The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as follows: choose a point p ∈ X, and let AX be the

: Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasi-parabolic G-bundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the corresponding spaces of global sections with the vacua spaces of Tsuchiya, Ueno and Yamada. The method uses the uniformization theorem which describes these stacks as double quotients of certain infinite dimensional algebraic groups. We describe also the dualizing bundle of the stack of G-bundles and show that it admits a unique square root, which we construct explicitly. If G is not simply connected, the square root depends on the choice of a theta-characteristic. These results about stacks allow to recover the Drezet-Narasimhan theorem (for the coarse moduli space) and to show an analogous statement when G = Sp 2r . We prove also that the coarse moduli spaces of semi-stable SOr -bundles are not loca...

These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in that we have

Introduction :\Gamma Let X be an algebraic curve over the field C of complex numbers which is assumed to be smooth, connected and projective. For simplicity, we assume that the genus of X is ? 2 . Let G be a simple simply connected group and MG (X) the coarse moduli scheme of semistable G-bundles on X . Any linear representation determines a line bundle \Theta on M and some non negative integer l (the Dynkin index of the representation, cf [KNR], [LS]). Its is known that the choice of a (closed) point x 2 X(C) (and, a priori, of a formal coordinate near x ) of X determines an isomorphism (see (5.4)) between the projective space of conformal blocks PB l (X) (for G ) of level l and the space PH 0 (MG (X); \Theta) of generalized theta

This paper is concerned with the moduli space of principal G-bundles on an algebraic curve, for G a complex semi-simple group. While the case G = SLr, which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it

this paper. We use this theorem to prove that b w \Gamma1 6= 0 if and only if v w, and in this case it has a pole of order exactly equal to `(w). Similarly \Gamma1 6= 0 if and only if v w (cf. Corollaries 3.2)

Let g be a simple complex Lie algebra, we denote by ̂g the affine Kac–Moody algebra associated to the extended Dynkin diagram of g. Let Λ0 be the fundamental weight of ̂g corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g–coweight λ ∨ , the Demazure submodule V−λ ∨(mΛ0) 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 ∨ = ∑ j λ ∨ j as a sum of dominant integral g–coweights, the Demazure module is (as g–module) isomorphic to ⊗ j V−λ ∨ j (mΛ0). For the “smallest ” case, λ ∨ = ω ∨ a fundamental coweight, we provide for g of classical type a decomposition of V−ω∨(mΛ0) into irreducible g–modules, so this can be viewed as a natural generalization of the decomposition formulas in [12] and [15]. A comparison with the Uq(g)–characters of certain finite dimensional U ′ q (̂g)–modules (Kirillov–Reshetikhin–modules) suggests furthermore that all quantized Demazure modules V−λ ∨,q(mΛ0) can be naturally endowed with the structure of a U ′ q(̂g)–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(̂g)–modules in [17]. For an integral dominant ̂g–weight Λ let V (Λ) be the corresponding irreducible ̂g–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.

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated finite Lie algebra.

We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra.

In the arithmetic theory of Shimura varieties it is of interest to have a model over