The classification of the unitary dual of a reductive algebraic group is one of the most important unsolved problems in Representation theory. Its origins go back to Wigner’s work on the representations of the Lorentz group in Physics. It is also a fundamental problem in harmonic analysis and the Langlands Program, since they appear as local components of automorphic representations. In this research project we are concerned with the unitary duals of classical groups SO(2n+1,F) or Sp(2n,F) over local non-Archimedean field F, which we uniformly denote by G_n.
Motivated by the conjectures of the Langlands Program, J. Arthur defined A-packets which are finite multisets of irreducible unitarizable representations of G_n. In a recent paper, M. Tadić showed that certain class of representations which are isolated in the unitary dual of G_n appear in the A-packets, thus indicating an important role of A-packets in the unitarizability problem.
The results of my doctoral thesis describe the composition series of induced representations whose study is motivated by the classification of the unitary duals of the general linear groups. The unitarizability of some of their irreducible subquotients is determined by studying the ends of complementary series. To deepen the connection to the unitary dual of considered groups G_n, we use C. Mœglin’s explicit characterisation of A-packets to detect more unitary irreducible subquotients.
Research stays: April 4 - July 3, 2022 and September 1 - October 1, 2022
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