Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Perverse Sheaves on affine Grassmannians and Langlands Duality   This is an expanded version of the text ``Perverse Sheaves on Loop
Grassmannians and Langlands Duality'', AG/9703010. The main new result is a
topological realization of algebraic representations of reductive groups over
arbitrary rings.
  We outline a proof of a geometric version of the Satake isomorphism. Given a
connected, complex algebraic reductive group G we show that the tensor category
of representations of the Langlands dual group is naturally equivalent to a
certain category of perverse sheaves on the loop Grassmannian of G. The above
result has been announced by Ginsburg in and some of the arguments are borrowed
from his approach. However, we use a more "natural" commutativity constraint
for the convolution product, due to Drinfeld. Secondly, we give a direct proof
that the global cohomology functor is exact and decompose this cohomology
functor into a direct sum of weights. The geometry underlying our arguments
leads to a construction of a canonical basis of Weyl modules given by algebraic
cycles and an explicit construction of the group algebra of the dual group in
terms of the affine Grassmannian. We completely avoid the use of the
decomposition theorem which makes our techniques applicable to perverse sheaves
with coefficients over an arbitrary commutative ring. We deduce the classical
Satake isomorphism using the affine Grassmannian defined over a finite field.
This note contains indications of proofs of some of the results. The details
will appear elsewhere.

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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph