Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103-113]. In the simpler case of $A_{1}^{1}$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154-156].


Introduction
Let G(A) be the complex Kac-Moody Lie algebra corresponding to an n × n symmetrizable Obviously J λ is a left ideal of U(N − ). It has a description in terms of Verma modules M (λ).
Let I λ be a one-dimensional (H ⊕ N + )-module with hu = λ(h)u, gu = 0 for g ∈ N + and arbitrary u ∈ I λ . The Verma module M (λ) is defined as the G(A)-module induced by I λ ; as a U(N − )-module, M (λ) is a free module with one generator u; this "vacuum vector" u is, with respect to the G(A)-module structure, a singular vector of type λ. It is easy to see that M (λ) has a unique maximal proper submodule and this submodule L(λ) is, actually, J λ u.
This observation demonstrates the fundamental importance of the following two problems. Problem 1 is solved, in a very exhaustive way, by Kac and Kazhdan [5]. They describe a subset S ⊆ H * such that the module M (λ) is reducible if and only if λ ∈ S; this subset is a countable union of hyperplanes. (See a precise statement in Section 2 below.) Actually, λ ∈ S if and only if M (λ) contains a singular vector not proportional to u.
A formula for such a singular vector in a wide variety of cases is given in the work of Feigin, Fuchs and Malikov [6]. This formula is short and simple, but it involves the generators f i raised to complex exponents; when reduced to the classical basis of U(N − ) the formula becomes very complicated (as shown in [6] in the example G(A) = sl n ). There remains a hope that the projection of these singular vectors onto reasonable quotients of U(N − ) will unveil formulas that possess a more intelligible algebraic meaning, and this was shown to be the case by Fuchs with the projection over the algebra A 1 1 into U(sl 2 ) and U(H), where H is the Heisenberg algebra [3], work which took its inspiration from the earlier investigation of Verma modules over the Virasoro algebra by Feigin and Fuchs [2].
In this note we extend these results by providing projections to U(sl 2 ) and U(H) of the singular vectors over the family of Kac-Moody Lie algebras G(A) of rank 2 (see Theorem 3 in Section 4 and a discussion in Section 5). As in [3] and [2], our formulas express the result in the form of an explicit product of polynomials of degree 2 in U(H) and U(sl 2 ).
It is unlikely that this work can be extended to algebras of larger rank.

Preliminaries
Let A = (a ij ) be an integral n × n matrix with a ij = 2 for i = j and a ij ≤ 0 for i = j. We assume that that A is symmetrizable, that is,  Fix an auxiliary n-dimensional complex vector space T with a basis α 1 , . . . , α n ; Let Γ denote a lattice generated by α 1 , . . . , α n , and let Γ + be the intersection of Γ with the (closed) positive octant. For an integral linear combination α = n i=1 m i α i , denote by G α the subspace of G(A) spanned by monomials in e i , h i , f i such that for every i, the difference between the number of occurrences of e i and f i equals m i . If α = 0 and G α = 0, then α is called a root of G(A). Every root is a positive, or a negative, integral linear combination of α i ; accordingly the root is called positive or negative (and we write α > 0 or α < 0). Obviously, Remark that Verma modules have a natural grading by the semigroup Γ + .
We can carry the inner product from H to T using the formula α, β = h α , h β . If α, α = 0, then we define a reflection s α : H * → H * by the formula The similar formula The Kac-Kazhdan criterion for reducibility of Verma modules M (λ), mentioned above, has precise statement: is reducible if and only if for some positive root α and some positive integer m, Moreover, if λ satisfies this equation for a unique pair α, m, then all non-trivial singular vectors of M (λ) are contained in M (λ) mα .
For m and α satisfying this criterion, Feigin, Fuchs and Malikov [6] give a description for the singular vector of degree mα in M (λ). In the case when α is a real root, that is, α, α = 0, their description is as follows. Let s α = s i N · · · s i 1 be a presentation of s α ∈ W (A) as a product of Notice that the exponents in the last formula are, in general, complex numbers. It is explained in [6] why the expression for F (s α ; λ) still makes sense.
Proof . The proof is by induction on n.
Obtaining explicit coordinates for the real orbits is straightforward in the affine case, because of the simpler geometry. For pq > 4 an explicit description of the sequence {a n } is possible using an argument familiar to Fibonnaci enthusiasts: Proof . Direct computation.
With these real roots now labelled by the sequence {a n }, we present the singular vectors indexed by them in the Verma modules over G(A). Write λ = xλ 1 + yλ 2 where λ i (h j ) = δ ij , so that λ(h 1 ) = x and λ(h 2 ) = y. Let us define the numbers Γ k 1 , Γ k 2 by the formulas: (Note that Γ 0 1 = Γ 0 2 = 0.) The formula from [6] takes in our case the following form.

Theorem 2. For the algebra G(A) with Cartan matrix
the singular vectors are as follows: 1. For the root α = (a 2n−1 , σa 2n−2 ), with m ∈ N, and t ∈ C arbitrary, 3. For α = (σ −1 a 2n−2 , a 2n−1 ), Proof . It must be checked that the vectors given above actually correspond to the Feigin-Fuchs-Malikov (FFM) procedure for obtaining singular vectors, and also that the Kac-Kazhdan criterion for reducibility is satisfied. For λ = xλ 1 + yλ 2 and the reflection s α = s i N · · · s i 1 (product of simple reflections) the algorithm requires successive application of the transformations s ρ 1 := s 1 (λ + ρ) − ρ and s ρ 2 := s 2 (λ + ρ) − ρ. One generates the list The algorithm then gives . So we first need to know the decomposition of s α into elementary reflections for α in the orbit of (1, 0) or (0, 1). Let S i (m) denote the word in H * beginning and ending with s i , and containing m s i 's. For example, S 1 (3) = s 1 s 2 s 1 s 2 s 1 .
Lemma 1. For real α as above, s α is the word Proof . This is an easy induction on n.
The coefficients of collinearity θ j have the following description.

Then for (i) it is verified that
The argument for (ii) is similar.
Let us put Γ k = Γ k 1 x + Γ k 2 y. We will also need Lemma 3.
Proof . These can be verified directly.
We are now in a position to show by induction that the FFM-exponents correspond to the Γ k in the statement of Theorem 2. It suffices by the second lemma to show that Γ 2n+1 = Λ 2n 1 + 1 and Γ 2n+2 = Λ 2n+1 2 + 1.
Inductively assume that for some (n − 1) > 0 Γ 2(n−1)+1 = Λ where the last equality comes from Lemma 3 and the inductive hypothesis is used in the preceding two lines. The same tack proves that Γ 2n+2 = Λ 2n+1 2 + 1. We now know that the FFM-exponents are as given Theorem 2. It only remains to check that the Kac-Kazhdan criterion (Theorem 1) is satisfied. For m and α satisfying this criterion, [6] give the prescription for the singular vector F (s α ; λ)u of degree mα in M (λ); so we need to verify the existence of such α and m.
For α = aα 1 + bα 2 and λ = xλ 1 + yλ 2 , h α = ad −1 1 h 1 + bd −1 2 = a p h 1 + b q h 2 , so the criterion can be restated as 2(xλ 1 + yλ 2 + ρ) a p h 1 + b q h 2 = m aα 1 + bα 2 , aα 1 + bα 2 . After the calculations this is So after change of variable x + 1 → x and y + 1 → y the Kac-Kazhdan criterion becomes We show that the integral exponent of the centermost element in the singular vectors in the statement of the theorem precisely meets the integrality requirement of the criterion. This is a case by case check, and somewhat tedious and technical; let us verify it for roots of type (a 2n+1 , σa 2n ), whose singular vector comprises 2n + 1 f 1 's and 2n f 2 's raised to appropriate powers; the centermost exponent is then the 2n + 1-st coefficient of collinearity in the FFM procedure, or what we have called Γ 2n+1 .
A remark, a lemma, and a corollary will show that Γ 2n+1 does what it is supposed to.
The next lemma will relate the root sequence {a n } to the exponents of the singular vectors. Proof . Induction on n.