Harmonic analysis and group representations (Q2756712)

The author begins by recalling that mathematical objects are often characterized by two parallel theories, e.g., conjugacy classes and irreducible characters for finite groups, logarithms of powers of prime numbers and zeros of \(\zeta (s)\) in number theory, rational conjugacy classes and automorphic representations for automorphic forms. In particular he sees Harmonic Analysis relating geometrical objects and spectral objects. NEWLINENEWLINENEWLINEThe 1979 Langlands program combined two kinds of spectral objects; it is a ``performed organizing scheme for fundamental arithmetic data in terms of highly structures analysis data''. NEWLINENEWLINENEWLINEThe main purpose of the article consists in giving a very sharp and accurate glimpse at the monumental achievements performed by Harish-Chandra, culmination of a 25 years' work, and its decisive influence on the study of the `Langlands program'. NEWLINENEWLINENEWLINEHarish-Chandra concentrated on semisimple Lie groups \(G\) and later more generally on reductive Lie groups, and also on \(p\)-adic groups. His Plancherel formula for \( {\mathcal C}_c^{\infty}(G),\) equals \(\|f\|_2^2\) with \(\|\widehat{f}\|_2 = \int _{\pi(G)} \|\widehat{f}(\pi)\|_2^2 d\pi\), where \( \pi\) runs over the space \(\pi (G)\) of equivalence classes of irreducible unitary representations of \(G\) and the Fourier transform \(\widehat{f}\) for \(f\) is given by NEWLINE\[NEWLINE \widehat{f}(\pi) = \int _G f(x)\pi(x) dx; NEWLINE\]NEWLINE \(\|\widehat{f}\|_2 \) is the Hilbert-Schmidt norm. The Plancherel measure \(d\pi\) exists and is unique. In particular, Harish-Chandra considers the discrete series, i.e., the family of representations \(\pi\) for which the Plancherel measure attaches positive mass. NEWLINENEWLINENEWLINEThe analytic aspect of the Langlands program is about automorphic functions and representation theory of reductive groups in the language introduced by Harish-Chandra; his powerful methods and constructions are largely applied to the study of automorphic functions.