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GEOMETRIC TAMELY RAMIFIED LOCAL THETA CORRESPONDENCE IN THE FRAMEWORK OF THE GEOMETRIC LANGLANDS PROGRAM
Part of:
Singularities
Algebraic groups
Lie groups
Topological $K$-theory
Families, fibrations
Representation theory of groups
Published online by Cambridge University Press: 18 February 2015
Abstract
This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non-Archimedean field 
$F$ of characteristic 
$p\neq 2$ in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of 
$(\mathbf{GL}_{1},\mathbf{GL}_{m})$ we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Given two reductive connected groups 
$G$ and 
$H$ over 
$F$, and a morphism 
${\check{G}}\times \text{SL}_{2}\rightarrow \check{H}$ of Langlands dual groups, we construct a bimodule over the affine extended Hecke algebras of 
$H$ and 
$G$ that should realize the geometric local Arthur–Langlands functoriality at the Iwahori level. Then, we propose a conjecture describing the geometric local theta correspondence at the Iwahori level constructed in the first part in terms of this bimodule, and we prove our conjecture for pairs 
$(\mathbf{GL}_{1},\mathbf{GL}_{m})$.
Keywords
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 3 , July 2016 , pp. 625 - 671
- Copyright
- © Cambridge University Press 2015
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