The contraction of K to \=NM
Let \(G=KAN\) be a connected semisimple Lie group of real rank one. Let \(H\in {\mathfrak a}^+\) be the unique element of norm 1. Define the dilation \(A_ t: G\to G\) \((t>0)\) and the contraction \(\pi_ t: \bar NM\to K\) \((t>0)\) as the following: \[ A_ t(g)=\exp (tH)\cdot g\cdot \exp (- tH),\quad g\in G,\quad \pi_ t(\bar nm)=k(A_ t(\bar n))\cdot m,\quad \bar n\in \bar N,\quad m\in M. \] The maps \(\pi_ t\) are diffeomorphisms onto a dense submanifold of K. Let \(X=G/K\) and \(\Omega =K/M\). The group K is the stabilizer of K-orbits in the symmetric space X and the group \(\bar NM\) is the stabilizer of every horocycle of the variety \(\Omega\). The group \(\bar NM\) is considered as a limit of the compact group K under the continuous family \((\pi_ t\); \(t>0)\) of contractions. Holomorphically induced irreducible unitary representations of the group \(\bar NM\), \(\sigma\) 's, and those of the group K, \(\rho\) 's, are constructed. The former are shown to be increasing limits of images under the maps \(\pi_ t\) of the latter, and using these facts, matrix coefficients of the representation \(\sigma\) are approximated by those of \(\rho\).
scientific article; zbMATH DE number 3944172
scientific article