LIE ALGEBRA COHOMOLOGY AND GENERATING FUNCTIONS SEMINAR ON SUPERMANIFOLDS-29 (A REMAKE)

ALEXEI TOLPYGO

Abstract. Let 𝔤 be a simple Lie algebra, V an irreducible 𝔤-module, W the Weyl group and 𝔟 the Borel subalgebra of 𝔤, 𝔫 = [𝔟, 𝔟], 𝔥 the Cartan subalgebra of 𝔤. The Borel-Weil-Bott’s theorem states that the dimension of H^{i}(𝔫; V) is equal to the cardinality of the set of elements of length i from W. Here a more detailed description of H^{i}(𝔫; V) as an 𝔥-module is given in terms of generating functions.

Results of Leger and Luks and Williams who described H^{i}(𝔫; 𝔫) for i ≤ 2 are generalized: dim H^{∗}(𝔫; Λ^{∗}(𝔫)) and dim H^{i}(𝔫; 𝔫) for i ≤ 3 are calculated and dim H^{i}(𝔫; 𝔫) as function of i and rank 𝔤 is described for the calssical series.

Introduction

The main field is ℂ and all the algebras and modules considered are finite dimensional over ℂ. It is well known, that the standard cochain complex C^{∗}(𝔫; V) = ⊕C^{k}(𝔫; V) is isomorphic to the space of linear maps from Λ^{∗}𝔫 into V; so

C^{∗}(𝔫; V) ≅ (Λ^{∗}𝔫)′ ⊕ V,  (0.4)

where prime denotes the dualization and the sign ^{∗} is reserved to denote the direct sum: for

example, H^{∗} denotes ⊕_{k} H^{k}, E_{1}^{∗,∗} = ⊕_{i,j}E_{1}^{ij}, E_{1}^{i,∗} = ⊕E_{1}^{i,j}, etc.

Let diag(a_{1}, …, a_{n}) denote the matrix with the numbers a_{i} on its main diagonal, the other elements being 0.

Let 𝔤 be a (semi)simple Lie algebra, V an irreducible 𝔤-module, W the Weyl group of 𝔤 and 𝔟 its Borel subalgebra; 𝔫 = [𝔟, 𝔟]. According to the Borel-Weil-Bott (BWB) theorem [5] (whose different proofs are presented, with increasing clarity, e.g., in [11], [1], [4]), we have

dim H^{i}(𝔫; V) = |W_{i}|,  (0.1)

where |S| denotes the cardinality of S and W_{i} is the set of elements from W of length i, see [6]. We hope there will be no confusion of the cardinality of a set with the modulus of a polynomial introcuded below.

In addition to 𝔤-modules, 𝔫 has a lot of natural modules which are not 𝔤-modules but which admit a 𝔟-module structure; and in this paper we will study such modules. The most interesting is the adjoint module 𝔫, and our main aim is to calculate cohomology with values in it. In particular, we will show (Theorem 7.6), that if 𝔤 = A_{r}, i.e., 𝔤 = 𝔰𝔩(r + 1), then

dim H^{∗}(𝔫; 𝔫) = (r + 1)!\frac{r^{2} + 9r + 2}{12}.

Our methods can be generalized to embrace other 𝔟-modules.

1991 Mathematics Subject Classification. 17B56, 22E25.

Key words and phrases. Lie algebra cohomology, Borel-Weil-Bott’s theorem.

Thanks are due to J. N. Bernstein and A. L. Onishchik for valuable advice and discussions of this paper, to D. A. Leites for editing and translating the manuscript and Stockholm University for financing the preprint of this text [20] and MPIM, Bonn, for the opportunity to make the paper avalable.

Let V be an arbitrary 𝔟-module; we will analyze the action of the Cartan subalgebra 𝔥 on the cochain complex C^{∗}(𝔫; V). This complex is a direct sum of its weight subspaces C_{ν}^{∗} where ν ∈ 𝔥′. In §2 we will derive a formula for the generating function:

F (t, x) = ∑_{k,ν} dim C_{ν}^{k} · t^{k} · x^{ν}.  (2.4)

The guiding idea in what follows is that, first, the coboundary operator d preserves each subcomplex C_{ν}^{∗} = ⊕_{k}C_{ν}^{k} and, second,

∑_{k}(−1)^{k} dim H_{ν}^{k} = ∑_{k}(−1)^{k} dim C_{ν}^{k}.  (0.2)

Therefore,

dim H^{∗} ≥ ∑_{ν} ∑_{k}(−1)^{k} dim C_{ν}^{k}.  (0.3)

Inequality (0.3) coincides with the inequality (2.5) and we will investigate for which modules this inequality becomes an equality. The modules for which the equality is attained are called blue ones.^{1}

In §4 we will show that any irreducible 𝔤-module V is blue, and in §6 we prove the same for V = 𝔫, if 𝔤 is of type A_{r}.

If V is a blue module, one can calculate its cohomology simply by counting the coefficients of the function G(x) = F (−1, x). In §7 we will calculate in this way dim H^{∗}(𝔫; 𝔫) and dim H^{k}(𝔫; 𝔫) for k ≤ 3. In §8 we will establish some properties of dim H^{k}(𝔫; 𝔫) regarded as a function of k and rk 𝔤.

In §9 we will discuss the results and problems for 𝔤 ≠ A_{r}.

I do not know any other investigations concerning dim H^{∗}(𝔫; V), where V is not a 𝔤module, except calculations of Leger and Luks [12] and Williams [21], where H^{2}(𝔫; 𝔫) and H^{1}(𝔫; V) for V equal to either 𝔫, or its dual, 𝔫′, or 𝔤/𝔫 are calculated.

Some results of this paper were announced in [15], see also [16]–[19]. They were delivered at A. L. Onishchik-É. B. Vinberg’s and D. A. Leites’ seminars in 1978/79.

§1. Generating Functions

1.1. Let x = (x_{1}, …, x_{r}) be an r-tuple. For ν = (ν_{1}, …, ν_{r}) ∈ ℝ^{r}, we set x^{ν} = x_{1}^{ν_{1}} … x_{r}^{ν_{r}}. In this paper, except for §§8–9, a polynomial is an expression P (x) = ∑_{ν} a_{ν} · x^{ν} with real or

complex coefficients, where ν runs over some finite set of real vectors.

For a polynomial P, we define its support as N_{P} = {ν ∈ ℝ^{r} | a_{ν} ≠ 0} and its modulus as |P (x)| = ∑_{ν∈N_{p}} |a_{ν}|. We call P a convex polynomial if N_{P} is a set of vertices of a convex

polyhedral.

If P (t, x) is a polynomial in two groups of indeterminates and t^{0} = (t_{1}^{0}, …, t_{s}^{0}) a vector with numerical coordinates, we denote by |P (t^{0}, x)| the modulus of the polynomial obtained from P (t, x) by replacing t with t^{0}. 1.2. Proposition . If P = ∑ a_{ν}x^{ν} and Q = ∑ b_{ν}x^{ν} are polynomials, then

(i) |P · Q| ≤ |P | · |Q|;

(ii) For any monomial Q = x^{ν}, we have |P · Q| = |P |.

(iii) If P is convex and b_{ν} are integers, then |P · Q| ≥ |P |.

1As opposed to yellow modules, the colours chosen to match the two colours of the banner of the independent Ukraine.

Proof. Statements (i) and (ii) are obvious. Now let P be convex and μ ∈ N_{P}. Then there exists a linear form φ ∈ (ℝ^{r})′ such that φ(μ) > φ(ν) for all ν ∈ N_{P}, ν ≠ μ. We can find λ ∈ N_{Q} such that φ(λ) ≥ φ(ν′) for any ν′ ∈ N_{Q}; then φ(λ + μ) > φ(ν + ν′) and, therefore, λ + μ ≠ ν + ν′ for any ν ∈ N_{P}, ν′ ∈ M_{Q}. So if P · Q = ∑ c_{ν}x^{ν}, then |c_{λ+μ}| = |a_{μ} · b_{λ}| ≥ |a_{μ}|.

Thus, for any μ ∈ N_{P} we have found a vector λ + μ ∈ N_{P ·Q} such that |c_{λ+μ}| ≥ |a_{μ}|. Clearly, if μ ≠ μ_{1} then μ + λ ≠ μ_{1} + λ_{1}. Hence,

|P · Q| = ∑_{ν∈N_{P ·Q}} |c_{ν}| ≥ ∑_{μ∈N_{P}} |c_{λ+μ}| ≥ ∑_{μ∈N_{P}} |a_{μ}| = |P |. □

1.3. If a polynomial Q is obtained from P by an affine change of indeterminates, we will write P ∼ Q. Proposition . If P ∼ Q, then |P | = |Q|.

Proof. The translation of the origin to the point ν corresponds to the multiplication of P by x^{ν}, so we may use Proposition 1.2(ii). Now, given a homogeneous change of indeteminates, let π = (π_{1}, …, π_{r}), where π_{i} = ∑_{j} a_{ij}ν_{j}. Then we may denote:

y_{i} = ∏_{j} x_{j}^{b_{ij}}, where (b_{ij}) = (a_{ij})^{−1}.

Clearly, y^{π} = x^{ν}. This means that the change of indeteminates transforms a monomial a_{ν}x^{ν} into some other monomial with the same coefficient a_{ν}. Now our statement is obvious. □

It is easy to see that the correspondence between the monomial x^{ν} and the δ-function δ_{ν} establishes an isomorphism of the polynomial ring and the group ring of ℝ^{r}. It will be more convenient for us to operate with the polynomial ring: to substitute numerical values of variables is more natural for polynomials.

1.4. Now let 𝔥 be an r-dimensional commutative Lie algebra and V an 𝔥-module. Let V be semisimple, i.e., let V have a basis v_{1}, …, v_{m} such that h · v_{i} = μ_{i}(h) · v_{i} for any h ∈ 𝔥 and some μ = (μ_{1}, …, μ_{m}) ∈ (𝔥′)^{m}.

We will also assume that 𝔥 has a basis h_{1}, …, h_{r} such that all the numbers μ_{i}(h_{j}) are real. Then we may consider the generating function of the 𝔥-module V:

A_{V} (x) = ∑_{i=1}^{m} x^{μ_{i}}  (1.1)

This generating function coincides with the character of module V in the terminology of [7]: if V = ⊕ V_{ν} is the decomposition of V into its weight subspaces, then, obviously,

A_{V} (x) = ∑_{ν} dim V_{ν} · x^{ν} = ch V.  (1.2)

If V = ⊕_{k∈ℤ}V^{k} is a ℤ-graded module, we will consider the generating function

A_{V} (t, x) = ∑_{k,ν} dim V_{ν}^{k} · t^{k} · x^{ν}  (1.3)

where t is one more (one-dimensional) indeterminate.

1.5. Proposition . Let U and V be two semisimple 𝔥-modules, A_{U}(x) and A_{V} (x) their generating functions (1.1). Then the 𝔥-modules U ⊕ V, U ⊗ V, V′ and Λ^{∗}V = ⊕Λ^{k}V are also semisimple and their generating functions are given by the formulas

A_{U⊕V} = A_{U} + A_{V} ,  (1.4)
A_{U⊗V} = A_{U} · A_{V} ,  (1.5)
A_{V′}(x) = A_{V} (x^{−1}) = ∑_{i} x^{−μ_{i}},  (1.6)

A_{Λ^{∗}V} (t, x) = ∏_{i}(1 + t · x^{μ_{i}}).  (1.7)

Proof is obvious.

§2. 𝔟-𝔫-modules

2.1. Now let 𝔫 be an arbitrary Lie algebra and V any 𝔫-module. A pair (D, A) is a derivation of V compatible with D if D is a derivation of 𝔫 and A : V ⟶ V is a linear transformation such that

A(n · v) = n · Av + Dn · v for any n ∈ 𝔫, v ∈ V.  (2.1)

Obviously such derivations form a Lie algebra, which we denote by 𝔡𝔢𝔯(V). There is a projection: (D, A) ↦ D from 𝔡𝔢𝔯(V) on 𝔡𝔢𝔯(𝔫); its kernel consists of all pairs (0, A), where A : V ⟶ V commutes with the 𝔫-action on V. This projection gives us the action of 𝔡𝔢𝔯(V) on 𝔫, and, therefore, we can define the action of 𝔡𝔢𝔯(V) on the cochain complex C^{∗}(𝔫; V) by the standard formula

((D, A)f)(n_{1}, …, n_{k}) = Af((n_{1}, …, n_{k}) − ∑ f((n_{1}, …, Dn_{i}, …, n_{k}).  (2.2)

The standard calculation shows that the coboundary operator d : C^{k} ⟶ C^{k+1} commutes with this action of 𝔡𝔢𝔯(V) on C^{∗}(𝔫; V). So this action induces an action of 𝔡𝔢𝔯(V) on H^{∗}(𝔫; V).

2.2. Now, let V be a 𝔟-module (so V is an 𝔫-module as well). For any b ∈ 𝔟 let D_{b} be the restriction of ad b onto 𝔫 and A_{b} the action of b on V. Then

b ↦ (D_{b}, A_{b})  (2.3)

is a homomorphism 𝔟 ⟶ 𝔡𝔢𝔯(V). So 𝔟 naturally acts on C^{∗}(𝔫; V) and on H^{∗}(𝔫; V). It is well known that every Lie algebra trivially acts on its (co)homology, so 𝔫 trivially acts on H^{∗}(𝔫; V), and we may regard the above 𝔟-action as the action of 𝔟/𝔫 = 𝔥 on H^{∗}(𝔫; V).

2.3. Henceforth we will assume that 𝔟 is a semidirect sum of its ideal 𝔫 and a commutative subalgebra 𝔥. Then any 𝔟-module is at the same time an 𝔥-module and an 𝔫-module. We will call V a 𝔟-𝔫-module if

(i) V is a 𝔟-module and

(ii) V has a basis of 𝔥-eigenvectors v_{1}, .., v_{m} with real weights in some fixed basis of 𝔥. The weight of v_{i} will be denoted by μ_{i} ∈ 𝔥′. We will assume that all the modules to be

studied are 𝔟-𝔫-modules; in particular, we will assume that the adjoint representation of 𝔥 on 𝔫 is semisimple; let λ_{1}, …, λ_{n} ∈ 𝔥′ be the weights of this representation.

2.4. Let C^{∗} = ⊕_{ν} C_{ν}^{∗} = ⊕_{k,ν} C_{ν}^{k} be the decomposintion of the graded 𝔥-module C^{∗} = C^{∗}(𝔫; V).

Set a_{kν} = dim C_{ν}^{k}.

Theorem . Let F_{V} (t, x) = ∑_{k,ν} a_{kν} · t^{k} · x^{ν} be the generating function of C^{∗}(𝔫; V). Then

F_{V} (t, x) = ∏_{i}(1 + t · x^{−λ_{i}}) · A_{V} (x).  (2.4)

Proof. Since C^{∗}(𝔫; V) is isomorphic to (Λ^{∗}𝔫)′ ⊗ V as a graded 𝔥-module (formula (0.3)), our theorem follows from Proposition1.5. □

2.5. Set b_{ν} = ∑_{0≤k≤n}(−1)^{k}a_{kν} and c_{kν} = a_{kν} −a_{k+1ν} −a_{k−1ν}; let G_{V} (x) = ∑ b_{ν} ·x^{ν}. Obviously,

G_{V} (x) = F_{V} (−1, x). Theorem . In the above notations

dim H^{∗}(𝔫; V) ≥ |G_{V} |;  (2.5)
dim H_{ν}^{k} ≥ c_{kν};  (2.6)
dim H^{k} ≥ ∑_{ν} max(0, c_{kν}).  (2.7)

Proof. Denote by H_{ν}^{k} is the cohomology of the complex

… ⟶ C_{ν}^{k−1} ⟶ C_{ν}^{k} ⟶ C_{ν}^{k+1} ⟶ …

Clearly,

dim H_{ν}^{k} ≥ dim C_{ν}^{k} − dim C_{ν}^{k−1} − dim C_{ν}^{k+1} = c_{kν},

and we obtain (2.6). It immediately implies (2.7).

To prove formula (2.5), let us apply (0.2) to the complex C_{ν}^{∗}. We get

|b_{ν}| = | ∑_{k}(−1)^{k} dim C_{ν}^{k}| = | ∑_{k}(−1)^{k} dim H_{ν}^{k}| ≤ ∑_{k} dim H_{ν}^{k} = dim H_{ν}^{k}.  (2.8)

Hence, dim H^{∗} = ∑_{ν} dim H_{ν}^{∗} ≥ ∑ |b_{ν}| = |G|. □

2.6. Corollary . If λ_{1} ≠ 0, …, λ_{n} ≠ 0 and V is a 𝔟-𝔫-module, then H^{∗}(𝔫; V) ≠ 0.

Proof. If H^{∗}(𝔫; V) = 0, then |G| = 0 and so G(x) ≡ 0. Since G(x) = F (−1, x), we see from (2.4) that some of the vectors λ_{1}, …, λ_{n} vanish. □

The assumption on V to be a 𝔟-𝔫-module is essential. If 𝔟 is a two-dimensional noncommutative Lie algebra, 𝔫 = [𝔟, 𝔟] is the one-dimensional Lie algebra and V is onedimensional non-trivial 𝔫-module, then H^{∗}(𝔫; V) = 0 even if λ ≠ 0. 2.7. Corollary . Let z^{0} = (z_{1}^{0}, …, z_{s}^{0}) be a set of complex numbers, |z_{i}^{0}| ≤ 1 for all i, and y = (y_{1}, …, y_{r−s}) some variables. Then dim H^{∗}(𝔫; V) ≥ |G(z^{0}, y)|.

§3. The spectral sequence. The term E_{1}^{ij}

3.1. If V = ℂ is the trivial 𝔟-module, then A_{V} (x) = 1. So F_{ℂ}(t, x) = ∏_{i}(1 + t · x^{λ_{i}}), and we

can write

F_{V} (t, x) = F_{ℂ}(t, x)A_{V} (x).  (3.1)

Our purpose in this section is to set an analogous formula (3.5) for the generating functions of cohomology.

3.2. Let V be a 𝔟-𝔫-module. Assume that V admits a filtration V = V_{m} ⊃ V_{m−1} ⊃ … ⊃ V_{1} ⊃ V_{0} = {0} such that dim V_{i} = i and

𝔟V_{i} ⊂ V_{i}; 𝔫V_{i} ⊂ V_{i−1}. (3.2)

If (3.2) is fulfilled, we call V a nilmodule. One can easily see that if there exists an h ∈ 𝔥 such that λ_{i}(h) > 0 for all i, then any such 𝔟-𝔫-module V is a nilmodule.

Denote: C^{(i)} = {φ ∈ C^{∗} | Imφ ⊂ V_{i}}. Then the sets C^{(i)} forms a filtration of C^{∗}, and this filtration generates a spectral sequence [9]. Let us calculate its first term E_{1}^{ij}. 3.3. Theorem . E_{1}^{ij} ≅ H^{i}(𝔫) as linear spaces for all j such that 0 ≤ j ≤ dim V − 1.

Proof. The main rule for calculating E_{1} is to replace the filtered complex with the graded one. In our case this means to substitute the graded module V_{gr} in place of the filtered module V. So E_{1}^{ij} ≅ H^{i}(𝔫; V_{j+1}/V_{j}) ≅ H^{i}(𝔫). □

Usually, the differentials d_{k} in the spectral sequence map E_{k}^{ij} to E_{k}^{i+1−k,j+k}, and ⊕_{j} E_{∞}^{i−j,j} is

the graded module associated with the filtered module H^{i}. But our method of filtration does not coincide with the standard one: here d_{k} maps E_{k}^{ij} to E_{k}^{i+1,j+k}, and the graded module

associated with H^{i}(𝔫; V) is ⊕_{j=0}^{m−1} E_{∞}^{ij} = E_{∞}^{i∗}.

3.4. The action of 𝔟 on the spectral sequence. Theorem . The action of 𝔟 on C^{∗} induces an action of 𝔟 on E_{k}^{ij} for all i, j, k. This action commutes with d_{k} and its restriction on 𝔫 is trivial.

Proof. Substitution of V_{gr} for V means that we regard the elements of V_{i−1} as “infinitesimals” as compared with the elements of V_{i}. (One can use the language of contracted representations [13], [2] to give the formal proof.) We know that d commutes with the action of 𝔟. So it commutes “even more so” if we neglect infinitesimals, and E_{1}^{∗} admits a 𝔟-module structure. The second of the formulas (3.2) shows that 𝔫 acts trivially on V_{gr} and, therefore, 𝔫 acts trivially on E_{1}^{∗} (hence, 𝔫 acts trivially on E_{k}^{∗}). Thus we have proved the theorem for k = 1. It remains to regard the infinitesimals up to the k-th order as negligible ones which gives the proof for any k. □

3.5. Main result of this section. Let h_{kν} = dim H_{ν}^{k}(𝔫) and let

P (t, x) = ∑_{k,ν} h_{kν} · t^{k} · x^{ν}  (3.3)

be the generating function for the graded 𝔥-module H^{∗}(𝔫). Theorem . Let V be a 𝔟-𝔫-nilmodule and V_{i} the terms of the corresponding filtration. Then the generating function D(t, s, x) of the graded 𝔟-module E_{1}^{∗} = ⊕E_{1ν}^{ij} is given by the formula

D_{V} (t, s, x) = P (t, x) · ∑_{j} s^{j} · x^{μ_{j}},  (3.4)

where μ_{j} is the weight of V_{j+1}/V_{j}.

Proof. Obviously, D(t, s, x) = ∑_{j} s^{j} ·D_{j}(t, x), where D_{j} is the generating function for E_{1}^{∗j}.

Now let u_{1} be the weight vector from E_{1}^{∗j} and u_{0} its representative in E_{0}^{∗j} = (Λ^{∗}𝔫)′⊗(V_{j+1}/V_{j}).

We see that the weights of u_{1} and u_{0} coincide, and are equal to λ = μ_{j}, where λ is the weight of some cohomology class from H^{∗}(𝔫). So, by multiplying P (t, x) by x^{μ_{j}}, we obtain D_{j}(t, x). This implies (3.4). □ 3.6. Corollary . E_{1}^{∗} is isomorphic to H^{∗}(𝔫) ⊗ V as 𝔥-modules.

Proof follows from the fact that the generating functions of these two 𝔥-modules coincide. □

3.7. The function D(t, s, x) is not invariant since one can consider filtration of V with respect to different s’s. So hereafter we will suppress the parameter s and use the function Q(t, x) = D(t, 1, x). Obviously,

Q_{V} (t, x) = P (t, x) · A_{V} (x).  (3.5)

So Q only depends on the algebra 𝔟 and module V over it. 3.8. Proposition . Q_{V} (−1, x) = G_{V} (x).

Proof. Applying formula (2.8) to the complex C^{∗}(𝔫) we see that P (−1, x) = F_{ℂ}(−1, x). Now it remains to use the definition of G(x) and formulas (3.1), (3.5). □

Therefore, hereafter we will use the function Q_{V} instead of F_{V}.

3.9. We will also consider the function R(x) = Q(1, x). Clearly, |R(x)| = |Q(t, x)| = dim E_{1}^{∗}. So we see that

|G_{V} (x)| ≤ dim H^{∗}(𝔫; V) ≤ |R_{V} (x)|.  (3.6)

Let r_{ν} = dim E_{1ν}^{∗}. Then R(x) = ∑ r_{ν} · x^{ν}. We will call r_{ν} = r_{ν}(V) the ν-multiplicity of the module V, and r(V) = max_{ν} r_{ν}(V) the multiplicity of V.

3.10. Recall [4] that for the nilradical 𝔫 of 𝔟, there exists a free resolution of the trivial 𝔟module such that Q_{V} (t, x) is the generating function for the corresponding cochain complex for any V. We will only use a much simpler, trivial fact that dim H_{ν}^{k} ≤ dim E_{1ν}^{k∗}; in particular, if E_{1ν}^{k∗} = 0, then H_{ν}^{k} = 0.

§4. The Blue Modules and the Borel-Weil-Bott Theorem

4.1. Now we will study the following question: when the inequality (2.5) becomes equality? Let us give an appropriate definition.

A 𝔟-𝔫-module V is a ν-blue module if

dim H_{ν}^{∗} = |b_{ν}|  (4.1)

and V is a blue module if it is ν-blue for all the ν.

So V is a blue module if and only if dim H^{∗}(𝔫; V) = |G_{V} |. 4.2. Proposition . Let V be a 𝔟-𝔫-module and ν ∈ 𝔥′ a weight of V. The module V is ν-blue if and only if H_{ν}^{k} = 0 for all even k or for all odd k.

Proof. Since b_{ν} = ∑_{k}(−1)^{k} dim H_{ν}^{k}, then V is ν-blue if and only if

∑_{k} dim H_{ν}^{k} = | ∑_{k}(−1)^{k} dim H_{ν}^{k}|.

So all the nonzero terms in the right-hand side must have the same sign. □ 4.3. Proposition . In the above notations any of the following conditions is sufficient for V to be ν-blue:

(i) C_{ν}^{k} ≠ 0 for no more than one k.

(ii) H_{ν}^{k} ≠ 0 for no more than one k.

(iii) dim C_{ν}^{∗} ≤ 1.

(iv) r_{ν} ≤ 1.

(v) dim H_{ν}^{∗} ≤ 1.

Proof. This is an easy corollary of Proposition 4.2 since any of the conditions (i)-(v) is stronger than the condition of Proposition 4.2.

4.4. Hereafter we will consider the situation discussed in Introduction: 𝔟 is the Borel subalgebra of some semisimple Lie algebra 𝔤, 𝔥 and 𝔫 are the Cartan subalgebra and the nilradical of 𝔟, respectively. Let R_{+} = \{α_{i}\}_{i∈I} be the set of positive roots of 𝔤, R_{−} = −R_{+}, ρ = \frac{1}{2} ∑_{i∈I} α_{i}

and W the Weyl group of 𝔤.

Clearly, if U is a 𝔤-module, then U is a 𝔟-𝔫-nilmodule. So we can apply all the previous results to U. The weights of U will be denoted by β_{j} for j = 1, …, m. Theorem . If U is an irreducible 𝔤-module, then U is a blue 𝔟-𝔫-module.

Proof will be given in sec. 4.5–4.8. (One can prove that any finite dimensional 𝔤-module is blue.)

4.5. We see that

G_{V} (x) = ∏_{i∈I}(1 − x^{−α_{i}}) · ∑_{j=1}^{m} x^{β_{j}}.  (4.2)

Let us change the variables by setting π = 2ν −ρ. Then G_{V} (x) transforms into an equivalent function G̃_{V}, and, clearly,

G̃_{V} (x) = ∏_{i∈I}(x^{α_{i}} − x^{−α_{i}}) · ∑_{j=1}^{m} x^{2β_{j}}.  (4.3)

In what follows we will operate in these new variables. 4.6. Lemma . If U = ℂ is the trivial 𝔤-module, then

G_{ℂ}(x) = ∑_{w∈W}(−1)^{l(w)} · x^{wρ},  (4.4)

where l(w) is the length of w ∈ W, see [6]. Moreover, |G_{ℂ}| = |W |.

Proof. One sees that the Weyl formula [7]

∏_{i∈I}(x^{α_{i}} − x^{−α_{i}}) = ∑_{w∈W}(−1)^{l(w)} · x^{wρ}  (4.5)

coincides with (4.4). To prove the second statement, we must show that w_{1}ρ ≠ w_{2}ρ if w_{1} ≠ w_{2}, because in this case there are |W | different monomials in the right hand side of (4.4). But this follows from the well known fact that ρ lies strictly inside a Weyl chamber. □ 4.7. Lemma . G_{V} ≥ |W | for any 𝔟-𝔫-module.

Proof. Clearly, |wρ| = |ρ| for any w. So, by Lemma 4.6 the set N_{G_{ℂ}} lies on the sphere of radius |ρ| and, therefore, G_{ℂ} is a convex polynomial. It remains to apply Proposition 1.2 (iii). □

4.8. Now, by the BWB theorem, if U is an irreducible 𝔤-module, then dim H^{∗}(𝔫; U) = |W |. With Lemma 4.7, we have:

|W | ≤ G_{U} ≤ dim H^{∗}(𝔫; U) = |W |  (4.6)

which implies that all the inequalities are equalities. So U is blue, and the proof of Theorem 4.4 is completed. □ 4.9. Corollary . dim H^{∗}(𝔫; V) ≥ |W | for any 𝔟-𝔫-module V.

Proof follows from Lemma 4.7. □

4.10. Now we can give a convenient formula for the introduced in sec. 3.5 function P (t, x). It follows from formula (4.4) and the fact that ℂ is a blue module that for a function m(w) we have

P (t, x) = ∑_{w∈W} x^{wρ} · t^{m(w)}.  (4.7)

Proposition . m(w) in formula (4.7) coincides with l(w).

Proof. It is well known [6] that the length of w coincides with card (wR_{+} ∩ R_{−}). So let l = l(w) and let w{α_{1}, …, α_{n}} = {−α_{1}, −α_{2}, …, −α_{l}, α_{l+1}, …, α_{n}}. Then

wρ = \frac{1}{2}(−α_{1} − … − α_{l} + α_{l+1} + … + α_{n}).

Let us open the parenthesis in the left hand side of (4.5). We can get the monomial x^{wρ} only if we take each second summand from the first l factors, and each first summand from all the other factors since the degree of any other monomial is less than that corresponding to wρ with respect to the Weyl chamber containing w(ρ). Now, consider the corresponding to G function F̃_{ℂ}(t, x) = ∏(x^{α_{i}} + tx^{−α_{i}}); similarly, if we want to get the term x^{wρ}t^{m}, we must take each second summand from the first l factors. Thus, we get the term x^{wρ}t^{l}. □

Finally, we obtain:

P (t, x) = ∑_{w} t^{l(w)}x^{wρ}.  (4.8)

4.11. Theorem . Let U be an irreducible 𝔤-module, ν an arbitrary weight. Then (i) dim H_{ν}^{∗}(𝔫; U) ≤ 1. (ii) H_{ν}^{∗}(𝔫; U) ≠ 0 ⟺ dim C_{ν}^{∗}(𝔫; U) = 1. So dim C_{ν}^{∗}(𝔫; U) is either even or is equal to 1

for any ν.

(iii) Inequality (2.7) is an equality.

(iv) Let λ_{+} be the highest weight of U, and σ = ρ + λ_{+}. Then the generating function P_{U}(t, x) := ∑_{k,ν} dim H_{ν}^{k}(𝔫; U) · t^{k} · x^{ν} is given by the formula

P_{U}(t, x) = ∑_{w∈W} x^{wσ} · t^{l(w)}.  (4.9)

Proof is an easy consequence of the proof of Theorem 4.4 and Proposition 4.10. □

4.12. It might be useful sometimes to have an explicit formula for P (t, x) when 𝔤 is any classical simple Lie algebra. Taking W and ρ from [6] we obtain Table 1.

If 𝔤 = A_{r}, we see that l(w) is equal to the number of inversions in the lower row of the

permutation w = (\begin{matrix}0 & 1 & … & r \\ i_{0} & i_{1} & … & i_{r} \end{matrix}).

4.13. Now we can find a convenient formula for the ν-multiplicity on any 𝔟-𝔫-module V. Let

M_{ν} = {(w, j) | w ∈ W, j = 1, …, m, wρ + β_{j} = ν.

Let L be the projection of M_{ν} on {1, …, m}. Proposition . r_{ν} = card M_{ν} = card L_{ν}.

Proof. The first equality immediately follows from formulas (3.5) and (4.3) because R(x) = Q(1, x). Now if w_{1}ρ + β_{j_{1}} = w_{2}ρ + β_{j_{2}} and w_{1} ≠ w_{2}, then, as we have seen, w_{1}ρ ≠ w_{2}ρ and so j_{1} ≠ j_{2}. This gives the second equality. □

Equivalently, we may describe r_{ν} as

r_{ν} = card {μ ∈ N_{A_{V}} | ν − μ = wρ for some w}.

§5. Subquotients

5.1. We will say that a module V is a subquotient of U if V is isomorphic to a quotient module of a submodule of U (in particular, if V is a submodule or quotient module of U). We clearly see that if V is any 𝔟-module in a 𝔟-subquotient of some 𝔤-module, then V is a 𝔟-𝔫-nilmodule.

Hereafter we will assume that all the modules are 𝔟-subquotients of a (usually irreducible) 𝔤-module. Let U be a 𝔤-module, V its subquotient; then we may regard C^{∗}(𝔤; U) as a 𝔤-module, and C^{∗}(𝔫; Ũ), E_{1}^{∗}(𝔫; Ũ), H^{∗}(𝔫; Ũ), where Ũ = U or V, are its subquotients.

5.2. In what follows the letter U will always denote some 𝔤-module and V its 𝔟-subquotient. Let V_{0} ⊃ V_{1} be 𝔟-submodules of U such that V = V_{0}/V_{1}. In U, we may find 𝔥-submodules V_{2} and V_{3}, complementary to V_{1} in V_{0} and to V_{0} in U, respectively. Evidently, A_{V} (x) = A_{V_{2}}(x) and V ≅ V_{2}, as 𝔥-modules; we will identify these modules, if necessary. We see also that the V_{i} are 𝔟-subquotients of U and U ≅ V_{1} + V_{2} + V_{3} as 𝔥-modules; so

A_{U}(x) = A_{V_{1}}(x) + A_{V_{2}}(x) + A_{V_{3}}(x).  (5.1)

The number cr_{ν}(V, U) = r_{ν}(U) − r_{ν}(V) is called the ν-comultiplicity of V with respect to U. Formula (5.1) implies that

E_{1ν}^{∗}(𝔫; U) ≅ E_{1ν}^{∗}(𝔫; V_{1}) + E_{1ν}^{∗}(𝔫; V_{2}) + E_{1ν}^{∗}(𝔫; V_{3}),

so

cr_{ν}(V, U) = r_{ν}(V_{1}) + r_{ν}(V_{3}).  (5.2)

5.3. Proposition . Let U be an irreducible 𝔤-module, V its 𝔟-subquotient and ν some weight such that r_{ν}(U) ≠ 1. Then

(i) dim H_{ν}^{∗}(𝔫; V) ≤ r_{ν}(V);

(ii) dim H_{ν}^{∗}(𝔫; V) ≤ cr_{ν}(V, U).

Proof. (i) is clear, because

dim H_{ν}^{∗}(𝔫; V) ≤ dim E_{1ν}^{∗}(𝔫; V) = r_{ν}(V).

In particular, this means that dim H_{ν}^{∗}(𝔫; V_{i}) ≤ r_{ν}(V_{i}). Now, consider the exact sequence 0 ⟶ V_{1} ⟶ V_{0} ⟶ V ⟶ 0. The corresponding exact cohomology sequence shows that

dim H_{ν}^{∗}(𝔫; V) ≤ dim H_{ν}^{∗}(𝔫; V_{1}) + dim H_{ν}^{∗}(𝔫; V_{0}).  (5.3)

But the exact sequence 0 ⟶ V_{0} ⟶ U ⟶ V_{3} ⟶ 0 shows that H_{ν}^{∗}(𝔫; V_{0}) ≅ H_{ν}^{∗}(𝔫; V_{3}), since H_{ν}^{∗}(𝔫; U) = 0 by Theorem 4.11 (ii). So dim H_{ν}^{∗}(𝔫; V_{0}) ≤ r_{ν}(V_{3}) and dim H_{ν}^{∗}(𝔫; V_{1}) ≤ r_{ν}(V_{1}). Now it only remains to apply formulas (5.3) and (5.2). □

If r_{ν}(U) = 1, then (ii) fails but we clearly see that in this case dim H_{ν}^{∗}(𝔫; V) = r_{ν}(V) for any V. 5.4. Proposition . Let U, V, and ν be as in Proposition 5.3.

(i) If r_{ν}(V) · cr(V, U) = 0, then dim H_{ν}^{∗}(𝔫; V) = 0.

(ii) If r_{ν}(V) or cr(V, U) is equal to 1, then dim H_{ν}^{∗}(𝔫; V) = 1.

(iii) V is ν-blue in all the cases mentioned in (i), (ii).

Proof. Since r_{ν}(U) is even, r_{ν}(V) and cr(V, U) are either simultaneously even or simultaneously odd and the same is true for r_{ν}(V) and dim H_{ν}^{∗}(𝔫; V).

So Proposition 5.3 implies (i) and (ii). Now (iii) is a consequence of Proposition 4.3 (v). □ 5.5. Proposition . If U is an irreducible 𝔤-module and r_{ν}(U) ≤ 2, then any 𝔟-submodule of U is ν-blue. In particular, if r(U) ≤ 2, then any 𝔟-submodule of U is blue.

Proof. If r_{ν}(U) = 1, we have to apply Proposition 4.3 (iv), otherwise we can apply Proposition 5.4 (iii), because r_{ν}(V) ≤ 1 or cr_{ν}(V, U) ≤ 1. □

We see that the proposition is true if r_{ν}(U) ≤ 3, but such generalization is inessential, because we know that dim E_{1ν}^{∗}(𝔫; U) is either even or is equal to 1.

5.6. Proposition . Let U be an irreducible 𝔤-module, V its 𝔟-submodule and Ṽ = U/V. Then V is ν-blue ⟺ Ṽ is ν-blue for any ν.

Proof. Let G_{U}(x) = ∑ b_{ν}x^{ν}, G_{V} (x) = ∑ b′_{ν}x^{ν} and GṼ = ∑ b′′_{ν}x^{ν} be the generating functions of the corresponding modules. As A_{U}(x) = A_{V} (x) + AṼ (x), we easily see that G(x) = G_{V} (x) + GṼ (x). So b_{ν} = b′_{ν} + b′′_{ν}. Now we have to study two cases:

(1) If b_{ν} ≠ 0, then dim C_{ν}^{∗}(𝔫; U) = 1 (Theorem 4.11 (ii)), and both V and Ṽ are ν-blue by Proposition 4.3 (ii).

(2) If b_{ν} = 0, then b′_{ν} = |b′′_{ν}|. But in this case H_{ν}^{∗}(𝔫, U) = 0, and the exact sequence 0 ⟶ V ⟶ U ⟶ Ṽ ⟶ 0 shows that H_{ν}^{∗}(𝔫; V) ≅ H_{ν}^{∗}(𝔫; Ṽ). So b′_{ν} = dim H_{ν}^{∗}(𝔫; V) if and only if b′′_{ν} = dim H_{ν}^{∗}(𝔫, Ṽ). □

5.7. Proposition . Let U be an irreducible 𝔤-module, ν one of its weights, V and V_{i}, i = 1, 2, 3, as in sec. 5.2. Suppose that V is ν-blue and dim H_{ν}^{∗}(𝔫; V) ≥ dim H_{ν}^{∗}(𝔫; V_{1}) + dim H_{ν}^{∗}(𝔫; V_{3}). Then both V_{1} and V_{3} are ν-blue.

Proof. Let G_{U}, G_{V_{1}}, G_{V}, and G_{V_{3}} be the generating functions for U, V_{1}, V ≅ V_{2} and V_{3} respectively. Then, as earlier, b_{ν} = b′_{ν} + b′′_{ν} + b′′′_{ν} = b′_{ν} + b′′′_{ν} ± dim H_{ν}^{∗}(𝔫; V). As the case b_{ν} ≠ 0 is trivial, we may suppose b_{ν} = 0, so

dim H_{ν}^{∗}(𝔫; V) = |b′_{ν} + b′′′_{ν}| ≤ |b′_{ν}| + |b′′′_{ν}| ≤ dim H_{ν}^{∗}(𝔫; V_{1}) + dim H_{ν}^{∗}(𝔫; V_{3}).

Hence, all the inequalities are equalities and (4.1) holds for V_{1}, V_{3}. □

5.8. Proposition . A 𝔟-𝔫-module V is blue if and only if so is V′. Moreover, r(V) = r(V′).

Proof. By formula (1.6) the generating functions corresponding to V and V′ are

G_{V} (x) = P (−1, x) · A_{V} (x) and G_{V′}(x) = P (−1, x) · A_{V} (x^{−1}).

So we can change variables x ↦ x^{−1} and write:

|G_{V′}(x)| = |P (−1, x) · A_{V} (x^{−1})| = |P (−1, x^{−1}) · A_{V} (x)|.

But P (−1, x) = G_{ℂ}(x) = ∏(x^{α_{i}} − x^{−α_{i}}), so we easily see that P (−1, x^{−1}) = ±P (−1, x). Therefore G_{V′}(x^{−1}) = ±G_{V} (x) and so |G_{V′}| = |G_{V} |. On the other hand, it is well known that (H^{∗}(𝔫; V))′ ≅ H^{∗}(𝔫; V′) so dim H^{∗}(𝔫; V) = dim H^{∗}(𝔫; V′). The rest is clear. □

Problem . Is it true that if V is ν-blue, then V′ is ν-blue?

In the rest of the paper we will use the methods developed above to calculate certain cohomology.

§6. The adjoint representation of 𝔫 is blue for 𝔤 = A_{r}

This section is devoted to the proof of the following theorem:

6.1. Theorem . Let 𝔤 = A_{r} and 𝔫 the nilradical of its Borel subalgebra 𝔟. Then 𝔫, the adjoint 𝔫-module, is blue.

Proof will be given in sec. 6.2–6.9.

6.2. Recall that for A_{r} = 𝔰𝔩(r + 1) we have: 𝔟 is the algebra of upper triangular matrices, 𝔥 the algebra of diagonal matrices and 𝔫 the algebra of stirictly upper triangular matrices. We use the natural coordinates: if h ∈ 𝔥, then h = diag(h_{0}, …, h_{r}) with ∑ h_{i} = 0; let ν = (ν_{0}, …, ν_{r}) be the dual coordinates on 𝔥′. So ν has r + 1 coordinates instead of r but such coordinates are more convenient.

We begin with a technical lemma. Let U_{r+1} be the standard representation of 𝔤 in the (r + 1)-dimensional space. Lemma . r(U_{r+1}) = 2.

Proof. Obviously, the weights of U_{r+1} are β_{j} = (0, …, 0, 1, 0, …, 0) with a 1 in the jth slot. Set x^{β_{j}} = x_{j}. Using Table 1, we obtain:

R_{U_{r+1}}(x) = ∑ x_{i_{0}}^{0} … x_{i_{r}}^{r}(x_{0} + x_{1} + … + x_{r}).  (6.1)

Simplifying, we get

R_{U_{r+1}}(x) = ∑ x_{i_{0}}^{0} … x_{i_{r−1}}^{r−1}x_{i_{r}}^{r+1} + 2 ∑_{0≤k≤r−1; i_{k}<i_{k+1}} x_{i_{0}}^{0} … x_{i_{k−1}}^{k−1}x_{i_{k}}^{k+1}x_{i_{k}}^{k+1}x_{i_{k+2}}^{k+2} · . . . · x_{i_{r}}^{r}.

So max_{ν} r_{ν}(U_{r+1}) = 2. □

6.3. According to Propositions 5.5 and 5.4, we have: Corollary . Any 𝔟-subquotient U_{r+1} is a blue module. Moreover, if V is a subquotient of U_{r+1} and r_{ν}(V) = 2, then H_{ν}^{∗}(V) = 0.

6.4. Now, let us return to the module 𝔫. We will have to consider also the modules 𝔤 and 𝔟, and the modules 𝔟_{−} = 𝔤/𝔫, 𝔫_{−} = 𝔤/𝔟 and 𝔥 = 𝔟/𝔫. Obviously, 𝔤 is an irreducible 𝔤-module and all the other modules are its subquotients. We have also the isomorphisms of 𝔥-modules:

𝔤 ≅ 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫, 𝔟 = 𝔥 ⊕ 𝔫, 𝔟_{−} ≅ 𝔫_{−} ⊕ 𝔥.

We see from formula (1.4) that it suffices to find the generating functions for 𝔫_{−}, 𝔥, and 𝔫. The only weight of 𝔥 is zero, so x^{ν} = 1. Now, if ν is a weight of 𝔫 or 𝔫_{−}, then x^{ν} = x_{i}x_{j}^{−1} for i ≠ j, and ν is the weight of 𝔫 if and

only if i > j, see [7]. Hence,

A_{𝔫}(x) = ∑_{i>j} x_{i}x_{j}^{−1};  (6.2)
A_{𝔥}(x) = r;  (6.3)
A_{𝔫_{−}}(x) = ∑_{i>j} x_{i}^{−1}x_{j} = A_{𝔫}(x^{−1}).  (6.4)

Now, consider the functions R_{V} (x) for V ∈ {𝔤, 𝔫, 𝔥, 𝔫_{−}}. Recall that L_{ν} = {j | ν = wρ + β_{j}} for any ν (Proposition 4.13), or, equivalently, x^{wρ}x^{β_{j}} = x^{ν}. We know (Table 1) that x^{wρ} = x_{i_{0}}^{0} … x_{i_{r}}^{r} and either x^{β_{j}} = x_{k}x_{1}^{−1} or x^{β_{j}} = 1. Multiplying these, we obtain the following monomials:

1) x_{i_{0}}^{0}x_{i_{1}}^{1} … x_{i_{r}}^{r}.

2) x_{i_{0}}^{−1}x_{i_{1}}^{1} … x_{i_{r−1}}^{r−1}x_{i_{r}}^{r+1}.

3) x_{i_{0}}^{0} … x_{i_{k−1}}^{k−1}x_{i_{k}}^{k}x_{i_{k+1}}^{k}x_{i_{k+2}}^{k+2} … x_{i_{r−1}}^{r−1}x_{i_{r}}^{r+1} for 0 ≤ k ≤ r − 2.

In what follows, for brevity, we will not write explicitly the “normal” factors, i.e., factors x_{i_{j}}^{k} with k = j.

4) x_{i_{0}}^{−1} … x_{i_{k}}^{k+1}x_{i_{k+1}}^{k+1} … for 1 ≤ k ≤ r − 1.

5) … x_{i_{k−1}}^{k}x_{i_{k}}^{k}x_{i_{k+1}}^{k} … for 1 ≤ k ≤ r − 1.

6) … x_{i_{k−1}}^{k}x_{i_{k}}^{k} … x_{i_{l}}^{l}x_{i_{l+1}}^{l} … for 1 ≤ k ≤ r and 0 ≤ l ≤ r−1 such that {k−1, k}∩{l, l+1} = ∅.

It is important to note that in cases 3, 4 we can assume that i_{k} < i_{k+1}; similarly, in case 5 we will assume i_{k−1} < i_{k} < i_{k+1} and in case 6 that i_{k−1} < i_{k} and l_{l} < i_{l+1}. We do not assume that k < l.

6.5. Let us study the ν-multiplicity of the above mentioned modules for any ν of types 1−6.

1) One can obtain the monomial x_{i_{0}}^{0} … x_{i_{r}}^{r} by multiplying the same monomial by 1 for V = 𝔥 or by multiplying … x_{i_{k+1}}^{k}x_{i_{k}}^{k+1} … by x_{i_{k+1}}x_{i_{k}}^{−1} (here 0 ≤ k ≤ r−1 and V is isomorphic to 𝔫 if i_{k+1} > i_{k} or 𝔫_{−} otherwise). It follows that r_{ν}(𝔥) = r and r_{ν}(𝔫_{−}) + r_{ν}(𝔫) = r. Since dim H_{ν}^{∗}(𝔫) = 1 and 𝔥 ≃ ℂ⊕⋯⊕ℂ, as 𝔟-module, it follows that dim H_{ν}^{∗}(𝔫; 𝔥) = r dim H_{ν}^{∗}(𝔫) = r. Now Proposition 5.7 shows that both 𝔫 and 𝔫_{−} are ν-blue.

Clearly, if ν is of types 2 − 6, then r_{ν}(𝔥) = 0.

Now, let X_{ν} = {x^{β} |∈ β ∈ L_{ν}}. Hence, we have a one-to-one correspondence between L_{ν} and X_{ν} (this does not hold for 𝔥, because the multiplicity of the weight zero is greater than 1), so r_{ν}(V) = card X_{ν}, where ν is of types 2-6 and V ∈ {𝔫, 𝔫_{−}, 𝔤}.

2)x_{i_{0}}^{−1} … x_{i_{r}}^{r+1} = x_{i_{0}}^{0} … x_{i_{r}}^{r}(x_{i_{0}}^{−1}x_{i_{r}}). Obviously, this is the only way to get this monomial; so X_{ν} = {x_{i_{0}}^{−1}x_{i_{r}}}, and r_{ν}(𝔤) = 1.

3) Clearly, … x_{i_{k}}^{k}x_{i_{k+1}}^{k} … x_{i_{r}}^{r+1} can be obtained by multiplying x^{wρ} by x_{i_{k}}^{−1}x_{i_{r}} or by x_{i_{k+1}}^{−1}x_{i_{r}}. So r_{ν}(𝔤) = card {x_{i_{k}}^{−1}x_{i_{r}}, x_{i_{k+1}}^{−1}x_{i_{r}}} = 2.

4) is similar to 3). So X_{ν} = {x_{i_{0}}^{−1}x_{i_{k}}, x_{i_{0}}^{−1}x_{i_{k+1}}}, and r_{ν}(𝔤) = 2.

5) Here X_{ν} = {x_{j_{1}}^{−1}x_{j_{1}}}, where j_{1}, j_{2} ∈ {i_{k−1}, i_{k}, i_{k+1}}. So r_{ν}(𝔤) = card X_{ν} = 6.

6)X_{ν} = {x_{j_{1}}^{−1}x_{j_{1}}}, where j_{2} ∈ {i_{k−1}, i_{k}} and j_{1} ∈ {i_{l}, i_{l+1}}. So r_{ν}(𝔤) = 4. 6.6. Lemma . 𝔫 is ν-blue if ν is of types 1 − 4.

Proof. Lemma was already proved for ν of type 1. For the types 2 − 4 it is a corollary of Proposition 5.5. □

6.7. We have not yet proved the theorem for the types 5 and 4. Here we will deal with type 5.

Let x^{ν} = … x_{i_{k−1}}^{k}x_{i_{k}}^{k}x_{i_{k+1}}^{k} …, where i_{k} < i_{k} < i_{k+1}. If x^{ν} = x^{wρ}x^{β}, then there are three weights β which belong to 𝔫:

1) x^{β} = x_{i_{k−1}}^{−1}x_{i_{k}};

2) x^{β} = x_{i_{k}}^{−1}x_{i_{k+1}};

3) x^{β} = x_{i_{k−1}}^{−1}x_{i_{k+1}}.

Thus, x^{wρ} is, respectively,

… x_{i_{k}}^{k−1}x_{i_{k+1}}^{k}x_{i_{k−1}}^{k+1} …  (6.5)
… x_{i_{k+1}}^{k−1}x_{i_{k−1}}^{k}x_{i_{k}}^{k+1} …  (6.6)
… x_{i_{k+1}}^{k−1}x_{i_{k}}^{k}x_{i_{k−1}}^{k+1} …  (6.7)

Let s be the number of inversions in the transposition (i_{0} … i_{r}). Then (i_{0} … i_{k}i_{k+1}i_{k−1} … i_{r}) has s + 2 inversions. So if x^{wρ} is given by (6.5), then l(w) = s + 2. Similarly, l(w) = s + 2 for (6.6) and l(w) = s+3 for (6.7). This means that dim E_{1ν}^{s+2,∗}(𝔫; 𝔫) = 2 and dim E_{1ν}^{s+3,∗}(𝔫; 𝔫) = 1. So there are two possibilities:

either dim H_{ν}^{s+2}(𝔫; 𝔫) = 2 and dim H_{ν}^{s+3}(𝔫; 𝔫) = 1;
or  dim H_{ν}^{s+2}(𝔫; 𝔫) = 1 and dim H_{ν}^{s+3}(𝔫; 𝔫) = 0.

But similar considerations show that dim E_{1ν}^{1,∗}(𝔫; 𝔫_{−}) = \begin{cases}2 & if l = s + 1, \\ 1 & if l = s, \\ 0 & if l ≠ s, s + 1. \end{cases}

In particular, E_{1ν}^{s+2,∗}(𝔫; 𝔫_{−}) = 0 and so H_{ν}^{s+2}(𝔫; 𝔫_{−}) = 0. Since E_{1ν}^{∗}(𝔫; 𝔥) = 0, we have H_{ν}^{s+2}(𝔫; 𝔟_{−}) = H_{ν}^{s+2}(𝔫; 𝔫_{−}) = 0.

Now it remains to use the exact sequence

0 ⟶ 𝔫 ⟶ 𝔤 ⟶ 𝔟_{−} ⟶ 0.

We see that H_{ν}^{∗}(𝔫; 𝔤) = 0 because r_{ν}(𝔤) = 6 ≠ 1, and the corresponding cohomology sequence

… ⟶ H^{s+2}(𝔫; 𝔟_{−}) ⟶ H_{ν}^{s+3}(𝔫; 𝔫) ⟶ H_{ν}^{s+3}(𝔫; 𝔤) ⟶ …

shows that H_{ν}^{s+3}(𝔫; 𝔫) = 0. Thus, the case 2) takes place and

dim H_{ν}^{∗}(𝔫; 𝔫) = dim H_{ν}^{s+2}(𝔫; 𝔫) = 1.  (6.8)

Hence, 𝔫 is ν-blue by Proposition 4.3 (v).

6.8. Finally, let ν be of type 6, x = … x_{i_{k−1}}^{k}x_{i_{k}}^{k} … x_{i_{l}}^{l}x_{i_{l+1}}^{l} . . .. We have to study 6 subcases:

\begin{matrix}(a) & i_{k−1} < i_{k} < i_{l} < i_{l+1} & (d) & i_{l} < i_{k−1} < i_{k} < i_{l+1} \\ (b) & i_{k−1} < i_{l} < i_{k} < i_{l+1} & (e) & i_{l} < i_{k−1} < i_{l+1} < i_{k} \\ (c) & i_{k−1} < i_{l} < i_{l+1} < i_{k} & (f) & i_{l} < i_{l+1} < i_{k−1} < i_{k} \end{matrix}

Recall that we have assumed that i_{l} < i_{l+1}, i_{k−1} < i_{k}. The elements of

X_{ν} = {x_{i_{k−1}}x_{i_{l}}^{−1}, x_{i_{k}}x_{i_{l}}^{−1}, x_{i_{k−1}}x_{i_{l+1}}^{−1}, x_{i_{k}}x_{i_{l+1}}^{−1}}

belong partly to X_{ν}(𝔫) and partly to X_{ν}(𝔫_{−}), namely:

(a) all the weights belong to 𝔫_{−}, so r_{ν}(𝔫) = 0, and r_{ν}(𝔫_{−}) = 4.

(b) ν belongs to 𝔫 if and only if x = x_{i_{k}}x_{i_{l}}^{−1}; the remaining three weights belong to 𝔫_{−} and so r_{ν}(𝔫) = 1, while r_{ν}(𝔫_{−}) = 3.

(c) X_{ν}(𝔫_{−}) = {x_{i_{k}}x_{i_{l}}^{−1}, and x_{i_{k}}x_{i_{l+1}}^{−1}}; hence, r_{ν}(𝔫) = 2, and r_{ν}(𝔫_{−}) = 2.

(d) X_{ν}(𝔫_{−}) = {x_{i_{k}}x_{i_{l}}^{−1}, and x_{i_{k−1}}x_{i_{l+1}}^{−1}}, hence, r_{ν}(𝔫) = r_{ν}(𝔫_{−}) = 2.

The cases (e) and (f) are opposite to (b) and (a), respectively, so we have:

(e) r_{ν}(𝔫) = 3 and r_{ν}(𝔫_{−}) = 1  (f) r_{ν}(𝔫) = 4 and r_{ν}(𝔫_{−}) = 0.

Clearly, r_{ν}(𝔫_{−}) = cr_{ν}(𝔫; 𝔤). So, by Proposition 5.4, 𝔫 is ν-blue in cases (a), (b), (e) and (f).

6.9. It remains to study cases 6(c), 6(d). Let ν be of type 6(c). The corresponding to X_{ν}(𝔫) eigenvectors in 𝔫 are the elementary matrices E_{i_{k}i_{l}} and E_{i_{k}i_{l+1}}.

Now let V_{j} be the 𝔟-submodule of 𝔫 which consists of matrices with first j rows zero and Ṽ_{j} = V_{j−1}/V_{j}. We may identify this subquotient with the linear space of matrices with all the rows zero, except the jth row. So this 𝔫-module is isomorphic to U′_{r+1}, cf. sec. 6.2. To demonstrate the 𝔟-isomorphism of these modules, one has to add the constant weight λ such that x^{λ} = x_{j}.

We see that r_{ν}(𝔫) = r_{ν}(Ṽ_{i_{k}}) = 2. Obviously, Lemma 6.2 holds for the dual module U′_{r+1}, so H_{ν}^{∗}(𝔫; Ṽ_{i_{k}}) = 0. Since dim E_{1ν}^{∗}(𝔫; 𝔫) = dim E_{1ν}^{∗}(𝔫; Ṽ_{i_{k}}), they are isomorphic. So H_{ν}^{∗}(𝔫; 𝔫) = H_{ν}^{∗}(𝔫; Ṽ_{i_{k}}) = 0, and 𝔫 is ν-blue.

Similar arguments (take columns instead of rows) show that 𝔫 is ν-blue if ν is of type 6(d).

Thus, we proved Theorem 6.1 for all the ν. □ 6.10. Corollary . 𝔟-𝔫-modules 𝔟, 𝔫_{−}, and 𝔟_{−} are blue.

Proof. As follows from (6.4), 𝔫_{−} is isomorphic to 𝔫′, 𝔟_{−} is isomorphic to 𝔤/𝔫, and 𝔫_{−} ≅ 𝔤/𝔟. Now apply Propositions 5.6, 5.8. □

§7. Dimensions of the cohomology with the coefficients in the adjoint module

7.1. In §§7 and 8 we still assume that 𝔤 = A_{r} and preserve the notations of the previous

sections. We will also need many new notations: for any w = (\begin{matrix}0 & … & r \\ i_{0} & … & i_{r} \end{matrix}) set

\begin{matrix}h_{1}(w) = |\{j | i_{j} < i_{j+1}\}|; \\ h_{2}(w) = \begin{cases}1 & if i_{0} < i_{r}, \\ 0 & if i_{0} > i_{r}; \end{cases} \\ h_{3}(w) = |\{j | i_{j} < i_{r} < i_{j+1}\}|; \\ h_{4}(w) = |\{j | i_{j} < i_{0} < i_{j+1}\}|; \\ h_{5}(w) = |\{j | i_{j−1} < i_{j} < i_{j+1}\}|; \\ h_{6}(w) = |\{(k, l) | i_{k−1} < i_{l} < i_{k} < i_{l+1}\}| = |\{(k, l) | i_{l} < i_{k−1} < i_{l+1} < i_{k}\}|. \end{matrix}  (7.1)

Further, for any N ∈ {1, …, 6} set h_{N} = ∑_{w∈W} h_{N}(w).

Finally, let W_{l} = {w ∈ W | l(w) = l}, and denote: h_{N}(l) = ∑_{N∈W_{l}} h_{N}(w). For l < 0 set

h_{N}(l) = 0. 7.2. Theorem . dim H^{∗}(𝔫; 𝔫) = h_{1} + h_{2} + h_{3} + h_{4} + h_{5} + 2h_{6}. 7.3. Theorem .

\begin{matrix}dim H^{k}(𝔫; 𝔫) = h_{1}(k − 1) + h_{2}(k) + h_{3}(k − 1)+ \\ h_{4}(k − 1) + h_{5}(k − 2) + h_{6}(k − 2) + h_{6}(k − 1). \end{matrix}

7.4. Proof of Theorems 7.2, 7.3. Consider, for example, type 1. If

x^{ν} = x_{i_{0}}^{0} … x_{i_{r}}^{r} = x_{j} · x_{k}^{−1} · x^{wρ}, where x^{βj} = x_{j} · x_{k}^{−1} ∈ 𝔫,

then, as we have seen, j = i_{s+1}, k = i_{s} and j > k. So w ∈ W_{l+1} for all j, k, and the number of admissible pairs (j, k) is equal to the number of indices s such that i_{s+1} > i_{s}, i.e., to h_{1}(w). So we see that ∑_{ν} dim H_{ν}^{∗}, where ν runs over type 1 vectors, is equal to h_{1}. If we want to

find ∑_{ν} dim H_{ν}^{k}, we must take l + 1 = k, so w ∈ W_{k−1}.

Similar considerations for other types lead us to Table 2. Theorems 7.2, 7.3 are its direct consequences. □

7.5. We have subdivided the set of weights into several subsets according to their types. Set H_{N}^{∗} = ⊕_{ν∈N} H_{ν}^{∗} for any type N from Table 2. Let us find dim H_{N}^{∗}.

1) for w = (\begin{matrix}0 & … & r \\ i_{0} & … & i_{r} \end{matrix}), set w̄ = (\begin{matrix}0 & 1 & … & r \\ i_{r} & i_{r−1} & … & i_{0} \end{matrix}). Then, clearly, h_{1}(w̄) = r − h_{1}(w).

Since the map w ↦ w̄ is a one-to-one,

h_{1} = ∑_{w∈W} h_{1}(w) = ∑_{w∈W} h_{1}(w̄) = ∑_{w∈W}(r − h_{1}(w)) = r · |W | − h_{1};

hence,

dim H_{1}^{∗}(𝔫; 𝔫) = h_{1} = \frac{1}{2}r · (r + 1)!  (7.2)

2) Clearly, i_{0} < i_{r} in the half of the cases, hence,

dim H_{2}^{∗}(𝔫; 𝔫) = h_{2} = \frac{1}{2}|W | = \frac{(r + 1)!}{2}  (7.3)

3) Let us fix j. Then |\{w | i_{j} < i_{r} < i_{j+1}\}| = \frac{1}{6}|W |. So dim H_{3}^{∗} = h_{3} is equal to the sum of such numbers for all j such that 1 ≤ j ≤ r − 1. Hence,

dim H_{3}^{∗}(𝔫; 𝔫) = h_{3} = ∑_{1≤j≤r−1} \frac{|W |}{6} = \frac{(r − 1)}{6} · (r + 1)!  (7.4)

4), 5) Obviously h_{3} = h_{4} (moreover, h_{3}(k) = h_{4}(k) for any k), hence,

dim H_{4}^{∗}(𝔫; 𝔫) = h_{4} = \frac{(r − 1)}{6} · (r + 1)!  (7.5)

The same formula is true for H_{5}:

dim H_{5}^{∗}(𝔫; 𝔫) = h_{5} = \frac{(r − 1)}{6} · (r + 1)!  (7.6)

because in all the cases 3 − 5 we have to order 3 indices in the transposition.

6) Similarly, fix k and l. Then there are \frac{1}{24}|W | transpositions which satisfy the condition i_{k−1} < i_{l} < i_{k} < i_{l+1}. So

dim H_{6(b)}^{∗} = ∑_{(k,l)} \frac{|W |}{24} = dim H_{6(e)}^{∗}.

There are (r − 1) · (r − 2) pairs (k, l) such that {k − 1, k} ∩ {l, l − 1} = ∅. So

dim H_{6}^{∗}(𝔫; 𝔫) = 2h_{6} = \frac{(r − 1)(r − 2)}{12}(r + 1)!  (7.7)

7.6. Theorem . dim H^{∗}(𝔫; 𝔫) = \frac{1}{12}(r^{2} + 9r + 2)(r + 1)!.

Proof: formulas (7.2)–(7.7). □

7.7. From Theorem 7.3 one can deduce dim H^{k}(𝔫; 𝔫) for small k: Theorem .

dim H^{k}(𝔫; 𝔫) = \begin{cases}1 & if k = 0 and r > 0 \\ 2r & if k = 1 and r > 1 \\ \frac{3}{2}(r^{2} + 3r − 4) & if k = 2 and r > 2 \\ \frac{2}{3}r^{3} + 2r^{2} − \frac{11}{3}r − 1 & if k = 3 and r > 3 \end{cases}

If r = 2, then dim H^{2} = 5, dim H^{3} = 2. If r = 3, then dim H^{3} = 21.

Proof of this theorem also consists in studying different types of transpositions with 1, 2 or 3 inversions. It is trivial, but cumbersome, so we omit it. □

§8. The behaviour of h(r, k) = dim H^{k}(𝔫; 𝔫) as a function of k and r = rk𝔤

8.1. Theorem . If k is fixed and r is sufficiently great, then for some rational in k functions c_{l}(k) we have

h(r, k) = ∑_{0≤l≤k} c_{l}(k) · r^{k−1}.  (8.1)

Proof will be given in sec. 8.2–8.8. In sec. 8.2–8.6 we fix a permutation w = (\begin{matrix}0 & … & r \\ i_{0} & … & i_{r} \end{matrix}).

Let l = l(w) and X = {0, 1, …, r}.

8.2. Let σ_{i} = (\begin{matrix}0 & … & i − 1 & i & … & r \\ 0 & … & i & i − 1 & … & r \end{matrix}), where i = 1, …, r, be the standard generators

of W. By definition of length, w is a product of l generators: w = σ_{i_{1}}σ_{i_{2}} … σ_{i_{l}}. Let K = {i_{1}, …, i_{l}}. Let us subdivide K into subsets K_{1}, …, K_{m} so that

(i) if |i_{a} − i_{b}| ≤ 1 (in particular, if i_{a} = i_{b}), then i_{a}, i_{b} belong to the same K_{j}; (ii) if on the contrary, there is an integer i such that i_{a} < i < i_{b} and none of i_{j} is equal to

i, then i_{a} and i_{b} belong to distinct sets K_{j}.

Clearly, (i) and (ii) define the subsets K_{t} uniquely, up to permutation. It is well known that if i and j belong to distinct sets K_{t}, then σ_{i}σ_{j} = σ_{j}σ_{i}. Thus we can write

w = w_{1}w_{2} … w_{m},  (8.2)

where w_{j} is the product of the generators which belong to K_{j}. Such a representation of w is a canonical one, and w_{i}w_{j} = w_{j}w_{i} for all i, j.

8.3. Let us fix j and consider K_{j} and the corresponding permutation w_{j}. If follows from the definition of K_{j} that there exist integers k < l such that K_{j} contains σ_{i} (perhaps, more than once) if and only if k < i ≤ l and the segments [k, l] have no common elements for distinct sets K_{j}. We will call B_{j} = {k, k + 1, …, l} the block corresponding to K_{j}. One easily sees that B_{j} is w-invariant and that

w_{j}(i) = \begin{cases}w(i) & if i ∈ B_{j} \\ i & if i ∉ B_{j} \end{cases} .  (8.3)

So w and w_{j} give the same permutation of {k, …, l}. Let us denote by w_{B} the corresponding permutation of {0, …, l − k}; so

w_{B} = (\begin{matrix}0 & … & l − k \\ i_{k} − k & … & i_{l} − k \end{matrix}) .  (8.4)

8.4. Lemma . Let B_{1}, …, B_{m} be the blocks of w, then i) l(w_{B_{j}}) ≥ \frac{1}{2}cardB_{j} for any j.

ii) l(w) = ∑_{1≤j≤m} l(w_{B_{j}}).

Proof. Since w = σ_{i_{1}} … σ_{i_{l}} is a minimal representation of w as the product of generators, we easily deduce that i ∈ X belongs to some block if and only if i forms an inversion with some i_{1} from the same block. This gives us (i).

Now, the elements of distinct blocks do not form inversions, and the number of inversions in the restriction of w onto B and in w_{B} coincide. This implies (ii). □ 8.5. Lemma . Let B_{1}, …, B_{m} be the blocks of w, and r_{j} = cardB_{j}. Then

h_{1}(w) = r + m − ∑_{1≤j≤m} r_{j} + ∑_{1≤j≤m} h_{1}(w_{B_{j}}).  (8.5)

Proof. Let Z = {j | i_{j} < i_{j+1}}, so h_{1}(w) = |Z|. Let Z_{1} = X\ ∪ B_{j} and let Z_{2} be the set of last elements of blocks: i ∈ Z_{2} if and only if there exists a B_{j} equal to {k, …, i}. Then, clearly

|Z_{1}| = r + 1 − ∑_{1≤j≤m} r_{j}, |Z_{2}| = m, Z_{1} ∩ (Z ∪ Z_{2}) = (Z_{1} ∪ Z_{2})\{r}  (8.6)

and it is always true that r ∈ Z_{1} ∪ Z_{2}. Finally, let i ∈ Z, i ∉ Z_{1} ∪ Z_{2}. Then i and i + 1 belong to some block B = {k, …, l}. So

|Z_{1}\(Z ∪ Z_{2})| = ∑_{B} h_{1}(w_{B}).  (8.7)

Now (8.5) is a direct consequence of (8.6) and (8.7). □

8.6. We can get similar formulas for functions h_{2}(w), …, h_{6}(w); for example, we see that if r > 2l, then h_{2}(w) = 1. We won’t need the exact formulas, so we shall formulate the corresponding result as follows.

Call B a block of type 1 if B ∋ 0, of type 2 if B ∋ r or of type 3 if 0, r ∉ B, respectively. (Lemma 8.4 (i) implies that if r > 2l, then B cannot simultaneously contain 0 and r.) Let Ω be the set of all the w_{B} for given types of blocks. We will say that Ω is a structure on the Weyl group W, and that w is a permutation of the structure Ω. Lemma . In what follows the c_{i} depend only on Ω and do not depend on w or r:

h_{2}(w) = c_{2} = 1, h_{3}(w) = c_{3}, h_{4}(w) = c_{4}, h_{5}(w) = r + c_{5}, h_{6}(w) = c_{6}.  (8.8)

Proof is similar to that of Lemma 8.5, and we omit it. □

Note, however, the difference between h_{1}, h_{5} and the other four functions. This difference is due to the fact that if i ∉ ∪B_{j}, then the inequalities i_{i} < i_{i+1} and i_{i−1} < i_{i} < i_{i+1} hold, and other inequalities in formulas (7.1) do not hold.

8.7. Let Ω be a structure on W, and W (r, Ω) the set of all permutations of {0, …, r} with the given structure Ω. We will consider f(r, Ω) = |W (r, Ω)| as a function of r, Ω for a fixed Ω. Lemma . f(r, Ω) = ∑_{1≤k≤N} α_{k}r^{k}, where α_{N} ≠ 0 and N is equal to the number of blocks of

type 3 in Ω.

Proof. First, suppose, that Ω = {w_{1}, …, w_{m}}, all the w_{i} are of type 3, and no two pairs of them coincide. Then there is a one-to-one correspondence between the permutations of the structure Ω and the arrangements of the sets B_{1}, …, B_{m} of given lengths a_{1}, …, a_{m} from {1, …, r − 1}. It is well-known that the number of such arrangements is equal to

\frac{(r − 1 + m − ∑_{j} a_{j})!}{(r − 1 − ∑_{j} a_{j})!}.  (8.9)

So it is a polynomial in r of degree m.

If we have blocks of type 1 or 2, then their places are fixed, and we have to place the remaining blocks. So f(r, Ω) is still a polynomial in r whose degree is equal to the number of blocks of type 3.

Finally, if w_{1} = w_{2} = … = w_{b_{1}}, w_{b_{1}+1} = … = w_{b_{1}+b_{2}}, and so on, we have to divide (8.9)

by ∏_{i}(b_{i})!. □

8.8. Now we are ready to finish the proof of the theorem. Let Ψ_{l} be the set of all structures with l inversions. Lemma 8.4 shows that Ψ_{l} is finite and does not depend on r if r is sufficiently large. We know (Lemmas 8.5, 8.6) that the functions h_{i}(w) depend only on Ω, i.e., h_{i}(w) = h_{i}(Ω).

So we may rewrite Theorem 7.3 so that every summand in it takes the form

h_{i}(l) = ∑_{Ω∈Ψ_{l}} h_{i}(Ω)f(t, Ω).  (8.10)

Every summand is a polynomial and their number does not depend on r. Applying Theorem 7.3 we complete the proof. □

8.9. Theorem . Let c_{l}(k) be the coefficients in (8.1). Then (k − l)! · c_{l}(k) is a polynomial in k.

Proof. It is clear from formulas (8.10), (8.9), that k! · c_{l}(k) is a polynomial. But if l > k then the term of degree k − l vanishes, so c_{l}(k) = 0. □

8.10. Let us find c_{0}(k). Clearly, we must consider only two terms in Theorem 7.3:

dim H^{k}(𝔫; 𝔫) = ∑_{Ω∈Ψ_{l}} f(r, Ω) · h_{2}(Ω) + ∑_{Ω∈Ψ_{l}} f(r, Ω) · h_{1}(Ω) + …  (8.11)

because the degrees of the terms denoted by dots are < k. We know, that h_{2}(Ω) = 1 and h_{1}(Ω) = r + C. Hence, the terms in (8.11) have degree k if and only if Ω consists of the maximal number of blocks (k for the first term and k − 1 for the second one, respectively).

But then every block has only one inversion, so Ω = ((\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , …, (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})) and |Ω| is

equal to k or k − 1. Then

f(r, Ω) = (\begin{matrix}r − |Ω| + 1 \\ |Ω| \end{matrix}) = \frac{r^{|Ω|}}{|Ω|!} + ….  (8.12)

So

c_{0}(k) = \frac{1}{k!} + \frac{1}{(k − 1)!} = \frac{1}{k!} · (k + 1).  (8.13)

8.11. Using the same method, one can prove that

c_{1}(k) = \frac{1}{(k − 1)!} · \frac{k^{2} − 1}{2}; c_{2}(k) = \frac{1}{(k − 2)!} · \frac{3k^{3} − k^{2} − 92k + 116}{24}.  (8.14)

For example,

dim H^{3}(𝔫; 𝔫) = \frac{4}{3!}r^{3} + \frac{4}{2!}r^{2} − \frac{88}{24}r + …

which coincides with Theorem 7.7;

dim H^{4}(𝔫; 𝔫) = \frac{5}{24}r^{4} + \frac{5}{4}r^{3} − \frac{19}{6}r^{2} + …

§9. Some other results and problems

9.1. The most natural generalization of the problems solved in this paper is to study 𝔤 ≠ A_{r}. I has studied the other classical algebras and G_{2}. Their study is much more complicated then that of A_{r} because one has to consider more cases. The values of |G_{𝔫}| are given in Table 3.

Moreover, I know, that dim H^{∗}(𝔫; 𝔫) = |G_{𝔫}| in all these cases except 𝔤 = D_{r}. The main difficulty is that a straightforward analogue of Lemma 6.2 fails for D_{r} (see below). Problem . Calculate dim H^{∗}(𝔫; 𝔫) for 𝔤 = D_{r} as well as for the exceptional algebras.

9.2. Now let us discuss other modules. The classical Lie algebras A_{r}, B_{r}, C_{r}, D_{r} have standard (identity) representations, whose dimensions are r+1, 2r+1, 2r and 2r, respectively. Their multiplicities (see 3.9) are equal to 2 for the types A, B, C and 4 for the type D. We will analyze these representations elsewhere; here I only give the main result. For simplicity we assume 𝔤 = A_{r}, the same results hold for the types B and C.

Let v_{0}, …, v_{r} be the standard basis of U_{r+1}. Then all the 𝔟-submodules are generated by V_{i} = Span(v_{i}, v_{i+1}, …, v_{r}) for 0 ≤ i ≤ r. Let V_{ij} = V_{i}/V_{j+1}; clearly, dim V_{ij} = j + 1 − i. Set l = dim V_{ij}. Then

dim H^{∗}(𝔫; V_{ij}) = |W | · \frac{l(r + 2 − l)}{r + 1}  (9.1)

and dim H^{k}(𝔫; V_{ij}) for a fixed k does not depend on i, j, l if i > k, l > k, r − j > k. This dimension is a polynomial in r and its degree is equal to k.

It is worth to note that dim H^{∗} grows as |W | · dim V as r ⟶ ∞ and the growth of dim H^{k} is a polynomial one. Moreover, dim H^{∗} has a single formula for all r, great and small, whereas dim H^{k} has such formula only for sufficiently large r.

9.3. Finally, Table 4 gives dim H(𝔫; Λ^{∗}𝔫) for different types of 𝔤.

Problem . Study the growth of dim H(𝔫; Λ^{∗}𝔫) as r ⟶ ∞ for various series of matrix algebras 𝔤; first, for 𝔤 = A_{r}.

Table 1

\begin{matrix}𝔤 & ρ & P (t, x) \\ A_{r} & (1 2 … r) & ∑ x_{i_{0}}^{0}x_{i_{1}}^{1} … x_{i_{r}}^{r}t^{l(w)} \\ B_{r} & (\frac{1}{2} \frac{1}{2} \frac{1}{2} … \frac{2r−1}{2}) & ∑ x_{i_{1}}^{±1/2} … x_{i_{r}}^{±(2r−1)/2}t^{l(w)} \\ C_{r} & (1 2 3 … r) & ∑ x_{i_{1}}^{±1} … x_{i_{r}}^{±r}t^{l(w)} \\ D_{r} & (0 1 2 … (r − 1)) & ∑ x_{i_{1}}^{0}x_{i_{2}}^{±1} … x_{i_{r}}^{±(r−1)}t^{l(w)} \end{matrix}

Table 2

N  x^{0} = x^{wρ} · x_{j}x_{k}^{−1}  (j, k)  dim C_{ν}^{∗}
1  x_{i_{0}}^{0}x_{i_{1}}^{1} … x_{i_{r}}^{r}  (i_{j}, i_{j+1}), if i_{j} > i_{j+1}  h_{1}(w)
2  x_{i_{0}}^{0}x_{i_{1}}^{1}x_{i_{2}}^{2} … x_{i_{r−1}}^{r−1}x_{i_{r}}^{r+1}  (i_{r}i_{0})  h_{2}(w)
3  x_{i_{0}}^{0} … x_{i_{j−1}}^{j−1}x_{i_{j}}^{j−1}x_{i_{j+1}}^{j+1} … x_{i_{r}}^{r+1}  (i_{r}i_{j})  2 if i_{j} < i_{r}
(i_{j−1} < i_{j})  (i_{r}, i_{j−1})  1 if i_{j−1} < i_{r} < i_{j}
0 if i_{r} < i_{j−1}
4 x_{i_{0}}^{−1} … x_{i_{j}}^{j+1}x_{i_{j+1}}^{j+1} … x_{i_{r}}^{r} (i_{j} < i_{j+1})  (i_{j}, i_{0}), (i_{j+1}, i_{0})  2 if i_{0} < i_{j}
1 if i_{j} < i_{0} < i_{j+1}
0 if i_{j+1} < i_{0}
5  x_{i_{0}}^{0} … x_{i_{j−1}}^{j}x_{i_{j}}^{j}x_{i_{j+1}}^{j} … x_{i_{r}}^{r}  (i_{j}, i_{j−1})
(i_{j−1} < i_{j} < i_{j+1})  (i_{j+1}, i_{j−1})  3

(i_{j+1}, i_{j})

6 x_{i_{0}}^{0} … x_{i_{k−1}}^{k}x_{i_{k}}^{k} … x_{i_{l}}^{l}x_{i_{l+1}}^{l} … x_{i_{r}}^{r}  (i_{k−1}, i_{l}), (i_{k}, i_{l})
(i_{k−1} < i_{k}, i_{l} < i_{l+1})  (i_{k−1}, i_{l+1}), (i_{k}, i_{l+1}) 0 ≤ dim C_{ν}^{∗} ≤ 4

Table 3

\begin{matrix}𝔤 & A_{r} & B_{r} & C_{r} & D_{r} & G_{2} \\ \frac{1}{|W |}|G_{𝔫}| & \frac{r^{2} + 9r + 2}{12} & \frac{2r^{2} + 10r − 1}{12} & \frac{2r^{2} + 8r + 3}{12} & \frac{2r^{2} + 7r + 3}{12} & 34 \end{matrix}

Table 4

\begin{matrix}𝔤 & A_{1} & A_{2} & A_{3} & B_{2} & G_{2} \\ dim H^{∗}(𝔫; Λ^{∗}𝔫) & 4 & 36 & 600 & 68 & 220 \end{matrix}

References

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1Correspondence: Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, SE-106 91, Stockholm, Sweden; mleites@mathematik.su.se; tolpygo@kcpik.org.ua