Endoscopy
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.
Return to conference
wepage.