New York Journal of Mathematics

New York J. Math. 15 (2009) 133–160.

On a symmetry of the category of integrable modules

William J. Cook and Christopher M. Sadowski

Abstract. Haisheng Li showed that given a module (W, Y_{W} (·, x)) for a vertex algebra (V, Y (·, x)), one can obtain a new V -module

W^{Δ} = (W, Y_{W} (Δ(x)·, x))

if Δ(x) satisfies certain natural conditions. Li presented a collection of such Δ-operators for V = L(k, 0) (a vertex operator algebra associated with an affine Lie algebra, k a positive integer). In this paper, for each irreducible L(k, 0)-module W, we find a highest weight vector of W^{Δ} when Δ is associated with a minuscule coweight. From this we completely determine the action of these Δ-operators on the set of isomorphism equivalence classes of L(k, 0)-modules.

Contents

1. Introduction 134 2. Definitions and background 136

2.1. Vertex algebras 136 2.2. Affine Lie algebras 137 2.3. Li’s Δ-operators 140

3. The sl_{2}(ℂ) case 142 4. The general case 143

4.1. The Weyl group elements’ actions 145 4.2. The main theorems 146

Appendix A. Weyl group calculations 151 A.1. Type B_{ℓ} 151 A.2. Type C_{ℓ} 152

Received December 30, 2008; revised March 20, 2009. Mathematics Subject Classification. 17B10, 17B67, 17B69. Key words and phrases. affine Lie algebras; vertex operator algebras. C. Sadowski acknowledges support from the the Rutgers Mathematics/DIMACS REU

Program during the summers of 2007 and 2008, and NSF grant DMS-0603745.

133^{ISSN 1076-9803/09}

A.3. Type D_{ℓ} (any rank) 153 A.4. Type D_{ℓ} (odd rank) 155 A.5. Type D_{ℓ} (even rank) 157 A.6. Exceptional types 157

References 159

1. Introduction

Haisheng Li introduced his Δ-operators in a very general setting in [Li1]. These operators allow one to obtain new vertex algebra modules from old ones by modifying the vertex algebra’s action on the module while leaving the underlying vector space unchanged. Thus, given a vertex algebra V and a collection of Δ-operators, we obtain (usually quite interesting) symmetries of the category of V -modules.

Consider the 𝔤̂-module L(k, 0) = L(kΛ_{0}) where k is a positive integer and 𝔤̂ is an untwisted affine Lie algebra. It is well-known that L(k, 0) has the structure of a vertex operator algebra (VOA), and that 𝔤̂-modules L(k, λ) for certain λ are modules for this VOA. Define

Δ(H, x) = x^{H(0)} exp(∑_{n≥1} \frac{H(n)}{−n}(−x)^{−n})

where H is a coweight of 𝔤 (the underlying finite-dimensional Lie algebra). If (W, Y (·, x)) is an L(k, 0)-module, then (W, Y (Δ(H, x)·, x)) (call this module W^{(H)}) is also an L(k, 0)-module. In fact, Li proved [Li2] that these two modules are equivalent if H is a coroot. However, when H is not a coroot, we may get a new (inequivalent) module. So, using Li’s Δ-operators we can induce an action on the equivalence classes of L(k, 0)-modules. In his thesis [C] (see also [CLM]), the first author was able to use these operators to obtain recurrence relations for characters of integrable highest weight 𝔤̂- modules which in turn led to interesting combinatorial identities. These recurrence relations were obtained by studying the effect of Δ(H, x) (for H a coroot) on characters. If we consider a coweight instead of a coroot, we obtain a relation between the characters of two different integrable 𝔤̂- modules.

In this paper, we completely determine the action of Δ(H, x) on equivalence classes of L(k, 0)-modules for all coweights H.

First, recall that L(k, 0) is a regular VOA. This implies (among other things) that its modules are completely reducible. This means that we just need to determine Δ(H, x)’s action on irreducible L(k, 0)-modules (these are precisely the integrable 𝔤̂-modules of level k).

Next, we know that the minuscule coweights provide a complete set of coset representatives for the coweight lattice modulo the coroot lattice. So, since the coroots give back equivalent modules, it is enough to study the action of Δ(H, x) on an irreducible L(k, 0)-module L(k, λ) where H is one of the minuscule coweights. To determine which module L(k, λ^{(H)}) is obtained from L(k, λ) via the Δ(H, x)-action, it is enough to identify a highest weight vector and “measure” its weight.

In this paper we explicitly determine a highest weight vector for L(k, λ)^{(H)} — that is, the module (L(k, λ), Y (Δ(H, x)·, x)) — if L(k, λ) is an L(k, 0)- module and H is a minuscule coweight (of 𝔤).

Recall that the 𝔤̂-modules L(k, λ) are induced up from 𝔤-modules L(λ) and in fact L(λ) (the “finite-dimensional part”) makes up the lowest conformal weight space of L(k, λ). Now let us change the action of L(k, λ) from Y (·, x) to Y (Δ(H, x)·, x) where H is a minuscule coweight. We get a new L(k, 0)-module action while leaving the underlying vector space fixed. It is interesting to note that in each case, the highest weight vector stays inside the lowest conformal weight space of L(k, λ). Li in [Li2] considered the special case, L(k, 0), and found that the old highest weight vector (which is the vaccuum vector) is also the new highest weight vector. This happens because the lowest conformal weight space is 1-dimensional — that is, the vector has nowhere to go. On the other hand, when we consider L(k, λ), λ ≠ 0, the lowest conformal weight space is bigger and so the highest weight vector moves and thus is harder to find.

One rather interesting feature of the action of Δ(H, x) is that the restriction on λ so that L(k, λ) is an L(k, 0)-module shows up explicitly in this action. We know that if L(k, λ) is an L(k, 0)-module, then ⟨λ, θ⟩ ≤ k where θ is the highest long root of 𝔤. In each case, when acting on L(k, λ) with Δ(H, x) where H is the j^{th} minuscule coweight, we see that the coefficent of λ_{j} (the j^{th} fundamental weight) is replaced by k − ⟨λ, θ⟩.

Without making any changes, the calculations and proofs involved in determining the new highest weight vectors of the L(k, 0)-modules L(k, λ)^{(H)} (H the j^{th} minuscule coweight) apply to any V (k, 0)-module as well (V (k, 0) is a generalized Verma module which also has VOA structure). It is wellknown that L(k, λ), where λ is a dominant integral weight of 𝔤, is a V (k, 0)- module. However, if ⟨λ, θ⟩ > k (so that L(k, λ) is not an L(k, 0)-module), then the Δ(H, x)-action will produce an (irreducible) weak V (k, 0)-module L(k, λ^{(H)}). But L(k, λ^{(H)}) is not a V (k, 0)-module (as a VOA) since the coffecient of λ_{j} in λ^{(H)} is negative (and thus not a dominant integral weight). So for V (k, 0)-modules, the Δ-action can send (strong) modules to weak modules.

It is interesting to note that Li’s Δ-operators also allow one to create new intertwining operators from old ones ([Li2], Proposition 2.12). In fact, since the Δ-operators are invertible, they give isomorphisms between spaces of intertwining operators. Therefore, using the results above, one can obtain symmetries of fusion rules. This is the topic of a future project of the authors with Sjuvon Chung and Yi-Zhi Huang.

The contents of the paper are organized as follows:

In the second section, we begin by reviewing the definition of a vertex operator algebra and its modules. Then, we set up all of the necessary notations related to finite-dimensional simple Lie algebras and untwisted affine Lie algebras and conclude by introducing Li’s Δ-operators and performing some preliminary calculations.

The third section illustrates our results in the most basic case — that of 𝔤̂ = ŝl_{2}. The fourth section tackles the general case where we must consider the effects of the Δ-operators one type at a time. In the course of deriving these results, we need to perform some tedious Weyl group calculations. The calculations for types B_{ℓ}, C_{ℓ}, and D_{ℓ} are given in an appendix (the fifth section). The appendix also contains a summary of results for types E_{6} and E_{7}. The calculations for types E_{6} and E_{7} were performed in Maple using a modified version of a worksheet developed by the first author for another project [CMS].

Acknowledgements. This paper grew out of the second author’s summer research experience for undergraduates (REU) mentored by the first author and Yi-Zhi Huang at Rutgers University during the summer of 2007. The authors would like to thank Yi-Zhi Huang for his encouragement and advice throughout this project.

2. Definitions and background

We will begin by reviewing the definition of a vertex operator algebra and its modules. Further details can be found in [LL].

2.1. Vertex algebras. A vertex algebra is a complex vector space V equipped with a linear map,

Y (·, x) : V → (End V)[[x, x^{−1}]]
v ↦ Y (v, x) = ∑_{n∈ℤ}v_{n}x^{−n−1}

and a distinguished vector 1 ∈ V (the vacuum vector) such that for u, v ∈ V, u_{n}v = 0 for n sufficiently large. The operator u_{n} is called the n^{th}-mode of u.

It is assumed that the vacuum vector behaves like an identity in that Y (1, x) = Id_{V} and for v ∈ V

Y (v, x)1 ∈ V [[x]] and lim_{x→0} Y (v, x)1 = v (the creation axiom).

Finally, we also must require that the Jacobi identity holds: for all u, v ∈ V

x_{0}^{−1}δ (\frac{x_{1} − x_{2}}{x_{0}} ) Y (u, x_{1})Y (v, x_{2}) − x_{0}^{−1}δ (\frac{x_{2} − x_{1}}{−x_{0}} ) Y (v, x_{2})Y (u, x_{1})

= x_{2}^{−1}δ (\frac{x_{1} − x_{0}}{x_{2}} ) Y (Y (u, x_{0})v, x_{2}).

Please note that δ(x) = ∑_{n∈ℤ} x^{n} is the formal delta function and we adopt the binomial expansion convention, namely, (x + y)^{n} should be expanded in nonnegative powers of y.

The vertex algebras that we will consider have additional structure making them vertex operator algebras. A vertex operator algebra is a vertex algebra V with the following additional data:

V is a ℤ-graded vector space V = ∐_{n∈ℤ} V_{(n)} (over ℂ) such that dim V_{(n)} < ∞ for all n ∈ ℤ and V_{(n)} = 0 for n sufficiently negative.

Elements of V_{(n)} are said to have conformal weight n. The vacuum must have conformal weight 0 (e.g., 1 ∈ V_{(0)}).

V has a second distinguished vector ω ∈ V_{(2)} (the conformal vector), where

Y (ω, x) = ∑_{n∈ℤ}ω_{n}x^{−n−1} = ∑_{n∈ℤ}L(n)x^{−n−2}.

The modes of the conformal vector satisfy the Virasoro relations:

(2.1) [L(m), L(n)] = (m − n)L(m + n) + \frac{m^{3} − m}{12}δ_{m+n,0} c_{V} for m, n ∈ ℤ.

The scalar c_{V} ∈ ℂ is called the central charge (or rank) of V. Finally, we must require that

Y (L(−1)v, x) = \frac{d}{dx}Y (v, x) for v ∈ V, (2.2)

V_{(n)} = {v ∈ V | L(0)v = nv} for n ∈ ℤ.

Let V be a vertex algebra. A V -module is a complex vector space W equipped with a linear map

Y_{W} (·, x) : V → (End W)[[x, x^{−1}]]
v ↦ Y (v, x) = ∑_{n∈ℤ}v_{n}x^{−n−1}

such that for v ∈ V and w ∈ W, v_{n}w = 0 for n sufficiently large. Also, Y_{W} (1, x) = Id_{W} and the Jacobi identity holds.

If V is a vertex operator algebra, we say W is a (strong) V -module if W is a module for V thought of as a vertex algebra and in addition W is a ℂgraded vector space W = ∐_{n∈ℤ} W_{(n)} where W_{(n)} = {w ∈ W | L(0)w = nw} such that for all n ∈ ℂ, dim W_{(n)} < ∞ and W_{(n+r)} = 0 for r sufficiently negative.

If W is a vertex algebra module for a vertex operator algebra V, we say that W is a weak V -module.

2.2. Affine Lie algebras. Following [H] and [K], we now establish some notation and review some basic definitions and facts concerning (untwisted) affine Lie algebras.

Let 𝔤 be a finite-dimensional simple Lie algebra of rank ℓ (over ℂ). Fix a Cartan subalgebra 𝔥 ⊂ 𝔤, and let ⟨·, ·⟩ be the standard form such that ∥α∥^{2} = ⟨α, α⟩ = 2 for any long root α. Let Δ be the set of roots of 𝔤. Fix a set of simple roots {α_{1}, …, α_{ℓ}}, simple coroots {H_{1}, …, H_{ℓ}}, and Chevalley generators {E_{i}, F_{i}, H_{i} | i = 1, …, ℓ}. Recall that α_{j}(H_{i}) = a_{ij} where C = (a_{ij})_{i,j=1}^{ℓ} is the Cartan matrix of 𝔤 Let Δ^{+} ⊂ Δ be the system of positive roots corresponding to the α’s. We have the triangular decomposition:

𝔤 = 𝔤_{+} ⊕ 𝔥 ⊕ 𝔤_{−}  where  𝔤_{±} = ∑_{α∈Δ^{+}} 𝔤_{±α}.

Let {λ_{1}, …, λ_{ℓ}} ⊂ 𝔥^{∗} be the fundamental weights and let {H^{(1)}, …, H^{(ℓ)}} be the fundmental coweights. Of course, the fundamental weights are dual to the simple coroots (i.e., λ_{i}(H_{j}) = δ_{ij}) and the fundamental coweights are dual to the simple roots (i.e., α_{i}(H^{(j)}) = δ_{ij}). We should also note that H_{i} = ∑_{i=1}^{ℓ} a_{ji}H^{(j)}.

Let Q and P denote the root lattice and weight lattice of 𝔤, respectively:

Q = ℤα_{1} + ⋯ + ℤα_{ℓ} and P = ℤλ_{1} + ⋯ + ℤλ_{ℓ}

and let Q^{∨} and P^{∨} denote the coroot lattice and coweight lattice of 𝔤, respectively:

Q^{∨} = ℤH_{1} + ⋯ + ℤH_{ℓ} and P^{∨} = ℤH^{(1)} + ⋯ + ℤH^{(ℓ)}.

Define the set of dominant integral weights by:

P_{+} = {λ ∈ 𝔥^{∗}|λ(H_{i}) ∈ ℤ_{≥0}, 1 ≤ i ≤ ℓ}.

For λ ∈ 𝔥^{∗}, define the Verma module of highest weight λ by:

(2.3)  V (λ) = U(𝔤) ⊗_{U(𝔥⊕𝔤_{+})} ℂ_{λ},

where ℂ_{λ} is a 1-dimensional 𝔥 ⊕ 𝔤_{+}-module given by:

h · 1 = λ(h)  for all h ∈ 𝔥
g · 1 = 0  for all g ∈ 𝔤_{+}.

Let J(λ) be the maximal proper submodule of V (λ). Then

L(λ) = V (λ)/J(λ)

is an irreducible (highest weight) 𝔤-module. Of course, L(λ) is finite-dimen-sional if and only if λ ∈ P_{+}.

Let α be a root, then the map σ_{α} : 𝔥^{∗} → 𝔥^{∗} defined by

σ_{α}(λ) = λ − \frac{2⟨α, λ⟩}{∥α∥^{2}}α = λ − \frac{2⟨α, λ⟩}{⟨α, α⟩}α

is called the reflection associated with α (notice that σ_{α}(α) = −α and σ_{α}^{2} = Id_{𝔥^{∗}}). The group W generated by these reflections is called the Weyl group of 𝔤. Let σ_{i} = σ_{α_{i}} for each simple root α_{i}. The σ_{i}’s are called simple reflections. It is well-known that W is generated by simple reflections. The elements of W are isometries (with respect to the standard form) of 𝔥^{∗}. Note: we can (and do) transport the action of W on 𝔥^{∗} to an action on 𝔥 using the standard form.

The (untwisted) affine Lie algebra associated with 𝔤 is given by

(2.4)  𝔤̂ = 𝔤 ⊗ ℂ[t, t^{−1}] ⊕ ℂc,

where for a, b ∈ 𝔤, m, n ∈ ℤ,

[a ⊗ t^{m}, b ⊗ t^{n}] = [a, b] ⊗ t^{m+n} + m⟨a, b⟩δ_{m+n,0}c,
[𝔤̂, c] = 0.

For a ∈ 𝔤, n ∈ ℤ, let a(n) denote the action of a ⊗ t^{n} ∈ 𝔤̂ on a 𝔤̂-module. Let θ = ∑_{i=1}^{ℓ} a_{i}α_{i} be the highest long root of 𝔤, and choose (nonzero) vectors E_{θ} ∈ 𝔤_{θ} and F_{θ} ∈ 𝔤_{−θ} such that ⟨E_{θ}, F_{θ}⟩ = 1. Let H_{θ} = [E_{θ}, F_{θ}]. Define

(2.5)  e_{0} = F_{θ} ⊗ t,  f_{0} = E_{θ} ⊗ t^{−1},  and  h_{0} = [e_{0}, f_{0}]

and for i = 1, …, ℓ define:

(2.6)  e_{i} = E_{i} ⊗ 1,  f_{i} = F_{i} ⊗ 1,  and  h_{i} = H_{i} ⊗ 1.

Then, by [K], {e_{i}, f_{i}, h_{i} | 0 ≤ i ≤ ℓ} is a set of Chevalley generators for 𝔤̂. Let 𝔤̂_{>0} = 𝔤 ⊗ tℂ[t] and 𝔤̂_{0} = 𝔤 ⊗ 1 ⊕ ℂc. Also, let

𝔤̂_{≥0} = 𝔤̂_{>0} ⊕ 𝔤̂_{0} = 𝔤 ⊗ ℂ[t] ⊕ ℂc

and fix a scalar k ∈ ℂ. We can make L(λ) into a 𝔤̂_{≥0}-module by extending the action of 𝔤 = 𝔤 ⊗ 1 as follows:

c · v = kv  for all v ∈ L(λ),
x · v = 0  for all x ∈ 𝔤_{>0}.

We now induce up to a 𝔤̂-module:

V (k, λ) = U(𝔤̂) ⊗_{U(𝔤̂_{≥0})} L(λ).

V (k, λ) is a generalized Verma module.

Unless k is the negative of the dual Coxeter number of 𝔤, V (k, 0) has the structure of a vertex operator algebra. For such k, V (k, λ) is a weak V (k, 0)- module for all λ ∈ 𝔥^{∗}. Moreover, V (k, λ) is a (strong) V (k, 0)-module if (and only if) λ ∈ P_{+}.

Let J(k, λ) be the maximal proper submodule of V (k, λ), and define

L(k, λ) = V (k, λ)/J(k, λ).

If k is not the negative of the dual Coxeter number of 𝔤, then L(k, 0) has the structure of a simple vertex operator algebra, and each L(k, λ) is an irreducible weak (unless λ ∈ P_{+}) V (k, 0)-module.

If k is a positive integer, then:

Theorem 2.1 ([DL]). L(k, λ) is an L(k, 0)-module if and only if ⟨λ, θ⟩ ≤ k, where θ is the highest long root of 𝔤.

2.3. Li’s Δ-operators. Now, let Let H ∈ P^{∨}. Set

(2.7)  Δ(H, x) = x^{H(0)} exp(∑_{n≥1} \frac{H(n)}{−n}(−x)^{−n})

(recall the H(n) is the action of H ⊗ t^{n} on a 𝔤̂-module).

Note that Δ(H, x) enjoys the following properties:

Δ(H_{1} + H_{2}, x) = Δ(H_{1}, x)Δ(H_{2}, x),
Δ(0, x) = Id.

We fix the notation

(L(k, λ)^{(H)}, Y_{L(k,λ)}^{(H)}(·, x)) = (L(k, λ), Y_{L(k,λ)}(Δ(H, x)·, x)).

For v ∈ L(k, 0), set

(2.8)  Y_{L(k,λ)}^{(H)}(v, x) = Y_{L(k,λ)}(Δ(H, x)v, x) = ∑_{n∈ℤ}v^{(H)}(n)x^{−n−1}.

Let us take care of some preliminary calculations involving Δ(H, x). For g ∈ 𝔤_{β}, β ∈ Δ ∪ {0}, so that [h, g] = β(h)g for all h ∈ 𝔥 (Here, 𝔤_{0} = 𝔥), we have:

[h(m), g(−1)] = [h, g](m − 1)1 + m⟨h, g⟩(m − 1)1 + δ_{m−1,0}k1

= β(h)g(m − 1)1 + ⟨h, g⟩δ_{m,1}k1

= \begin{cases}β(h)g(−1)1 & m = 0 \\ ⟨h, g⟩kδ_{m,1}1 & m > 0. \end{cases}

Note that g(m − 1)1 = 0 for m > 0 by the creation axiom.

We also have (for any g ∈ 𝔤):

(∑_{m≥1} \frac{H(m)}{−m}(−x)^{−m})g = ⟨H, g⟩k1x^{−1}.

and for n ≥ 2

(2.9) (∑_{m≥1} \frac{H(m)}{−m}(−x)^{−m})^{n}g = k⟨H, g⟩x^{−1}(∑_{m≥1} \frac{H(m)}{−m}(−x)^{−m})^{n−1}1

= 0

(again using the creation axiom, H(m)1 = 0 for m ≥ 1).

Therefore,

(2.10)  Δ(H, x)g = x^{H(0)}g + x^{H(0)}⟨H, g⟩k1x^{−1}
= x^{β(H)}g + ⟨H, g⟩k1x^{−1}

if g ∈ 𝔤_{β}.

Applying this to our vertex operator map, we have:

(2.11)  Y^{(H)}(g, x) = Y (Δ(H, x)g, v)
= x^{β(H)}Y (g, x) + ⟨H, g⟩k(Id)x^{−1}.

Let k be a positive integer and λ ∈ P_{+} (a dominant integral weight). In addition, assume that ⟨λ, θ⟩ ≤ k so that L(k, λ) is an (irreducible) L(k, 0)- module. It was proved in ([Li2], Proposition 2.9) that (L(k, λ)^{(H)}, Y_{L(k,λ)}^{(H)}) carries the structure of an irreducible (weak) L(k, 0)-module. However, since L(k, 0) is a regular vertex operator algebra, its weak modules are in fact (strong) modules. So we have that:

Proposition 2.2. There exists a (unique) λ^{(H)} ∈ P_{+} such that ⟨λ^{(H)}, θ⟩ ≤ k and (L(k, λ))^{(H)} ≅ L(k, λ^{(H)}) as L(k, 0)-modules.

Also, Li established that:

Theorem 2.3 ([Li2], Proposition 2.25). For H ∈ Q^{∨}, (L(k, λ))^{(H)} and L(k, λ) are isomorphic as L(k, 0) modules.

That is, in our notation, λ^{(H)} = λ when H ∈ Q^{∨}.

Our objective is to see what happens when we allow H to be any element of P^{∨} (not just Q^{∨}).

Because Δ(H′ + H′′, x) = Δ(H′, x)Δ(H′′, x), we have the following:

L(k, λ^{(H′+H′′)}) ≅ L(k, λ)^{(H′+H′′)}
= (L(k, λ)^{(H′)})^{(H′′)}
≅ L(k, λ^{(H′)})^{(H′′)}
≅ L(k, (λ^{(H′)})^{(H′′)}).

Therefore, λ^{(H′+H′′)} = (λ^{(H′)})^{(H′′)}.

The coweights {H^{(1)}, …, H^{(ℓ)}} form a basis for P^{∨} and so, by the observation above, all we need to find is action of Δ(H^{(i)}, x). Let us fix the notation (L(k, λ))^{(i)} = (L(k, λ))^{(H^{(i)})} and λ^{(i)} = λ^{(H^{(i)})} for i = 1, …, ℓ that is L(k, λ^{(i)}) ≅ (L(k, λ))^{(i)} = (L(k, λ))^{(H^{(i)})}.

Actually, since the operators associated with coroots act trivially, we only need to consider one representative for each coset of P^{∨}/Q^{∨}. This pares down the list of coweights considerably.

We know that the irreducible L(k, 0)-modules are precisely the irreducible integrable highest weight modules (i.e., standard modules) for 𝔤̂ ([DL]). Now, an irreducible highest weight module is completely determined by its highest weight. Therefore, if we can locate a highest weight vector in (L(k, λ))^{(i)} and measure its weight, we have determined λ^{(i)} and thus the action of Δ(H^{(i)}, x) on L(k, λ).

Now, for our Chevalley generators e_{i}, f_{i}, h_{i} (0 ≤ i ≤ ℓ), we define e_{i}^{(j)}, f_{i}^{(j)}, and h_{i}^{(j)} to be the actions of e_{i}, f_{i}, and h_{i} on L(k, λ)^{(j)}. Let us calculate these actions:

Y^{(j)}(H_{i}, x) = Y (Δ(H^{(j)}, x)H_{i}, x)

= Y (H_{i}, x) + \frac{2k}{∥α_{i}∥^{2}} δ_{i,j}x^{−1}

and so we have that

(2.12)  (H_{i})^{(j)}(0) = H_{i}(0) + \frac{2k}{∥α_{i}∥^{2}} δ_{i,j}.

For e_{i}, 1 ≤ i ≤ ℓ, we have

Y^{(j)}(E_{i}, x) = Y (Δ(H^{(j)}, x)E_{i}, x)
= x^{α_{i}(H^{(j)})}Y (E_{i}, x)
= x^{δ_{i,j}}Y (E_{i}, x)

and so

(2.13)  e_{i}^{(j)} = E_{i}^{(H^{(j)})}(0) = E_{i}(δ_{i,j}).

Finally, for e_{0}, we have:

Y^{(j)}(F_{θ}, x) = Y (Δ(H^{(j)}, x)F_{θ}, x)
= x^{−θ(H^{(j)})}Y (F_{θ}, x)
= x^{−a_{j}}Y (F_{θ}, x),

(recall that θ = ∑_{i=1}^{ℓ} a_{i}α_{i}) and so

(2.14)  e_{0}^{(j)} = F_{θ}^{(j)}(1) = F_{θ}(1 − a_{j}).

3. The sl_{2}(ℂ) case

Before answering our question for all finite-dimensional simple Lie algebras let us consider the simplest case — that of 𝔤 = sl_{2}(ℂ).

Let E = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}), F = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}), and H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}). We know that E,

F, H are Chevalley generators of sl_{2}(ℂ). Let α be the fundamental root of sl_{2}(ℂ) (that is, α(H) = 2), and λ = \frac{1}{2}α be the fundamental weight of sl_{2}(ℂ). Since α is the only positive root, it is the highest long root — that is α = θ.

Then, H is the coroot corresponding to α and H^{(1)} = \frac{1}{2}H is the coweight corresponding to λ.

In the sl_{2}(ℂ) case, we denote L(k, nλ) by L(k, n). Let k be a positive integer and n ∈ ℤ such that 0 ≤ n ≤ k.

Theorem 3.1. (L(k, n))^{(1)} ≅ L(k, k −n). Moreover, if v is a highest weight vector for L(k, n), then F (0)^{n}v is a highest weight vector for (L(k, n))^{(1)} (with weight Λ = (k, (k − n)λ)).

Proof. Since −θ = −α, we have F = F_{θ} and so, recalling (2.5), we have e_{0} = F_{θ} ⊗ t = F ⊗ t. By (2.14), we have that e_{0}^{(1)} · w = F (0) · w. Therefore,

e_{0}^{(1)} · F (0)^{n}v = F (0) · F (0)^{n}v = F (0)^{n+1} · v = 0

by the representation theory of sl_{2}(ℂ).

Next, we have

e_{1}^{(1)} · F (0)^{n}v = E(1)F (0)^{n}v.

Since L(k, n) is a vertex operator algebra module, we can consider conformal weights. Notice that as operators wt F (0) = wt (F) − 0 − 1 = 0 and wt E(1) = wt (E) − 1 − 1 = −1. This implies that e_{1}^{(1)} · F (0)^{n}v has a lower (conformal) weight than v. However, the highest weight vector occupies the lowest (conformal) weight space of L(k, n). Thus e_{1}^{(1)}F (0)^{n}v = 0. Therefore, F (0)^{n}v is a highest weight vector for (L(k, n))^{(1)}.

Now let us determine the weight of F (0)^{n}v. We will use the fact that [H, F^{n}] = −2nF^{n} (in U(sl_{2})).

(H)^{(1)} · F (0)^{n}v = H(0)F (0)^{n}v + kF (0)^{n}v
= F (0)^{n}H(0)v + [H(0), F (0)^{n}]v + kF (0)^{n}v
= nF (0)^{n}v − 2nF (0)^{n}v + kF (0)^{n}v
= (k − n)F (0)^{n}v.

Hence, (L(k, n))^{(1)} ≅ L(k, k − n). □

Remark 3.2. The lowest conformal weight space of L(k, n) is a copy of L(nλ) (the finite-dimensional sl_{2}-module with highest weight nλ). The highest weight vector of L(k, n) is located in this (lowest) conformal weight space.

Observe that the new highest weight vector for (L(k, n))^{(1)} is also located in this copy of L(nλ). In fact, we can obtain the new highest weight vector from the old one by applying the (only) reflection in the Weyl group of sl_{2}.

4. The general case

Recall that, given an irreducible L(k, 0)-module W and coroot H, W^{(H)} and W are isomorphic as L(k, 0)-modules. This, along with the fact that W^{(H′+H′′)} is isomorphic to (W^{(H′)})^{(H′′)}, implies that we only need to consider one representative from each distinct coset of P^{∨}/Q^{∨}. It is well-known that H = 0 (which acts as the identity — W^{(0)} = W), along with the minuscule coweights, give us a complete set of coset representatives. We give a list of such coweights in Table 4.1.

Table 4.1.

Type  A_{ℓ}  B_{ℓ} C_{ℓ}  D_{ℓ}  E_{6}  E_{7}
Coweights H^{(1)}, …, H^{(ℓ)} H^{(1)} H^{(ℓ)} H^{(1)}, H^{(ℓ−1)}, H^{(ℓ)} H^{(1)}, H^{(6)} H^{(7)}

Types E_{8}, F_{4}, and G_{2} have no minuscule coweights, so the action of Δ(H, x) is always trivial.

Given any coweight H from this list, we wish to determine λ^{(H)} such that (L(k, λ))^{(H)} is isomorphic to L(k, λ^{(H)}). To do this, we need to identify a highest weight vector for (L(k, λ))^{(H)} and then measure its weight. We will see that in each case our “new” highest weight vector (for (L(k, λ))^{(H)}) is located in the lowest (conformal) weight space. This lowest (conformal) weight space is a copy of L(λ) — the finite-dimensional 𝔤-module from which L(k, λ) is built. In fact, our “new” highest weight vector can be obtained from our “old” highest weight vector by applying the Weyl group elements given in Table 4.2. (For each type X_{ℓ} and coweight H^{(j)}, define an element σ_{X}^{(j)} as in the table.)

Table 4.2.

Type  Weyl Group Element
A_{ℓ}  1 ≤ j ≤ ℓ, σ_{A}^{(j)} = (σ_{1}σ_{2} ⋯ σ_{ℓ})^{j}
B_{ℓ}  σ_{B}^{(1)} = σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}σ_{1}
C_{ℓ}  σ_{C}^{(ℓ)} = (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1})(σ_{ℓ})
D_{ℓ}  σ_{D}^{(1)} = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1}
ℓ even σ_{D}^{(ℓ−1)} = (σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{3}σ_{2})
(σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{3}) ⋯ (σ_{ℓ−1})
σ_{D}^{(ℓ)} = (σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1})(σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{3}σ_{2})
(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{3}) ⋯ (σ_{ℓ})
ℓ odd σ_{D}^{(ℓ−1)} = (σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{3}σ_{2})
(σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{4}σ_{3}) ⋯ (σ_{ℓ})
σ_{D}^{(ℓ)} = (σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1})(σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{3}σ_{2})
(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{4}σ_{3}) ⋯ (σ_{ℓ−1})
E_{6}  σ_{E}^{(1)} = σ_{1}σ_{3}σ_{4}σ_{2}σ_{5}σ_{4}σ_{3}σ_{1}σ_{6}σ_{5}σ_{4}σ_{2}σ_{3}σ_{4}σ_{5}σ_{6}
E_{6}  σ_{E}^{(6)} = σ_{6}σ_{5}σ_{4}σ_{2}σ_{3}σ_{4}σ_{5}σ_{6}σ_{1}σ_{3}σ_{4}σ_{2}σ_{5}σ_{4}σ_{3}σ_{1}
E_{7}  σ_{E}^{(7)} = σ_{7}σ_{6}σ_{5}σ_{4}σ_{3}σ_{2}σ_{4}σ_{5}σ_{6}σ_{7}σ_{1}σ_{3}σ_{4}σ_{5}σ_{6}σ_{2}σ_{4}σ_{5}σ_{3}σ_{4}σ_{1}σ_{3}σ_{2}σ_{4}σ_{5}σ_{6}σ_{7}

Let w be a Weyl group element and v ∈ L(λ)_{μ} (a vector of weight μ). Then w(v) ∈ L(λ)_{w(μ)}. We begin by calculating the effects of the Weyl group elements defined above on the fundamental weights and simple roots of 𝔤. To simplify computations, we will use the following conventions:

λ_{0} = 0  λ_{j} = λ_{j(mod(l+1))}
α_{0} = −θ  α_{j} = α_{j(mod(l+1))}.

We will repeatedly use the fact that σ_{i}(λ_{j}) = λ_{j} − δ_{i,j}α_{j}.

4.1. The Weyl group elements’ actions. Let us begin by calculating the action of σ_{A}^{(1)} on the fundamental weights and then use that to determine the action of our type A_{ℓ} Weyl group elements.

For λ_{1}, we have: σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{1}) = σ_{1}(λ_{1}) = −λ_{1} + λ_{2}. For λ_{j}, with 1 < j < ℓ:

σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{j}) = σ_{1}σ_{2} ⋯ σ_{j}(λ_{j})
= σ_{1}σ_{2} ⋯ σ_{j−1}(λ_{j−1} − λ_{j} + λ_{j+1})
= σ_{1}σ_{2} ⋯ σ_{j−2}(λ_{j−2} − λ_{j−1} + λ_{j+1})
= ⋯
= σ_{1}(λ_{1} − λ_{2} + λ_{j+1})
= −λ_{1} + λ_{j+1}.

Finally, for λ_{ℓ}:

σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}(λ_{ℓ−1} − λ_{ℓ})
= σ_{1}σ_{2} ⋯ σ_{ℓ−2}(λ_{ℓ−2} − λ_{ℓ−1})
= ⋯
= σ_{1}(λ_{1} − λ_{2})
= −λ_{1}.

Adhering to our above convention, we conclude that for 1 ≤ j ≤ l:

(4.1)  σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{j}) = −λ_{1} + λ_{j+1}.

Therefore, for 1 ≤ j < ℓ (recall that λ_{0} = 0),

σ_{1}σ_{2} ⋯ σ_{ℓ}(α_{j}) = σ_{1}σ_{2} ⋯ σ_{ℓ}(−λ_{j−1} + 2λ_{j} − λ_{j+1})

= −(−λ_{1} + λ_{j−1+1}) + 2(−λ_{1} + λ_{j+1}) − (−λ_{1} + λ_{j+1+1})

= −λ_{j} + 2λ_{j+1} − λ_{j+2} = α_{j+1}.

For α_{ℓ}, we have that

σ_{1}σ_{2} ⋯ σ_{ℓ}(α_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ}(−λ_{ℓ−1} + 2λ_{ℓ})
= −(−λ_{1} + λℓ − 1 + 1) + 2(−λ_{1} + λ_{ℓ+1})
= −λ_{1} − λ_{ℓ} = −θ (= α_{0}).

For α_{0} = −θ, we have:

σ_{1}σ_{2} ⋯ σ_{ℓ}(−θ) = σ_{1}σ_{2} ⋯ σ_{ℓ}(−λ_{1} − λ_{ℓ})
= −(−λ_{1} + λ_{2}) − (−λ_{1} + λ_{ℓ+1})
= 2λ_{1} − λ_{2} = α_{1}.

So we have found that for all 0 ≤ i ≤ ℓ, σ_{A}^{(1)}(α_{i}) = α_{i+1} and therefore

(4.2)  σ_{A}^{(j)}(α_{i}) = (σ_{1}σ_{2} ⋯ σ_{ℓ})^{j}(α_{i}) = α_{i+j}.

The calculations for types B_{ℓ}, C_{ℓ}, and D_{ℓ} are similar and can be found in the Appendix. Calculations for types E_{6} and E_{7} were done with the help of a Maple worksheet written and updated by the first author which was orginally written for [CMS]. This worksheet is available at http://dimax.rutgers.edu/∼sadowski/LieAlgebraCalculations/index.html. The Weyl group elements’ actions are summed up in Table 4.3 (recall that

α_{0} = −θ).

4.2. The main theorems. Now we can apply our Weyl group calculations to determine the highest weight vector of (L(k, λ))^{(i)} (which is simultaneously an L(k, 0) and 𝔤̂ module). Suppose our underlying finite-dimensional simple Lie algebra 𝔤 is of type X_{ℓ}. Recall that θ = ∑_{j} a_{j}α_{j} is the highest long root of 𝔤, k is a positive integer, λ ∈ P_{+} with ⟨λ, θ⟩ ≤ k, and H^{(i)} is one of the coweights appearing in Table 4.1. Also, note that for each valid choice of index i, we have a_{i} = 1.

The irreducible 𝔤̂-module L(k, λ) was built up from the irreducible 𝔤module L(λ). In fact, the lowest (conformal) weight space of L(k, λ) is merely a copy of L(λ). In what follows, let us identify L(λ) with this subspace of L(k, λ).

Let v be a highest weight vector for L(k, λ). Such a vector is also homogeneous vector of lowest conformal weight in L(k, λ). So we have that v ∈ L(λ) ⊂ L(k, λ).

Theorem 4.1. σ_{X}^{(i)}(v) is a highest weight vector for (L(k, λ))^{(i)}.

Proof. Let w = σ_{X}^{(i)}(v) and μ = σ_{X}^{(i)}(λ).

Since σ_{X}^{(i)} is an invertible map, w ≠ 0. The weight of v (as an element of L(λ)) is λ, so the weight of w is μ. To establish that w is a highest weight vector for (L(k, λ))^{(i)}, we need to show that (e_{j})^{(i)} · w = 0 for j = 0, …, ℓ.

By (2.13), we have (e_{j})^{(i)} · w = E_{j}(δ_{ij})(w) for 1 ≤ j ≤ ℓ.

When j = i, (e_{i})^{(i)} · w = E_{i}(1)(w). Now recall that the operator E_{i}(1) lowers (conformal) weights by −1. Since w is already a lowest (conformal) weight vector, (e_{i})^{(i)} · w = 0.

Next, when j ≠ i, (e_{j})^{(i)} ·w = E_{j}(0)(w) which is a vector of weight μ+α_{j} (in the 𝔤-module L(λ)).

Table 4.3.

A_{ℓ} : σ_{A}^{(j)}(α_{j}) = α_{j+i (mod ℓ+1)} (0 ≤ j ≤ ℓ and 1 ≤ i ≤ ℓ).
B_{ℓ} : σ_{B}^{(1)}(α_{0}) = α_{1}, σ_{B}^{(1)}(α_{1}) = α_{0}, and σ_{B}^{(1)}(α_{j}) = α_{j} (1 < j ≤ ℓ).
C_{ℓ} : σ_{C}^{(ℓ)}(α_{0}) = α_{ℓ}, σ_{C}^{(ℓ)}(α_{j}) = α_{ℓ−j} (1 ≤ j < ℓ), and σ_{C}^{(ℓ)}(α_{ℓ}) = α_{0}.
D_{ℓ} : σ_{D}^{(1)}(α_{0}) = α_{1}, σ_{D}^{(1)}(α_{1}) = α_{0}, σ_{D}^{(1)}(α_{j}) = α_{j} (1 < j < ℓ − 1),

σ_{D}^{(1)}(α_{ℓ−1}) = α_{ℓ}, and σ_{D}^{(1)}(α_{ℓ}) = α_{ℓ−1}.

ℓ odd σ_{D}^{(ℓ−1)}(α_{0}) = α_{ℓ−1}, σ_{D}^{(ℓ−1)}(α_{1}) = α_{ℓ}, and

σ_{D}^{(ℓ−1)}(α_{j}) = α_{ℓ−j} (1 < j ≤ ℓ).

σ_{D}^{(ℓ)}(α_{0}) = α_{ℓ}, σ_{D}^{(ℓ)}(α_{j}) = α_{ℓ−j} (1 ≤ j < ℓ − 1),

σ_{D}^{(ℓ)}(α_{ℓ−1}) = α_{0}, and σ_{D}^{(ℓ)}(α_{ℓ}) = α_{1}.

ℓ even σ_{D}^{(ℓ−1)}(α_{0}) = α_{ℓ−1}, σ_{D}^{(ℓ−1)}(α_{1}) = α_{ℓ},

σ_{D}^{(ℓ−1)}(α_{j}) = α_{ℓ−j} (1 < j < ℓ − 1),

σ_{D}^{(ℓ−1)}(α_{ℓ−1}) = α_{0}, and σ_{D}^{(ℓ−1)}(α_{ℓ}) = α_{1}.

σ_{D}^{(ℓ)}(α_{j}) = α_{ℓ−j} (0 ≤ j ≤ ℓ).

E_{6} : σ_{E}^{(1)}(α_{0}) = α_{1}, σ_{E}^{(1)}(α_{1}) = α_{6}, σ_{E}^{(1)}(α_{2}) = α_{3}, σ_{E}^{(1)}(α_{3}) = α_{5},

σ_{E}^{(1)}(α_{4}) = α_{4}, σ_{E}^{(1)}(α_{5}) = α_{2}, and σ_{E}^{(1)}(α_{6}) = α_{0}.

σ_{E}^{(6)}(α_{0}) = α_{6}, σ_{E}^{(6)}(α_{1}) = α_{0}, σ_{E}^{(6)}(α_{2}) = α_{5}, σ_{E}^{(6)}(α_{3}) = α_{2},

σ_{E}^{(6)}(α_{4}) = α_{4}, σ_{E}^{(6)}(α_{5}) = α_{3}, and σ_{E}^{(6)}(α_{6}) = α_{1}.

E_{7} : σ_{E}^{(7)}(α_{0}) = α_{7}, σ_{E}^{(7)}(α_{1}) = α_{6}, σ_{E}^{(7)}(α_{2}) = α_{2}, σ_{E}^{(7)}(α_{3}) = α_{5},

σ_{E}^{(7)}(α_{4}) = α_{4}, σ_{E}^{(7)}(α_{5}) = α_{3}, σ_{E}^{(7)}(α_{6}) = α_{1}, and σ_{E}^{(7)}(α_{7}) = α_{0}.

By (2.14), we have (e_{0})^{(i)} · w = F_{θ}(1 − a_{i})(w) = F_{θ}(0)(w) which has weight μ − θ = μ + α_{0}.

Therefore, we have reduced our problem to establishing that μ + α_{j} for j ≠ i, 0 ≤ j ≤ ℓ are not weights of the 𝔤-module L(λ).

Since Weyl group elements permute the weights of L(λ), if μ + α_{j} is a weight, then (σ_{X}^{(i)})^{−1}(μ + α_{j}) = λ + (σ_{X}^{(i)})^{−1}(α_{j}) must be a weight as well.

However, we can see by inspecting Table 4.3 that σ_{X}^{(i)} permutes the set of simple roots along with α_{0} = −θ. Notice that each case σ_{X}^{(i)}(α_{0}) = α_{i}.

Therefore, (σ_{X}^{(i)})^{−1} maps each α_{j} where j ≠ i, 0 ≤ j ≤ ℓ to some α_{k} where 1 ≤ k ≤ ℓ. But λ is a highest weight vector for L(λ), therefore λ + α_{k} (1 ≤ k ≤ ℓ) is not a weight. This implies that μ + α_{j} (for j ≠ i, 0 ≤ j ≤ ℓ) is not a weight. Therefore, w is annihilated by the action of each (e_{j})^{(i)} (for 0 ≤ j ≤ ℓ) and thus is a highest weight vector for (L(k, λ))^{(i)}. □

Remark 4.2. Elements of the Weyl group are linear transformations, so they always send the linear functional λ = 0 to itself. This implies that a highest weight vector for L(k, 0) (i.e., a nonzero scalar multiple of the vaccuum vector) is still a highest weight vector for (L(k, 0))^{(i)}. This special case was discussed in [Li2].

Theorem 4.3. Recall (L(k, λ))^{(i)} ≅ L(k, λ^{(i)}). Let λ = ∑_{j=1}^{ℓ} m_{j}λ_{j}. Then we have the following for the displayed types:

A_{ℓ}  λ^{(i)} = ∑_{j=1}^{ℓ} m_{j}λ_{j+i} + (k − ⟨λ, θ⟩)λ_{i}

B_{ℓ}  λ^{(1)} = (k − ⟨λ, θ⟩)λ_{1} + ∑_{j=2}^{ℓ} m_{j}λ_{j}

C_{ℓ}  λ^{(ℓ)} = ∑_{j=1}^{ℓ} m_{j}λ_{ℓ−j} − (k − ⟨λ, θ⟩)λ_{ℓ}

D_{ℓ}  λ^{(1)} = (k − ⟨λ, θ⟩)λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{j} + m_{ℓ−1}λ_{ℓ} + m_{ℓ}λ_{ℓ−1}

(ℓ odd) λ^{(ℓ−1)} = m_{ℓ−1}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} + (k − ⟨λ, θ⟩)λ_{ℓ−1} + m_{1}λ_{ℓ}

(ℓ even) λ^{(ℓ−1)} = m_{ℓ}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} + (k − ⟨λ, θ⟩)λ_{ℓ−1} + m_{1}λ_{ℓ}

(ℓ odd)  λ^{(ℓ)} = m_{ℓ}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} + m_{1}λ_{ℓ−1} + (k − ⟨λ, θ⟩)λ_{ℓ}

(ℓ even) λ^{(ℓ)} = m_{ℓ−1}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} + m_{1}λ_{ℓ−1} + (k − ⟨λ, θ⟩)λ_{ℓ}
E_{6}  λ^{(1)} = (k − ⟨λ, θ⟩)λ_{1} + m_{5}λ_{2} + m_{2}λ_{3} + m_{4}λ_{4} + m_{3}λ_{5} + m_{1}λ_{6}
λ^{(6)} = m_{6}λ_{1} + m_{3}λ_{2} + m_{5}λ_{3} + m_{4}λ_{4} + m_{2}λ_{5} + (k − ⟨λ, θ⟩)λ_{6}

E_{7}  λ^{(7)} = m_{1}λ_{6} + m_{2}λ_{2} + m_{3}λ_{5} + m_{4}λ_{4} + m_{5}λ_{3} + m_{6}λ_{1}
+ (k−⟨λ, θ⟩)λ_{7}.

Proof. Let 𝔤 be of type X_{ℓ} and let H^{(i)} be a coweight from Table 4.1. Consider a vector u ∈ L(λ)_{β} ⊂ L(λ) ⊂ L(k, λ) (i.e., the 𝔥-weight of u is β). Consider u as a vector in (L(k, λ))^{(i)} (which is the same vector space as L(k, λ)). We have a new associated 𝔤̂-action and thus a new 𝔥-action. In particular, by (2.12), (H_{j})^{(i)}(0) = H_{j}(0) + \frac{2k}{∥α_{i}∥^{2}} δ_{i,j}. If we restrict i to the indices appearing in Table 4.1, we see that in each case ∥α_{i}∥^{2} = 2. Thus (H_{j})^{(i)}(0) = H_{j}(0) + kδ_{i,j}. Therefore, when u is thought of as a weight vector under the new 𝔥-action, the (new) 𝔥-weight of u is β + kλ_{i}.

Let v be a highest weight vector for L(k, λ). Theorem 4.1 states that w = σ_{X}^{(i)}(v) is a highest weight vector for (L(k, λ))^{(i)}. As an element of the 𝔤-module L(λ), w has 𝔥-weight μ = σ_{X}^{(i)}(λ). Therefore, the 𝔥-weight of w in terms of the new 𝔥-action is λ^{(i)} = μ + kλ_{i}.

For X_{ℓ} = A_{ℓ}, using (4.1), we have

μ = ∑_{j=1}^{ℓ} m_{j}(λ_{j+i} − λ_{i}) = ∑_{j=1}^{ℓ} m_{j}λ_{j+i} − (∑_{j=1}^{ℓ} m_{j})λ_{i}

= ∑_{j=1}^{ℓ} m_{j}λ_{j+i} − ⟨λ, θ⟩λ_{i}.

Therefore, λ^{(i)} = ∑_{j=1}^{ℓ} m_{j}λ_{j+i} − ⟨λ, θ⟩λ_{i} + kλ_{i}.

For type B_{ℓ}, by (A.1) and (A.2), we have

μ = ∑_{j=1}^{ℓ−1} m_{j}(−2λ_{1} + λ_{j}) + m_{ℓ}(−λ_{1} + λ_{ℓ})

= ∑_{j=1}^{ℓ} m_{j}λ_{j} − 2(∑_{j=1}^{ℓ−1} m_{j})λ_{1} + m_{ℓ}λ_{1}

= ∑_{j=2}^{ℓ} m_{j}λ_{j} − ⟨λ, θ⟩λ_{1}.

Therefore, λ^{(1)} = ∑_{j=2}^{ℓ} m_{j}λ_{j} − ⟨λ, θ⟩λ_{1} + kλ_{1}.

For type C_{ℓ}, by (A.3), we have

μ = ∑_{j=1}^{ℓ} m_{j}(λ_{ℓ−j} − λ_{ℓ}) = ∑_{j=1}^{ℓ} m_{j}λ_{ℓ−j} − (∑_{j=1}^{ℓ} m_{j})λ_{ℓ}

= ∑_{j=1}^{ℓ} m_{j}λ_{ℓ−j} − ⟨λ, θ⟩λ_{ℓ}.

Therefore, λ^{(ℓ)} = ∑_{j=1}^{ℓ} m_{j}λ_{ℓ−j} − ⟨λ, θ⟩λ_{ℓ} + kλ_{ℓ}.

For type D_{ℓ} and i = 1, by (A.4), (A.5), and (A.6), we have

μ = ∑_{j=1}^{ℓ−2} m_{j}(−2λ_{1} + λ_{j}) + m_{ℓ−1}(−λ_{1} + λ_{ℓ}) + m_{ℓ}(−λ_{1} + λ_{ℓ−1})

= ∑_{j=1}^{ℓ−2} m_{j}(−2λ_{1} + λ_{j}) + m_{ℓ−1}(−λ_{1} + λ_{ℓ}) + m_{ℓ}(−λ_{1} + λ_{ℓ−1})

= −⟨λ, θ⟩λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{j} + m_{ℓ−1}λ_{ℓ} + m_{ℓ}λ_{ℓ−1}.

Therefore, λ^{(1)} = −⟨λ, θ⟩λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{j} + m_{ℓ−1}λ_{ℓ} + m_{ℓ}λ_{ℓ−1} + kλ_{1}.

Now, consider type D_{ℓ} when ℓ is odd and i = ℓ − 1. By (A.9), (A.10), (A.11), and (A.12), we have

μ = m_{1}(−λ_{ℓ−1} + λ_{ℓ}) + ∑_{j=2}^{ℓ−2} m_{j}(λ_{ℓ−j} − 2λ_{ℓ−1} + m_{ℓ−1}(λ_{1} − λ_{ℓ−1})

+ m_{ℓ}(−λ_{ℓ−1})

= m_{ℓ−1}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} − (m_{1} + 2 ∑_{j=2}^{ℓ−2} m_{j} + m_{ℓ−1} + m_{ℓ})λ_{ℓ−1} + m_{1}λ_{ℓ}

= m_{ℓ−1}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} − ⟨λ, θ⟩λ_{ℓ−1} + m_{1}λ_{ℓ}.

Therefore, λ^{(ℓ−1)} = m_{ℓ−1}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} − ⟨λ, θ⟩λ_{ℓ−1} + m_{1}λ_{ℓ} + kλ_{ℓ−1}. Considering a Dynkin diagram symmetry (interchanging the roles of nodes ℓ−1 and ℓ), we also have λ^{(ℓ)} = m_{ℓ}λ_{1}+∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j}+m_{1}λ_{ℓ−1}−⟨λ, θ⟩λ_{ℓ}+kλ_{ℓ}.

When ℓ is even and i = ℓ − 1, by (A.14), we have

μ = m_{1}(−λ_{ℓ−1} + λ_{ℓ}) + ∑_{j=2}^{ℓ−2} m_{j}(λ_{ℓ−j} − 2λ_{ℓ−1}) + m_{ℓ−1}(−λ_{ℓ−1})

+ m_{ℓ}(λ_{1} − λ_{ℓ−1})

= m_{ℓ}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} − (m_{1} + 2 ∑_{j=2}^{ℓ−2} m_{j} + m_{ℓ−1} + m_{ℓ})λ_{ℓ−1} + m_{1}λ_{ℓ}

= m_{ℓ}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} − ⟨λ, θ⟩λ_{ℓ−1} + m_{1}λ_{ℓ}.

Therefore, λ^{(ℓ−1)} = m_{ℓ}λ_{1}+∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} −⟨λ, θ⟩λ_{ℓ−1}+m_{1}λ_{ℓ}+kλ_{ℓ−1}. Again, considering a Dynkin diagram symmetry (interchanging the roles of nodes ℓ − 1 and ℓ), we also have

λ^{(ℓ)} = m_{ℓ−1}λ_{1} + ∑_{j=2}^{ℓ−2} m_{j}λ_{ℓ−j} + m_{1}λ_{ℓ−1} − ⟨λ, θ⟩λ_{ℓ} + kλ_{ℓ}.

Finally, let us consider types E_{6} and E_{7}. For type E_{6} and i = 1, by (A.16), we have

μ = m_{1}(−λ_{1} + λ_{6}) + m_{2}(−2λ_{1} + λ_{3}) + m_{3}(−2λ_{1} + λ_{5}) + m_{4}(−3λ_{1} + λ_{4})

+ m_{5}(−2λ_{1} + λ_{2}) + m_{6}(−λ_{1}) = −(m_{1} + 2m_{2} + 2m_{3} + 3m_{4} + 2m_{5} + m_{6})λ_{1} + m_{5}λ_{2} + m_{2}λ_{3} + m_{4}λ_{4}

+ m_{3}λ_{5} + m_{1}λ_{6} = −⟨λ, θ⟩λ_{1} + m_{5}λ_{2} + m_{2}λ_{3} + m_{4}λ_{4} + m_{3}λ_{5} + m_{1}λ_{6}.

Therefore, λ^{(1)} = −⟨λ, θ⟩λ_{1}+m_{5}λ_{2}+m_{2}λ_{3}+m_{4}λ_{4}+m_{3}λ_{5}+m_{1}λ_{6}+kλ_{1}. Now we can use the symmetry of the Dynkin diagram of E_{6} (interchanging nodes 1 and 6 and also interchanging nodes 3 and 5). Therefore, for type E_{6} with i = 6, we have λ^{(6)} = m_{6}λ_{1} + m_{3}λ_{2} + m_{5}λ_{3} + m_{4}λ_{4} + m_{2}λ_{5} − ⟨λ, θ⟩λ_{6} + kλ_{6}.

For type E_{7} and i = 7, by (A.18), we have

μ = m_{1}(−2λ_{7} + λ_{6}) + m_{2}(−2λ_{7} + λ_{2}) + m_{3}(−3λ_{7} + λ_{5}) + m_{4}(−4λ_{7} + λ_{4})

+ m_{5}(−3λ_{7} + λ_{3}) + m_{6}(−2λ_{7} + λ_{1}) + m_{7}(−λ_{7}) = m_{1}λ_{6} + m_{2}λ_{2} + m_{3}λ_{5} + m_{4}λ_{4} + m_{5}λ_{3} + m_{6}λ_{1}

− (2m_{1} + 2m_{2} + 3m_{3} + 4m_{4} + 3m_{5} + 2m_{6} + m_{7})λ_{7} = m_{1}λ_{6} + m_{2}λ_{2} + m_{3}λ_{5} + m_{4}λ_{4} + m_{5}λ_{3} + m_{6}λ_{1} − ⟨λ, θ⟩λ_{7}.

Therefore,

λ^{(7)} = m_{1}λ_{6} + m_{2}λ_{2} + m_{3}λ_{5} + m_{4}λ_{4} + m_{5}λ_{3} + m_{6}λ_{1} − ⟨λ, θ⟩λ_{7} + kλ_{7}. □

Appendix A. Weyl group calculations

All of these calculations repeatedly use the fact that σ_{i}(λ_{j}) = λ_{j} − δ_{i,j}α_{j}, so in particular, σ_{j−1}σ_{j−2} ⋯ σ_{2}σ_{1}(λ_{j}) = λ_{j}. In addition, if C = (a_{ij}) is the Cartan matrix of our simple Lie algebra 𝔤, then α_{i} = ∑_{j} a_{ij}λ_{j}. Also, recall our conventions that λ_{0} = 0, α_{0} = −θ (the negative of the highest long root of 𝔤), and λ_{j} = λ_{j (mod ℓ+1)} for all j ∈ ℤ.

A.1. Type B_{ℓ}. Looking at the Cartan matrix of type B_{ℓ}, we see that α_{i} = −λ_{i−1} + 2λ_{i} − λ_{i+1} for i ≠ ℓ − 1 and α_{ℓ−1} = −λ_{ℓ−2} + 2λ_{ℓ−1} − 2λ_{ℓ}. Also, recall that σ_{B}^{(1)} = σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}σ_{1}.

For λ_{j}, 1 ≤ j ≤ ℓ − 1, we have:

σ_{B}^{(1)}(λ_{j}) (A.1)

= σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}σ_{1}(λ_{j})
= σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j+1}σ_{j}(λ_{j})

= σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j+2}σ_{j+1}(λ_{j−1} − λ_{j} + λ_{j+1}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j+3}σ_{j+2}(λ_{j−1} − λ_{j+1} + λ_{j+2}) = ⋯ = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−1}(λ_{j−1} − λ_{ℓ−2} + λ_{ℓ−1}) = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ}(λ_{j−1} − λ_{ℓ−1} + 2λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}(λ_{j−1} + λ_{ℓ−1} − 2λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−2}(λ_{j−1} + λ_{ℓ−2} − λ_{ℓ−1}) = σ_{1}σ_{2} ⋯ σ_{ℓ−3}(λ_{j−1} + λ_{ℓ−3} − λ_{ℓ−2}) = ⋯ = σ_{1}σ_{2} ⋯ σ_{j}(λ_{j−1} + λ_{j} − λ_{j+1}) = σ_{1}σ_{2} ⋯ σ_{j−1}(2λ_{j−1} − λ_{j}) = σ_{1}σ_{2} ⋯ σ_{j−2}(2λ_{j−2} − 2λ_{j−1} + λ_{j}) = ⋯ = σ_{1}(2λ_{1} − 2λ_{2} + λ_{j}) = −2λ_{1} + λ_{j}.

For λ_{ℓ}, we have:

σ_{B}^{(1)}(λ_{ℓ}) (A.2)

= σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}σ_{1}(λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}σ_{ℓ}(λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}(λ_{ℓ−1} − λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−2}(λ_{ℓ−2} − λ_{ℓ−1} + λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−3}(λ_{ℓ−3} − λ_{ℓ−2} + λ_{ℓ}) = ⋯ = σ_{1}(λ_{1} − λ_{2} + λ_{ℓ}) = −λ_{1} + λ_{ℓ}.

Applying these results to the fundamental roots and highest long root, we have:

σ_{B}^{(1)}(α_{0}) = σ_{B}^{(1)}(−θ) = σ_{B}^{(1)}(−λ_{2}) = 2λ_{1} − λ_{2} = α_{1}
σ_{B}^{(1)}(α_{1}) = σ_{B}^{(1)}(2λ_{j} − λ_{j+1}) = −2λ_{1} + 2λ_{1} − λ_{2} = −θ = α_{0}
σ_{B}^{(1)}(α_{j}) = α_{j}  for 1 < j ≤ ℓ.

A.2. Type C_{ℓ}. Looking at the Cartan matrix of type C_{ℓ}, we see that

α_{i} = −λ_{i−1} + 2λ_{i} − λ_{i+1} (1 ≤ i < ℓ),
α_{ℓ} = −2λ_{ℓ−1} + 2λ_{ℓ}.

Also, recall that σ_{C}^{(ℓ)} = (σ_{ℓ} ⋯ σ_{2}σ_{1})(σ_{ℓ} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1})(σ_{ℓ}).

σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1}(λ_{j}) = σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j}(λ_{j}) = σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j+1}(λ_{j−1} − λ_{j} + λ_{j+1})
= σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j+2}(λ_{j−1} − λ_{j+1} + λ_{j+2}) = ⋯
= σ_{ℓ}(λ_{j−1} − λ_{ℓ−1} + λ_{ℓ}) = λ_{j−1} + λ_{ℓ−1} − λ_{ℓ}

where 1 ≤ j < ℓ. Also, σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1}(λ_{ℓ}) = σ(λ_{ℓ}) = 2λ_{ℓ−1} − λ_{ℓ}, so the above formula works for all j. Applying this result multiple times, we have (for 1 ≤ j < ℓ):

σ_{C}^{(ℓ)}(λ_{j}) = (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1})(σ_{ℓ})(λ_{j}) = ⋯ = (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j})(λ_{j})

+ (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j} − 1)(λ_{j−1} + λ_{ℓ−1} − λ_{ℓ}) = (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{j} − 2)(λ_{j−2} + λ_{ℓ−2} − λ_{ℓ}) = ⋯ = (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(λ_{1} + λ_{ℓ−(j−1)} − λ_{ℓ}) = λ_{ℓ−j} − λ_{ℓ}.

For λ_{ℓ}, we have:

σ_{C}^{(ℓ)}(λ_{ℓ}) = (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1})(σ_{ℓ})(λ_{ℓ})
= (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1})(2λ_{ℓ−1} − λ_{ℓ})

= (σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−1} ⋯ σ_{2}) ⋯ (σ_{ℓ}σ_{ℓ−1}σ_{ℓ−2})(2λ_{ℓ−2} − λ_{ℓ}) = ⋯

= σ_{ℓ}σ_{ℓ−1} ⋯ σ_{1}(2λ_{1} − λ_{ℓ}) = 2λ_{ℓ−1} − 2λ_{ℓ} − 2λ_{ℓ−1} + λ_{ℓ} = −λ_{ℓ}.

Adhering to our convention (i.e., λ_{0} = 0), we have that

(A.3)  σ_{C}^{(ℓ)}(λ_{j}) = λ_{ℓ−j} − λ_{ℓ}

for all j = 1, …, ℓ.

A quick calculation now shows that

σ_{C}^{(ℓ)}(α_{0}) = σ_{C}^{(ℓ)}(−θ) = α_{ℓ}
σ_{C}^{(ℓ)}(α_{j}) = α_{ℓ−j} (for 1 ≤ j < ℓ)
σ_{C}^{(ℓ)}(α_{ℓ}) = −θ = α_{0}.

A.3. Type D_{ℓ} (any rank). Looking at the Cartan matrix of type D_{ℓ}, we see that

α_{i} = −λ_{i−1} + 2λ_{i} − λ_{i+1} (1 ≤ i < ℓ − 2),
α_{ℓ−2} = −λ_{ℓ−3} + 2λ_{ℓ−2} − λ_{ℓ−1} − λ_{ℓ},
α_{ℓ−1} = −λ_{ℓ−2} + 2λ_{ℓ−1},
α_{ℓ} = −λ_{ℓ−2} + 2λ_{ℓ}.

First, recall that σ_{D}^{(1)} = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1}.

As a first step, we determine that action of σ_{1}σ_{2} ⋯ σ_{ℓ} on λ_{j}. For 1 ≤ j < ℓ − 2 we have:

σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{j}) = σ_{1}σ_{2} ⋯ σ_{j}(λ_{j})
= σ_{1}σ_{2} ⋯ σ_{j−1}(λ_{j−1} − λ_{j} + λ_{j+1}) = ⋯
= σ_{1}(λ_{1} − λ_{2} + λ_{j+1}) = −λ_{1} + λ_{j+1}.
σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{ℓ−2}) = σ_{1}σ_{2} ⋯ σ_{ℓ−2}(λ_{ℓ−2})
= σ_{1}σ_{2} ⋯ σ_{ℓ−3}(λ_{ℓ−3} − λ_{ℓ−2} + λ_{ℓ−1} + λ_{ℓ})
= σ_{1}σ_{2} ⋯ σ_{ℓ−4}(λ_{ℓ−4} − λ_{ℓ−3} + λ_{ℓ−1} + λ_{ℓ}) = ⋯
= σ_{1}(λ_{1} − λ_{2} + λ_{ℓ−1} + λ_{ℓ}) = −λ_{1} + λ_{ℓ−1} + λ_{ℓ}.
σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{ℓ−1}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}(λ_{ℓ−1}) = σ_{1}σ_{2} ⋯ σ_{ℓ−2}(λ_{ℓ−2} − λ_{ℓ−1})

154 William J. Cook and Christopher M. Sadowski = σ_{1}σ_{2} ⋯ σ_{ℓ−3}(λ_{ℓ−3} − λ_{ℓ−2} + λ_{ℓ})

= σ_{1}σ_{2} ⋯ σ_{ℓ−4}(λ_{ℓ−4} − λ_{ℓ−3} + λ_{ℓ})
= ⋯ = σ_{1}(λ_{1} − λ_{2} + λ_{ℓ}) = −λ_{1} + λ_{ℓ}.
σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−1}(λ_{ℓ−2} − λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ−2}(λ_{ℓ−2} − λ_{ℓ})
= σ_{1}σ_{2} ⋯ σ_{ℓ−3}(λ_{ℓ−3} − λ_{ℓ−2} + λ_{ℓ−1})
= σ_{1}σ_{2} ⋯ σ_{ℓ−4}(λ_{ℓ−4} − λ_{ℓ−3} + λ_{ℓ−1})
= ⋯ = σ_{1}(λ_{1} − λ_{2} + λ_{ℓ−1}) = −λ_{1} + λ_{ℓ−1}.

For 1 ≤ j ≤ ℓ − 2 we have: σ_{D}^{(1)}(λ_{j}) (A.4)

= σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1}(λ_{j}) = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2} ⋯ σ_{j}(λ_{j}) = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2} ⋯ σ_{j+1}(λ_{j−1} − λ_{j} + λ_{j+1}) = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2} ⋯ σ_{j+2}(λ_{j−1} − λ_{j+1} + λ_{j+2}) = ⋯ = σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2}(λ_{j−1} − λ_{ℓ−3} + λ_{ℓ−2}) = σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{j−1} − λ_{ℓ−2} + λ_{ℓ−1} + λℓ) = (−λ_{1} + λ_{j}) − (−λ_{1} + λ_{ℓ−1} + λ_{ℓ}) + (−λ_{1} + λ_{ℓ}) + (−λ_{1} + λ_{ℓ−1}) = −2λ_{1} + λ_{j}.

σ_{D}^{(1)}(λ_{ℓ−1}) (A.5)

= σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1}(λ_{ℓ−1}) = σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{ℓ−1}) = −λ_{1} + λ_{ℓ}.

σ_{D}^{(1)}(λ_{ℓ}) (A.6)

= σ_{1}σ_{2} ⋯ σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2}σ_{1}(λ_{ℓ}) = σ_{1}σ_{2} ⋯ σ_{ℓ}(λ_{ℓ}) = −λ_{1} + λ_{ℓ−1}.

A quick calculation shows:

σ_{D}^{(1)}(α_{0}) = σ_{D}^{(1)}(−θ) = α_{1}
σ_{D}^{(1)}(α_{1}) = −θ = α_{0}
σ_{D}^{(1)}(α_{j}) = α_{j} (for 2 ≤ j ≤ ℓ − 2)
σ_{D}^{(1)}(α_{ℓ−1}) = α_{ℓ}
σ_{D}^{(1)}(α_{ℓ}) = α_{ℓ−1}.

To help determine the actions of σ_{D}^{(ℓ−1)} and σ_{D}^{(ℓ)} on each λ_{j}, we will first consider σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}:

σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{j}) = σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{j}(λ_{j})

= σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{j+1}(λ_{j−1} − λ_{j} + λ_{j+1})
= σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{j+2}(λ_{j−1} − λ_{j+1} + λ_{j+2}) = ⋯
= σ_{ℓ−2}(λ_{j−1} − λ_{ℓ−3} + λ_{ℓ−2})
= λ_{j−1} − λ_{ℓ−2} + λ_{ℓ−1} + λ_{ℓ}, (1 ≤ j ≤ ℓ − 2),
σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{ℓ−1}) = λ_{ℓ−1},
σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{ℓ}) = λ_{ℓ}.

Next, we consider σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}:

(A.7) σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{j}) = σ_{ℓ−1}(λ_{j−1} − λ_{ℓ−2} + λ_{ℓ−1} + λ_{ℓ})
= λ_{j−1} − λ_{ℓ−1} + λ_{ℓ}, (1 ≤ j ≤ ℓ − 2),
σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{ℓ−1}) = λ_{ℓ−2} − λ_{ℓ−1},
σ_{ℓ−1}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{ℓ}) = λ_{ℓ}.

Finally, consider σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}:

(A.8)  σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{j}) = σ_{ℓ}(λ_{j−1} − λ_{ℓ−2} + λ_{ℓ−1} + λ_{ℓ})
= λ_{j−1} + λ_{ℓ−1} − λ_{ℓ}, (1 ≤ j ≤ ℓ − 2),
σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{ℓ−1}) = λ_{ℓ−1},
σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{1}(λ_{ℓ}) = λ_{ℓ−2} − λ_{ℓ}.

A.4. Type D_{ℓ} (odd rank). Now consider the case when ℓ is odd and recall that

σ_{D}^{(ℓ−1)} = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2})(σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{3}) ⋯ (σ_{ℓ}).

First, consider the case of λ_{1}.

σ_{D}^{(ℓ−1)}(λ_{1}) (A.9)

= (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2})(σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{3}) ⋯ (σ_{ℓ})(λ_{1})

= ⋯ = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(λ_{1}) = −λ_{ℓ−1} + λ_{ℓ}.

(We obtain the last step by using (A.7) above.)

Next, consider the case of λ_{j}, 1 < j ≤ ℓ − 2 and j odd. By applying (A.7) and (A.8) successively, we obtain the following:

σ_{D}^{(ℓ−1)}(λ_{j}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2})(σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{3}) ⋯ (σ_{ℓ})(λ_{j})
= ⋯ = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{j})(λ_{j})
= (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ}σ_{ℓ−2} ⋯ σ_{j−1})(λ_{j−1} − λ_{ℓ−1} + λ_{ℓ})
= (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{j−2})(λ_{j−2} + λ_{ℓ−2} − 2λ_{ℓ})
= (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ}σ_{ℓ−2} ⋯ σ_{j−3})(λ_{j−3} + λ_{ℓ−3} − 2λ_{ℓ−1})
= ⋯ = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(λ_{1} + λ_{ℓ−(j−1)} − 2λ_{ℓ})
= −λ_{ℓ−1} + λ_{ℓ} + λ_{ℓ−j} − λ_{ℓ−1} + λ_{ℓ} − 2λ_{ℓ}
= λ_{ℓ−j} − 2λ_{ℓ−1}.

A similar calculation shows that the same holds for j even and 2 ≤ j ≤ ℓ − 3. Therefore, we have that

(A.10)  σ_{D}^{(ℓ−1)}(λ_{j}) = λ_{ℓ−j} − 2λ_{ℓ−1} (for 1 < j ≤ ℓ − 2).

This leaves the cases j = ℓ − 1 and j = ℓ.

σ_{D}^{(ℓ−1)}(λ_{ℓ−1}) (A.11)

= (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2})(σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{3})

⋯ (σ_{ℓ})(λ_{ℓ−1}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ−1}σ_{ℓ−2})(λ_{ℓ−1}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3})(λ_{ℓ−2} − λ_{ℓ−1}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{ℓ−4})(λ_{ℓ−3} − λ_{ℓ}) = ⋯ = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(λ_{2} − λ_{ℓ}) = λ_{1} − λ_{ℓ−1} + λ_{ℓ} − λ_{ℓ} = λ_{1} − λ_{ℓ−1}.

σ_{D}^{(ℓ−1)}(λ_{ℓ}) (A.12)

= (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3} ⋯ σ_{2})(σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{3}) ⋯ (σ_{ℓ})(λ_{ℓ}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ−1}σ_{ℓ−2})(λ_{ℓ−2} − λ_{ℓ}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ}σ_{ℓ−2}σ_{ℓ−3})(λ_{ℓ−3} − λ_{ℓ−1}) = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1}) ⋯ (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{ℓ−4})(λ_{ℓ−4} − λ_{ℓ}) = ⋯ = (σ_{ℓ−1}σ_{ℓ−2} ⋯ σ_{1})(λ_{1} − λ_{ℓ}) = −λ_{ℓ−1} + λ_{ℓ} − λ_{ℓ} = −λ_{ℓ−1}.

A quick calculation shows:

σ_{D}^{(ℓ−1)}(α_{0}) = σ_{D}^{(ℓ−1)}(−θ) = α_{ℓ−1}
σ_{D}^{(ℓ−1)}(α_{1}) = α_{ℓ}
σ_{D}^{(ℓ−1)}(α_{j}) = α_{ℓ−j} (for 2 ≤ j ≤ ℓ − 1)
σ_{D}^{(ℓ−1)}(α_{ℓ}) = −θ = α_{0}.

Using a Dynkin diagram symmetry, we see that all of these results should still hold if we interchange the labels ℓ − 1 and ℓ. So we also have that

(A.13)  σ_{D}^{(ℓ)}(λ_{1}) = λ_{ℓ−1} − λ_{ℓ}
σ_{D}^{(ℓ)}(λ_{j}) = λ_{ℓ−j} − 2λ_{ℓ} (for 2 ≤ j ≤ ℓ − 2)
σ_{D}^{(ℓ)}(λ_{ℓ−1}) = −λ_{ℓ}
σ_{D}^{(ℓ)}(λ_{ℓ}) = λ_{1} − λ_{ℓ},

and also

σ_{D}^{(ℓ)}(α_{0}) = σ_{D}^{(ℓ)}(−θ) = α_{ℓ}

σ_{D}^{(ℓ)}(α_{j}) = α_{ℓ−j} (for 1 ≤ j ≤ ℓ − 2)
σ_{D}^{(ℓ)}(α_{ℓ−1}) = −θ = α_{0}
σ_{D}^{(ℓ)}(α_{ℓ}) = α_{1}.

A.5. Type D_{ℓ} (even rank). Almost identical calculations reveal that for even ℓ we have the following:

(A.14)  σ_{D}^{(ℓ−1)}(λ_{1}) = −λ_{ℓ−1} + λ_{ℓ}
σ_{D}^{(ℓ−1)}(λ_{j}) = λ_{ℓ−j} − 2λ_{ℓ−1} (for 2 ≤ j ≤ ℓ − 2)
σ_{D}^{(ℓ−1)}(λ_{ℓ−1}) = −λ_{ℓ−1}
σ_{D}^{(ℓ−1)}(λ_{ℓ}) = λ_{1} − λ_{ℓ−1}

and it follows that:

σ_{D}^{(ℓ−1)}(α_{0}) = σ_{D}^{(ℓ−1)}(−θ) = α_{ℓ−1}
σ_{D}^{(ℓ−1)}(α_{1}) = α_{ℓ}
σ_{D}^{(ℓ−1)}(α_{j}) = α_{ℓ−j} (for 2 ≤ j ≤ ℓ − 2)
σ_{D}^{(ℓ−1)}(α_{ℓ−1}) = −θ = α_{0}
σ_{D}^{(ℓ−1)}(α_{ℓ}) = α_{1}.

Again using a Dynkin diagram symmetry, we see that all of these results should still hold if we interchange the labels ℓ − 1 and ℓ. So we also have that:

(A.15)  σ_{D}^{(ℓ)}(λ_{1}) = λ_{ℓ−1} − λ_{ℓ}
σ_{D}^{(ℓ)}(λ_{j}) = λ_{ℓ−j} − 2λ_{ℓ} (for 2 ≤ j ≤ ℓ − 2)
σ_{D}^{(ℓ)}(λ_{ℓ−1}) = λ_{1} − λ_{ℓ}
σ_{D}^{(ℓ)}(λ_{ℓ}) = −λ_{ℓ}

and it follows that (recall −θ = α_{0}):

σ_{D}^{(ℓ)}(α_{j}) = α_{ℓ−j} (for 1 ≤ j ≤ ℓ).

A.6. Exceptional types. Recall that if 𝔤 is of type E_{8}, F_{4} or G_{2}, then 𝔤 has no minuscule weights and so P^{∨} = Q^{∨}. Thus the action of Δ(H, x) is always trivial. So we only need to consider types E_{6} and E_{7}.

The following calculations for types E_{6} and E_{7} were done with the help of a Maple worksheet which is available at http://dimax.rutgers.edu/∼sadowski/LieAlgebraCalculations/index.html. For type E_{6} using H^{(1)}, we have:

(A.16)  σ_{E}^{(1)}(λ_{1}) = −λ_{1} + λ_{6}

σ_{E}^{(1)}(λ_{2}) = −2λ_{1} + λ_{3}
σ_{E}^{(1)}(λ_{3}) = −2λ_{1} + λ_{5}
σ_{E}^{(1)}(λ_{4}) = −3λ_{1} + λ_{4}
σ_{E}^{(1)}(λ_{5}) = −2λ_{1} + λ_{2}
σ_{E}^{(1)}(λ_{6}) = −λ_{1}

and so

σ_{E}^{(1)}(α_{0}) = σ_{E}^{(1)}(−θ) = α_{1}
σ_{E}^{(1)}(α_{1}) = α_{6}
σ_{E}^{(1)}(α_{2}) = α_{3}
σ_{E}^{(1)}(α_{3}) = α_{5}
σ_{E}^{(1)}(α_{4}) = α_{4}
σ_{E}^{(1)}(α_{5}) = α_{2}
σ_{E}^{(1)}(α_{6}) = −θ = α_{0}.

For type E_{6} using H^{(6)}, we have:

(A.17)  σ_{E}^{(6)}(λ_{1}) = −λ_{6}
σ_{E}^{(6)}(λ_{2}) = −2λ_{6} + λ_{5}
σ_{E}^{(6)}(λ_{3}) = −2λ_{6} + λ_{2}
σ_{E}^{(6)}(λ_{4}) = −3λ_{6} + λ_{4}
σ_{E}^{(6)}(λ_{5}) = −2λ_{6} + λ_{3}
σ_{E}^{(6)}(λ_{6}) = −λ_{6} + λ_{1}

and so

σ_{E}^{(6)}(α_{0}) = σ_{E}^{(6)}(−θ) = α_{6}
σ_{E}^{(6)}(α_{1}) = α_{5}
σ_{E}^{(6)}(α_{2}) = α_{4}
σ_{E}^{(6)}(α_{3}) = α_{3}
σ_{E}^{(6)}(α_{4}) = α_{2}
σ_{E}^{(6)}(α_{5}) = α_{1}
σ_{E}^{(6)}(α_{6}) = −θ = α_{0}.

Symmetry of the category of integrable modules 159

For type E_{7} using H^{(7)}, we have:

(A.18)  σ_{E}^{(7)}(λ_{1}) = −2λ_{7} + λ_{6}
σ_{E}^{(7)}(λ_{2}) = −2λ_{7} + λ_{2}
σ_{E}^{(7)}(λ_{3}) = −3λ_{7} + λ_{5}
σ_{E}^{(7)}(λ_{4}) = −4λ_{7} + λ_{4}
σ_{E}^{(7)}(λ_{5}) = −3λ_{7} + λ_{3}
σ_{E}^{(7)}(λ_{6}) = −2λ_{7} + λ_{1}
σ_{E}^{(7)}(λ_{7}) = −λ_{7}

and so

σ_{E}^{(7)}(α_{0}) = σ_{E}^{(7)}(−θ) = α_{7}
σ_{E}^{(7)}(α_{1}) = α_{6}
σ_{E}^{(7)}(α_{2}) = α_{2}
σ_{E}^{(7)}(α_{3}) = α_{5}
σ_{E}^{(7)}(α_{4}) = α_{4}
σ_{E}^{(7)}(α_{5}) = α_{3}
σ_{E}^{(7)}(α_{6}) = α_{1}
σ_{E}^{(7)}(α_{7}) = −θ = α_{0}.

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Department of Mathematical Sciences, Appalachian State University, Boone, NC 28608 cookwj@appstate.edu

Department of Mathematics, Rutgers University, Piscataway, NJ 08854 sadowski@math.rutgers.edu

This paper is available via http://nyjm.albany.edu/j/2009/15-7.html.