Biomedical subjects
Publications and source records attributed to A W Knapp.
For a wide class of linear connected semisimple Lie groups, one obtains formulas limiting the Langlands parameters of irreducible unitary representations obtained from maximal parabolic subgroups. The formulas relate unitarity to the number of roots satisfying certain conditions. Some evidence is presented that the formulas are sharp. The results confirm aspects of conjectures that relate unitary parameters to cohomological induction.
For each connected real semisimple matrix group, one obtains a constructive list of the irreducible tempered unitary representations and their characters. These irreducible representations all turn out to be instances of a more general kind of representation, here called basic. The result completes Langland's classification of all irreducible admissible representations for such groups. Since not all basic representations are irreducible, a study is made of character identities relating different basic representations and of the commuting algebra for each basic representation.
The intertwining operators that have been constructed for all the series of unitary representations appearing in the Plancherel formula of a connected real semisimple Lie group of matrices are given a new normalization and then applied in two ways. The first is to obtain dimension formulas for the commuting algebras of the unitary representations in question. The second is to establish the existence of complementary series. These bear the same relationship to the unitary representations under study that the complementary series found earlier bear to the principal series.
The intertwining operators for nonunitary principal series representations of a real semisimple Lie group are used to construct intertwining operators for all the series of unitary representations appearing in the Plancherel formula. The resulting machinery reduces computations with Harish-Chandra's c-functions and the Plancherel measure to computations with those c-functions and measures obtained from a minimal parabolic subgroup.
A further study is made of the intertwining operators for the representations of the simple Lie groups of real rank one. We extend and apply this to the higher rank case, obtaining various results dealing with reducibility of principal series and existence of complementary series.
The L(2) theory of singular integral operators on nilpotent Lie groups is studied, extending known results for IR(n). The intertwining operators for the representations of the simple Lie groups of real rank one turn out to be of this type. As a result we determine which representations of the principal series of these groups are irreducible.
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