Demazure flags, Chebyshev polynomials, partial and mock theta functions

We study the level m-Demazure flag of a level ℓ  -Demazure module for sl2[t]sl2[t]. We define the generating series Anℓ→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=1q=1 is a rational function involving the Chebyshev polynomials. We give a closed form for Anℓ→ℓ+1(x,q) and prove that it is given by a rational function. In the case when m=ℓ+1m=ℓ+1 and ℓ=1,2ℓ=1,2, we relate the generating series to partial theta series. We also study the specializations An1→3(qk,q) and relate them to the fifth order mock-theta functions of Ramanujan.