Some functorial constructions for pro-p-Iwahori cohomology
A classical result of Borel and Bernstein shows that the category of smooth complex representations of a p-adic reductive group G which are generated by their Iwahori-invariant vectors is equivalent to the category of modules over a certain Iwahori-Hecke algebra H. This makes the algebra H an extremely useful tool in studying the representation theory of G, and thus in the Local Langlands Program. When the field of complex numbers is replaced by a field of characteristic p, this equivalence no longer holds. However, Schneider has recently shown that one can recover an equivalence by passing to derived categories and upgrading H to a certain differential graded algebra. We will attempt to shed some light on this derived equivalence by relating several functorial constructions on both sides. If time permits, we'll show how these methods can be used to calculate the H-module structure on all (pro-p-)Iwahori cohomology spaces with coefficients in irreducible representations, when G = GL_2(Q_p).