Hilbert-substructure of real measurable spaces on reductive groups. I: Basic theory. (English) Zbl 1458.43003

Oyadare, O. O.

J. Gen. Lie Theory Appl. 10, No. 1, Article ID 1000242, 4 p. (2016).

Summary: This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, \(\mathfrak{F}_n(G)\), of real \(L^p(G)\)-spaces (when \(G\) is a reductive group in the Harish-Chandra class and \(p=2n\)), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, \(L^2(G)\). This success opens the door for harmonic analysis of unitary representations, \(G\to \text{End}(\mathfrak{F}_n(G))\), of \(G\) on the Hilbert-substructure \(\mathfrak{F}_n(G)\), which has hitherto been considered impossible.

MSC:

43A15	\(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A80	Analysis on other specific Lie groups

Keywords:

reductive groups; Hilbert spaces; orthogonal polynomials