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12
Conformal blocks and generalized theta functions
 Comm. Math. Phys
, 1994
"... 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 foll ..."
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Cited by 112 (11 self)
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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
The line bundles on the moduli of parabolic Gbundles over curves and their sections
, 1997
"... : 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 quasiparabolic Gbundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the correspond ..."
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Cited by 69 (4 self)
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: 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 quasiparabolic Gbundles 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 Gbundles 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 thetacharacteristic. These results about stacks allow to recover the DrezetNarasimhan 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 semistable SOr bundles are not loca...
Infinite Grassmannians and moduli spaces of Gbundles
 Math. Annalen
, 1994
"... 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 th ..."
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Cited by 57 (2 self)
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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
Hitchin's and WZW connections are the same
, 1998
"... 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 Gbundles on ..."
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Cited by 27 (2 self)
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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 Gbundles 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
The Nil Hecke Ring And Singularity Of Schubert Varieties
, 1995
"... 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) ..."
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Cited by 20 (0 self)
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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)
Birational Weyl group action arising from a nilpotent Poisson algebra
 in Proc. of the Nagoya 1999 International Workshop “Physics and Combinatorics 1999” (Nagoya), World Sci. Publ., River Edge, NJ
"... We propose a general method to realize an arbitrary Weyl group of KacMoody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra. ..."
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Cited by 15 (5 self)
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We propose a general method to realize an arbitrary Weyl group of KacMoody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra.
Cohomology of line bundles on Schubert varieties in the KacMoody setting
, 2006
"... In this paper, we describe the indices of the top and the least nonvanihing cohomologies H i (X(w),Lλ) of line budles on Schubert varieties X(w) given by nondominant weights in the KacMoody setting. We also prove some surjective Theorem for maps between some cohomology modules. 1 ..."
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Cited by 3 (0 self)
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In this paper, we describe the indices of the top and the least nonvanihing cohomologies H i (X(w),Lλ) of line budles on Schubert varieties X(w) given by nondominant weights in the KacMoody setting. We also prove some surjective Theorem for maps between some cohomology modules. 1
POSITIVITY IN EQUIVARIANT SCHUBERT
"... We prove a positivity property for the cup product in the Tequivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of S. Billey. The result for the flag variety follows from a more general result about algebraic varieties with an action ..."
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We prove a positivity property for the cup product in the Tequivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of S. Billey. The result for the flag variety follows from a more general result about algebraic varieties with an action of a solvable linear algebraic group such that the unipotent radical acts with finitely many orbits. The methods are those used by S. Kumar and M. Nori. 1.