Affine Deligne-Lusztig theory

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The classical Deligne-Lusztig theory gives a complete description of representations of connected reductive groups over finite fields in terms of algebraic geometry. The goal of this project is to develop a complete analog of the classical Deligne-Lusztig theory for connected reductive groups over local fields. This should contribute to the automorphic induction and a purely local geometric proof of the local Langlands correspondence. Especially in the ramified case this leads to substantially new insights. A purely local proof of the local Langlands correspondence is very desirable: it has the same importance for the contemporary arithmetic geometry as the purely local approach to the local class field theory had for the algebraic number theory in the 1930s.