Traces of intertwiners for quantum groups and Difference Equations II

P. Etingof, O. Schiffmann, and A. Varchenko

Abstract

In this paper we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights λ, μ, three scalar parameters q, ω, k, and spectral parameters z_{1}, ..., z_{N}, which may be regarded as q-analogs of conformal blocks of the Wess-Zumino-Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined in [FV2]. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations – the Macdonald-Ruijsenaars equation, the q-Knizhnik-Zamolodchikov-Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation λ ↔ μ, k ↔ ω. Thus, our results generalize those of [ES1], dealing with the case q = 1, and [EV, ES2], dealing with the finite dimensional case.

1 Introduction

Let us recall the results of [EV, ES1, ES2]. In [EV], the main objects of study are suitably normalized traces of products of intertwining operators for the quantum group U_{q}(𝔤) corresponding to a finite dimensional simple Lie algebra 𝔤. These traces are functions of two weights of 𝔤, which take values in the endomorphism algebra of the zero weight space of a tensor product of finite dimensional U_{q}(𝔤)-modules. It is shown in [EV] that these traces satisfy four systems of difference equations – the Macdonald-Ruijsenaars (MR), dual MR, quantum Knizhnik-Zamolodchikov-Bernard (qKZB), and dual qKZB equations. It is also shown that the traces are symmetric with respect to the two weights on which they depend.

In [ES2], the results of [EV] are generalized to the “twisted” case, i.e. with traces involving an arbitrary diagram automorphism T of 𝔤. In this case, the four systems of difference equations can still be derived, but the symmetry no longer holds (unless T = 1).

It is an interesting problem to prove similar results in the case when 𝔤 is replaced by the affine Lie algebra 𝔤̃. For the classical case q = 1, this was done earlier. Namely, the untwisted case T = 1 goes back to Bernard [Be], while the twisted case was first considered in [E1] in a special case, and then in [ES1] in general. In the quantum case, an attempt to attack this problem was made in [E2], but only partial results were obtained.

The goal of this paper is to give a complete solution of this problem. More specifically, we derive the Macdonald-Ruijsenaars and qKZB equations and their dual versions for traces of products of intertwining operators for a quantum affine algebra, twisted by a diagram automorphism T of this algebra, and show that the trace functions are symmetric if T = 1. In particular, this gives a representation-theortic proof of the result from [FTV] that the dual difference equation to qKZB equation for 𝔤 = 𝔰𝔩_{2} is also the qKZB equation, with the step and elliptic modulus interchanged, and allows us to generalize this result to other simple Lie algebras.

It turns out that the methods of [EV] and [ES2] apply with only minor modifications, related to the fact that certain completions need to be taken in the affine case. Therefore, we will omit proofs of the statements, referring the reader to appropriate statements in [EV, ES2], and explaining only the required modifications.

To simplify notation, we will work with simple Lie algebras of type A,D,E. The results can be generalized to other types.

The structure of the paper is as follows. Section 2 serves to make the main definitions and set notations. In Section 3, we remind the basic facts about dynamical twists and R-matrices, which are essential ingredients in the paper. In Section 4 we introduce the trace functions – the main characters of the paper. In Section 5 we discuss the MR and dual MR equations, in Section 6 – the qKZB and dual qKZB equations. The untwisted case and the symmetry identity are discussed in Section 7.

The content of this paper has the following connection with other literature. For T = 1, the trace functions are q-analogs of conformal blocks on an elliptic curve, which are elliptic deformations of the q-conformal blocks on ℙ^{1}, defined by Frenkel and Reshetikhin [FR]. If 𝔤 = 𝔰𝔩(2), it is expected that the trace function F_{T}^{V_{1},...,V_{N}} (z_{1}, ..., z_{n}, λ, ω, μ, k) introduced here is (up to simple renomalizations) the elliptic hypergeometric function u(z_{1}, ..., z_{n}, λ, τ, μ, p) introduced in [FV1, FV2, FTV] by an explicit integral. This conjecture is motivated by the fact that both functions satisfy qKZB and dual qKZB equations and the symmetry relations. The conjecture has been checked in the trigonometric limit ([EV], section 8), and in the classical limit q → 1, where it turns into the integral formula for conformal blocks on the elliptic curve. We plan to discuss the general case in a subsequent paper.

Another interesting case is 𝔤 = 𝔰𝔩_{n}, and T is the rotation of the affine Dynkin diagram (which is an n-gon) by r/n of a full circle, where (r, n) = 1. This is the only case when the trace function does not depend on the dynamical parameters λ and μ, and is a function of z_{1}, ..., z_{n} and two “elliptic moduli” ω and k (more precisely, μ takes a finite set of values). The qKZB equation is in this case the elliptic qKZ equation considered in [JM, JMN], which involves Belavin’s elliptic R-matrix. Thus we show that the dual difference equation to this equation (i.e., its “monodromy”) is the difference equation involving special values of Felder’s R-matrix at finitely many points.

Acknowledgements. P.E. is grateful to Igor Frenkel, who proposed to him the problem to derive difference equations for trace functions for quantum affine algebra in 1992 as a topic for a Ph.D. thesis. The work of P.E. was done in part for the Clay Mathematics Institute, as a CMI Prize fellow, and partially supported by the NSF grant DMS-9988796. The work of O.S. was done in part for the Clay Mathematics Institute. A.V. was supported by the NSF grant DMS-9801582. P.E. and A.V. are grateful to MPIM for hospitality. O.S. is grateful to MIT for hospitality.

2 Notations

2.1 Simple Lie algebras

We will use the following notations:

𝔤: a simple complex Lie algebra of type A,D,E.

𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}: its Cartan decomposition.

Γ = {α_{1}, …, α_{r}}: the Dynkin diagram.

Δ ⊂ 𝔥^{∗}: the root system.

Δ_{+}: the set of positive roots.

( , ): the nondegenerate invariant bilinear form on 𝔤, normalized in such a way that the induced form on 𝔥^{∗} (also denoted by ( , )) satisfies the equation (α, α) = 2 for roots α.

θ ∈ Δ^{+}: the maximal root.

ρ = \frac{1}{2} ∑_{α∈Δ^{+}} α: the half sum of positive roots.

h: the Coxeter (=dual Coxeter) number of 𝔤.

e_{i}, f_{i}, h_{i}, i = 1, ..., r: the Chevalley-Serre generators of 𝔤.

Ω_{𝔥} ∈ 𝔥 ⊗ 𝔥: the inverse element to the form ( , ) on 𝔥.

2.2 Affine Lie algebras

Let 𝔤̃ = 𝔤[t, t^{−1}] ⊕ ℂ∂ ⊕ ℂc be the affine algebra Lie associated to 𝔤, where c is the central element and ∂ is the grading element. That is, for Laurent polynomials a(t), b(t) one has

[a(t), b(t)] = [ab](t) + Res_{0}(a′(t), b(t))c,

and [∂, a(t)] = ta′(t), [∂, c] = 0. Let 𝔟̃_{±} be the Borel subalgebras of 𝔤̃ and let 𝔫̃_{±} be their nilpotent radicals. The Cartan subalgebra of 𝔤̃ is 𝔥̃ = 𝔥⊕ℂc⊕ℂ∂. The dual of the Cartan subalgebra is given by 𝔥̃^{∗} = 𝔥^{∗} ⊕ ℂΛ_{0} ⊕ ℂδ where Λ_{0}, δ are defined by the relations ⟨Λ_{0}, 𝔥⟩ = ⟨Λ_{0}, ∂⟩ = ⟨δ, 𝔥⟩ = ⟨δ, c⟩ = 0 and ⟨Λ_{0}, c⟩ = ⟨δ, ∂⟩ = 1.

Set 𝔤̂ = 𝔤[t, t^{−1}] ⊕ ℂc = [𝔤̃, 𝔤̃]. The Lie algebra 𝔤̂ is a Kac-Moody Lie algebra. Let Γ̃ and Δ̃ be the Dynkin diagram and root system of this algebra, respectively. Thus Δ̃ = (Δ+ℤδ)∪(ℤ\{0})δ and Γ̃ = Γ∪{α_{0}} where α_{0} = δ−θ is the affine simple root. Let (a_{ij}) be the Cartan matrix of 𝔤̃. Let e_{i}, f_{i}, h_{i}, i = 0, ..., r, be the standard Kac-Moody generators of 𝔤̂. The span of h_{i} (i.e. the space 𝔥 ⊕ ℂc = 𝔤̂ ∩ 𝔥̃) will be denoted by 𝔥̂.

Let D ∈ 𝔥 ⊕ ℂ∂ be the principal gradation element: [D, e_{i}] = e_{i} and [D, f_{i}] = −f_{i} for all i.

The Lie algebra 𝔤̃ carries a nondegenerate invariant form (,), which restricts to the form on 𝔤 defined above. Such a form is not unique: it depends on one free parameter. We will uniquely determine the form by the condition that (D, D) = 0. (We note that this form is somewhat different from the commonly used form, which satisfies (∂, ∂) = 0.) The restriction of this form to 𝔥̃ is nondegenerate, so it defines a form on the dual space.

Let 𝔥′ be the orthogonal complement to ℂc ⊕ ℂD in 𝔥̃. It is easy to see that the map x → x − \frac{(ρ,x)}{h}c defines an orthogonal isomorphism 𝔥 → 𝔥′.

Let ρ̃ = ρ + hΛ_{0}. Then (ρ̃, α_{i}) = 1 for i = 0, …, r. Thus, ρ̃ corresponds to D under the identification 𝔥̃^{∗} → 𝔥̃ defined by the inner product.

Finally, for any z ∈ ℂ^{∗}, we define an automorphism D_{z} ∈ Aut U(𝔤̂) by D_{z}(e_{i}) = ze_{i} and D_{z}(f_{i}) = z^{−1}f_{i} for any i ∈ Γ̃, and D_{z}c = c. In other words, we have D_{z}(x) = z^{D}xz^{−D}.

2.3 Quantum groups

Let q be a nonzero complex number, such that |q| < 1. We fix ℏ such that q = e^{ℏ}. For any operator A, the expression q^{A} will stand for e^{ℏA}.

Let U_{q}(𝔤̃) be the quantized affine algebra corresponding to 𝔤̃. It is a Hopf algebra over ℂ with generators E_{i}, F_{i}, i = 0, 1, ..., r and q^{h}, h ∈ 𝔥̃ and with relations

q^{x+y} = q^{x}q^{y}, x, y ∈ 𝔥̃, q^{0} = 1

q^{h}E_{i}q^{−h} = q^{α_{i}(h)}E_{i}, q^{h}F_{i}q^{−h} = q^{−α_{i}(h)}F_{i}

E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{q^{h_{i}} − q^{−h_{i}}}{q − q^{−1}} ,

∑_{k=0}^{1−a_{ij}}(−1)^{k} [\begin{matrix}1 − a_{ij} \\ k \end{matrix}]_{q} E_{i}^{1−a_{ij}−k}E_{j}E_{i}^{k} = 0, i ≠ j,

and

∑_{k=0}^{1−a_{ij}}(−1)^{k} [\begin{matrix}1 − a_{ij} \\ k \end{matrix}]_{q} F_{i}^{1−a_{ij}−k}F_{j}F_{i}^{k} = 0, i ≠ j.

[\begin{matrix}n \\ k \end{matrix}]_{q} = \frac{[n]_{q}!}{[k]_{q}![n − k]_{q}!}, [n]_{q}! = [1]_{q} · [2]_{q} · · · · · [n]_{q}, [n]_{q} = \frac{q^{n} − q^{−n}}{q − q^{−1}}

The comultiplication Δ, antipode S and counit ϵ in U_{q}(𝔤̃) are given by

Δ(E_{i}) = E_{i} ⊗ q^{h_{i}} + 1 ⊗ E_{i}, Δ(F_{i}) = F_{i} ⊗ 1 + q^{−h_{i}} ⊗ F_{i},
Δ(q^{h}) = q^{h} ⊗ q^{h},
S(E_{i}) = −E_{i}q^{−h_{i}}, S(F_{i}) = −q^{h_{i}}F_{i}, S(q^{h}) = q^{−h},
ϵ(E_{i}) = ϵ(F_{i}) = 0, ϵ(q^{h}) = 1.

The quantum group U_{q}(𝔤) is the Hopf subalgebra of U_{q}(𝔤̃) generated by E_{i}, F_{i}, i ≥ 1, and q^{h}, h ∈ 𝔥.

If λ̃ ∈ 𝔥̃^{∗}, then we will write qλ̃ for the element q^{hλ̃}, where hλ̃ is the image of λ̃ under the identification 𝔥̃^{∗} → 𝔥̃.

Let U_{q}(𝔤̂) be the subalgebra of U_{q}(𝔤̃) generated by E_{i}, F_{i}, α_{i} ∈ Γ̃ and q^{h}, h ∈ 𝔥 ⊕ ℂc. Let U_{q}(𝔫̂_{±}) be the subalgebra generated by (E_{i})_{α_{i}∈Γ̃} and (F_{i})_{α_{i}∈Γ̃} respectively, and U_{q}(𝔟̂_{±}) be generated by U_{q}(𝔫̂_{±}) and elements q^{h}, h ∈ 𝔥 ⊕ ℂc.

It is known that U_{q}(𝔤̃) has a (generalized) quasitriangular structure, given by the R-matrix

R = R_{0}q^{(c⊗D+D⊗c)/h+Ω_{𝔥′}},

where R_{0} belongs to an appropriate completion of U_{q}(𝔟̂_{+}) ⊗ U_{q}(𝔟̂_{−}), and its degree zero part (under the principal gradation of the first component of the tensor product) is 1. (Here Ω_{𝔥′} denotes the inverse to the form on 𝔥′).

The automorphism D_{z} of U_{q}(𝔤̃) is defined as in the classical case. Given a finite dimensional representation V of U_{q}(𝔤̂), and any z ∈ ℂ^{∗}, define the shifted representation V (z) to be the vector space V with the new action of U_{q}(𝔤̂) defined by π_{V (z)}(a) = π_{V} (D_{z}(a)).

2.4 Dynamical notation

We will use the following notations throughout the paper. Let V_{1}, …, V_{N} be U_{q}(𝔤̂)-modules. For an element A of U_{q}(𝔤̂) and i ∈ {1, …, N}, we denote by A^{i} the action of A on the ith component of the tensor product V_{1} ⊗ ⋯ ⊗ V_{N}. Now let 𝔩 ⊂ 𝔥̂ and suppose that V_{1}, …, V_{N} are 𝔩-semisimple. Let S(λ) : 𝔩^{∗} → U_{q}(𝔤̂) be a function. Then for any i, j ∈ {1, …, N} we denote by S^{i}(λ+h^{(j)}) the endomorphism of V_{1} ⊗ ... ⊗ V_{N} defined by

S^{i}(λ + h^{(j)})(v_{1} ⊗ ⋯ ⊗ v_{N}) = S_{i}(λ + μ_{j})(v_{1} ⊗ ⋯ ⊗ v_{N})

if v_{j} is of weight μ_{j} ∈ 𝔩^{∗}.

3 Dynamical twists and quantum dynamical R-matrices

In this section we construct dynamical twists and R-matrices arising from a Dynkin diagram authomorphism ([JKOS, ESS, ES2]).

Let T ∈ Aut (Γ̃) be an automorphism of order N. Let us denote by B_{T} : U_{q}(𝔤̃) → U_{q}(𝔤̃) the automorphism defined by

B_{T} (E_{i}) = E_{T (i)}, B_{T} (F_{i}) = F_{T (i)}, B_{T} (h_{i}) = h_{T (i)},

and B_{T} (D) = D. For brevity, we will denote B_{T} simply by B, assuming that T has been fixed.

It is easy to see that B is an orthogonal automorphism of 𝔥̃, which preserves 𝔥′, c and D. In particular, it preserves the principal grading, i.e D_{z}B = BD_{z} for any z ∈ ℂ^{∗}.

Let 𝔩̃ = ( ∑_{α∈Γ̃} ℂ(α − T α))^{⊥} = 𝔥̃^{B} ⊂ 𝔥̃. Set 𝔩 = 𝔩̃ ∩ 𝔥′. Since B is orthogonal, the form is nondegenerate on 𝔩̃ and on 𝔩. We identify 𝔩 (resp. 𝔩̃) with their duals using ( , ). Let 𝔥_{0} ⊂ 𝔥′ be the orthogonal complement of 𝔩̃ in 𝔥̃, and let Ω_{𝔥_{0}} ∈ 𝔥_{0} ⊗ 𝔥_{0} and Ω𝔩̃ ∈ 𝔩̃⊗ 𝔩̃ be the inverse elements to the restriction of ( , ) to 𝔥_{0} and 𝔩̃ respectively.

Define the Cayley transform C_{T} : 𝔥_{0} → 𝔥_{0} of T by C_{T} = \frac{B+1}{B−1}. This is a skew-symmetric operator on 𝔥_{0}.

Let I_{±} ⊂ U_{q}(𝔟̃_{±}) be the kernel of the projection U_{q}(𝔟̃_{±}) → U_{q}(𝔥̃) on the elements of zero degree. Set

A_{λ} = ℂ[q^{±2(λ,α_{1})}, …, q^{±2(λ,α_{r})}].

We define an automorphism B ∈ Aut (U_{q}(𝔤̃)) in the same fashion as in the classical case. Then BD_{z} = D_{z}B for any z ∈ ℂ^{∗}. Let

Z = −\frac{1}{2}((1 + C_{T}) ⊗ 1)Ω_{𝔥_{0}}.

Let q^{−2ω} be a formal parameter. More precisely, we will work over the ring of power series in the expression q^{−2ω}.

Let λ̃ = λ + ωρ̃/h, λ ∈ 𝔩^{∗}. (This makes sense, since λ̃ will occur only in the expression q^{−2(λ,α_{i})}, which expresses via the formal parameter q^{−2ω}).

Theorem 3.1 ([ESS]). There exists a unique 𝔩̃-invariant element J_{T} (λ̃) = ∑_{m≥0} J_{T,m}(λ̃), such that

J_{T,0} = q^{Z}, J_{T,m} ∈ ((I_{−}[−m] ⊗ I_{+}[m])𝔩̃⊗ A_{λ})[[q^{−2ω}]]

(where I_{±}[m] = {x ∈ I_{±}, [ρ, x] = mx}), satisfying the ABRR equation ([ABRR])

R^{21}(Ad(q^{2λ̃}B) ⊗ 1)J_{T} (λ̃) = J_{T} (λ̃)q^{Ω𝔩̃}.  (3.1)

Moreover, J_{T} (λ̃) satisfies the following shifted 2-cocycle condition :

J_{T}^{12,3}(λ̃)J_{T}^{12}(λ̃ + \frac{1}{2}h^{(3)}) = J_{T}^{1,23}(λ̃)J_{T}^{23}(λ̃ − \frac{1}{2}h^{(1)}).  (3.2)

Below we will need a slightly modified version of J. Namely, we set

𝕁_{T} (λ̃) = J_{T} (λ̃ + \frac{1}{2}(h^{(1)} + h^{(2)})).

Remark. At q^{−2ω} = 0, both J_{T} and 𝕁_{T} specialize to R_{0}^{21}.

Now define the exchange operator

ℝ_{T} (λ̃) = 𝕁_{T} (λ̃)^{−1}R^{21}𝕁_{T}^{21}(λ̃),  (3.3)

Proposition 3.1. The element ℝ_{T} (z, λ̃) satisfies the quantum dynamical Yang-Baxter equation :

\begin{aligned}ℝ_{T}^{12} & (λ̃)ℝ_{T}^{13}(λ̃ + h^{(2)})ℝ_{T}^{23}(λ̃) \\ & = ℝ_{T}^{23}(λ̃ + h^{(1)})ℝ_{T}^{13}(λ̃)ℝ_{T}^{12}(λ̃ + h^{(3)}). \end{aligned}  (3.4)

Proof. By definition, ℝ_{T} (λ̃)^{21} is the twist of the quantum R-matrix R by the noncommutative dynamical 2-cocycle 𝕁_{T} (λ̃). This implies the statement.

3.1 Spectral parameter

Now let us introduce a spectral parameter. Consider the expression

𝕁_{T} (z, λ̃) = (D_{z} ⊗ 1)𝕁_{T} (λ̃).

It is clear that we have

𝕁_{T} (z, λ̃) ∈ ((U_{q}(𝔟̃_{−}) ⊗ U_{q}(𝔟̃_{+}))𝔩̃[[z^{−1}]] ⊗ A_{λ})[[q^{−2ω}]].

Since BD_{z} = D_{z}B, we have

(D_{z_{1}} ⊗ D_{z_{2}})𝕁_{T} (λ̃) = (D_{z_{1}/z_{2}} ⊗ 1)𝕁_{T} (λ̃) = 𝕁_{T} (z_{1}/z_{2}, λ̃).  (3.5)

Now set

ℝ_{T} (z, λ̃) = (D_{z} ⊗ 1)ℝ_{T} (λ̃)  (3.6)

Note that ℝ_{T} (z, λ̃) belongs to the space ^{1}

(U_{q}(𝔤̃)⊗U_{q}(𝔤̃)((z^{−1}))ˆ⊗A_{λ})[[q^{−2ω}]].

Proposition 3.2. The element ℝ_{T} (z, λ̃) satisfies the quantum dynamical Yang-Baxter equation with spectral parameters :

\begin{aligned}ℝ_{T}^{12} & (z_{1}/z_{2}, λ̃)ℝ_{T}^{13}(z_{1}/z_{3}, λ̃ + h^{(2)})ℝ_{T}^{23}(z_{2}/z_{3}, λ̃) \\ & = ℝ_{T}^{23}(z_{2}/z_{3}, λ̃ + h^{(1)})ℝ_{T}^{13}(z_{1}/z_{3}, λ̃)ℝ_{T}^{12}(z_{1}/z_{2}, λ̃ + h^{(3)}). \end{aligned}  (3.7)

Proof. Equation (3.7) is obtained by applying D_{z_{1}} ⊗ D_{z_{2}} ⊗ D_{z_{3}} to (3.4) and using (3.5). ■

Now let V be a finite dimensional representation of U_{q}(𝔤̂). Since the level of such a representation is always zero (see e.g. [CP]), we can evaluate the element R and hence ℝ_{T} (z, λ̃) in the tensor product V ⊗V. The obtained operator ℝ_{T}^{V V} (z, λ, ω) is a finite dimensional solution of the quantum dynamical Yang-Baxter equation with a spectral parameter z and dynamical parameter λ, which belongs to (End(V ⊗ V)((z^{−1})) ⊗ A_{λ})[[q^{−2ω}]].

In fact, the series defining ℝ_{T}^{V V} has a nonempty region of convergence. More precisely, the coefficients to all powers of q^{−2ω} are convergent series in z^{−1} outside a common disk, and for any point of this disk the resulting series in q^{−2ω} (with numerical coefficients) is convergent if the real part of ω is small enough.

1Here and below, the sign ˆ⊗ will denote a completed tensor product. The nature of completion in each situation should be clear from the context.

To show this, for any two finite-dimensional U_{q}(𝔤̂)-modules V and W we set

R^{V,W} (z) = (π_{V} ⊗ π_{W})((D_{z} ⊗ 1)R).

This is a power series in z. Then we have the following result.

Theorem 3.2 ([KS]). The power series R^{V,W} (z) converges to a holomorphic function in the neighborhood of z = 0, which extends to a meromorphic function on ℂ.

Using this theorem and the Birkhoff’s theory of q-difference equations (see e.g. [EFK], Chapter 10), we get from the ABRR equation that the series ℝ_{T}^{V V} defines a meromorphic function of z, λ, ω in the region z ≠ 0, Re(ω) < 0.

Remark 1. Another proof of Theorem 3.2 can be found in [EM].

Remark 2. We note that after evaluation in V ⊗ W, in the case T = 1, the defining equation 3.1 for 𝕁 becomes the quantum KZ equation of Frenkel and Reshetikhin [FR]. In the case T ≠ 1, this is a twisted version of the quantum KZ equation.

Remark 3. If 𝔤 = 𝔰𝔩_{n}, V is its vector representation, and T = 1, then the dynamical R-matrix ℝ_{T}^{V V} (z, λ, ω) is gauge equivalent to Felder’s elliptic R-matrix (see [FR, Mo]). If T is the rotation of the Dynkin diagram of 𝔤̂ by r/n of a full circle, (r, n) = 1, then R_{T}^{V V} does not depend on λ (as 𝔩 = 0), and is a usual R-matrix with a spectral parameter. Namely, as was shown in [T], it is the Belavin R-matrix.

4 Twisted traces of intertwiners

4.1 Intertwiners

If W is a graded space, we will denote by W [ν] the weight subspace of weight ν in W.

For any μ ∈ 𝔥^{′∗} and k ∈ ℂ, let μ̃ = μ + kρ̃/h. Let Mμ̃ be the U_{q}(𝔤̃)-Verma module of weight μ̃ (it has level k). Let xμ̃ be a highest weight vector in this module. Let M ∗ ; μ̃ be the restricted dual module of Mμ̃, and x∗ ; μ̃ ∈ M ∗ ; μ̃[−μ̃] be such that ⟨x∗ ; μ̃, xμ̃⟩ = 1.

Let V be a finite-dimensional U_{q}(𝔤̂)-module. Let β ∈ 𝔥̂^{∗}.

Lemma 4.1. [FR] Assume that M_{μ̃+β}^{∗} is irreducible. Then

Hom_{U_{q}(𝔤̂)}(Mμ̃, M_{μ̃+β} ˆ⊗V) ≃ V [β]

where the isomorphism is as follows : to Φ : Mμ̃ → M_{μ̃+β} ˆ⊗V we associate the element ⟨Φ⟩ = ⟨x_{μ̃+β}^{∗}, Φ(xμ̃)⟩.

We will denote the operator corresponding to v ∈ V [β] under the correspondence of Lemma 4.1 by Φv ; μ̃. Define also the intertwining operator Φv ; μ̃(z) : Mμ̃ → M_{μ̃+β} ˆ⊗V (z). This operator can be regarded as an infinite in both directions Laurent series in z with operator coefficients (for the coefficients, the completion of the tensor product is unnecessary).

Let V_{1}, …, V_{N} be finite-dimensional U_{q}(𝔤̂)-modules and let v_{1} ∈ V_{1}[ν_{1}], …, v_{N} ∈ V_{N}[ν_{N}]. For generic values of μ ∈ 𝔥^{′∗} and k we can consider the composition

Φv_{1},...v_{N} ; μ̃ (z_{1}, …, z_{N}) = Φ_{μ̃−ν_{2}−...−ν_{N}}^{v_{1}} (z_{1}) ⋯ Φv_{N} ; μ̃ (z_{N}) :

Mμ̃ → M_{μ̃−ν_{1}−...−ν_{N}} ˆ⊗V_{1} ˆ⊗ ⋯ ˆ⊗V_{N},

It is easy to see that, for a given μ̃, the matrix coefficients of Φv_{1},... ,v_{N} ; μ̃ (z_{1}, …, z_{N}) belong to the space z_{1}^{l_{1}} ⋯ z_{N}^{l_{N}} ℂ[[\frac{z_{2}}{z_{1}} , …, \frac{z_{N}}{z_{N−1}} ]] for some l_{1}, …, l_{n} ∈ ℤ.

Theorem 4.1 ([FR],[EFK]). The matrix elements of the operator valued

series Φv_{1},... ,v_{N} ; μ̃ (z_{1}, …, z_{N}) converge in the the region z_{1} ≫ z_{2} ≫ ⋯ ≫ z_{N}

and extend to meromorphic functions on (ℂ^{∗})^{N}.

4.2 Trace functions

For any μ ∈ 𝔥^{′∗} we define the linear operator B : Mμ̃ → M_{Bμ̃} by setting B(u · xμ̃) = B(u) · x_{Bμ̃} for any u ∈ U_{q}(𝔫̃_{−}).

Consider the following trace function, which will be the main object of study of this paper.

Ψ_{T}^{v_{1},... ,v_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) = Tr_{|Mμ̃}(Φ_{Bμ̃}^{v_{1},... ,v_{N}} (z_{1}, …, z_{N})Bq^{2λ̃}),

where λ̃ = λ+ωρ̃/h, and λ ∈ 𝔩^{∗}. Note that Ψ_{T}^{v_{1},... ,v_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) = 0 unless ∑_{i} ν_{i} = Bμ − μ, and that Ψ is independent of z_{i} if N = 1. Moreover, given ν_{1}, …, ν_{N}, the set of all μ ∈ 𝔥^{′∗} satisfying this condition is an 𝔩^{∗}- homogeneous space. By Theorem 4.1, for (almost) every such fixed value of μ and k, the series Ψ_{T}^{v_{1},... ,v_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) converges in the region z_{1} ≫ ⋯ ≫ z_{N} and extends to a meromorphic function on (ℂ^{∗})^{N} with values in q^{2(λ,μ)}A_{λ}[[q^{−2ω}]].

Now we will consider k as a formal parameter. Namely, we set

A_{μ}^{−} = ℂ[q^{−2(μ,α_{1})}, …, q^{−2(μ,α_{r})}], A_{μ,k} = A_{μ}^{−}[[q^{−2k}]].

The following lemma is easy to prove.

Lemma 4.2. The function Ψ_{T}^{v_{1},... ,v_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) has a series expansion, which lies in (V_{1} ⊗⋯⊗V_{N})^{𝔩}⊗q^{2(λ,μ)}A_{λ,ω} ˆ⊗A_{μ,k}, where μ̃ = μ+kρ̃/h and λ̃ = λ + ωρ̃/h.

We also define the following universal trace function, which takes values in (V_{1} ⊗ ⋯ ⊗ V_{N})^{𝔩} ⊗ (V_{N}^{∗} ⊗ ⋯ ⊗ V_{1}^{∗})^{𝔩} ⊗ q^{2(λ,μ)}A_{λ,ω} ˆ⊗A_{μ,k} :

\begin{aligned}Ψ_{T}^{V_{1},... ,V_{N}} & (z_{1}, …, z_{N}, λ, ω, μ, k) \\ & = ∑_{v_{i}∈B_{i}} Ψ_{T}^{v_{1},... ,v_{N}} (z_{N}, …, z_{1}, λ, ω, μ, k) ⊗ v_{N}^{∗} ⊗ ⋯ ⊗ v_{1}^{∗} \end{aligned}

where B_{i} is a homogeneous basis of V_{i} and {v_{i}^{∗}}_{v_{i}∈B_{i}} is the dual basis of V_{i}^{∗}. This function is defined for μ ∈ 𝔥^{′∗}, but has support on finitely many shifted copies of 𝔩^{∗}.

It is more convenient to renormalize the function Ψ_{T}^{V_{1},... ,V_{N}} as follows. Define

𝕁_{T}^{1···N}(λ̃) = 𝕁_{T}^{1,2···N}(λ̃) ⋯ 𝕁_{T}^{N−1,N}(λ̃),
𝕁_{T}^{1···N}(z_{1}, …, z_{N}, λ̃) = (D_{z_{1}} ⊗ ⋯ ⊗ D_{z_{N}})𝕁_{T}^{1···N}(λ̃).

It is easy to check that

𝕁_{T}^{1···N}(z_{1}, …, z_{N}, λ̃) ∈ U_{q}(𝔤̃)^{⊗N}[[\frac{z_{2}}{z_{1}} , …, \frac{z_{N}}{z_{N−1}} ]]ˆ⊗A_{λ,ω}.

Further, put ℚ_{T} (λ̃) = m_{21}((1 ⊗ S)[(R^{21})^{−1}𝕁_{T} (λ̃)]) (here m_{21} is the multiplication of components, i.e. m_{21}(a ⊗ b) := ba). This is a formal power series lying in U_{q}(𝔤̃)ˆ⊗A_{λ,ω}. Note that ℚ_{T} (λ̃) is B-invariant and D_{z}-invariant. If T = 1, we will drop the subscript T and denote the corrseponding elements 𝕁_{T} , ℚ_{T}, ℝ_{T} simply by 𝕁, ℝ, ℚ.

Finally, let

δ_{q}^{T} (λ, ω) = (Tr_{|M_{−ρ̃}}(Bq^{2(λ+ωρ̃/h)}))^{−1}

be the twisted Weyl denominator. It can be written as an infinite product of binomial factors (see [ES1]).

Define the renormalized trace function by

F_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k)

= [ℚ^{−1}(μ + h^{(∗1···∗N)}, k)^{(∗N)} ⊗ ⋯ ⊗ ℚ^{−1}(μ + h^{(∗1)}, k)^{(∗1)}]
φ_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, −μ − ρ, −k − h)

where

\begin{aligned}φ_{T}^{V_{1},... ,V_{N}} & (z_{1}, …, z_{N}, λ, ω, μ, k) \\ & = 𝕁_{T}^{1···N}(z_{1}, …, z_{N}, λ, ω)^{−1}Ψ_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k)δ_{q}^{T} (λ, ω). \end{aligned}

(here ℚ(λ, ω) := ℚ(λ̃), 𝕁(λ, ω) := 𝕁(λ̃))). Note that F_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) is in the space

E = (V_{1} ⊗ ⋯ ⊗ V_{N})^{𝔩} ⊗ (V_{N}^{∗} ⊗ ⋯ ⊗ V_{1}^{∗})^{𝔩}((\frac{z_{2}}{z_{1}} , …, \frac{z_{N}}{z_{N−1}} ))ˆ⊗q^{−2(λ,μ)}A_{λ,ω} ˆ⊗A_{μ,k}.

Remark 1. Here we regard the function F_{T} as a formal power series. It is expected, but, as far as we know, not proved in general, that this series is actually convergent in some region, defining an analytic function. In the case T = 1, it is expected that F_{T} has an integral representation coming from the free bosonic realization, which could be used for a proof of analyticity. Such a presentation has not been worked out in general. However, in the case of 𝔤 = 𝔰𝔩_{2}, such presentation is known ([Ko1, Ko2]), which means that in this case the function F_{T} for T = 1 is analytic.

Remark 2. It is important to note that the element ℚ_{T} (λ̃) defined here is somewhat different from the one used in [EV, ES2]. This is done because the formula m_{21}((1 ⊗ S^{−1})(𝕁_{T} (λ))) used in [EV, ES2] does not make sense for affine Lie algebras due to appearance of infinite summations. Indeed, at q^{−2ω} = 0 we have 𝕁 = R_{0}^{21}, so already the computation of the leading term of ℚ_{T} as in [EV, ES2] in a finite dimensional representation V would involve evaluating the R-matrix of V, V^{∗} at z = 1 and multiplying the dual to the second component by the first component. However, this operation contains infinite summations, which are manifestly illegal, since the R-matrix R_{V,V^{∗}}(z) has a pole at z = 1. On the other hand, the definition of ℚ_{T} (λ) given here makes sense, because the expression (R_{0}^{21})^{−1}𝕁 is a Taylor series in q^{−2ω}, whose coefficients involve only finite summations.

Now we explain the connection between our definition and that of [ES2] in more detail.

In the finite dimensional case, both definitions make sense, and the relationship is given by the formula ℚ_{T} (λ) = S(u)^{−1}ℚ_{T}^{ES2}(λ), where u is the Drinfeld element [D], u = m_{21}((1 ⊗ S)(R)), The trace functions F_{T}^{V_{1},...,V_{N}} can be defined using either definition: the difference would be just a constant factor.

In the affine case, neither of the factors S(u)^{−1} and ℚ_{T}^{ES2}(λ̃) makes sense because of infinite summations, but the product S(u)^{−1}ℚ_{T} (λ̃) does. So we use this product as a substitute for the non-existenct ℚ_{T}^{ES2}(λ̃).

Remark 3. One may also consider traces over integrable modules rather than Verma modules. In the particular case 𝔤 = 𝔰𝔩_{n}, N = 1, V_{1} = S^{kn}ℂ^{n} they are affine analogues of Macdonald’s polynomials and were considered in [EK1].

5 The twisted Macdonald-Ruijsenaars equations

5.1 The equations

Let W be an integrable lowest weight U_{q}(𝔤̃)-module of level η (so η is a nonpositive integer). Consider the following difference operator with respect to the variable λ acting on functions in the space E:

\begin{matrix}D_{W}^{T} \\ = ∑_{ν∈𝔩^{∗},a∈ℂ}Tr_{|W [ν+aδ+ηρ̃/h]}(ℝ_{T}^{W V_{1}}(z_{1}^{−1}, λ + h^{(2···N)}, ω) ⋯ ℝ_{T}^{W V_{N}} (z_{N}^{−1}, λ, ω))q^{−2ka}𝕋_{ν} \end{matrix} (5.1)

where 𝕋_{ν}f(λ) = f(λ + ν), and ℝ(z, λ, ω) := ℝ(z, λ̃). In the above, we only consider the trace of the “diagonal block” of the product ℝ_{T}^{W V_{1}}(z_{1}^{−1}, λ + h^{(2···N)}, ω) ⋯ ℝ_{T}^{W V_{N}} (z_{N}^{−1}, λ, ω), i.e the part that preserves W [ν + aδ + ηρ̃/h].

Theorem 5.1 (Macdonald-Ruijsenaars equations).

\begin{aligned}D_{W}^{T} F_{T}^{V_{1},... ,V_{N}} & (z_{1}, … z_{N}, λ, ω + η, μ, k) \\ & = χ_{W} (q^{−2μ̃})F_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k), \end{aligned}  (5.2)

where χ_{W} (x) = ∑ dim W [ν]x^{ν} is the character of W and where D_{W}^{T} acts on the variable λ.

Remark 1. Note that equation (5.2) is an infinite difference equation. That is, the difference operator on the left hand side involves an infinite linear combination of shifts by different weights. Nevertheless, it makes sense as an operator on power series in the formal parameters q^{−2ω} and q^{−2k}, since the coefficient of any fixed power of q^{−2k} is an honest (finite) difference operator. Such operators were previously considered in [E3]. Abusing terminology, we will call them “difference operators”, without specifying that they are infinite.

Remark 2. We do not know a simple explicit formula for the operator 5.1 in any nontrivial case.

Let us denote by W^{B} the twist of W by B: as a vector space W = W^{B} and the U_{q}(𝔤̃)-action is given by π_{W^{B}}(u) = π_{W} (B^{−1}(u)). Now suppose that W ≃ W^{B} as U_{q}(𝔤̃)-modules and let us consider the intertwiner W → W^{B} which preserves the lowest weight vector (abusing notation, we will denote it by the same letter B). This endows W with the structure of a module over ℂ[⟨B⟩] ⋉ U_{q}(𝔤̃), where ⟨B⟩ is the cyclic group generated by B.

Consider the following difference operator with respect to the variable μ̃ acting on functions in the space E:

\begin{matrix}D_{W}^{∨,T} \\ = ∑_{ν,a} Tr_{|W [ν+aδ+ηρ̃/h]}(ℝ^{W V_{N}^{∗}} (z_{N}^{−1}, μ + h^{(∗1···∗N−1)}, k) ⋯ ℝ^{W V_{1}^{∗}}(z_{1}^{−1}, μ, k)B_{W} )q^{−2ωa}𝕋_{ν}^{∨} \end{matrix} (5.3)

where the sign ∨ means that the operators act on functions of μ rather than λ.

Theorem 5.2 (Dual Macdonald-Ruijsenaars equations).

\begin{aligned}D_{W}^{∨,T} F_{T}^{V_{1},... ,V_{N}} & (z_{1}, …, z_{n}, λ, ω, μ, k + η) \\ & = Tr_{|W} (q^{−2λ̃}B)F_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k). \end{aligned}  (5.4)

Remark. Any B-invariant integrable lowest weight U_{q}(𝔤̃)-module is a direct sum of modules \overline{V} − ; ν̃_{0} := ⨁_{ν̃∈⟨B⟩ν̃_{0}} V − ; ν̃, where V − ; ν̃ is the irreducible lowest weight module of lowest weight −ν̃, and ν̃_{0} is a dominant integral weight. It is easy to see that both sides of (5.4) identically vanish when W = \overline{V} ν̃_{0} and ν̃_{0} ∉ 𝔩^{∗} (i.e when B(ν̃_{0}) ≠ ν̃_{0}).

5.2 The proofs

The proof of Theorem 5.1 is an extension of the proofs of Theorem 2.1 of [ES2] and Theorem 1.1 of [EV]. We first introduce the notion of (twisted) radial part.

Proposition 5.1. (see [E3]) Let V be a finite-dimensional U_{q}(𝔤̂)-module. For any element X ∈ U_{q}(𝔤̂) there exists a unique difference operator (with respect to the variable λ) D_{X}^{T} acting on power series in the space V^{𝔩}⊗q^{2(λ,μ)}A_{λ,ω} such that we have

Tr_{|M_{μ}}(Φ_{Bμ}^{V}Xq^{2λ̃}B) = D_{X}^{T}Tr_{|M_{μ}}(Φ_{Bμ}^{V}q^{2λ̃}B).

The operator D_{X}^{T} is called the twisted radial part of X.

Let W be any lowest-weight integrable U_{q}(𝔤̃)-module. Consider the expression

C_{W} = Tr_{|W} (1 ⊗ π_{W})(R^{21}R(1 ⊗ q^{2ρ̃})).

Set M_{W}^{T} = D_{C_{W}}^{T}.

Proposition 5.2. We have

1. M_{W}^{T} M_{W′}^{T} = M_{W′}^{T}M_{W}^{T} for any two integrable modules W, W′,

2. For any finite-dimensional U_{q}(𝔤̂)-module V we have M_{W}^{T} Ψ_{T}^{V} (z, λ, ω, μ, k) = χ_{W} (q^{2(μ̃+ρ̃)})Ψ_{T}^{V} (λ, ω, μ, k) where χ_{W} (x) = ∑_{ν} dim W [ν]x^{ν} is the character of W.

Proof. See [EK], [EV].

The computation of the operator M_{W}^{T} is now identical to that in [ES2], Section 3. We obtain

\begin{aligned}M_{W}^{T} Ψ_{V}^{T} & (λ, ω, μ, k) = ∑_{ν,a} \{q^{2(k+h)a} \frac{δ_{q}^{T} (λ + ν, ω + η)}{δ_{q}^{T} (λ, ω)} × \\ & Tr_{|W [ν+aδ+ηρ̃/h]}(ℝ_{T}^{W V} (z^{−1}, λ, ω))Ψ_{T}^{V} (z, λ + ν, ω + η, μ, k)\}. \end{aligned} (5.5)

Theorem 5.1 now follows from (5.5) and from the fusion identity

𝕁_{T}^{1···N}(λ̃)^{−1}ℝ_{T}^{0,1···N}(λ̃))𝕁_{T}^{1···N}(λ̃ + h^{(0)} = ℝ_{T}^{01}(λ̃ + h^{(2···N)}) ⋯ (ℝ_{T}^{0N}(λ̃)). (5.6)

The proof of Theorem 5.2 is completely parallel to the proofs of Theorem 2.3 in [ES2] and Theorem 1.2 in [EV].

6 The twisted qKZB and dual qKZB equations

Let V_{1}, …, V_{N} be irreducible finite-dimensional U_{q}(𝔤̂)-modules. For each j ∈ {1, …, N} we define the following operators acting on the space E :

D_{j}^{T} =q_{∗j}^{−2μ−C_{𝔥′}} q_{∗j,∗1}^{−2Ω_{𝔥′}} ⋯ q_{∗j,∗j−1}^{−2Ω_{𝔥′}},  (6.1)
K_{j}^{T} =ℝ_{T}^{j+1,j}(\frac{z_{j+1}}{z_{j}} , λ + h^{(j+2,... ,N)}, ω)^{−1} ⋯ ℝ_{T}^{Nj}(\frac{z_{N}}{z_{j}} , λ, ω)^{−1}Γ_{j} ×

ℝ_{T}^{j1}(\frac{z_{j}}{z_{1}} , λ + h^{(2... ,j−1)} + h^{(j+1... ,N)}, ω) × ⋯ ℝ_{T}^{j,j−1}(\frac{z_{j}}{z_{j−1}} , λ + h^{(j+1... ,N)}, ω) (6.2)

where C_{𝔥′} = m_{12}(Ω_{𝔥′}) ∈ U(𝔥′) is the quadratic Casimir element for 𝔥′, and where Γ_{j}f(λ) = f(λ + h^{(j)}).

Theorem 6.1 (qKZB equations). For all j = 1, …, N, we have :

\begin{aligned}F_{T}^{V_{1},... ,V_{N}} & (z_{1}, …, pz_{j}, …, z_{N}, λ, ω, μ, k) \\ & = (D_{j}^{T} ⊗ K_{j}^{T})F_{T}^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k), \end{aligned}  (6.3)

where p = q^{2k}.

Similarly, consider the operators

\begin{matrix}D_{j}^{∨,T} = & q_{j}^{−2λ−C_{𝔩}}q_{j,j+1}^{−Ω_{𝔩}} ⋯ q_{j,N}^{−Ω_{𝔩}}, \\ K_{j}^{∨,T} = & ℝ^{∗j−1,∗j}(\frac{z_{j−1}}{z_{j}} , μ + h^{(∗1···∗j−2)}, k)^{−1} ⋯ ℝ^{∗1,∗j}(\frac{z_{1}}{z_{j}} , μ, k)^{−1}Γ_{B^{−1}(j)}^{∗} × \\ ℝ^{∗j,∗N}(\frac{z_{j}}{z_{N}} , μ + h^{(∗j+1···∗N−1)} + h^{(∗1···∗j−1)}, k) × ⋯ × \\ ℝ^{∗j,∗j+1}(\frac{z_{j}}{z_{j+1}} , μ + h^{(∗1···∗j−1)}, k), \end{matrix} (6.4)

where C_{𝔩} = m_{12}(Ω_{𝔩}) ∈ U(𝔩) and where (Γ_{B^{−1}(j)}^{∗}f)(μ) = f(μ + B^{−1}(h^{(∗j)})).

Theorem 6.2 (Dual qKZB equations). For all j = 1 …, N, we have :

\begin{matrix}F_{T}^{V_{1},... ,V_{j},... ,V_{N}} (z_{1}, …, sz_{j}, …, z_{N}, λ, , ω, μ, k) \\ = (D_{j}^{∨,T} ⊗ K_{j}^{∨,T})F_{T}^{V_{1},... ,V_{j}^{B},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k), \end{matrix}  (6.5)

where s = q^{2ω}, and we identify V_{j} with V_{j}^{B}.

The proof of Theorem 5.1 is very similar to the proof of Theorem 2.2 of [ES2] and Theorem 1.2 of [EV] (with modifications described in Remark 2 in Section 4). The proof of the dual qKZB equation runs parallel to [ES2], Section 6.

7 The symmetry identity

In this section we assume that T = 1. Thus 𝔩 = 𝔥′ and the renormalized trace function F^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) takes values in the space

(V_{1} ⊗ ⋯ ⊗ V_{N})[0] ⊗ (V_{N}^{∗} ⊗ ⋯ ⊗ V_{1}^{∗})[0]((\frac{z_{2}}{z_{1}} , …, \frac{z_{N}}{z_{N−1}} ))ˆ⊗q^{−2(λ,μ)}A_{λ,ω} ˆ⊗A_{μ,k}.

As in the case of finite-dimensional simple Lie algebra 𝔤 (see [EV]), we have the following symmetry identity :

Theorem 7.1.

F^{V_{1},... ,V_{N}} (z_{1}, …, z_{N}, λ, ω, μ, k) = F_{∗}^{V_{N}^{∗} ,... ,V_{1}^{∗}}(z_{N}, …, z_{1}, μ, k, λ, ω),

where F_{∗} is the result of interchanging the two factors (V_{1} ⊗ ⋯ ⊗ V_{N})[0] and (V_{N}^{∗} ⊗ ⋯ ⊗ V_{1}^{∗})[0].

This result follows from Theorem 4.1 and Theorem 4.2 and the arguments in [EV], Section 5.

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Pavel Etingof, MIT Mathematics Dept., 77 Massachusetts Ave., Cambridge 02139 MA., USA

etingof@math.mit.edu

Olivier Schiffmann, Yale Mathematics Dept., 10 Hillhouse Ave., New Haven, 06510 CT., USA

schiffmann@math.yale.edu

Alexander Varchenko, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA

anv@email.unc.edu