Tensor product structure of affine Demazure modules and limit constructions

Let \Lg be a simple complex Lie algebra, we denote by \Lhg the corresponding affine Kac--Moody algebra. Let $\Lambda_0$ be the additional fundamental weight of \Lhg. For a dominant integral \Lg--coweight \lam^\vee, the Demazure submodule V_{-\lam^\vee}(m\Lam_0) is a \Lg--module. For any partition of \lam^\vee=\sum_j \lam_j^\vee as a sum of dominant integral \Lg--coweights, the Demazure module is (as \Lg--module) isomorphic to \bigotimes_j V_{-\lam^\vee_j}(m\Lam_0). For the ``smallest'' case, \lam^\vee=\om^\vee a fundamental coweight, we provide for \Lg of classical type a decomposition of V_{-\om^\vee}(m\Lam_0) into irreducible \Lg--modules, so this can be viewed as a natural generalization of the decomposition formulas in \cite{KMOTU} and \cite{Magyar}. A comparison with the U_q(\Lg)--characters of certain finite dimensional U_q'(\Lhg)--modules (Kirillov--Reshetikhin--modules) suggests furthermore that all quantized Demazure modules V_{-\lam^\vee,q}(m\Lam_0) can be naturally endowed with the structure of a U_q'(\Lhg)--module. Such a structure suggests also a combinatorially interesting connection between the LS--path model for the Demazure module and the LS--path model for certain U_q'(\Lhg)--modules in \cite{NaitoSagaki}. For an integral dominant \Lhg--weight $\Lambda$ let V(\Lam) be the corresponding irreducible \Lhg--representation. Using the tensor product decomposition for Demazure modules, we give a description of the \Lg--module structure of V(\Lam) as a semi-infinite tensor product of finite dimensional \Lg--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.Comment: 24 pages, in the current version we added the case of twisted affine Kac--Moody algebra

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

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 {\tt 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 \Lgc 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 \Lgc-module structure of the Kac-Moody algebra \Lhg-module V(\ell\Lam_0) as a semi-infinite fusion product of finite dimensional \Lgc--modules

One-skeleton galleries, the path model and a generalization of Macdonald's formula for Hall-Littlewood polynomials

We give a direct geometric interpretation of the path model using galleries in the $1-$skeleton of the Bruhat-Tits building associated to a semi-simple algebraic group. This interpretation allows us to compute the coefficients of the expansion of the Hall-Littlewood polynomials in the monomial basis. The formula we obtain is a "geometric compression" of the one proved by Schwer, its specialization to the case ${\tt A}_n$ turns out to be equivalent to Macdonald's formula.Comment: 43 pages, 3 pictures, some improvements in the presentation, semistandard tableaux for type B and C define

Richardson Varieties and Equivariant K-Theory

We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get closer to a geometric interpretation of the standard monomial theory. Our methods show that in order to develop a SMT for a certain class of subvarieties in G/B (which includes G/B), it suffices to have the following three ingredients, a basis for the space of sections of an effective line bundle on G/B, compatibility of such a basis with the varieties in the class, certain quadratic relations in the monomials in the basis elements. An important tool will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above. This provides a direct connection to the equivariant K-theory, where the combinatorially defined notion of standardness gets a geometric interpretation.Comment: 38 page

Equations defining symmetric varieties and affine Grassmannians

Let $\sigma$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $\sigma$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the restricted root system is of type $B$ and $G$ is adjoint, then we describe a standard monomial theory and the equations for the coordinate ring $k[G/H]$ using the standard monomial theory and the Pl\"ucker relations of an appropriate (maybe infinite dimensional) Grassmann variety.Comment: 48 page