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99
Geometric Langlands duality and representations of algebraic groups over commutative rings
, 2004
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Whittaker patterns in the geometry of moduli spaces of bundles on curves
 ANN OF MATH
, 2001
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LSgalleries, the path model and MVcycles
 Duke Math. J
"... Let G be a complex semisimple algebraic group. We give an interpretation of the path model of a representation [17] in terms of the geometry of the affine Grassmannian for G. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. The ..."
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Cited by 39 (2 self)
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Let G be a complex semisimple algebraic group. We give an interpretation of the path model of a representation [17] in terms of the geometry of the affine Grassmannian for G. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. The connection with geometry is obtained as follows: consider a Demazure–Hansen–Bott–Samelson desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The points of this variety can be viewed as galleries of a fixed type in the affine Tits building associated to G. The retraction with center − ∞ of the Tits building onto the affine Coxeter complex induces, in this way, a stratification of the G(C[[t]])–orbit (identified with an open subset of ˆ Σ(λ)), indexed by certain folded galleries in the Coxeter complex. Each strata can be viewed as an open subset of a Bia̷lynicki– Birula cell of ˆ Σ(λ). The connection with representation theory is given by the fact that the closures of the strata associated to LSgalleries are the MV–cycles [23].
The quantization conjecture revisited
 Ann. of Math
"... Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies ove ..."
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Cited by 39 (7 self)
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Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies over X. This generalizes the theorem of [GS], which concerned the spaces of global sections, and strengthens its subsequent extensions ([M], [V]) to RiemannRoch numbers. A remarkable byproduct is the rigidity of cohomology of certain vector bundles over the GIT quotient under a small change in the defining polarization. One application is a new proof of the BorelWeilBott theorem of [T] for the moduli stack M of Gbundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Also studied are equivariant holomorphic forms and the equivariant Hodgetode Rham spectral sequences for X and its strata, whose collapse at E1 is shown. Collapse of the Hodgetode Rham sequence for M is hence deduced.
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 38 (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 Arason invariant and mod 2 algebraic cycles
 J. A.M.S
, 1998
"... 2. The special Clifford group 5 3. Kcohomology of split reductive algebraic groups 7 ..."
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Cited by 33 (8 self)
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2. The special Clifford group 5 3. Kcohomology of split reductive algebraic groups 7
The Picard group of the moduli of Gbundles on a curve
 Compositio Math. 112
, 1998
"... This paper is concerned with the moduli space of principal Gbundles on an algebraic curve, for G a complex semisimple 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 rec ..."
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Cited by 33 (3 self)
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This paper is concerned with the moduli space of principal Gbundles on an algebraic curve, for G a complex semisimple 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
Parabolic Bundles, Products of Conjugacy Classes, and Quantum Cohomology
, 2001
"... We prove a condition for the existence of flat bundles on the punctured twosphere with prescribed holonomies around the punctures, involving GromovWitten invariants of generalized flag varieties. This generalizes the case of special unitary connections described by Agnihotri and the second author ..."
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Cited by 23 (3 self)
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We prove a condition for the existence of flat bundles on the punctured twosphere with prescribed holonomies around the punctures, involving GromovWitten invariants of generalized flag varieties. This generalizes the case of special unitary connections described by Agnihotri and the second author [1] and Belkale [5].