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

Farang-Hariri, Banafsheh

doi:10.1090/S1088-4165-2013-00448-X

Geometric local theta correspondence for dual reductive pairs of type II at the Iwahori level
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by Banafsheh Farang-Hariri

Represent. Theory 17 (2013), 610-646

DOI: https://doi.org/10.1090/S1088-4165-2013-00448-X

Published electronically: December 9, 2013

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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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