Rogers-Ramanujan type identities and Nil-DAHA

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.