Jean-Pierre Labesse (Marseilles)

Many questions about noncommutative Lie groups boil down to questions in invariant harmonic analysis, the study of distributions on the group that are invariant by conjugacy. The fundamental objects of invariant harmonic analysis are orbital integrals and characters, respectively the geometric and spectral sides of the trace formula.

In the Langlands program a cruder form of conjugacy called stable conjugacy plays a role. The study of Langlands functoriality often leads to correspondences that are defined only up to stable conjugacy. Endoscopy is the name given to a series of techniques aimed to investigate the difference between ordinary and stable conjugacy.

Let G be a reductive group over a field F. Recall that one says that g and g' in G(F) are conjugate if there exists x in G(F) such that g' =x g x-1. Roughly speaking, stable conjugacy amounts to conjugacy over the algebraic closure F of F: at least for strongly regular semisimple elements, one says that g and g' in G(F) are stably conjugate if there is x in G(F) such that g' =x g x-1.

On the geometric side, the basic objects of stably invariant harmonic analysis are stable orbital integrals. On the spectral side, the notion of L-packets of representations is the stable analogue for characters of tempered representations.  The case of non-tempered representations is the subject of conjectures of Arthur that will be examined in Vogan's lectures.

The word "endoscopy" has been coined to express that we want to see ordinary conjugacy inside stable conjugacy. We shall introduce the basic notions of local endoscopy: \kappa-orbital integrals, endoscopic groups, endoscopic transfer of orbital integrals and its dual for characters with an emphasis on the case of real groups, following the work of Diana Shelstad.

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