Research Projects

In Mathematics, the Langlands program is a grand unifying theory, which started with a quest for finding a non-abelian Class field theory. Building on the existing Harish-Chandra's philosophy of cusp forms, it was formulated in 1967 by Robert Langlands. Very broadly, Langlands program is a scheme for organizing fundamental arithmetic objects (akin to Galois representations) in terms of some highly structured analytic objects called automorphic representations. Local Langlands conjectures is a specialization of the Langlands program. Given an algebraic group G over a local field F, the local Langlands conjecture (LLC) stipulates that the irreducible admissible objects of the category R(G(F)) of smooth representations of G(F) can partitioned into finite sets called L-packets, which are parameterized by arithmetic objects called the Langlands parameters, in a certain natural way. The category R(G(F)) decomposes into a product of indecomposable sub-categories called Bernstein blocks. Each Bernstein block is equivalent to the module category of an assciative C-algebra called Hecke algebra. Bushnell-Kutzko theory gives a strategy of constructing the Hecke algebras associated to Bernstein blocks. When the residue characteristic is not too small, a construction of Ju-Lee Kim and Jiu-Kang Yu accomplished a part of the strategy, namely the construction of “types”. The first goal of this project (Project (A)) is to complete the program of Bushnell-Kutzko by describing all Hecke algebras coming from Kim-Yu types. The second goal of this project (Project (B)) is to specialize the result in Project A to principal series blocks and obtain Langlands classification of irreducible principal series representations for non-split groups.