Demazure Flags, Chebyshev polynomials, Partial and Mock theta functions

TL;DR

This paper investigates Demazure modules for sl_2[t], deriving recursive formulas for their generating series, connecting them to Chebyshev polynomials, partial theta functions, and Ramanujan's mock theta functions.

Contribution

It introduces new recursive formulas for Demazure module generating series and links these series to classical special functions, including Chebyshev, partial theta, and mock theta functions.

Findings

Recursive formulas for generating series of Demazure modules.

Specialization at q=1 yields rational functions involving Chebyshev polynomials.

Connections established between generating series and Ramanujan's mock theta functions.

Abstract

We study the level $m$--Demazure flag of a level $\ell$--Demazure module for $\frak{sl}_2[t]$. We define the generating series $A_n^{\ell \rightarrow m}(x,q)$ 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 $A_n^{\ell \rightarrow \ell+1}(x,q)$ 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 $A_n^{1\rightarrow 3}(q^k,q)$ and relate them to the fifth order mock-theta functions of Ramanujan.