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

Banafsheh Farang-Hariri (1)

1 International Research Training Group GRK 1800, moduli and automorphic forms

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 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 of action Iwahori-Hecke algebras $\iwahorihecke{G}$ and $\iwahorihecke{H}$ 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 $\iwahorihecke{L}$-subelgebra of $\iwahorihecke{G}$ for some Levi subgroup $L$.

Mots clés: affine flag varieties; geometric representations theory; local theta correspondence; Iwahori-Hecke algebra; perverse sheaves; affine flag varieties.

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]

Banafsheh Farang-Hariri. Geometric local theta correspondence for dual reductive pairs of type II at the Iwahori level. 2013. ⟨hal-00796584v2⟩