Pro-p-Iwahori-Hecke Algebras in the mod-p Local Langlands Program

• Main Author: Koziol, Karol
• Language: English
• Published: 2014
• Subjects: Mathematics
• Online Access: https://doi.org/10.7916/D89C6VKV

Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. This thesis is dedicated to the study of the pro-p-Iwahori-Hecke algebra H_{F_p}(G, I(1)) in the mod-p Local Langlands Program, where G is the group of F-points of a connected, reductive group, and I(1) is a pro-p-Iwahori subgroup of G. When G = U(2,1)(E/F) is an unramified unitary group in three variables, we first describe the structure and simple modules of the algebra H_{F_p}(G, I(1)). We then adapt methods of Schneider-Stuhler and Paskunas to construct, for each supersingular H_{F_p}(G, I(1))-module, a supersingular representation of G. These are exactly the representations which are expected to correspond to irreducible Galois parameters. When G = U(1,1)(Q_{p^2} /Q_p) is an unramified unitary group in two variables, we use the pro-p-Iwahori-Hecke algebra H_{F_p}(G_S , I_S(1)) of the derived subgroup G_S to classify the supersingular representations of G. Combining this with previous results, we obtain a classification of all irreducible representations of G, and then construct a correspondence between representations of G and Galois parameters. Finally, when G = GL_n(F) and G_S = SL_n(F), we show how to relate the two algebras H_{F_p}(G, I(1)) and H_{F_p}(G_S, I_S(1)). Using this interplay, we prove a numerical correspondence between L-packets of supersingular H_{F_p}(G_S , I_S(1))-modules and irreducible projective n-dimensional Galois representations, and prove that this correspondence is induced by a functor when F = Q_p.