Demazure Flags, Chebyshev polynomials, Partial and Mock theta functions

We study the level $m$–Demazure flag of a level $\mathrm{\ell }$–Demazure module for ${\mathfrak{s}\mathfrak{l}}_2[t]$. We define the generating series $A_n^{\ell \rightarrow m}(x,q)$whichencodesthe$q$--multiplicityofthelevel$m$ 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 $A_n^{\mathrm{\ell }\to \mathrm{\ell }+1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\mathrm{\ell }+1$ and $\mathrm{\ell }=1,2$, 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.