Ordinary mod p representations of the metaplectic cover of p-adic SL2

We classify the genuine ordinary mod p representations of the metaplectic group SL2&d15; (F), where F is a p-adic field, and compute its genuine mod p spherical and Iwahori Hecke algebras. The motivation is an interest in a possible correspondence between genuine mod p representations of SL2&d15; (F) and mod p representations of the dual group PGL2(F), so we also compare the two Hecke algebras to the mod p spherical and Iwahori Hecke algebras of PGL2(F). We show that the genuine mod p spherical Hecke algebra of SL2&d15; (F) is isomorphic to the mod p spherical Hecke algebra of PGL2(F), and that one can choose an isomorphism which is compatible with a natural, though partial, correspondence of unramified ordinary representations via the Hecke action on their spherical vectors. We then show that the genuine mod p Iwahori Hecke algebra of SL2&d15; (F) is a subquotient of the mod p Iwahori Hecke algebra of PGL2(F), but that the two algebras are not isomorphic. This is in contrast to the situation in characteristic 0, where by work of Savin one can recover the local Shimura correspondence for representations generated by their Iwahori fixed vectors from an isomorphism of Iwahori Hecke algebras.