Demazure Flags, Chebyshev polynomials, Partial and Mock theta functions

Abstract

We study the level $m$--Demazure flag of a level $\ell$--Demazure module for ${\mathfrak{sl}}_2[t]$. We define the generating series $A_n^{\ell \to m}(x,q)$ which encodes the $q$--multiplicity of the level $m$ Demazure module of weight $n$. We establish two recursive formulae for these functions. We show that the specialization to $q=1$ is a rational function involving the Chebyshev polynomials. We give a closed form for $A_n^{\ell \to \ell +1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\ell +1$ and $\ell =1,2$, we relate the generating series to partial theta series. We also study the specializations $A_n^{1\to 3}(q^k,q)$ and relate them to the fifth order mock-theta functions of Ramanujan.

31 pages