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...point is the set of genus one Knizhnik--Zamolodchikov--Bernard (KZB) equations, obtained by Bernard [B1, B2] as generalization of the KZ equations. These equations have been studied recently in [FG], =-=[EK]-=-, [FW], [I]. Let g be a simple complex Lie algebra with invariant bilinear form normalized in such a way that long roots have square length 2. Fix a Cartan subalgebra h . The KZB equations are equatio...

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... polynomials. The extension of the full 2-variable Macdonald theory to affine root systems has not yet been developed. The affine analogs of Jack polynomials have been studied by Etingof and Kirillov =-=[EK]-=-. For future reference we will need the following fact. (7.2.6) Proposition. We have the equality Pb(t; l) = K(t; l) ∑ w∈ ̂ W l l(w)( l −̂ρ w(l ̂ρ t) ) b ∏ α∈ ̂ ∆+∩w −1 ( ̂ ∆−) 1 − t α∨ 1 − l 2 t α∨ ....

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...that, in terms of Weierstrass’ ℘ function with periods 1, τ, ρ ′ = −℘ + η1 for some η1(τ). Let us describe explicitly the eigenvectors in a special case, first considered by Etingof and Kirillov [6], =-=[7]-=- in which the equation reduces to a scalar equation. We take g = slN with h = C N /C(1, . . ., 1) and V = S pN C N , the symmetric power of the defining representation C N . Thus V can be realized as ...

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...e argument approaches affine root hyperplanes. We conjecture that these conditions characterize completely the image. This conjecture is confirmed in some cases, including a special case which arises =-=[6]-=- in the theory of quantum integrable many body problems (see 4.1): we describe explicitly the space of conformal blocks in the case of slN, n = 1, where the representation is any symmetric power of th...

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...rtwiners form an eigenbasis for this operator in the space of equivariant functions. In a special case, they coincide (up to a certain factor) with the affine Jack’s polynomials which were defined in =-=[EK3]-=-. At the critical value of the degree (minus the dual Coxeter number of the underlying simple Lie algebra) there exist higher order Laplace operators coming from the corresponding central elements of ...

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...ical functions on the Kac–Moody group. Introduce two transformations λ2 −πiκ (2) Tu(λ, τ) = u(λ, τ + 1), Su(λ, τ) = e where we fix arg τ ∈ (0, π). 2τ τ −1 2 − p(p+1) κ ( ) λ u , −1 , τ τ Theorem 2.1. =-=[EK]-=- If u(λ, τ) is a solution of the KZB-heat equation, then Tu(λ, τ) and Su(λ, τ) are solutions too. Moreover, the transformations T and S preserve the properties (i)-(iv). The proof of Theorem 2.1 is by...