Conformal blocks and generalized theta functions

Beauville, Arnaud; Laszlo, Yves

doi:10.1007/bf02101707

Cited by 230 publications

(288 citation statements)

References 15 publications

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“…It also follows from Theorem 2.10 that to conclude the desired formula it suffices to consider the case that C is totally degenerate; we choose the totally degenerate curve obtained by gluing two copies of P 1 to each other three times. By Proposition 3.5, and the fact [18, p. 206] that a torally indigenous bundle on P 1 with three marked points is determined by its radii, we find that this number is the same as the number of dormant torally indigenous bundles on P 1 with three marked points, which is p 3 Remark 4.8. One natural question which is not addressed by these techniques, due to the necessity of working level by level, is the relationship between the strata of different levels.…”

Section: Theorem 47mentioning

confidence: 61%

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

Osserman

2007

Publ. Res. Inst. Math. Sci.

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Mochizuki's work on torally crys-stable bundles [18] has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications: first, one can conclude almost immediately certain results classifying Frobenius-unstable vector bundles on curves; and second, it follows from the results of [22] that one also obtains results on rational functions with prescribed ramification in positive characteristic. §1. IntroductionMochizuki's theory of torally crys-stable bundles and torally indigenous bundles developed in [18] has, after appropriate translation, immediate implications for logarithmic connections on vector bundles of rank 2 on curves. This in turn has immediate implications to a subject which has recently been studied by a number of different people (see, for instance,

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Section: Theorem 47mentioning

confidence: 61%

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

Osserman

2007

Publ. Res. Inst. Math. Sci.

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“…8.2] we see that G-invariant line bundles onR are in bijection with line bundles on M r (C), sô L must be a power of the pullback of L . Now, the sections of L ⊗n on M r (C) are in bijection with the sections on M ss r (C) [2,Prop. 8.3], which are in bijection with G-invariant sections of the pullback onR ss = R [2, Lem.…”

Section: Appendix a Some General Results On The Verschiebungmentioning

confidence: 99%

The generalized Verschiebung map for curves of genus 2

Osserman

2006

Math. Ann.

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Abstract. Let C be a smooth curve, and Mr(C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map Vr : Mr(C (p) ) Mr(C) induced by pulling back under Frobenius. Our main result is a computation of the degree of V 2 for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.

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“…For each simple root α i , we choose a non-trivial additive subgroup x i of U + such that a λ x i (b)a −λ = x i a α i ,λ b holds for all λ ∈ Λ, a ∈ C × , b ∈ C. Then there is a unique morphism ϕ i : SL 2 Let N G (T ) be the normalizer of T in G and let W = N G (T )/T be the Weyl group of (G, T ). Each element s i normalizes T ; its class s i modulo T is called a simple reflection.…”

Section: Preliminariesmentioning

confidence: 99%

“…The affine Grassmannian H is the set of C-points of an ind-scheme defined over C (see [2] for H = GL n or SL n and Chapter 13 of [20] for H simple). This means, in particular, that H is the direct limit of a system…”

Section: The Affine Grassmannianmentioning

confidence: 99%

On Mirković-Vilonen cycles and crystal combinatorics

Baumann

Gaussent

2008

Represent. Theory

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Abstract. Let G be a complex connected reductive group and let G ∨ be its Langlands dual. Let us choose a triangular decomposition n −,∨ ⊕ h ∨ ⊕ n +,∨ of the Lie algebra of G ∨ . Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannianis a crystal isomorphic to the crystal of the canonical basis of U (n +,∨ ). Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that the varieties involved in Lusztig's algebraic-geometric parametrization of the canonical basis are closely related to MV cycles. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals.

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Conformal blocks and generalized theta functions

Cited by 230 publications

References 15 publications

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

The generalized Verschiebung map for curves of genus 2

On Mirković-Vilonen cycles and crystal combinatorics

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