INdAM Program on Serre Conjectures and the p-adic local Langlands program
Let H be Vignéras pro-p Iwahori-Hecke algebra with mod p coefficients of a p-adic GLn. Its finite part acts on the cohomology of the mod p dual flag variety by means of classical Demazure operators. We explain how this finite representation can be uniquely extended to a natural representation of the whole algebra H and give some of its properties. Compatibility of the construction with various operations on the dual group give applications to the mod p local Langlands correspondence. If time permits, we discuss the cases of n = 2 and n = 3 in some detail. This is joint work with C. Ppin.