Bernstein–Gel’fand–Gel’fand Resolution

Let us say that condition (2.2) isalmost fulfilledif it is satisfied form= 0, that is, when ++1is on the walls of the dominant Weyl chamber [109]. First we discuss the situation when condition (2.2) is fulfilled for"and is fulfilled or almost fulfilled for any subroot of". Consider the explicit expansion of" ∈ B⁺into simple roots" = ∑^ℓk=1n_k!k, with n_k∈ ℤ+, and defineJ_"≡ {k|n_k= 0̸ }.

In this situation we can give the formula for the singular vector for arbitrary positive roots, that is, not only for straight roots. We have:

Proposition 2 ([208]).LetG be a complex simple Lie algebra. Let+∈H^∗,"∈B⁺and m∈ ℕbe such that(2.2)is fulfilled. Let also:

[(++1,!^∨_k) –m_k]_q

k = 0, k∈J_", m_k∈ ℤ+. (2.84)

Assume also the presentation(2.35)of". Then the singular vector of the Verma module V⁺over U_q(G)corresponding to",m is given by:

2.6 Singular Vectors for Nonstraight Roots 57 that (2.85) follows from the explicit formulae in the previous subsection after suitably renormalizing the coefficients in formula (2.37). Then because of (2.84)all terms but onein (2.37) will vanish and we obtain the monomial expression in (2.85a).

We now state the main result of this subsection.

Proposition 3 ([208]).LetGbe a complex simple Lie algebra. Let+∈A+. Then all singu-lar vectors of the Verma module V⁺over U_q(G)when q is not a root of 1 are parametrized Proof.It is almost obvious that (2.86) is a singular vector. For fixedk= 1,. . .rformula (2.86) means that condition (2.2) is fulfilled with respect to the root!i_k in a Verma moduleV^+k with highest weight shifted by the Weyl dot reflection+k = s_i

k–1. . .s_i

1⋅+. For this we have to prove that m̃_k ∈ ℕ. Actually from the previous proposition we know thatm̃_k ∈ ℤ+. Suppose now that for somekwe havem̃_k = 0. This means that

1⋅+, which would contradict the fact [109] that the Weyl group acts transitively on the Weyl chambers. Finally we have to prove that (2.86) provides all singular vectors ofV⁺. For this we use the fact that whenqis not a root of 1 the structure of the Verma moduleV⁺is the same as forq= 1 [441, 531]. In the caseq = 1 and+∈A+the submodule structure ofV⁺is completely described by the Weyl group;

namely, there is a one-to-one correspondence between the submodules ofV⁺and the

elementsw∈W,w≠e. ◼

Corollary: LetG be a complex simple Lie algebra. Let++1∈A+. Then formula (2.86) describes a singular vector of the Verma moduleV⁺overU_q(G) whenqis not a root of

1. We also havem̃_k∈ ℤ+. ◊

Remark 2.1.The above corollary follows from either of the propositions in this section.

Note that if++1∈A+and+∉A+, that is, when++1is on the walls of the dominant Weyl chamber, the submodule structure ofV⁺is not completely described by the singular vectors in (2.86), and furthermore singular vectors corresponding to different elements ofWmay coincide (the action ofWbeing not transitive). ◊ The results presented so far provide an explicit realization of the Bernstein—

Gel’fand—Gel’fand resolution [96]. In the multiplet classification approach [193, 194, 196–198], the submodule structure ofV⁺for+integral dominant was described

by the maximal multipletM⁺, the elements of which are Verma modulesV^+󸀠which are in one-to-one correspondence with the elementsw∈W, namely,+^󸀠=w⋅+. Let us define the following submodules ofV⁺

C_n⁺≡ ⊕

w∈W,ℓ(w) =nV^w⋅+. (2.87) Note thatC0⁺=V⁺. Recall [109] that the maximal length of an element ofWis equal to the number of positive roots, that is,ℓ(w₀) =|B⁺|, wherew₀is the longest element of W. By [28] there exists a resolution ofL₊for+∈A₊in terms of the above submodules, that is, an exact sequence:

0←L₊←V⁺←C₁⁺← ⋅ ⋅ ⋅ ←C_ℓ⁺_(w0)←0 (2.88) The mapV⁺→L₊is the natural surjection, while for fixedn= 1,. . . ℓ(w₀)

the mapdⁿ : C_n⁺ → C_n–1⁺ is a collection of the maps embedding the components V^w⋅+,ℓ(w) =n, ofCn⁺into the componentsV^w⋅+,ℓ(w) =n– 1, ofCn–1⁺ . One has to check d^n–1∘dⁿ= 0. In [28] this was proved by using general properties of the Weyl group and the uniqueness of the embedding between two Verma modules. (The BGG resolution in a similar context was considered in [110] with explicit expressions in theA₂case using singular vectors in the Poincaré—Birkhoff—Witt basis.)

Here we would like to present an explicit realization of the above uniqueness using our results on the singular vectors. The main ingredient is the commutativity of certain embedding diagrams which involve only subalgebras of rank 2. The reason is that any multiplet of Verma modules, in particular, the maximal one, may be viewed as consisting of submultiplets containing four and six members (for the simply laced algebras), also with eight members (for the nonsimply laced algebras) and with 12 members (forG₂). More explicitly, letV ≡ V^+󸀠,+^󸀠 = w⋅+; for somew ∈ W be such that condition (2.2) is fulfilled for+^󸀠for at least two simple roots; say!pand!r,p≠r.

ThenVis contained in a submultiplet with four members ifa_rpa_pr = 0 with weights V^w⋅+󸀠,w = {e,s_p,s_r,s_ps_r = s_rs_p} ≅ W(A₁⊕A₁); with six members ifa_rpa_pr = 1 with weights V^w⋅+󸀠,w ∈ {e,s_p,s_r,s_ps_r,s_rs_p,s_ps_rs_p = s_rs_ps_r} ≅ W(A₂); with eight mem-bers ifa_rpa_pr = 2 with weightsV^w⋅+󸀠,w ∈ {e,s_p,s_r,s_ps_r,s_rs_p,s_ps_rs_p,s_rs_ps_r,s_ps_rs_ps_r = s_rs_ps_rs_p} ≅ W(B₂); with twelve members ifa_rpa_pr = 3 with weightsV^w⋅+󸀠,W(G₂) ≅ {e,s_p,s_r,s_ps_r,s_rs_p,s_ps_rs_p,s_rs_ps_r,s_ps_rs_ps_r,s_rs_ps_rs_p,s_rs_ps_rs_ps_r,s_ps_rs_ps_rs_p,s_ps_rs_ps_rs_ps_r = s_rs_ps_rs_ps_rs_p}.

Remark 2.2.Naturally, if (2.2) is fulfilled forn > 2 simple roots, thenVwill play the same role in(ⁿ₂)submultiplets of the type just described. If (2.2) is fulfilled only with respect to 0, 1, simple roots thanVis a member of such a multiplet with weight+^󸀠 = w₀⋅+. respectively,+^󸀠 =w⋅+,w≠e,w₀, wherew₀is the longest element of the rank two subalgebras used above. Thus no new submultiplets of the type described above

arise. ◊

2.6 Singular Vectors for Nonstraight Roots 59

We have to establish commutativity of the embedding diagrams describing the above submultiplets. In the four-member submultiplet this is trivial since [X^–_p,X^–_r] = 0.

For the six-member submultiplet we use the caseA₂for"=!₁+!₂,m_j= (++1,!^∨_j)∈ ℤ₊, (2.89) are used to prove commutativity of certain embedding diagrams, in particular, the hexagon diagram ofU_q(sl(3,ℂ)) [198] (or, forq= 1, the hexagon diagram ofsl(3,ℂ) [195]).

For the eight- and twelve-member submultiplets we need the following:

Lemma: Assume the above setting and also that!pis shorter than!r, thusa_rp = –1,

1 =q^1+%,q, respectively; the weight of the highest-weight vectorv₀is+^󸀠in (2.90a,b), s_r⋅+^󸀠in (2.90c),s_p⋅+^󸀠in (2.90d), (s_rs_p)^%⋅+^󸀠in (2.90e), (s_ps_r)^%⋅+^󸀠in (2.90f).

Proof.Direct calculation using Serre relations and formulae of the type of (2.40). ◼ The above lemma ensures the desired propertyd^n–1∘dⁿ = 0 by just choosing prop-erly the constants in (2.90). Only the caseG₂requires further work since we have to establish the following relations: vectors corresponding to the nonstraight roots𝛾,𝛾^󸀠:

v^𝛾₊^,m=P₊^𝛾,m⊗v₀=

2.6 Singular Vectors for Nonstraight Roots 61

For+obeying the assumptions of Proposition 2, the above polynomials should reduce to monomials as in (2.85); this is one justification for the above conjecture.

ExampleA₃

In the situation when (2.2) is almost fulfilled there are also available mixed forms of the singular vectors. We considered the exampleA₂above. Analogously let us have forA₃

[(++1,!^∨_j) –m_j]_q= 0, j= 1, 2, 3, m_j∈ ℤ₊, m=m₁+m₂+m₃∈ ℕ, (2.94) Denotingm_ij=m_i+m_jwe write down the reduction of formula (2.37):

v^m,_s ^"=c^󸀠₁(X^–₁)^m23(X^–₂)^m3(X₃^–)^m(X^–₂)^m12(X^–₁)^m1⊗v₀=

=c^󸀠₂(X₁^–)^m23(X₃^–)^m12(X₂^–)^m(X^–₃)^m3(X₁^–)^m1⊗v₀=

=c^󸀠₃(X^–₃)^m12(X^–₂)^m1(X₁^–)^m(X₂^–)^m23(X^–₃)^m3⊗v₀, (2.95) and several other expressions which analogously to (2.89b) use the polynomials cor-responding to roots which are the sum of two simple roots (and some expressions which use the trivial commutativity [X₁^–,X₃^–] = 0).

Remark 2.3.Most results above may be extended to affine Lie algebras. Consider, for example,U_q(A⁽¹⁾₁ ) and let!1,!2be the simple roots ofA⁽¹⁾₁ , so that (!1,!1) = (!2,!2) = 2 = –(!1,!2). There are two nonsimple straight roots:"ij = !i+ 2!j=s_j(!i) for (i,j) = (1, 2), (2, 1). The singular vector for"12is given by formula forB₂and for"21 by the

interchange of indices 1 and 2. ◊