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

We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of sln+1 having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBWgraded Demazure module and further, that the Demazure module is a favourable module, e.g. interesting geometric properties are governed by combinatorics of convex polytopes. Thus, we obtain for any minuscule Schubert variety a flat degeneration into a toric projective variety which is projectively normal and arithmetically Cohen-Macaulay. We provide a necessary and sufficient condition on the Weyl group element such that the toric variety associated to the chain polytope and the toric variety associated to the order polytope are isomorphic.

We study a family of finite-dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type 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 ≥ 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 hypergeometric series and that they are related to the q-Fibonacci polynomials defined by Carlitz.

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For a finite-dimensional Hopf algebra A, the McKay matrix M V of an A-module V encodes the relations for tensoring the simple A-modules with V. We prove results about the eigenvalues and the right and left (generalized) eigenvectors of M V by relating them to characters. We show how the projective McKay matrix Q V obtained by tensoring the projective indecomposable modules of A with V is related to the McKay matrix of the dual module of V. We illustrate these results for the Drinfeld double D n of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of M V and Q V in terms of several kinds of Chebyshev polynomials. For the matrix N V that encodes the fusion rules for tensoring V with a basis of projective indecomposable D n-modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.