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Boundary values of holomorphic functions and spectra of some unitary representations


These notes are based on my lectures on discrete spectra given in Tambov in August 1996 (school “Analysis on homogeneous spaces”). Now spectra in various problems of noncommutative harmonic analysis are completely or partially evaluated. It is well-known that sometimes such spectra contain discrete increments. Quite often such discrete increments are singular (“exotic”) unitary representations and it is very difficult to construct these unitary representations by other way, see [Puk], [Nai], [Boy], [Ism], [Mol1-3], [Str], [Far], [F-J], [Sch], [Kob1], [Kob2], [RSW], [Tsu], [How], [Ada], [Li], [Pat], [BO]. It was observed in [Ner1], [Ols2], [Ols3], [NO], [Ner2] that very often discrete increments to spectra in various problems of noncommunicative harmonic analysis (decomposition of tensor products, decomposition of restrictions, decomposition of induced representation) are related to some functional-theoretical phenomena, namely to so-called “trace theorems about existence of restrictions of discontinuous functions to submanifolds, for this type theorems see [RS], IX.3,IX.9 and references to this sections, [Bar], chapter 5, [NR], [Rud], 11.2).

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