Rogers-Ramanujan type identities and Nil-DAHA

TL;DR

This paper explores Rogers-Ramanujan type identities within Nil-DAHA, linking q-Hermite polynomials, modular functions, and affine root systems, revealing new interpretations and connections to Kac-Moody modules and coset theory.

Contribution

It introduces a family of modular functions satisfying Rogers-Ramanujan identities for affine root systems and connects q-Hermite polynomials to Demazure characters and level one Kac-Moody modules.

Findings

Derived expansions of theta function products in terms of q-Hermite polynomials.

Established Rogers-Ramanujan type identities for arbitrary affine root systems.

Connected q-series to Demazure characters and level-rank duality.

Abstract

In the theory of the Nil-DAHA Fourier transform, the inner products of q-Hermite polynomials for the measure function multiplied by a level one theta function are the key. They are used to obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers-Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to Rogers dilogarithm and Nahm's conjecture is discussed. Some of our q-series can be identified with known ones, but their interpretation seems new. Using that the q-Hermite polynomials are closely related to the Demazure level one characters in the twisted case (Sanderson, Ion), we outline a connection of our formulas to the level one integrable Kac-Moody modules and the coset theory. Several instances of the level-rank duality are provided.