Restricting representations of completely solvable Lie groups.

*(English)*Zbl 0724.22007Let G be a connected and simply connected Lie group with Lie algebra \({\mathfrak g}\), and let H be an analytic subgroup of G with Lie algebra \({\mathfrak h}\). It is an interesting problem to know how decomposes an irreducible unitary representation of G when restricted to H. G being nilpotent, a complete answer was given by L. Corwin and F. Greenleaf [Pac. J. Math. 135, 233-267 (1988; Zbl 0628.22007)] in terms of Kirillov’s orbit method. This fundamental result is generalized in this paper for completely solvable Lie groups.

Suppose that G is completely solvable. Then starting from \(\phi\in {\mathfrak g}^*\), we can construct an irreducible unitary representation \(\nu_{\phi}\) of G. This procedure permits us to parametrize the unitary dual \(\hat G\) of G by the orbit space \({\mathfrak g}^*/G\) under coadjoint action of G. For \(\pi\in \hat G\), we denote by \(\Omega_{\pi}\) the corresponding G-orbit. Then we have the following decomposition formula for the restriction \(\pi |_ H:\) \[ \pi |_ H=\int^{\oplus}_{p(\Omega_{\pi})/H}n_{\psi}\nu_{\psi}d\lambda ({\dot \phi}), \] where p: \({\mathfrak g}^*\to {\mathfrak h}^*\) is the canonical projection, \(\lambda\) is the push-forward of the canonical invariant measure on \(\Omega_{\pi}\) under \({\mathfrak g}^*\to {\mathfrak h}^*/H\), and the multiplicity \(n_{\psi}\) is given by the number of H- orbits contained in \(\Omega_{\pi}\cap p^{-1}(H\cdot \phi)\).

Suppose that G is completely solvable. Then starting from \(\phi\in {\mathfrak g}^*\), we can construct an irreducible unitary representation \(\nu_{\phi}\) of G. This procedure permits us to parametrize the unitary dual \(\hat G\) of G by the orbit space \({\mathfrak g}^*/G\) under coadjoint action of G. For \(\pi\in \hat G\), we denote by \(\Omega_{\pi}\) the corresponding G-orbit. Then we have the following decomposition formula for the restriction \(\pi |_ H:\) \[ \pi |_ H=\int^{\oplus}_{p(\Omega_{\pi})/H}n_{\psi}\nu_{\psi}d\lambda ({\dot \phi}), \] where p: \({\mathfrak g}^*\to {\mathfrak h}^*\) is the canonical projection, \(\lambda\) is the push-forward of the canonical invariant measure on \(\Omega_{\pi}\) under \({\mathfrak g}^*\to {\mathfrak h}^*/H\), and the multiplicity \(n_{\psi}\) is given by the number of H- orbits contained in \(\Omega_{\pi}\cap p^{-1}(H\cdot \phi)\).

Reviewer: H.Fujiwara (Iizuka)

##### MSC:

22E27 | Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) |

22E25 | Nilpotent and solvable Lie groups |