zbMATH — the first resource for mathematics

Restricting representations of completely solvable Lie groups. (English) Zbl 0724.22007
Let 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 $$\Omega_{\pi}\cap p^{-1}(H\cdot \phi)$$.

MSC:
 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 2.2e+26 Nilpotent and solvable Lie groups
Full Text: