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New Developments in Representation Theory
(6 - 31 March 2016)


Organizing Committee · Visitors and Participants · Overview · Activities · Venue

 

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 Overview


The programme will focus on the representation theory of reductive groups over local fields and their related Hecke algebras. It covers the following three aspects of the subject:


(i) Classification of irreducible representations in terms of L-parameters:


This is part of the well-known Langlands program and it has been a major theme in number theory and representation theory for nearly half a century. Recent work of Arthur, Mok and Moeglin-Waldspurger has led to the proof of the local Langlands correspondence for quasi-split classical groups in characteristic 0.  The subsequent work of Kaletha-Minguez-Shin-White has extended this to inner forms of unitary groups. The very recent work of Kaletha on rigid inner forms has laid the necessary  foundation for the case of general inner forms. The recent spectacular work of V. Lafforgue has greatly advanced the subject in positive characteristic. A more refined problem is the classification of the unitary dual. Here, the Atlas project initiated by Adams, Vogan and their collaborators has seen significant progress in the last few years, at least for real groups. The work of Barbasch-Moy, Barbashc-Ciubotaru, Ciubotaru-He and Lapid-Minguez have provided some progress in the p-adic case. Yet another recent development is an attempt, initiated by Weissman, to extend the Langlands program to nonlinear covering groups.


(ii) Understanding and computation of invariants of representations:


Vast amount of techniques have been developed to understand representations internally through the use of invariants. Among the key harmonic analytic invariants are matrix coefficients and their growth properties, characters and their asymptotics (leading to wavefront sets and wavefront cycles), as well as models of representations with good properties (such as Fourier coefficients and periods with multiplicity at most one). In the case of real reductive groups, there are additional invariants of algebraic/geometric nature, foremost the K-types and their asymptotics (leading to associated varieties and associated cycles). This internal approach has a  long history, and has been an equally powerful driving force for critical advances and new developments in representation theory. One recent active area of research is concerned with the Gross-Prasad conjecture.  Another recent advance is the extension of the theory of derivatives for GL(n) from the p-adic case to the case of real groups.


(iii)  Explicit constructions of representations:


Explicit constructions of representations continue to be the best way of enhancing one’s understanding of representation theory. The study of classical invariant theory continues to be a fruitful avenue of research in this direction. Its transcendental version  is the theory of theta correspondence initiated by R. Howe. It is a well-studied subject that has seen significant progress in the last two years, with the proof of various longstanding conjectures. Another is a generalisation, due to D. Jiang and his collaborators, of the theory of automorphic descent (or backward lifting) pioneered by Ginzburg-Rallis-Soudry. This has the potential  of giving explicit constructions of Langlands functoriality beyond the generic case.


The program will examine not just recent developments  in each of these three areas, but also to highlight the fruitful interactions among them. For example, there has been much work on how various harmonic analytic invariants behave under Langlands functorial lifting or theta lifting. Moreover, the generalisation of automorphic descent requires one to have a good understanding of the possible Fourier coefficients or the wavefront set of  representations. Uncovering such interactions, especially unexpected ones,  will be a main goal of this program.



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Organizing Committee · Visitors and Participants · Overview · Activities · Venue

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