Related papers: Traces of intertwining operators and Macdonald's p…

In [H5] (q-alg/9512024) and [H7] (q-alg/9704008), the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing…

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps…

Operators that intertwine representations of a degenerate version of the double affine Hecke algebra are introduced. Each of the representations is related to multi-variable orthogonal polynomials associated with Calogero-Sutherland type…

We study effective gravitational F-terms, obtained by integrating an $U(N)$ adjoint chiral superfield $\Phi$ coupled to the ${\cal N}=1$ gauge chiral superfield $W_\alpha$ and supergravity, to arbitrary orders in the gravitational…

Let $G$ be a (split) reductive group over $\mathbb{F}_q$, and let $M$ be a standard Levi subgroup of $G$. Consider $P$ and $P'$ parabolics in $G$, containing $M$, with Levi factor $M$. we let $U = R_u(P)$ (resp., $U' = R_u(P')$) denote the…

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular…

We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A…

We consider a system of three commuting difference operators in three variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$. This system and its eigenfunctions generalize the trigonometric $A_1$ Ruijsenaars-Schneider…

We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations…

We give a differentially closed description of the uniformizing representation to the analytical apparatus on Riemann surfaces and orbifolds of finite analytic type. Apart from well-known automorphic functions and Abelian differentials it…

We extend the modular invariance property of the trace functions of vertex operator algebra on the set of irreducible modules (Zhu's theory) to the case of trace functions of intertwining operators.

Let $\hat{\mathfrak{g}}$ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type $\widehat{BC}_n=A^{(2)}_{2n}$). We present an affine Pieri rule for a basis of periodic Macdonald…

In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining…

We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew…

We derive a formula for the adjoint $\overline{A}$ of a square-matrix operation of the form $C=f(A)$, where $f$ is holomorphic in the neighborhood of each eigenvalue. We then apply the formula to derive closed-form expressions in particular…

Let $(\pi_{\mathbf{z}},V_{\mathbf{z}})$ be an unramified principal series representation of a reductive group over a nonarchimedean local field, parametrized by an element $\mathbf{z}$ of the maximal torus in the Langlands dual group. If…

For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of…

We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in…

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the…

The fusion ring for $\widehat{\mathfrak{su}}(n)_m$ Wess-Zumino-Witten conformal field theories is known to be isomorphic to a factor ring of the ring of symmetric polynomials presented by Schur polynomials. We introduce a deformation of…