We study finite dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type ADE, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module.For the current algebra Cg of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the Cg-module structure of the Kac-Moody algebra g-module V (ℓΛ 0 ) as a semi-infinite fusion product of finite dimensional Cg-modules. *

We study the PBW filtration on the highest weight representations V (λ) of sl n+1 . This filtration is induced by the standard degree filtration on U(n − ). We give a description of the associated graded S(n − )-module gr V (λ) in terms of generators and relations. We also construct a basis of gr V (λ). As an application we derive a graded combinatorial character formula for V (λ), and we obtain a new class of bases of the modules V (λ) conjectured by Vinberg in 2005.

Abstract. Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper, we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.

We study the PBW filtration on the highest weight representations V (λ) of sp 2n . This filtration is induced by the standard degree filtration on U (n − ). We give a description of the associated graded S(n − )-module grV (λ) in terms of generators and relations. We also construct a basis of grV (λ). As an application we derive a graded combinatorial formula for the character of V (λ) and obtain a new class of bases of the modules V (λ).

Abstract. We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce flat degenerations of the corresponding flag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.