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Geometric local theta correspondence for dual reductive pairs of type II at the Iwahori level

Author: Banafsheh Farang-Hariri
Journal: Represent. Theory 17 (2013), 610-646
MSC (2010): Primary 14D24, 11F27; Secondary 22E57, 20C08
Published electronically: December 9, 2013
MathSciNet review: 3139267
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Abstract: In this paper we are interested in the geometric local theta correspondence at the Iwahori level for dual reductive pairs $(G,H)$ of type II over a non-Archimedean field of characteristic $p\neq 2$ in the framework of the geometric Langlands program. We consider the geometric version of the $I_{H}\times I_{G}$-invariants of the Weil representation $\mathcal {S}^{I_{H}\times I_{G}}$ as a bimodule under the action of Iwahori-Hecke algebras $\mathcal {H}_{I_{G}}$ and $\mathcal {H}_{I_{H}}$ and we give some partial geometric description of the corresponding category under the action of Hecke functors. We also define geometric Jacquet functors for any connected reductive group $G$ at the Iwahori level and we show that they commute with the Hecke action of the $\mathcal {H}_{I_{L}}$-subelgebra of $\mathcal {H}_{I_{G}}$ for a Levi subgroup $L$.

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Banafsheh Farang-Hariri
Affiliation: Humboldt-Universitët zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany
Address at time of publication: Université de Paris XI, Laboratoire de Mathématiques, Bât 425, 91405 Orsay Cedex, France

Keywords: Geometric representations theory, local theta correspondence, Iwahori-Hecke algebra, perverse sheaves, affine flag varieties
Received by editor(s): February 26, 2013
Received by editor(s) in revised form: September 24, 2013
Published electronically: December 9, 2013
Article copyright: © Copyright 2013 American Mathematical Society