Demazure flags, $q$--Fibonacci polynomials and hypergeometric series

Abstract

We study a family of finite--dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $\tt A_2^{(2)}$. We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $m'$ Demazure module admits a filtration by level $m$ Demazure modules for all $m\ge m'$. We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish a recursive formulae. In the case when $m'=1,2$ and $m=2,3$ we determine these generating functions completely and show that they define hypergeoemetric series and that they are related to the $q$--Fibonacci polynomials defined by Carlitz.