The Hilbert scheme of points in the plane Hn = Hilb n (C2) is an algebraic variety which parametrizes finite subschemes S of length n in C2. To each such subscheme S corresponds an n-element multiset, or unordered n-tuple with possible repetitions, σ(S) =[P1,...,Pn] of points in C2,wherethePiare the points of S, repeated with

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This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation conjecture (now a theorem) is in fact a q; t-generalization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the so-called q-dimensions are undoubtedly important. It is likely that we can incorporate the Kac-Moody case as well. The necessary technique was developed in [C4]. As to the duality theorem (in its complete form), it states that the generalized trigonometric-difference zonal Fourier transform is self-dual (at least formally). We define this q; t-transform in terms of double affine Hecke algebras. The most natural way to check the self-duality is to use the connection of these algebras with the so-called elliptic braid groups (the Fourier involution will turn into the transposition of the periods of an elliptic curve). The classical trigonometric-differential Fourier transform (corresponding to the limit q = t as t ! 1 for certain special k) plays one of the main roles in the harmonic analysis on symmetric spaces. It sends symmetric trigonometric polynomials to the corresponding radial parts of Laplace operators (HarishChandra, Helgason) and is not self-dual. The calculation of its inverse (the Plancherel theorem) is always challenging and involving. * Partially supported by NSF grant DMS--9301114 In the rational-differential setting, Charles Dunkl introduced the generalized Hankel transform which appeared to be self-dual [D,J]. We demonstrate in this paper that one can save this very important property if trigonometric polynomials come together with difference operators. At the moment, it is mostly an algebraic observation (the difference-analitical aspects were not touched upon). The root systems of ...

Abstract. We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials. The orthogonality of Schur functions (sµ, sλ) = δµλ, is the orthogonality relation for characters of the unitary group U(n). The orthogonality of characters means that a character (as a function on the group) vanishes in all but one irreducible representation.

hep-th 9407047

Abstract. We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland’s integrable N-body Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof–Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik–Zamolodchikov– Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a “Hermite–Bethe ” variety, a generalization of the spectral variety of the Lamé operator. We also give the q-deformed version of our first formula. In the scalar slN case, this gives common eigenfunctions of the commuting Macdonald–Rujsenaars difference operators. 1.

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We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.