Quasiinvariants of Coxeter groups and m-harmonic polynomials

The space of m-harmonic polynomials related to a Coxeter group G and a multiplicity function m on its root system is defined as the joint kernel of the properly gauged invariant integrals of the corresponding generalised quantum Calogero-Moser problem. The relation between this space and the ring of all quantum integrals of this system (which is isomorphic to the ring of corresponding quasiinvariants) is investigated.Comment: 23 page

Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations

We study the conformal vertex algebras which naturally arise in relation to the Nakajima-Yoshioka blow-up equations.Comment: 23 pages v2. 24 pages, references added, proofs in section 3 are expanded, many typos correcte

Multidimensional Baker-Akhiezer functions and Huygens' Principle

A notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C^n is introduced. It is proved that BA function exists only for very special configurations (locus configurations), which satisfy certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrodinger equations, so any locus configuration determines such an equation. Some results towards the classification of all locus configurations are presented. This theory is applied to the famous Hadamard's problem of description of all hyperbolic equations satisfying Huygens' Principle. We show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the BA functions.Comment: 35 pages, LATEX, 2 figures included in graphicx. Submitted to Comm.Math.Phys. (Dec. 1998

Algebra of screening operators for the deformed WnW_n algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed $W_n$ algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed $W_3$ algebra.Comment: 46 page

Degenerate flag varieties: moment graphs and Schr\"oder numbers

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schr\"oder number and the Poincar\'e polynomial is given by a natural statistics counting the number of diagonal steps in a Schr\"oder path. As an application we obtain a new combinatorial description of the large and small Schr\"oder numbers and their q-analogues.Comment: 25 page

A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P^1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra U(g,e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of arXiv:math/0401409 when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of certain shifted Yangians.Comment: minor change

Quantum Algebraic Approach to Refined Topological Vertex

We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining operators Phi and Phi^* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Phi and Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Phi and Phi^*. The spectral parameters attached to Fock spaces play the role of the K"ahler parameters.Comment: 27 page

Free Boson Realization of Uq(slN^)U_q(\widehat{sl_N})

We construct a realization of the quantum affine algebra $U_q(\widehat{sl_N})$ of an arbitrary level $k$ in terms of free boson fields. In the $q\!\rightarrow\! 1$ limit this realization becomes the Wakimoto realization of $\widehat{sl_N}$. The screening currents and the vertex operators(primary fields) are also constructed; the former commutes with $U_q(\widehat{sl_N})$ modulo total difference, and the latter creates the $U_q(\widehat{sl_N})$ highest weight state from the vacuum state of the boson Fock space.Comment: 24 pages, LaTeX, RIMS-924, YITP/K-101