Rachel Ollivier

Let F be a finite extension of ℚₚ, k the algebraic closure of ?ₚr for some r, and G = SL₂(F). In the context of the mod-p Local Langlands correspondence, it is natural to study the pro-p Iwahori Hecke algebra H := End(k[G/I],k[G/I]) attached to the pro-p Iwahori subgroup I ≤ G. One reason is that when F = ℚₚ, there is an equivalence between the category of H-modules and the category of smooth k-representations of G generated by their I-fixed vectors. Unfortunately, this equivalence fails when F is a proper extension of ℚₚ.We overcome this obstacle somewhat by passing to the derived setting. When F is a proper extension and I is a torsion free group, it was shown by Schneider that we can obtain an equivalence between the derived category of smooth representations and a certain derived category associated to H. Relatively little is known about this equivalence. In understanding more, we can consider the cohomology algebra E* := Ext*(k[G/I],k[G/I]). The goal of this thesis is to study the related algebra EP* := Ext*(k[G/P],k[G/P]) when P is taken to be either the Iwahori subgroup J of G or the maximal compact subgroup K = SL₂(ℤₚ), both of which contain I. We are able to give explicit descriptions of these algebras, including the full product. Surprisingly, we deduce that EK* is commutative (not graded commutative) and that it is isomorphic to the centre of EJ*.

View record