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

TL;DR

This paper explores the structure of certain finite-dimensional modules over a twisted affine Lie algebra, revealing their filtrations by Demazure modules, and connects their generating functions to hypergeometric series and $q$-Fibonacci polynomials.

Contribution

It establishes a filtration structure for these modules and links their generating functions to hypergeometric series and $q$-Fibonacci polynomials, providing new insights into their combinatorial and algebraic properties.

Findings

Modules admit filtrations by Demazure modules of different levels.

Generating functions are hypergeometric series related to $q$-Fibonacci polynomials.

Complete formulas for specific cases connect representation theory with special functions.

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.