Continuing [C3,C4], we study the intertwining operators of double affine Hecke algebras H. They appeared in several papers (especially in [C2,C4,C6]). However for the first time here we apply them systematically to create the nonsymmetric [M3,C4] and symmetric [M2] Macdonald polynomials for arbitrary root systems and to start the theory of induced and co-spherical H-modules. The importance of t...

We study intertwining operators among the twisted modules for rational VOAs and show the modular invariance property of the space spanned by the trace functions concerning to some representations of rational orbifold VOAs. Our result generalizes the results obtained in [DLM2] and [M3].

We establish an isomorphism between the space of logarithmic intertwining operators among suitable generalized modules for a vertex operator algebra and the space of homomorphisms between suitable modules for a generalization of Zhu’s algebra given by Dong-Li-Mason.

Certain combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level k standard modules for affine Lie algebra sl(r, C)̃ are parametrized by (k, r)-admissible configurations. In this note we use Capparelli-LepowskyMilas’ method to give a new proof of linear independence of these bases, the main ingredient in the proof being the use of Dong-Lepowsky’s intertwining operators for fundamental...

The infinite dimensional Lie algebra ŝln = A n−1 can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of ŝln corresponding to all partitions of n can be described very simply by using combinatorial constructions.

We establish a useful identity for intertwining a creation or annihilation operator with the heat kernel of a self-interacting bosonic field theory. I Background Consider creation operators a(f) and annihilation operators a(h), both linear in their respective test functions f, h ∈ L(R, dx), acting on the Fock Hilbert space H, and satisfying the canonical commutation relations [a(h), a(f)] = 〈

0. Introduction 1. Affine Weyl groups (Reduction modulo W ) 2. Double Hecke algebras (Automorphisms, Demazure-Lusztig operators) 3. Macdonald polynomials (Intertwining operators) 4. Fourier transform on polynomials (Basic transforms) 5. Jackson integrals (Macdonald’s η-identities) 6. Semisimple representations (Main Theorem, GLn and other applications) 7. Spherical representations (Semisimple s...

We propose a generalization of Laplace transformations to the case of linear partial differential operators (LPDOs) of arbitrary order in R . Practically all previously proposed differential transformations of LPDOs are particular cases of this transformation (intertwining Laplace transformation, ILT ). We give a complete algorithm of construction of ILT and describe the classes of operators in...

0. Introduction 1. Affine Weyl groups (Reduction modulo W ) 2. Double Hecke algebras (Automorphisms, Demazure-Lusztig operators) 3. Macdonald polynomials (Intertwining operators) 4. Fourier transform on polynomials (Four transforms) 5. Jackson integrals (Macdonald’s η-identities) 6. Semisimple and pseudo-unitary representations (Main Theorem) 7. Spherical representations (Semisimple spherical r...