Ghislain Fourier

Tensor product structure of affine Demazure modules and limit constructions

Let g be a simple complex Lie algebra, we denote by ĝ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g . Let Λ 0 be the fundamental weight of ĝ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g -coweight λ ∨ , the Demazure submodule V _ λ∨ (mΛ 0 ) is a g -module. We provide a description of the g -module structure as a tensor product of “smaller” Demazure modules. More precisely, for any partition of λ ∨ = λ∑ j as a sum of dominant integral g -coweights, the Demazure module is (as g -module) isomorphic to ⊗ j V _ (mΛ 0 ). For the “smallest” case, λ ∨ = ω ∨ a fundamental coweight, we provide for g of classical type a decomposition of V_ ω∨ (mΛ 0 ) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the U q (g)-characters of certain finite dimensional -modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V_ λ∨,q (mΛ 0 ) can be naturally endowed with the structure of a -module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the “smallest” Demazure modules are, when viewed as g -modules, isomorphic to some KR-modules. For an integral dominant ĝ-weight Λ let V ( Λ ) be the corresponding irreducible ĝ-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g -module structure of V ( Λ ) as a semi-infinite tensor product of finite dimensional g -modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

Kirillov-Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov--Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types D_n^(1), B_n^(1), A_{2n-1}^(2) we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type C_n^(1) we use a Dynkin diagram folding and for types A_{2n}^(2), D_{n+1}^(2) a similarity construction. We also show that for types C_n^(1) and D_{n+1}^(2) the analog of the Dynkin diagram automorphism exists on the level of crystals.

Demazure structure inside Kirillov-Reshetikhin crystals

The conjecturally perfect Kirillov–Reshetikhin (KR) crystals are known to be isomorphic as classical crystals to certain Demazure subcrystals of crystal graphs of irreducible highest weight modules over affine algebras. Under some assumptions we show that the classical isomorphism from the Demazure crystal to the KR crystal, sends zero arrows to zero arrows. This implies that the affine crystal structure on these KR crystals is unique.

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional represen- tations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defin- ing twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological prop- erties and a tensor product property) extend to the more general setting considered in the current paper.

Demazure modules and Weyl modules: The twisted current case

We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure modules for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover, we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight. These results were previously known for untwisted current algebras and are new for all twisted types.

Global Weyl Modules for Equivariant Map Algebras

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra A that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of A-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e. untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that A is finitely generated and the global Weyl module is a finitely generated A-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in a previous paper. Finally, we show that A is the algebra of coinvariants of the analogous algebra in the untwisted case.

PBW-degenerated Demazure modules and Schubert varieties for triangular elements

We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sl n + 1 . We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements.These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.

Posets, tensor products and Schur positivity

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set

PBW-filtration over ℤ and compatible bases for V ℤ (λ) in type A n and C n

P(\lambda, k)

Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces

of k-tuples of dominant weights which add up to \lambda. Let