ON THE STRUCTURE OF PRINCIPAL SUBSPACES OF STANDARD MODULES FOR AFFINE LIE ALGEBRAS OF TYPE A

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1 ON THE STRUCTURE OF PRINCIPAL SUBSPACES OF STANDARD MODULES FOR AFFINE LIE ALGEBRAS OF TYPE A BY CHRISTOPHER MICHAEL SADOWSKI A dissertation submitted to the Graduate School New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Yi-Zhi Huang and James Lepowsky And approved by New Brunswick, New Jersey May, 2014

2 ABSTRACT OF THE DISSERTATION On the structure of principal subspaces of standard modules for affine Lie algebras of type A by Christopher Michael Sadowski Dissertation Directors: Yi-Zhi Huang and James Lepowsky Using the theory of vertex operator algebras and intertwining operators, we obtain presentations for the principal subspaces of all the standard sl(3)-modules. Certain of these presentations had been conjectured and used in work of Calinescu to construct exact sequences leading to the graded dimensions of certain principal subspaces. We prove the conjecture in its full generality for all standard sl(3)-modules. We then provide a conjecture for the case of ŝl(n), n 4. In addition, we construct completions of certain universal enveloping algebras and provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators, along with conjecturally assumed presentations for certain principal subspaces, to construct exact sequences among principal subspaces of certain standard ŝl(n)-modules, n 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W (k 1 Λ 1 + k 2 Λ 2 ) and W (k n 2 Λ n 2 + k n 1 Λ n 1 ). This generalizes earlier work by Calinescu on principal subspaces of standard sl(3)-modules, where similar assumptions were made. ii

3 Acknowledgements I would like to give my deepest thanks to my advisors, Yi-Zhi Huang and James Lepowsky, for their support, guidance, and encouragement during my time at Rutgers University. I thank them for all their time and patience, and for the inspiration they provided me with throughout the years. I would also like to express my thanks to people who have contributed to this thesis. I would like to thank Corina Calinescu for conversations which led to ideas found in Chapter 4 of this thesis, and to Francesco Fiordalisi and Shashank Kanade, for letting me bounce so many ideas off of them as this work was in progress. I thank James Lepowsky, Antun Milas, and Mirko Primc for suggested improvements to various parts of the papers whose results are included in this work. During my time at Rutgers, I was exposed to a wide range of topics in representation theory through the excellent special topics courses offered by the department. I would like to thank Lisa Carbone, Roe Goodman, Yi-Zhi Huang, James Lepowsky, and Siddartha Sahi for these courses. I would also like to thank William Cook, who first introduced me to Lie algebra theory and vertex operator algebra theory as an undergraduate, and who, along with Yi-Zhi Huang, gave me my start in mathematics research. I would like to acknowledge support from Yi-Zhi Huang and NSF grant PHY during the Summer of 2011, and from the National Science Foundation s East Asia and Pacific Summer Institute program and the Beijing International Center for Mathematical Research during the Summer of I would also like to thank my dissertation defense committee, consisting of Yi-Zhi Huang, James Lepowsky, Haisheng Li, and Antun Milas, for serving as committee members and for taking the time to review this work and provide comments. iii

4 Lastly, I would like to thank my family: my parents Henryk and Zofia, and my brother Thomas, for all their love, support, and encouragement. iv

5 Dedication To my parents. v

6 Table of Contents Abstract ii Acknowledgements iii Dedication v 1. Introduction Preliminaries Constructions in the sl(n + 1) case The case of g = sl(3) Principal subspaces of standard modules General definitions Details for the sl(3) case Presentations of the principal subspaces of the standard sl(3)-modules A proof of the presentations Presentations of principal subspaces of standard modules and a completion of U( n) A reformulation of the presentation problem Exact sequences and multigraded dimensions Exact sequences Multigraded dimensions Appendix vi

7 7.1. A completion of the universal enveloping algebra of certain nilpotent Lie algebras References vii

8 1 Chapter 1 Introduction There is a long-standing connection between the theories of vertex operators and vertex operator algebras ([B], [FLM], [LL], etc.) and affine Lie algebras (cf. [K]) on the one hand, and Rogers-Ramanujan-type combinatorial identities (cf. [A]) on the other hand ([LM], [LW1] [LW4], [LP1] [LP2], and many other references). In this introduction, we briefly sketch the results found in this thesis about several closely related problems. The first of these problems concerns finding presentations for principal subspaces of standard modules. In particular, we give new results concerning presentations of the principal subspaces of the standard sl(3)-modules. This set of results is the subject of Chapter 4 and can also be found in [S1]. The second of these problems concerns providing a natural setting for families of operators from which such presentations arise. Using a completion of certain universal enveloping algebras, we provide such a setting, and reformulate all known and conjectured presentations in this context. This is carried out in Chapter 5 and the Appendix. The third and final of these problems concerns finding families of exact sequences and q-difference equations. In Chapter 6, we derive such exact sequences, and obtain the multigraded dimensions of certain principal subspaces, which are related to the sum sides of Rogers-Ramanujan-type combinatorial identities. The results concerning the latter two problems can also be found in [S2]. We begin this introduction by first sketching a brief history of some of the connections between affine Lie algebras, vertex operator algebras, and Rogers-Ramanujan-type combinatorial identities. Many difference-two type partition conditions have been interpreted and obtained by the study of certain natural substructures of standard (i.e., integrable highest weight) modules for affine Lie algebras. In particular, in [FS1] [FS2], Feigin and Stoyanovsky,

9 2 motivated by the earlier work by Lepowsky and Primc [LP2], introduced the notion of principal subspace of a standard module for an affine Lie algebra, and in the case of A (1) 1 (= sl(2)) and A (1) 2 (= sl(3)) obtained, under certain assumptions (presentations for these principal subspaces in terms of generators and relations) the multigraded dimensions ( characters ) of the principal subspaces of the vacuum standard modules. Interestingly enough, these multigraded dimensions were related to the Rogers- Ramanujan partition identities, and more generally, the Gordon-Andrews identities, but in a different setting than the original vertex-algebraic interpretation of these identities in [LW2] [LW4]. A more general case was considered by Georgiev in [G], where combinatorial bases were constructed for the principal subspaces associated to certain standard A (1) n -modules. Using these bases, Georgiev obtained the multigraded dimensions of these principal subspaces. More recently, combinatorial bases have been constructed for principal subspaces in more general lattice cases ([P], [MiP]), for the principal subspaces of the vacuum standard modules for the affine Lie algebras B (1) 2 [Bu], for principal subspaces in the quantum sl(n + 1)-case [Ko], and for certain natural substructures of principal subspaces ([Pr], [J1] [J3], [T1] [T4], [Ba], [JPr]). In [CLM1] [CLM2], the authors addressed the problem of vertex-algebraically interpreting the classical Rogers-Ramanujan recursion and, more generally, the Rogers- Selberg recursions (cf. [A]) by using intertwining operators among modules for vertex operator algebras to construct exact sequences leading to these recursions. In particular, the solutions of these recursions gave the graded dimensions of the principal subspaces of the standard A (1) 1 -modules. In [CLM1] [CLM2] (as in [FS1] [FS2]), the authors assumed certain presentations for the principal subspaces of the standard sl(2)-modules (presentations that can be derived from [LP2]; the nontrivial part is the completeness of the relations). In [CalLM1] [CalLM2], the authors gave an a priori proof, again using intertwining operators, of the completeness of the presentations assumed in [FS1]-[FS2] and [CLM1] [CLM2]. These results were extended to the level 1 standard sl(n + 1)- modules by Calinescu in [C4], and later to the level 1 standard modules for the untwisted affine Lie algebras of type ADE in [CalLM3]. The desired presentations were proved, and exact sequences were obtained leading to recursions and the graded dimensions of

10 3 the principal subspaces of the level 1 standard modules. In [CalLM4], the authors have initiated the study of principal subspaces for standard modules for twisted affine Lie algebras, extending the past work of [CLM1] [CLM2], [CalLM1] [CalLM3] to the case of the level 1 standard module for the twisted affine Lie algebra A (2) 2. In the work [C3], Calinescu considered the principal subspaces of certain higher level standard sl(3)-modules. In this work, she conjecturally assumed presentations for certain principal subspaces, and using the theory of vertex operator algebras and intertwining operators, she constructed exact sequences among these principal subspaces. Using these exact sequences, along with the multigraded dimensions in [G], Calinescu was able to find the multigraded dimensions of principal subspaces which had not previously been studied. A different variant of principal subspace was considered in [AKS] and [FFJMM]. In [AKS], the authors cite well-known presentations for standard modules, and use these to provide (without proof) a set of defining relations for each principal subspace. In [FFJMM], in which the authors consider A (1) 2, they do indeed prove that certain relations form a set of defining relations for their variant of principal subspace. the case of the vacuum modules, the principal subspaces in [FFJMM] are essentially identical to the principal subspaces considered in the present work, and their defining In relations indeed agree with those in Chapter 4 of the present work. For the nonvacuum modules, the principal subspaces considered in the present work can be viewed as proper substructures of those considered in [FFJMM], and correspondingly, the defining relations we obtain are different. Our method for proving the completeness of our defining relations is completely different from the method in [FFJMM]. We now give a brief overview of the structure of this thesis. In Chapter 2, we recall certain vertex-algebraic constructions of standard sl(n + 1)-modules and of intertwining operators among these modules. In Chapter 3, we recall the notion of principal subspace of a standard module and prove certain useful properties of principal subspaces. We now very briefly recall some of these notions. Given a complex semisimple Lie algebra g, a fixed Cartan subalgebra h, a fixed set of positive roots +, and a root vector x α for each α +, consider the subalgebra n = α + Cx α g spanned by the positive

11 4 root vectors. The affinization n = n C[t, t 1 ] of n is a subalgebra of the affine Lie algebra ĝ = g C[t, t 1 ] Cc. Let L(Λ) be the standard module of ĝ with highest weight Λ and level k, a positive integer, and let v Λ L(Λ) be a highest weight vector. The principal subspace of L(Λ) is defined by W (Λ) = U( n) v Λ, (1.1) where U( ) is the universal enveloping algebra. We also have natural surjective maps f Λ : U( n) W (Λ). (1.2) By presentation of W (Λ), we mean a complete description of kerf Λ in terms of its generators. Chapter 4 in this thesis is another step forward in the spirit of [CalLM1]-[CalLM3]. We exploit intertwining operators among vertex operator algebra modules to solve the problem of giving an a priori proof of presentations for the principal subspaces of all the standard modules for A (1) 2 (= sl(3)), including those assumed conjecturally and used in [C3]. The methods used in the proof of these presentations are similar to those in [CalLM1] [CalLM3], in that certain minimal counterexamples are postulated and shown not to exist. However, in the general case, we needed to introduce certain new ideas to prove our presentations. We then proceed to formulate the presentations for principal subspaces of all the standard modules for A (1) n as a conjecture. In particular, we take g = sl(3). We precisely determine Kerf Λ in terms of certain natural left ideals of U( n). Specifically, in terms of the fundamental weights of A (1) 2, which we label Λ 0, Λ 1, and Λ 2, we may express Λ as Λ = k 0 Λ 0 + k 1 Λ 1 + k 2 Λ 2, for some nonnegative integers k 0, k 1, and k 2. We define an ideal I kλ0 in terms of left ideals generated by the coefficients of certain vertex operators associated with singular vectors in a natural way. This left ideal is then used to define a larger left ideal I Λ = I kλ0 + U( n)x α1 ( 1) k 0+k U( n)x α2 ( 1) k 0+k U( n)x α1 +α 2 ( 1) k 0+1,

12 5 where we use x(n) to denote the action of x t n ĝ for x g and n Z. We then proceed to show that Kerf Λ = I Λ. The proof of this result is similar in structure to the proof of the presentations in [CalLM2]. Considering all dominant integral weights together, we choose minimal counterexamples (certain elements in Kerf Λ \ I Λ ) and show that a contradiction is reached for each Λ. Certain maps used in [CalLM3] are also generalized and used in the proof, but these ideas do not extend to the most general case. We develop a method for reaching the desired contradictions for each Λ which rebuilds the minimal counterexample to show that it is in fact an element of I Λ. This rebuilding technique can also be used to show all of the presentations proved in the works [CalLM1]-[CalLM3] in the type A case with suitable modifications (see remarks at the end of Section 4). In [C3], certain of these presentations were conjectured and used to construct exact sequences among principal subspaces. Using these exact sequences, Calinescu obtained the previously unknown graded dimensions for principal subspaces whose highest weights are of the form k 1 Λ 1 + k 2 Λ 2, where k 1, k 2 are positive integers. The problem of constructing exact sequences for more general highest weights is still unsolved. Chapter 5 of this thesis, along with the Appendix, focuses on providing a more natural setting for the annihilating ideals which give presentations of the principal subspaces of the standard modules. In [CLM1] [CLM2] and [CalLM1] [CalLM3], the annihilator of the highest weight vector of each principal subspace is written in terms of certain elements of U(ĝ) which, when viewed as operators, annihilate the highest weight vector. An important set of these operators arises from certain null vector identities given by powers of vertex operators and are written as infinite formal sums of elements of U(ĝ) also viewed as operators. The ideals which annihilate the highest weight vectors can be expressed using operators defined by certain truncations of these formal sums, in order to view these operators as elements of U(ĝ). We provide the details of the construction of a completion of the universal enveloping algebra U( n) to give more natural presentations (without such truncations) for the defining annihilating

13 6 ideals of principal subspaces. This completion was discussed in [C1] [C2] and [CalLM3], but the details of this construction were not supplied. We prove various properties of this completion and the defining ideals for principal subspaces, including their more natural definition inside this completion. These completions may be generalized to the twisted setting used in [CalLM4] (as in [LW3], where similar completions were originally constructed in a general twisted or untwisted setting). Our main result in Chapter 6 is a natural generalization of [C3] to the case of sl(n + 1), n 2. Although our methods recover the same information as in [CLM1] [CLM2] when n = 1, we take n 2 for notational convenience. In the case where n = 2, we recover the results in [C3] with a slight variant of the methods. As in [C3], we conjecturally assume presentations for certain principal subspaces, and use these to provide exact sequences among principal subspaces of certain standard sl(n + 1)- modules. Using these exact sequences, along with the multigraded dimensions found in [G], we give previously unknown multigraded dimensions of principal subspaces. To state the main result of this chapter, we let Λ 0,..., Λ n denote the fundamental weights of sl(n + 1). The dominant integral weights Λ of sl(n + 1) are k 0 Λ k n Λ n for k 0,..., k n N, and we use L(Λ) to denote the standard module with highest weight Λ, W (Λ) to denote its principal subspace, and χ W (Λ) (x 1,..., x n, q) to denote its multigraded dimension. Our result states: Theorem Let k 1. For any i with 1 i n 1 and k i, k i+1 N such that k i + k i+1 = k, the sequences W (k i Λ i + k i+1 Λ i+1 ) φ i (1.3) W (k i Λ 0 + k i+1 Λ i ) 1 k i 1 Y c(e λ i,x) 1 k i+1 W ((k i 1)Λ 0 + (k i+1 + 1)Λ i ) 0 when k i 1, and W (k i Λ i + k i+1 Λ i+1 ) ψ i (1.4) W (k i+1 Λ 0 + k i Λ i+1 ) 1 k i+1 1 Y c(e λ i+1,x) 1 k i W ((k i+1 1)Λ 0 + (k i + 1)Λ i+1 ) 0

14 7 when k i+1 1, are exact. The maps φ i, ψ i, and Y c (e λ i, x) are maps naturally arising from the lattice construction of the level 1 standard modules and intertwining operators among these modules. As a consequence of this theorem, we obtain results about multigraded dimensions when the first map φ i or ψ i is injective, and we have the following theorem and its corollary: Theorem Let k 1. Let k 1, k 2, k n 1, k n N with k 1 1 and k n 1, such that k 1 + k 2 = k and k n 1 + k n = k. Then χ W (k 1 Λ 1 +k 2 Λ 2 ) (x 1,..., x n, q) = = x k 1 1 χ W ((k 1 1)Λ 0 +(k 2 +1)Λ 1 ) (x 1q 1, x 2 q, x 3..., x n, q) (1.5) x k 1 1 χ W (k 1 Λ 0 +k 2 Λ 1 ) (x 1q 1, x 2 q, x 3,..., x n, q) and χ W (k n 1 Λ n 1 +k nλ n) (x 1,..., x n, q) = = x kn n χ W ((k n 1)Λ 0 +(k n 1 +1)Λ n) (x 1,..., x n 1 q, x n q 1, q) (1.6) xn kn χ W (k nλ 0 +k n 1 Λ (x n) 1,..., x n 1 q, x n q 1, q). Theorem immediately gives us: Corollary In the setting of Theorem 1.0.2, we have that χ W (k 1 Λ 1 +k 2 Λ 2 ) (x 1,..., x n, q) = and = ( q r ( q r (1) n (1) 1 2 (k) +...+r 1 ( q r (1) k t=k 1 +1 r(t) 1 + k t=1 r(t) 2 r(t) 1 (1 q r(k 1 ) (q) (1) r... (q) 1 r(2) (k 1) 1 r 2 (k) +...+r 2 2 r (1) 2 r(1) (q) (1) r... (q) 2 r(2) (k 1) 2 r 1 r (k) 1 (q) r (k) r(k) 2 r(k) 1 2 r (k) 2 2 (k) 2 (1) +...+r n r n r (1) n 1... r(k) n r (k) n 1 (q) (1) r n r n (2)... (q) r (k 1) n r n (k) (q) r (k) n ) (q) r (k) 2 ) x k 1+ k i=1 r(i) 1 1 ) 1 x ) n i=1 r(i) n n χ W (k n 1 Λ n 1 +k nλ n) (x 1,..., x n, q) =

15 8 = ( q r(1) 1 (q) (1) r 1 r(2) 1 2 (k) r 1... (q) r (k 1) ( q r (1) n q k t=1 r(t) 1 r (k) 1 (q) r (k) 1 )( q r (1) 2 2 (k) +...+r 2 2 r (1) 2 r(1) (q) (1) r... (q) 2 r(2) (k 1) 2 r 2 (k) 2 (1) +...+r n r n r (1) n 1... r(k) n r (k) n 1 + k t=kn+1 r(t) n n 1 r(t) n (q) (1) r n r n (2) (1 q r(kn)... (q) r (k 1) n k n i=1 )x r(i) 1 r n (k) (q) r (k) n 1 x kn+ n n 1... r(k) 2 r(k) 1 2 r (k) 2 i=1 r(i) n ) (q) r (k) 2 ) where the sums are taken over decreasing sequences r (1) j r (2) j each j = 1,..., n. r (k) j 0 for The expressions in Corollary can also be written as follows: As in [G], for s = 1,..., k 1 and i = 1,..., n, set p (s) i = r (s) i r (s+1) i, and set p (k) i = r (k) i. Also, let (A lm ) n l,m=1 be the Cartan matrix of sl(n + 1) and Bst := min{s, t}, 1 s, t k. Then, χ W (k 1 Λ 1 +k 2 Λ 2 ) (x 1,..., x n, q) = q 1 s,t=1,...,l 2 l,m=1,...,n A lmb st p (s) l p (t) m n k i=1 s=1 (q) p (s) i q p 1 q k t=1 p(t) 2 + +p(k) 2 p(t) 1 p(k) 1 p (1) 1,...,p(k) 1 0 p (1) n.,...,p n (k) 0 (1 q p(k 1 ) 1 + +p (k) 1 )x k 1 1 where p 1 = p (k 1+1) 1 + 2p (k 1+2) k 2 p (k) 1 and n i=1 k s=1 x sp(s) i i χ W (k n 1 Λ n 1 +k nλ n) (x 1,..., x n, q) = q 1 s,t=1,...,l 2 l,m=1,...,n A lmb st p (s) l p (t) m n k i=1 s=1 (q) p (s) i q pn q k t=1 p(t) n 1 + +p(k) n 1 p(t) n p (k) n p (1) 1,...,p(k) 1 0 p (1) n.,...,p n (k) 0 where p n = p (kn+1) n (1 q p(kn) n + 2p (kn+2) n + +p (k) n )x kn n + + k n 1 p (k) 1. n i=1 k s=1 x sp(s) i i

16 9 Similar multigraded dimensions for different variants of principal subspaces have been studied in [AKS] and [FFJMM]. Modularity properties of certain multigraded dimensions, in the context of principal subspaces of standard modules, have been studied in [St], [WZ], and more recently in [BCFK].

17 10 Chapter 2 Preliminaries 2.1 Constructions in the sl(n + 1) case We begin by recalling certain vertex-algebraic constructions for the untwisted affine Lie algebra [LL]. sl(n + 1), n a positive integer. We shall be working in the setting of [FLM] and Fix a Cartan subalgebra h of sl(n + 1). Also fix a set of roots, a set of simple roots {α 1,..., α n }, and a set of positive roots +. Let, denote the Killing form, rescaled so that α, α = 2 for each α. Using this form, we identify h with h. Let λ 1,..., λ n h h denote the fundamental weights of sl(n + 1). Recall that λ i, α j = δ ij for each i, j = 1,..., n. Denote by Q = n i=1 Zα i and P = n i=1 Zλ i the root lattice and weight lattice of sl(n + 1), respectively. For each root α, we have a root vector x α sl(n + 1) (recall that [h, x α ] = α, h x α for each h h). We define a nilpotent subalgebra of sl(n + 1). n = α + Cx α, We have the corresponding untwisted affine Lie algebra given by sl(n + 1) = sl(n + 1) C[t, t 1 ] Cc, where c is a non-zero central element and [x t m, y t p ] = [x, y] t m+p + m x, y δ m+p,0 c for any x, y sl(n + 1) and m, p Z. If we adjoin the degree operator d, where [d, x t m ] = mx t m

18 11 [d, c] = 0, we obtain the affine Kac-Moody Lie algebra define two important subalgebras of sl(n + 1): sl(n + 1) = sl(n + 1) Cd (cf. [K]). We ĥ = h C[t, t 1 ] Cc and the Heisenberg subalgebra ĥ Z = h t m Cc m Z\{0} (in the notation of [FLM], [LL]). We extend our form, to h Cc Cd by defining c, c = 0 d, d = 0 c, d = 1. Using this form, we may identify h Cc Cd with (h Cc Cd). The simple roots of sl(n + 1) are α 0, α 1,..., α n and the fundamental weights of sl(n + 1) are Λ0, Λ 1,..., Λ n, given by α 0 = c (α 1 + α α n ) and Λ 0 = d, Λ i = Λ 0 + λ i for each i = 1,..., n. An sl(n + 1)-module V is said to have level k C if the central element c acts as multiplication by k (i.e. c v = kv for all v V ). Any standard (i.e. irreducible integrable highest weight) module L(Λ) with Λ (h Cc Cd) has nonnegative integral level, given by Λ, c (cf. [K]). Let L(Λ 0 ), L(Λ 1 ),..., L(Λ n ) denote the standard sl(n + 1)-modules of level 1 with v Λ0, v Λ1,..., v Λn as highest weight vectors, respectively. Continuing to work in the setting of [FLM] and [LL], we now recall the lattice vertex operator construction of the level 1 standard modules for sl(n + 1). We use U( )

19 12 to denote the universal enveloping algebra. The induced module M(1) = U(ĥ) U(h C[t] Cc) C has a natural ĥ-module structure, where h C[t] acts trivially and c acts as identity on the one-dimensional module C. Let s = 2(n + 1) 2. We fix a primitive s th root of unity ν s, and a central extension P of the weight lattice P by the finite cyclic group κ = κ κ s = 1 of order s, 1 κ P P 1 with associated commutator map c 0 : P P Z/sZ, defined by aba 1 b 1 = κ c 0(ā, b) for a, b P. Let c : P P C denote the alternating Z-bilinear map defined by c(λ, µ) = ν c 0(λ,µ) s for λ, µ P. We require that c(α, β) = ( 1) α,β for α, β Q. Such a central extension P of P does indeed exist (see Remark in [LL]). We define the faithful character χ : κ C by χ(κ) = ν s. Let C χ be the one dimensional κ -module, where the action of κ is given by κ 1 = ν s, and form the induced P -module C{P } = C[ P ] C[ κ ] C χ. For any subset E P, we define Ê = {a P ā E}, and we form C{E} in the obvious way. Then, the space V Q = M(1) C{Q} carries a natural vertex operator algebra structure, with 1 as vacuum vector, and the space V P = M(1) C{P } is naturally a V Q -module. We now recall some important details of this construction (cf. [LL]).

20 13 Choose a section e : P P (2.1) α e α, (i.e. a map which satisfies e = 1) such that e 0 = 1. Let ɛ 0 : P P Z/sZ the corresponding 2-cocycle, defined by the condition e α e β = κ ɛ 0(α,β) e α+β for α, β P and define the map ɛ : P P C by ɛ(α, β) = ν ɛ 0(α,β) s For any α, β P we have ɛ(α, β)/ɛ(β, α) = c(α, β) (2.2) and ɛ(α, 0) = ɛ(0, α) = 1. (2.3) We use this choice of section (2.1) identify C{P } and the group algebra C[P ]. In particular, we have a vector space isomorphism given by C[P ] C{P } (2.4) e α ι(e α ) for α P, where, for a P, we set ι(a) = a 1 C{P }. By restriction, we also have the identification C[Q] C{Q}. There is a natural action P on C[P ] given by e α e β = ɛ(α, β)e α+β, κ e β = ν s e β for α, β P. As operators on C[P ] C{P } we have e α e β = ɛ(α, β)e α+β. (2.5) We make the identifications V P = M(1) C[P ], V Q = M(1) C[Q] and we set V Q e λ i = M(1) C[Q]e λ i, i = 1,..., n

21 14 Given a Lie algebra element a t m denote its action on an sl(n + 1), where a sl(n + 1), m Z, we will sl(n + 1)-module using the notation a(m). In particular, for h h and m Z, we have the operators h(m) on V P : h(0)(v ι(e α )) = h, α (v ι(e α )) h(m)(v ι(e α )) = (h(m)v ι(e α )). For a formal variable x and λ P, we define the operator x λ by x λ (v ι(e µ )) = x λ,µ (v ι(e µ )) for v M(1) and µ P. For each λ P, we define the vertex operators Y (ι(e λ ), x) = E ( λ, x)e + ( λ, x)e λ x λ, (2.6) where E ± ( λ, x) = exp ( ±n>0 ) λ(n) n x n (End V P )[[x, x 1 ]] Using the identification (2.4) we write Y (e λ, x) for Y (ι(e λ ), x). In particular, for any root α we have the operators x α (m) defined by Y (e α, x) = m Z x α (m)x m 1. (2.7) It is easy to see that and x λ e µ = x λ,µ e µ x λ (2.8) λ(m)e µ = e µ λ(m) (2.9) for all λ, µ P and m Z. Using (2.2), (2.5) and (2.6)-(2.9) we obtain, for α, µ P, x α (m)e µ = c(α, µ)e µ x α (m + α, µ ). (2.10) Along with the action of ĥ, the operators x α (m), m Z, give V P a structure. In particular, we have that sl(n + 1)-module V P = V Q V Q e λ 1 V Q e λn

22 15 and that V Q, V Q e λ 1,..., V Q e λn are the level 1 basic representations of sl(n + 1) with highest weights Λ 0, Λ 1,..., Λ n and highest weight vectors v Λ0 = 1 1, v Λ1 = 1 e λ 1,..., v Λn = 1 e λn, respectively. We make the identifications L(Λ 0 ) = V Q for each i = 1,..., n. Moreover, taking L(Λ i ) = V Q e λ i n ω = 1 2 i=1 u (i) ( 1) 2 v Λ0 to be the standard conformal vector, where {u (1),..., u (n) } is an orthonormal basis of h, the operators L(m) defined by Y (ω, x) = m Z L(m)x m 2 (2.11) provide a representation of the Virasoro algebra of central charge n. The vertex operators (2.6) and (2.22) give L(Λ 0 ) the structure of a vertex operator algebra whose irreducible modules are precisely L(Λ 0 ),L(Λ 1 ),..., L(Λ n ). We shall write v Λ0 = 1, v Λ1 = e λ 1,..., v Λn = e λn. (2.12) As in [G], [CLM1] [CLM2], [C3] [C4], and [CalLM1] [CalLM3], we need certain intertwining operators among standard modules. We recall some facts from [FHL] and [DL] about intertwining operators and, in particular, the intertwining operators between L(Λ 0 ), L(Λ 1 ),..., L(Λ n ). Given modules W 1, W 2 and W 3 for the vertex operator algebra V, an intertwining operator of type is a linear map W 3 W 1 W 2 Y(, x) : W 1 Hom(W 2, W 3 ){x} w Y(w, x) = n Q w n x n 1

23 16 such that all the axioms of vertex operator algebra which make sense hold (see [FHL]). The main axiom is the Jacobi identity: ( ) x 1 0 δ x1 x 2 x 0 Y (u, x 1 )Y(w (1), x 2 )w (2) x 1 0 δ ( x2 x 1 x 0 = x 1 2 δ ( x1 x 0 x 2 ) Y(w (1), x 2 )Y (u, x 1 )w (2) ) Y(Y (u, x 0 )w (1), x 2 )w (2) for u V, w (1) W 1 and w (2) W 2. Define the operators e iπλ and c(, λ) on V P by: e iπλ (v e β ) = e iπ λ,β v e β, for v M(1) and β, λ P. We have that c(, λ)(v e β ) = c(β, λ)v e β, Y(, x) : L(Λ r ) Hom(L(Λ s ), L(Λ p )){x} (2.13) w Y(w, x) = Y (w, x)e iπλr c(, λ r ) defines an intertwining operator of type L(Λ p ) L(Λ r ) L(Λ s ) (2.14) if and only if p r + s mod (n + 1) (cf. [DL]). If we take u = e α and w 1 = e λr apply Res x0 (for r = 1,..., n) in the Jacobi identity (2.13) and (the formal residue operator, giving us the coefficient of x 1 0 ), we have [Y (e α, x 1 ), Y(e λr, x 2 )] = 0, (2.15) whenever α +, which means that each coefficient of the series Y(e λr, x) commutes with the action of x α (m) for positive roots α. Given such an intertwining operator, we define a map Y c (e λr, x) : L(Λ s ) L(Λ p )

24 17 by Y c (e λr, x) = Res x x 1 λr,λs Y(e λr, x) and by (2.15) we have [Y (e α, x 1 ), Y c (e λr, x 2 )] = 0, (2.16) which implies [x α (m), Y c (e λr, x 2 )] = 0 (2.17) for each m Z. Consider the space V k P = V P V }{{ P. (2.18) } k times We extend the operators e λ, λ P, to operators on Vp k, k a positive integer, by defining: For any standard Λ is of the form e k λ = e λ e λ : V k P V k P. sl(n + 1)-module L(Λ) of positive integral level k, its highest weight Λ = k 0 Λ k n Λ n for some nonnegative integers k 0,..., k n satisfying k k n = k. Any standard sl(n + 1)-module L(Λ) of positive integral level k, may be realized as an submodule of V k P. Indeed, let sl(n + 1)- v i1,...,i k = v Λi1 v Λik V k P, (2.19) where exactly k i indices are equal to i for each i = 0,..., n. Then, we have that v i1,...,i k is a highest weight vector for (cf. [K]). Here, the action of sl(n + 1), and a Lie algebra on a tensor product of modules: L(Λ) U( sl(n + 1)) vi1,...,i k V k P (2.20) sl(n + 1) on V k P is given by the usual diagonal action of a v = (a)v = (a a)v (2.21)

25 18 for a sl(n + 1), v V k P and is extended to U( sl(n + 1)) in the usual way. also have a natural vertex operator algebra structure on L(kΛ 0 ) and L(kΛ 0 )-module structure on L(Λ) given by: We Theorem ([FZ], [DL], [Li1]; cf. [LL]) The standard module L(kΛ 0 ) has a natural vertex operator algebra structure. The level k standard a complete list of irreducible L(kΛ 0 )-modules. sl(n + 1)-modules provide Let ω denote the Virasoro vector in L(kΛ 0 ). We have a natural representation of the Virasoro algebra on each L(Λ) given by Y L(Λ) (ω, x) = m Z L(m)x m 2 (2.22) The operators L(0) defined in (2.22) provide each L(Λ) of level k with a grading, which we refer to as the weight grading: L(Λ) = s Z L(Λ) (s+hλ ) (2.23) where h Λ Q and depends on Λ. In particular, we have the grading L(kΛ 0 ) = s Z L(Λ) (s). (2.24) We denote the weight of an element a v Λ W (Λ) by wt(a v Λ ). We will also write wt(x α (m)) = m, where we view x α (m) both as an operator and as an element of U( n). We also have n distinct charge gradings on each L(Λ) of level k, given by the eigenvalues of the operators λ i (0) for i = 1,..., n: L(Λ) = L(Λ) [ri + λ i,λ ]. (2.25) r i Z We call these the λ i -charge gradings. An element of L(Λ) with λ i -charges n i for i = 1,..., n has total charge n i=1 n i. The gradings (2.23) and (2.25) are compatible, and we have that L(Λ) = r 1,...,r n,s Z L(Λ) r1 + λ 1,Λ,...,r n+ λ nλ ;s+h Λ. (2.26)

26 The case of g = sl(3) In chapter 4 we work in the case where n = 2, and we recall some important details. The finite-dimensional simple Lie algebra sl(3) has a standard basis {h α1, h α2, x ±α1, x ±α2, x ±(α1 +α 2 )}; we do not need to normalize the root vectors. We fix the Cartan subalgebra h = Ch α1 Ch α2 of sl(3). Under our identification of h with h, we have α 1 = h α1 and α 2 = h α2. We also have the fundamental weights λ 1, λ 2 h of sl(3), given by the condition λ i, α j = δ i,j for i, j = 1, 2. In particular, we have and λ 1 = 2 3 α α 2 and λ 2 = 1 3 α α 2 α 1 = 2λ 1 λ 2 and α 2 = λ 1 + 2λ 2. The level 1 standard modules of sl(3) are L(Λ0 ), L(Λ 1 ), and L(Λ 2 ). Given the intertwining operators (2.13), we have that Y c (e λ i, x)v Λ0 = r 1 v Λi (2.27) Y c (e λ i, x)v Λi = r 2 x αi ( 1) v Λj = r 2e λi v Λi (2.28) Y c (e λ i, x)v Λj = r 3 x α1 +α 2 ( 1) v Λ0 = r 3e λi v Λj (2.29) for i, j = 1, 2, i j and some constants r 1, r 2, r 3, r 2, r 3 C. For any level k standard sl(3)-module L(Λ), its highest weight Λ is of the form Λ = k 0 Λ 0 + k 1 Λ 1 + k 2 Λ 2 for some nonnegative integers k 0, k 1, k 2 satisfying k 0 + k 1 + k 2 = k. We now give a realization of these modules. Consider the space V k P = V P V }{{ P, (2.30) } k times

27 20 and let v i1,...,i k = v Λi1 v Λik V k P, (2.31) where exactly k 0 indices are equal to 0, k 1 indices are equal to 1 and k 2 indices are equal to 2. Then, we have that v i1,...,i k (cf. [K]). is a highest weight vector for sl(3), and L(Λ) U( sl(3)) v i1,...,i k V k P (2.32) The operators L(0) defined in (2.22) provide each L(Λ) of level k with a grading, which we refer to as the weight grading: L(Λ) = s Z L(Λ) (s+hλ ) (2.33) where In particular, we have the grading h Λ = Λ, Λ + α 1 + α 2. 2(k + 3) L(kΛ 0 ) = s Z L(Λ) (s). (2.34) We denote the weight of an element a v Λ W (Λ) by wt(a v Λ ). We will also write wt(x α (m)) = m, where we view x α (m) both as an operator and as an element of U( n). We also have two different charge gradings on each L(Λ) of level k, given by the eigenvalues of the operators λ 1 (0) and λ 2 (0): L(Λ) = L(Λ) [ri + λ i,λ ] (2.35) r i Z for each i = 1, 2. We call these the λ 1 -charge and λ 2 -charge gradings, respectively. An element of L(Λ) with λ 1 -charge n 1 and λ 2 -charge n 2 is said to have total charge n 1 +n 2. The gradings (2.23) and (2.25) are compatible, and we have that L(Λ) = L(Λ) r1 + λ 1,Λ,r 2 + λ 2 Λ ;s+h Λ. (2.36) r 1,r 2,s Z

28 21 Chapter 3 Principal subspaces of standard modules 3.1 General definitions We are now ready to define our main object of study. Consider the sl(n + 1)-subalgebra n = n C[t, t 1 ]. (3.1) The Lie algebra n has the following important subalgebras: n = n t 1 C[t 1 ] and n + = n C[t] Let U( n) be the universal enveloping algebra of n. We recall that U( n) has the decomposition: Given a U( n) = U( n ) U( n) n +. (3.2) sl(n + 1)-module L(Λ) of positive integral level k with highest weight vector v Λ, the principal subspace of L(Λ) is defined by: W (Λ) = U( n) v Λ. W (Λ) inherits the grading (2.36), and we have that W (Λ) = W (Λ) r1 + λ 1,Λ,...,r n+ λ nλ ;s+h Λ (3.3) r 1,...,r n,s Z For convenience, we will use the notation W (Λ) r 1,...,r n;s = W (Λ) r1 + λ 1,Λ,...,r n+ λ nλ ;s+h Λ

29 22 As in [CLM1]-[CLM2], [CalLM3], [C1]-[C2], define the multigraded dimension of W (Λ) by: and its modification χ W (Λ) (x 1,..., x n, q) = tr W (Λ) x λ 1 1 xλn n q L(0) χ W (Λ) (x 1,..., x n, q) = x λ 1,Λ... x λn,λ q h Λ χ W (Λ) (x 1,..., x n, q) C[[x 1,... x n, q]] In particular, we have that χ W (Λ) (x 1,..., x n, q) = dim(w (Λ) r 1,...,r n;s)x r1 x rn q s. r 1,...,r n,s N For each such Λ, we have a surjective map F Λ : U(ĝ) L(Λ) (3.4) a a v Λ and its surjective restriction f Λ : f Λ : U( n) W (Λ) (3.5) a a v Λ. A precise description of the kernels Kerf Λ for every each Λ = n i=0 k iλ i gives a presentation of the principal subspaces W (Λ) for sl(n + 1), as we will now discuss. For each λ P and character ν : Q C, we define a map τ λ,ν on n by τ λ,ν (x α (m)) = ν(α)x α (m λ, α ) for α + and m Z. It is easy to see that τ λ,ν is an automorphism of n. In the special case when ν is trivial (i.e., ν = 1), we set τ λ = τ λ,1. The map τ λ,ν extends canonically to an automorphism of U( n), also denoted by τ λ,ν, given by τ λ,ν (x β1 (m 1 ) x βr (m r )) = ν(β 1 + +β r )x β1 (m 1 λ, β 1 ) x βr (m r λ, β r ) (3.6)

30 23 for β 1,..., β r + and m 1,..., m r Z. Notice that if λ = λ i for i = 1,..., n, we have that wt(τ λ (a)) wt(a) for each a U( n). We will use this fact frequently without mention. Define the formal sums Rt i = x αi (m 1 )x αi (m 2 ) x αi (m k+1 ) (3.7) and their truncations m 1 + +m n= t R i M,t = m m k+1 = t, m 1,..., m k+1 M x αi (m 1 ) x αi (m k+1 ) (3.8) for t Z, M Z and i = 1,..., n. Note that each R i M,t U( n) and the infinite sum R i t / U( n), but R i t is still well-defined as an operator on W (Λ), since, when acting on any element of W (Λ), only finitely many of its terms are nonzero. Let J be the left ideal of U( n) generated by the elements R i 1,t for t k + 1 and i = 1, 2: J = n U( n)r 1,t. i (3.9) i=1 t k+1 Define a left ideal of U( n) by: I kλ0 = J + U( n) n + and for each Λ = n i=0 k iλ i, define I Λ = I kλ0 + α + U( n)x α ( 1) k+1 α,λ. Conjecture For each Λ = k 0 Λ k n Λ n with k 0,..., k n, k N, k 1, and k k n = k, we have that In particular, Kerf Λ = I Λ Kerf k0 Λ 0 +k i Λ i = I kλ0 + U( n)x αi ( 1) k 0+1 (3.10)

31 24 In the case that g is of type ADE and k = 1 or g = sl(2) or g = sl(3) and k 1, this conjecture has been proved. The presentations (3.10) are suggested by the bases found in [G], but an a priori proof is lacking. This proof will be the focus of future work. 3.2 Details for the sl(3) case We define certain operators that will be needed for the proof of Conjecture when n = 2. These operators have natural generalization for n 2 and generalize the τ λ,ν maps above. Define the injective maps τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν : U( n) U( n) (3.11) a τ λ1,ν(a)x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2. and τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 2,ν : U( n) U( n) (3.12) a τ λ2,ν(a)x α2 ( 1) k 2 x α1 +α 2 ( 1) k 1. Let ω i = α i λ i P for i = 1, 2. Generalizing the idea of [CalLM3], we define, for each character ν : Q C, injective linear maps σ k 1Λ 1 +k 2 Λ 2 ω 1,ν : U( n) U( n) (3.13) a τ ω1,ν(a)x α1 ( 1) k 1. and σ k 1Λ 1 +k 2 Λ 2 ω 2,ν : U( n) U( n) (3.14) a τ ω2,ν(a)x α2 ( 1) k 2. The following facts about U( n) will be useful: Lemma Given r, k N and root vectors x α, x β sl(3) with α, β, α + β +

32 25 and [x α, x β ] = C α,β x α+β for some constant C α,β C, we have x β (m 1 )... x β (m r )x α ( 1) k k r = x α ( 1) k p p=0 j 1,,j p=1 j 1 < <j p C j1,...,j p x β (m 1 ) (3.15) x α+β (m j1 1)... x α+β (m jp 1) x β (m r ) for some constants C j1,...j p C. The constants C j1,...j p are understood to be 0 when p > r. Proof: We induct on k N. For k = 1 we have: x β (m 1 ) x β (m r )x α ( 1) = x α ( 1)x β (m 1 ) x β (m r ) (3.16) r + C β,α x β (m 1 ) x α+β (m j 1) x β (m r ) (3.17) j=1 and so our claim is true for k = 1. Assume that our claim is true for some k 1. Then

33 26 we have: x β (m 1 ) x β (m r )x α ( 1) k+1 k r = x α ( 1) k p = + p=0 k x α ( 1) k p+1 p=0 k x α ( 1) k p p=0 j 1,...,j p=1 j 1 < <j p r j 1,...,j p=1 j 1 < <j p ( C j1,...,j p x β (m 1 ) x α+β (m j1 1) ) x α+β (m jp 1) x β (m r )x α ( 1) ( C j1,...,j p x β (m 1 ) )... x α+β (m j1 1) x α+β (m jp 1) x β (m r ) r s j q,s=1 q=1,...,p r j 1,...,j p=1 j 1 < <j p ( C j1,...,j p C β,α x β (m 1 ) x α+β (m j1 1)... x α+β (m s 1) )... x α+β (m jp 1) x β (m r ) = k+1 x α ( 1) k+1 p p=0 r j 1,...,j p=1 j 1 < <j p C j 1,...,j p x β (m 1 ) x α+β (m j1 1) x α+β (m jp 1) x β (m r ) for some constants C j 1,...,j p C, concluding our proof. Corollary For 0 m k and simple roots α i, α j + such that α i +α j +, we have R 1,tx i αj ( 1) m = x αj ( 1) m R 1,t i + r 1 x αj ( 1) m 1 [R 1,t+1, i x αj (0)] r m [... [R 1,t+m, i x αj (0)],..., x αj (0)] + bx αi +α j ( 1) + c for some r 1... r m C, b U( n), and c U( n) n +. In particular, we have that R 1,tx i αj ( 1) m I kλ0 + U( n)x αi +α j ( 1). Moreover, if a I kλ0 then ax αj ( 1) m I kλ0 + U( n)x αi +α j ( 1).

34 27 The next two lemmas show that the maps τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ i,ν and σ k 1Λ 1 +k 2 Λ 2 ω i,ν, i = 1, 2, allow us to move between the left ideals we have defined. Lemma For every character ν, we have that and τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν (I k0 Λ 0 +k 1 Λ 1 +k 2 Λ 2 ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2 τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 2,ν (I k0 Λ 0 +k 1 Λ 1 +k 2 Λ 2 ) I k1 Λ 0 +k 2 Λ 1 +k 0 Λ 2. Proof: We prove the claim for τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν. The claim for τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 2,ν similarly. Since I k0 Λ 0 +k 1 Λ 1 +k 2 Λ 2 for ν = 1. We have that follows is a homogeneous ideal, it suffices to prove our claim τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (R 1,t) 1 ( = τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 = = m 1 + +m k+1 = t, m i 1 m 1 + +m k+1 = t, m i 1 m m k+1 = t, m i 1 ) x α1 (m 1 ) x α1 (m k+1 ) ) τ λ1 (x α1 (m 1 ) x α1 (m k+1 ) x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 x α1 (m 1 1) x α1 (m k+1 1)x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 R 1 1,t+(k+1) + ax α 1 ( 1) k 1+1 x α1 +α 2 ( 1) k 2 = x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 R 1 1,t+(k+1) + b[x α 2 (0),... [x α2 (0), x α1 ( 1) k 1+k 2 +1 ]... ] for some a, b U( n). Clearly x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 ( 1)R 1 1,t+(k+1) +s[x α2 (0),... [x α2 (0), x α1 ( 1) k 1+k 2 +1 ]... ] I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2 and so τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (R 1 1,t) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2.

35 28 We also have τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (R 1,t) 2 ( = τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 = = m 1 + +m k+1 = t, m i 1 m 1 + +m k+1 = t, m i 1 = R 2 1,tx α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = a + bx α1 +α 2 ( 1) k 2+1 m m k+1 = t, m i 1 ) x α2 (m 1 )...x α2 (m k+1 ) ) τ λ1 (x α2 (m 1 ) x α2 (m k+1 ) x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 x α2 (m 1 ) x α2 (m k+1 )x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 for some a I kλ0 and b U( n), with the last equality following from Corollary So we have that τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν (R 2 1,t) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. Since J is the left ideal of U( n) generated by R 1,t 1 and R2 1,t, we have that τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (J) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. We now show that τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (U( n) n + ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. If m N, we have have that τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α1 (m)) = x α1 (m 1)x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 x α1 (m 1) U( n) n + if m > 0 and τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α1 (m)) = x α1 ( 1) k 1+1 x α1 +α 2 ( 1) k 2 = r[x α2 (0),..., [x α2 (0), x α1 ( 1) k 1+k 2 +1 ]... ] I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2 for some r C if m = 0. We also have τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α2 (m)) = x α2 (m)x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 x α2 (m) +rx α1 ( 1) k 1 1 x α1 +α 2 ( 1) k 2 x α1 +α 2 (m 1)

36 29 for some r C and so Finally, we have that, for m 0, τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α2 (m)) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α1 +α 2 (m)) = x α1 +α 2 (m 1)x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 x α1 +α 2 (m 1) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. Since U( n) n + is a left ideal of U( n), we have that and so we have τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν (U( n) n + ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2 τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν (I kλ0 ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. We now check the remaining terms. We have τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α1 ( 1) k 0+k 2 +1 ) = x α1 ( 2) k 0+k 2 +1 x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = cx α1 +α 2 ( 1) k 2 R 1 1,2(k 0 +k 2 +1)+k 1 + a 1 x α1 ( 1) k 1+1 x α1 +α 2 ( 1) k 2 = cx α1 +α 2 ( 1) k 2 R 1 1,2(k 0 +k 2 +1)+k 1 + a 2 [x α2 (0),... [x α2 (0), x α1 ( 1) k 1+k 2 +1 ] I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2 for some c, a 1, a 2 U( n). So, since U( n)x α1 ( 1) k 0+k 2 +1 is the left ideal of U( n) generated by x α1 ( 1) k 0+k 2 +1, we have that τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,ν (U( n)x α ( 1) k 0+k 2 +1 ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2.

37 30 By Lemma we have τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α2 ( 1) k 0+k 1 +1 ) = x α2 ( 1) k 0+k 1 +1 x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = x α1 ( 1) k 1 x α2 ( 1) k 0+k 1 +1 x α1 +α 2 ( 1) k 2 +r 1 x α1 ( 1) k 1 1 x α1 +α 2 ( 2)x α2 ( 1) k 0+k 1 x α1 +α 2 ( 1) k r k1 x α1 +α 2 ( 2) k 1 x α2 ( 1) k 0+1 x α1 +α 2 ( 1) k 2 = r 0x α1 ( 1) k 1 [x α1 (0),... [x α1 (0), R 2 1,k+1 ]... ] +r 1x α1 ( 1) k 1 1 [x α1 (0),... [x α1 (0), R 2 1,k+2 ]... ] +... r k 1 [x α1 (0),... [x α1 (0), R 1,2k1 +k 0 +k 2 +1]... ] + ax α1 +α 2 ( 1) k 2+1 I k2 Λ 0 +k 0 Λ 1 k 1 Λ 2 for some a U( n) and r 0, r 1, r 1,..., r k 1, r k 1 C. So, since U( n)x α2 ( 1) k 0+k 1 +1 is the left ideal of U( n) generated by x α2 ( 1) k 0+k 1 +1, we have that Finally, we have that τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (U( n)x α2 ( 1) k 0+k 1 +1 ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (x α1 +α 2 ( 1) k 0+1 ) = x α1 +α 2 ( 2) k 0+1 x α1 ( 1) k 1 x α1 +α 2 ( 1) k 2 = r[x α2 (0),... [x α2 (0), R 1 1,2k 0 +2+k 1 +k 2 ],... ] + ax α1 +α 2 ( 1) k 2+1 for some a U( n) and some constant r C. So, since U( n)x α1 +α 2 ( 1) k 0+1 is the left ideal of U( n) generated by x α1 +α 2 ( 1) k 0+1, we have that This concludes our proof. τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1 (U( n)x α1 +α 2 ( 1) k 01 ) I k2 Λ 0 +k 0 Λ 1 +k 1 Λ 2. Lemma For every character ν, we have that and σ k 1Λ 1 +k 2 Λ 2 ω 1,ν (I k1 Λ 1 +k 2 Λ 2 ) I k1 Λ 0 +k 2 Λ 1 σ k 1Λ 1 +k 2 Λ 2 ω 2,ν (I k1 Λ 1 +k 2 Λ 2 ) I k2 Λ 0 +k 1 Λ 2.

38 31 Proof: We prove the claim for σ k 1Λ 1 +k 2 Λ 2 ω 1,ν. The claim for σ k 1Λ 1 +k 2 Λ 2 ω 2,ν Since I k1 Λ 1 +k 2 Λ 2 have that follows similarly. is a homogeneous ideal, it suffices to prove our claim for ν = 1. We σ k 1Λ 1 +k 2 Λ 2 ω 1 (R 1,t) 1 ( = σ k 1Λ 1 +k 2 Λ 2 ω 1 = = m 1 + +m k+1 = t, m i 1 m 1 + +m k+1 = t,m i 1 m 1 + +m k+1 = t, m i 1 = R 1 1,t+(k+1) + ax α 1 ( 1) k 1+1 ) x α1 (m 1 ) x α1 (m k+1 ) ) σ ω1 (x α1 (m 1 ) x α1 (m k+1 ) x α1 ( 1) k 1 x α1 (m 1 1) x α1 (m k+1 1)x α1 ( 1) k 1 for some a U( n) and so σ k 1Λ 1 +k 2 Λ 2 ω 1 (R 1,t 1 ) I k 1 Λ 0 +k 2 Λ 1. We also have, by Lemma 3.2.1, that σ k 1Λ 1 +k 2 Λ 2 ω 1 (R 1,t) 2 ( = σ k 1Λ 1 +k 2 Λ 2 ω 1 = = = m 1 + +m k+1 = t, m i 1 m 1 + +m k+1 = t, m i 1 m 1 + +m k+1 = t, m i 1 m 1 + +m k+1 = t, m i 1 p=0 ) x α2 (m 1 ) x α2 (m k+1 ) ) σ ω1 (x α2 (m 1 ) x α2 (m k+1 ) x α1 ( 1) k 1 x α2 (m 1 + 1) x α2 (m k+1 + 1)x α1 ( 1) k 1 k 1 x α1 ( 1) k 1 p k+1 j 1,...,j p=1 j 1 < <j p ( C j1,...,j p x α2 (m 1 + 1) ) x α1 +α 2 (m j1 ) x α1 +α 2 (m jp ) x α2 (m k+1 + 1) = k 1 p=0 x α1 ( 1) k 1 p [... [R 2 1,t (k+1 p), x α 1 (0)],..., x α1 (0)] + bx α2 (0) for some b U( n) and constants C j1,...,j p C. Since J is the left ideal of U( n) generated by R 1,t 1 and R2 1,t, we have that σ k 1Λ 1 +k 2 Λ 2 ω 1,ν (J) I k1 Λ 0 +k 2 Λ 1. We now show that σ k 1Λ 1 +k 2 Λ 2 ω 1 (U( n) n + ) I k1 Λ 0 +k 2 Λ 1. We have σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α1 (m)) = x α1 (m 1)x α1 ( 1) k 1 U( n) n + + U( n)x α1 ( 1) k 1+1

39 32 σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α2 (m)) = x α2 (m + 1)x α1 ( 1) k 1 = cx α1 ( 1) k 1 1 x α1 +α 2 (m) + x α1 ( 1) k 1 x α2 (m + 1) U( n) n + and σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α1 +α 2 (m)) = x α1 +α 2 (m)x α1 ( 1) k 1 = x α1 ( 1) k 1 x α1 +α 2 (m) U( n) n + for m 0. Since U( n) n + is the left ideal of U( n) generated by n +, we have that σ k 1Λ 1 +k 2 Λ 2 ω 1 (U( n) n + ) I k1 Λ 0 +k 2 Λ 1 and so we have σ k 1Λ 1 +k 2 Λ 2 ω 1,ν (I kλ0 ) I k1 Λ 0 +k 2 Λ 1. We now check the remaining terms. We have σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α1 ( 1) k 2+1 ) = x α1 ( 2) k 2+1 x α1 ( 1) k 1 = rr 1 1,2k 2 +2+k 1 + ax α1 ( 1) k 1+1 for some a U( n) and r C, and so σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α1 ( 1) k 2+1 ) I k1 Λ 0 +k 2 Λ 1. We also have, by Lemma 3.2.1, σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α2 ( 1) k 1+1 ) = x α2 (0) k 1+1 x α1 ( 1) k 1 = x α1 ( 1) k 1 x α2 (0) k 1+1 +r 1 x α1 ( 1) k 1 1 x α1 +α 2 ( 1)x α2 (0) k r k1 x α1 +α 2 ( 1) k 1 x α2 (0) for some constants r 1,..., r k1 C, and so σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α2 ( 1) k 1+1 ) U( n) n + I k1 Λ 0 +k 2 Λ 1.

40 33 Finally, σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α1 +α 2 ( 1)) = x α1 +α 2 ( 1)x α1 ( 1) k 1 = r[x α2 (0), x α1 ( 1) k 1+1 ] for some constant r C. So we have that σ k 1Λ 1 +k 2 Λ 2 ω 1 (x α1 +α 2 ( 1) k 1+1 ) U( n)x α1 ( 1) k 1+1 I k1 Λ 0 +k 2 Λ 1. This concludes our proof. Remark Lemmas and here directly generalize Lemma 3.1 and Lemma 3.2 in [CalLM3], respectively. Lemma in this paper does not have an analogue for I k0 Λ 0 +k 1 Λ 1 +k 2 Λ 2, and will be the main reason our proof of the presentations needs ideas other than those found in [CalLM1]-[CalLM3]. Remark Note that τ kλ 0 λ i,ν = τ λ i,ν, so that, as in [CalLM1]-[CalLM3], we have τ λi,ν(i kλ0 ) I kλi. For any λ P we have the linear isomorphism e λ : V P V P. In particular, for i, j = 1, 2 with i + j = 3 we have e λi v λ0 = v λi e λi v λi = ɛ(λ i, λ i )x αi ( 1) v λj e λi v λj = ɛ(λ i, λ j )x α1 +α 2 ( 1) v λ0 Since e λi x α (m) = c(α, λ i )x α (m α, λ i )e λi for α + and m Z

41 34 we have that e λi (a v λ0 ) = τ λi,c λi (a) v λi, a U( n). (3.18) For any λ P, we define linear isomorphisms on V k P by e k λ = e λ 1 e }{{ λ1 : V k } P V k P. k times In particular, we have e k λ 1 (v Λ0 v }{{ Λ0 v } Λ1 v Λ1 v Λ2 v Λ2 ) k 0 times k 1 times = ɛ(λ 1, λ 1 ) k 1 ɛ(λ 1, λ 2 ) k k 1! k 2! x α 1 ( 1) k 1 x α1 +α 2 ( 1) k 2 (v Λ1 v }{{ Λ1 v } Λ2 v Λ2 v Λ0 v Λ0 ). k 0 times k 1 times and e k λ 2 (v Λ0 v }{{ Λ0 v } Λ1 v Λ1 v Λ2 v Λ2 ) k 0 times k 1 times = ɛ(λ 2, λ 1 ) k 1 ɛ(λ 2, λ 2 ) k k 1! k 2! x α 2 ( 1) k 2 x α1 +α 2 ( 1) k 1 (v Λ2 v }{{ Λ2 v } Λ0 v Λ0 v Λ1 v Λ1 ). k 0 times k 1 times This, along with the fact that e k λ i x α (m) = c(α, λ i )x α (m α, λ i )e k λ i for α +, i = 1, 2, and m Z gives us e k λ 1 (a (v Λ0 v }{{ Λ0 v } Λ1 v Λ1 v Λ2 v Λ2 )) k 0 times k 1 times = ɛ(λ 1, λ 1 ) k 1 ɛ(λ 1, λ 2 ) k k 1! k 2! τ k 0Λ 0 +k 1 Λ 1 +k 2 Λ 2 λ 1,c λ1 (a) (v Λ1 v }{{ Λ1 v } Λ2 v Λ2 v Λ0 v Λ0 ). k 0 times k 1 times