Abstract
We state a conjecture on the geometric local Arthur–Langlands functoriality at the Iwahori level. Given a morphism Gˇ×SL2→Hˇ of Langlands dual groups over Q¯ℓ, we construct a bimodule over the affine extended Hecke algebras of G and H which should realize the geometric local Langlands functoriality at the Iwahori level. We state a second conjecture relating this bimodule and the bimodule arising in the geometric Howe correspondence at the Iwahori level. This conjecture is verified for all dual reductive pairs (G=GL1,H=GLm).
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