Local Langlands correspondence for depth-zero representations of classical groupsDesde el 13 de Enero, 2020 Hasta el 17 de Enero, 2020
Let F be non-archimedean local field and G the group of F-points of a connected reductive algebraic group defined over F. A representation of G has depth zero if it admits non-zero invariant vectors by the pro-p unipotent radical of a parahoric subgroup of G. We will first describe several well-known properties of these representations, including the construction of the supercuspidal ones, and illustrate them in the case where G is a classical group.
Next, we will introduce the enhanced Langlands parameters for G, and outline their main properties, in particular for the tame ones, that is, such that the restriction of the Langlands parameter to the wild inertia subgroup of F is trivial.
The depth-zero representations are expected to correspond to the tame enhanced Langlands parameters via the Langlands correspondence. We will explain a purely local construction of a candidate for the correspondence in several special cases.
When G is a quasi-split symplectic or special orthogonal group, and the characteristic of F is 0, the Langlands correspondence for G was established by J. Arthur for representations of arbitrary depth. In the last part of the course, we will outline a strategy in proving that the previous construction coincides with Arthur’s construction.
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