Geometric structure in the representation theory of p-adic groups

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Maarten Solleveld: Local Langlands and the ABPS conjecture

[pdf] Draft lecture notes

Part I. Introduction to the local Langlands correspondence

The local Langlands correspondence provides a close relation between irreducible representations of a reductive group over a local field F and certain representations of the absolute Galois group of F. We will discuss this for split reductive groups over local non-archimedean fields, and in particular for split tori and for unramified representations. We will also show how an extended version of the Langlands classification allows one to reduce the problem of the LLC to essentially square integrable representations.

Part II. The ABPS conjecture

We will present some conjectures on the geometric structure in the dual space of a reductive p-adic group G. Roughly speaking, they say that the space of irreducible G-representations contained in one Bernstein component s is in canonical bijection with an extended quotient of a torus T_s by a finite group W_s. The conjectures include a description of the infinitesimal central characters of irreducible representations and of the intersections of L-packets with one Bernstein component. We will illustrate these aspects with examples from special linear groups.