PropIwahoriHecke Algebras in the modp Local Langlands Program
 Title:
 PropIwahoriHecke Algebras in the modp Local Langlands Program
 Author(s):
 Koziol, Karol
 Thesis Advisor(s):
 Ollivier, Rachel
 Date:
 2014
 Type:
 Theses
 Degree:
 Ph.D., Columbia University
 Department(s):
 Mathematics
 Persistent URL:
 https://doi.org/10.7916/D89C6VKV
 Abstract:
 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 propIwahoriHecke algebra H_{F_p}(G, I(1)) in the modp Local Langlands Program, where G is the group of Fpoints of a connected, reductive group, and I(1) is a propIwahori 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 SchneiderStuhler 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 propIwahoriHecke 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 Lpackets of supersingular H_{F_p}(G_S , I_S(1))modules and irreducible projective ndimensional Galois representations, and prove that this correspondence is induced by a functor when F = Q_p.
 Subject(s):
 Mathematics
 Item views
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 Suggested Citation:
 Karol Koziol, 2014, PropIwahoriHecke Algebras in the modp Local Langlands Program, Columbia University Academic Commons, https://doi.org/10.7916/D89C6VKV.