Harmonic analysis on classical p-adic groups and L-packets

ABSTRACT
Local Langlands correspondences for reductive groups generalise
the Artin reciprocity law from the local class field theory. These
correspondences are expected to give natural partitions of irreducible
representations into finite sets, called L-packets. There were recently
big breakthroughs regarding them in the case of classical p-adic groups
(other then GL-groups which were settled earlier).

From the other side, the square integrable packets emerge naturally
considering some very basic problems of (pure) harmonic analysis of these groups.

In our lecture we shall discuss this connection, and how crucial data of
one theory correspond to the crucial data of the other theory (this is
nice instance of unity which we sometimes meet in mathematics). We shall
discuss how one can describe elements of packets and related questions,
like for example, given an irreducible  square integrable representation, what are the other elements of the packet etc.. All this is directly related to the classification of tempered representations and non-unitary dual in terms of cuspidal representations. Such classification in the case of GL-groups is given by Bernstein-Zelevinsly theory.

The other topic that we shall discuss in the lectures is the
unitarizability problem for classical groups. We recall that soon after
completion of Bernstein-Zelevinsky theory, the unitary duals in the
GL-case were classified (giving the same answer in the archimedean case). We expect pretty explicit picture of the unitary duals of classical groups, although much more complicated then in the GL-case (recall that similarly the classification of the irreducible square integrable representations is substantially more complicated for these groups then for GL-groups). For getting the answer, several difficult questions remains to be settled. We shall discuss what shape of classification we can expect and possible strategy for solving the problem.