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Localization of modules for a semisimple Lie algebra in prime characteristic
Representation Theory · Submitted 14 May 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers.
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The abstract, sentence by sentence
We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. | result 0.99 | 0.74 | 0.84 | asserted 1.00 |
| s1 | The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. | method 0.78 | 0.64 | 0.06 | asserted 1.00 |
| s2 | Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. | interpretation 0.57 | 0.65 | 0.26 | asserted 1.00 |
| s3 | Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. | method 0.96 | 0.57 | 0.08 | asserted 1.00 |
| s4 | As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character. | result 0.97 | 0.94 | 0.07 | asserted 1.00 |
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Original abstract
We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.
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How the literature uses it
6 arXiv papers cite this in 12 sentences
Papers citing it · 9 sentences without a confident label
Uses its method or tool 2 papers
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The proof of this Theorem relies on the powerful techniques introduced in [ref] and [ref] .
Moduli of flat connections in positive characteristic · Introduction · 1.00
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These discrepancies can be bridged via localization on partial flag varieties [ref] .
Beilinson-Bernstein localization over the Harish-Chandra center · Introduction · 0.81
Cites it as background 1 paper
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More detailed expositions can be found in [ref] and [ref] .
Moduli of flat connections in positive characteristic · D-modules · 1.00
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