Filtrations of Nilpotent Lie Groups by Square Integrable Subgroups

M. L. BARBERIS, I.DOTTI, J.A. WOLF

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

FaMAF-UNC

A connected simply connected Lie group $N$ with center $Z$ is called {em square integrable} if it has unitary representations $pi$ whose coefficients $f_{u,v}(x) = langle u, pi(x)v angle$ satisfy $|f_{u,v}| in L^2(N/Z)$. There is a well developed theory of square integrable representations of nilpotent Lie groups cite{MW1973}, based on their general representation theory cite{K1962}. We study the conditions for a nilpotent Lie group to decompose into square integrable subgroups. For those groups admitting such a decomposition we show that the Plancherel formula is easily described in terms of the given square integrable subgroups. We also present some examples of interest in harmonic analysis and differential geometry. %(It should not exceed 15 lines. Please, avoid the use of nonstandard symbols/fonts) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {small egin{thebibliography}{99} ibitem{K1962} A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspekhi Math. Nauk {f 17} (1962), 57--110 (English: Russian Math. Surveys {f 17} (1962), 53--104). ibitem{MW1973} C. C. Moore & J. A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. {f 185} (1973), 445--462.