The local Shimura correspondence relates representations of PGL_2
to those of the metaplectic cover of SL_2, where both groups are p-adic and
the representations are on C-vector spaces. Many pieces of the usual
construction break down, due to non-semisimplicity, when the
representations are instead taken over a field of positive characteristic.
The mod p case is especially problematic. In the specific case of the pair
(tilde{SL}_2, PGL_2), I will discuss an alternate approach, which involves
a comparison of certain Iwahori Hecke algebras of both groups, as well as
progress towards a classification of the mod p representations of the
metaplectic group.