilinear form L

k≥0

λ_k :S(n−)×S(n₊)→C. It is very easy to see that this form is of degree 0 (where the grading on S(n−) and S(n₊) is not the one that gives the k-th symmetric power the degree k for every k ∈ N, but is the one induced by the grading on n− and n+). Denote this form by (·,·)^◦_λ.

2.6.5. Proof of Theorem 2.6.6: Generic nondegeneracy of (·,·)^◦_λ

51Proof. The vector spaceS^k(n₋) is spanned by products of the formα₁α₂...α_k withα₁, α₂, ..., α_k ∈ n₋, whereas the vector space S^k(n₊) is spanned by products of the form β₁β₂...β_k with β₁, β₂, ..., β_k ∈n₊. Hence, the equation (36) makes it possible to compute the value of λ_k(A, B) for any A∈S^k(n₋) andB ∈S^k(n+). Thus, the equation (36) uniquely determinesλk. In other words, there exists at most oneC-bilinear formλk:S^k(n₋)×S^k(n+)→Csatisfying (36).

Lemma 2.6.12. Letλ ∈h^∗ be such that theC-bilinear formn−×n₊ →C,(α, β)7→

λ([α, β]) is nondegenerate. Then, the form (·,·)^◦_λ is nondegenerate.

Proof of Lemma 2.6.12. Let k ∈ N. Introduce the same notations as in the Second proof of Proposition 2.6.9.

The C-bilinear form n− ×n₊ → C, (α, β) 7→ λ([α, β]) is nondegenerate. Thus, the k-th tensor power of this form is also nondegenerate (since all tensor powers of a nondegenerate form are always nondegenerate). But thek-th tensor power of this form is B. Thus, B is nondegenerate. Hence, Lemma 2.6.7 (b) yields that the form B⁰ is nondegenerate. Due to (38), this yields that the formλ_k is nondegenerate.

Forget that we fixed k. We thus have shown that for every k ∈ N, the form λ_k is nondegenerate. Thus, the direct sum L

k≥0

λ_k of these forms is also nondegenerate. Since L

k≥0

λ_k = (·,·)^◦_λ, this yields that (·,·)^◦_λ is nondegenerate. This proves Lemma 2.6.12.

For every n ∈ N, define (·,·)^◦_λ,n : S(n−) [−n]×S(n₊) [n] → C to be the restriction of this form (·,·)^◦_λ = L

k≥0

λk :S(n−)×S(n+)→C to S(n−) [−n]×S(n+) [n]. We now need the following strengthening of Lemma 2.6.12:

Lemma 2.6.13. Let n∈N and λ∈h^∗ be such that the bilinear form g−k×g_k→C, (a, b)7→λ([a, b])

is nondegenerate for every k ∈ {1,2, ..., n}. Then, the form (·,·)^◦_λ,n must also be nondegenerate.

Proof of Lemma 2.6.13. For Lemma 2.6.12 to hold, we did not need gto be a graded Lie algebra; we only needed thatgis a graded vector space with a well-defined bilinear map [·,·] :g−k×gk →g0 for every positive integer k. This is a rather weak condition, and holds not only forg, but also for the graded subspace g−n⊕g−n+1⊕...⊕g_n of g.

Denote this graded subspace g−n⊕g−n+1⊕...⊕g_n byg⁰, and let n⁰₋⊕h⁰ ⊕n⁰₊ be its triangular decomposition (thus, n⁰₋ = g−n⊕g−n+1⊕...⊕g−1, h⁰ = g0 =h and n⁰₊ = g₁⊕g₂⊕...⊕g_n). TheC-bilinear formn⁰₋×n⁰₊ →C,(α, β)7→λ([α, β]) is nondegenerate (because the bilinear formg−k×g_k →C, (a, b)7→λ([a, b]) is nondegenerate for every k ∈ {1,2, ..., n}). Hence, by Lemma 2.6.12, the form (·,·)^◦_λ defined for g⁰ instead of g is nondegenerate. Since this form is of degree 0, the restriction (·,·)^◦_λ,n of this form toS n⁰₋

[−n]×S n⁰₊

[n] must also be nondegenerate⁵². But since S n⁰₊ [n] =

52This is because ifV andW are two graded vector spaces, and φ:V ×W →Cis a nondegenerate bilinear form of degree 0, then for everyn∈Z, the restriction ofφtoV [−n]×W[n] must also be nondegenerate.

S(n₊) [n] ⁵³ and S n⁰₋

[−n] =S(n₋) [−n] ⁵⁴, this restriction is exactly our form (·,·)^◦_λ,n : S(n−) [−n]×S(n₊) [n] → C (in fact, the form is clearly given by the same formula). Thus we have shown that our form (·,·)^◦_λ,n:S(n₋) [−n]×S(n₊) [n]→C is nondegenerate. Lemma 2.6.13 is proven.

2.6.6. Proof of Theorem 2.6.6: (·,·)^◦_λ is the “highest term” of (·,·)_λ

Before we go on, let us sketch the direction in which we want to go. We want to study how, for a fixed n ∈ N, the form (·,·)_λ,n changes with λ. If V and W are two finite-dimensional vector spaces of the same dimension, and if we have chosen bases for these two vector spaces V and W, then we can represent every bilinear form V ×W →Cas a square matrix with respect to these two bases, and the bilinear form is nondegenerate if and only if this matrix has nonzero determinant. This suggests that we study how the determinant det

(·,·)_λ,n

of the form (·,·)_λ,n with respect to some bases of M_λ⁺[−n] andM_−λ⁻ [n] changes with λ(and, in particular, show that this determinant is nonzero for generic λ when g is nondegenerate). Of course, speaking of this determinant det

(·,·)_λ,n

only makes sense when the bases of M_λ⁺[−n] and M_−λ⁻ [n] have the same size (since only square matrices have determinants), but this is automatically satisfied if we have dim (gn) = dim (g−n) for every integer n > 0 (this condition is automatically satisfied when g is a nondegenerate Z-graded Lie algebra, but of course not only then).

Unfortunately, the spaces M_λ⁺[−n] and M_−λ⁻ [n] themselves change with λ. Thus,

53Proof. Sincen+=P

i≥1

gi, we haveS(n+) = P

k∈N

P

(i1,i2,...,ik)∈N^k; eachi_j≥1

gi₁gi₂...gi_k and thus

S(n+) [n] =X

k∈N

X

(i₁,i₂,...,i_k)∈N^k; eachij≥1;

i₁+i₂+...+i_k=n

gi₁gi₂...gi_k

(since gi₁gi₂...gi_k⊆S(n+) [i1+i2+...+ik] for all (i1, i2, ..., ik)∈N^k). Similarly, S n⁰₊

[n] =X

k∈N

X

(i1,i2,...,ik)∈N^k; eachi_j≥1;

each|ij|≤n;

i1+i2+...+ik=n

gi₁gi₂...gi_k

(becauseg⁰ is obtained fromgby removing all g_i with|i|> n). Thus, S n⁰₊

[n] =X

k∈N

X

(i1,i2,...,ik)∈N^k; eachi_j≥1;

each|ij|≤n;

i1+i2+...+ik=n

g_i1g_i2...g_ik =X

k∈N

X

(i1,i2,...,ik)∈N^k; eachi_j≥1;

i1+i2+...+ik=n

g_i1g_i2...g_ik





here, we removed the condition (each |i_j| ≤n) , because it was redundant (since every (i₁, i₂, ..., i_k)∈N^k satisfyingi₁+i₂+...+i_k=nautomatically

satisfies (each |i_j| ≤n) )





=S(n₊) [n], qed.

54for analogous reasons

if we want to pick some bases of M_λ⁺[−n] and M_−λ⁻ [n] for all λ ∈ h^∗, we have to pick new bases for every λ. If we just pick these bases randomly, then the determi-nant det

(·,·)_λ,n

can change very unpredictably (because the determinant depends on the choice of bases). Thus, if we want to say something interesting about how det

(·,·)_λ,n

changes with λ, then we should specify a reasonable choice of bases for all λ. Fortunately, this is not difficult: It is enough to choose Poincar´e-Birkhoff-Witt bases for U(n−) [−n] and U(n₊) [n], and thus obtain bases M_λ⁺[−n] and M_−λ⁻ [n] due to the isomorphisms M_λ⁺[−n] ∼= U(n−) [−n] and M_−λ⁻ [n] ∼= U(n₊) [n]. (See Conven-tion 2.6.21 for details.) With bases chosen this way, the determinant det

(·,·)_λ,n will depend on λ polynomially, and we will be able to conclude some useful properties of this polynomial.

So much for our roadmap. Let us first make a convention:

Convention 2.6.14. If V and W are two finite-dimensional vector spaces of the same dimension, and if we have chosen bases for these two vector spaces V and W, then we can represent every bilinear form B : V ×W → C as a square matrix with respect to these two bases. The determinant of this matrix will be denoted by detB and called the determinant of the form B. Of course, this determinant detB depends on the bases chosen. A change of either basis induces a scaling of detB by a nonzero scalar. Thus, while the determinant detB itself depends on the choice of bases, the property of detB to be zero or nonzero does notdepend on the choice of bases.

Let us now look at how the form (·,·)_λ,n and its determinant det

(·,·)_λ,n

depend onλ. We want to show that this dependence is polynomial. In order to make sense of this, let us define what we mean by “polynomial” here:

Definition 2.6.15. LetV be a finite-dimensional vector space. A functionφ:V → C is said to be a polynomial function (or just to bepolynomial – but this is not the same as being a polynomial) if one of the following equivalent conditions holds:

(1) There exist a basis (β₁, β₂, ..., β_m) of the dual space V^∗ and a polynomial P ∈C[X1, X2, ..., Xm] such that

every v ∈V satisfiesφ(v) =P (β₁(v), β₂(v), ..., β_m(v)).

(2)For every basis (β₁, β₂, ..., β_m) of the dual space V^∗, there exists a polynomial P ∈C[X₁, X₂, ..., X_m] such that

every v ∈V satisfiesφ(v) =P (β₁(v), β₂(v), ..., β_m(v)).

(3) There exist finitely many elements β1, β2, ..., βm of the dual space V^∗ and a polynomial P ∈C[X₁, X₂, ..., X_m] such that

every v ∈V satisfiesφ(v) =P (β₁(v), β₂(v), ..., β_m(v)).

Note that this is exactly the meaning of the word “polynomial function” that is used in Classical Invariant Theory. In our case (where the field is C), polynomial functions

V →C can be identified with elements of the symmetric algebra S (V^∗), and in some sense are an “obsoleted version” of the latter.⁵⁵ For our goals, however, polynomial functions are enough. Let us define the notion of homogeneous polynomial functions:

Definition 2.6.16. Let V be a finite-dimensional vector space.

(a) Let n ∈ N. A polynomial function φ : V → C is said to be homogeneous of degree n if and only if

every v ∈V and every λ∈C satisfy φ(λv) = λⁿφ(v).

(b) A polynomial function φ : V → C is said to be homogeneous if and only if there exists some n∈N such thatφ is homogeneous of degreen.

(c) It is easy to see that for every polynomial function φ : V → C, there exists a unique sequence (φ_n)_n∈

N of polynomial functions φ_n : V → C such that all but finitely many n ∈ N satisfy φ_n = 0, such that φ_n is homogeneous of degree n for every n ∈ N, and such that φ = P

n∈N

φ_n. This sequence is said to be the graded decomposition ofφ. For every n∈N, its member φ_n is called the n-th homogeneous component of φ. If N is the highest n ∈N such that φn 6= 0, then φN is said to be the leading term of φ.

Note that Definition 2.6.16 (c) defines the “leading term” of a polynomial as its highest-degree nonzero homogeneous component. This “leading term” may (and usu-ally will) contain more than one monomial, so this notion of a “leading term” is not the same as the notion of a “leading term” commonly used, e. g., in Gr¨obner basis theory.

We now state the following crucial fact:

Proposition 2.6.17. Let n ∈ N. Assume that g is a nondegenerate Z-graded Lie algebra. As a consequence, dimh = dim (g0) 6=∞, so that dim (h^∗) 6=∞, and thus the notion of a polynomial function h^∗ →Cis well-defined.

There is an appropriate way of choosing bases of the vector spaces S(n−) [−n]

and S(n+) [n] and bases of the vector spaces M_λ⁺[−n] and M_−λ⁻ [n] for all λ ∈ h^∗ such that the following holds:

(a) The determinants det

(·,·)_λ,n

and det

(·,·)^◦_λ,n

(these determinants are defined with respect to the chosen bases of S(n−) [−n], S(n+) [n], M_λ⁺[−n] and M_−λ⁻ [n]) depend polynomially onλ. By this, we mean that the functions

h^∗ →C, λ7→det

(·,·)_λ,n and

h^∗ →C, λ7→det

(·,·)^◦_λ,n are polynomial functions.

55The identification of polynomial functionsV → Cwith elements of the symmetric algebra S (V^∗) works similarly over any infinite field instead of C. It breaks down over finite fields, however (because different elements of S (V^∗) may correspond to the same polynomial function over a finite

55The identification of polynomial functionsV → Cwith elements of the symmetric algebra S (V^∗) works similarly over any infinite field instead of C. It breaks down over finite fields, however (because different elements of S (V^∗) may correspond to the same polynomial function over a finite

Im Dokument Pavel Etingof: 18.747: Infinite-dimensional Lie algebras (Spring term 2012 at MIT) (Seite 73-118)