##
**Orbital parameters for induced and restricted representations.**
*(English)*
Zbl 0683.22009

Let G be a connected Lie group and H a closed Lie subgroup. The author is concerned with two problems in representation theory: finding direct integral decompositions of \(Ind^ G_ H\nu\), \(\nu \in H^{\wedge}\), and of \(\pi |_ H\), \(\pi \in G^{\wedge}\). The current paper is concerned with the case where G is a simply connected exponential solvable Lie group. Let \({\mathfrak g}\) be the Lie algebra of G, \({\mathfrak h}\) that of H. Elements of \(G^{\wedge}\) can be identified with \(Ad^*(G)\) orbits in \({\mathfrak g}^*\), and similarly for \(H^{\wedge}\), \(H^*.\)

Suppose first that G is nilpotent. The reviewer, F. P. Greenleaf, and G. Grelaud gave a solution to the first problem that can be described as follows: let p: \({\mathfrak g}^*\to {\mathfrak h}^*\) be the canonical projection, let \({\mathcal O}_{\nu}\) be the orbit in \({\mathfrak h}^*\) corresponding to \(\nu\), and let \(\mu\) be a finite measure on \(p^{- 1}({\mathcal O}_{\nu})\subseteq {\mathfrak g}^*\) equivalent to Lebesgue measure \((p^{-1}({\mathcal O}_{\nu})\approx {\mathbb{R}}^ k\) for some k). Let \(\mu_*\) be the push-forward of \(\mu\) to \(G^{\wedge}\approx {\mathfrak g}^*/Ad^*(G)\). Then \(Ind^ G_ H\cong \int^{\oplus}_{G^{\wedge}}h_{\pi}\pi d\mu_*(\pi)\), where \(h_{\pi}\) is the number of \(Ad^*(H)\)-orbits in \(p^{-1}({\mathcal O}_{\nu})\cap {\mathcal O}_{\pi}\). (\({\mathcal O}_{\pi}\subseteq {\mathfrak g}^*\) is the orbit corresponding to \(\pi\).) For the second problem, let \(\lambda\) be a finite measure on \({\mathcal O}_{\pi}\) equivalent to the G- invariant measure there, and let \(\lambda_*\) be its push-forward to \(H^{\wedge}\cong {\mathfrak h}^*/Ad^*(H)\) under p. Then \(\pi |_ H\cong \int^{\oplus}_{H^{\wedge}}m_{\nu}\nu d\lambda_*(\nu)\), where \(m_{\nu}\) is the number of \(Ad^*(H)\)-orbits in \(p^{-1}({\mathcal O}_{\nu})\cap {\mathcal O}_{\pi}\). This result was obtained by F. P. Greenleaf and the reviewer.

The author begins by giving a simplified proof for these combined results. He also states the results in a different, but quite natural, form. (An account that is similar in some respects is given by the reviewer and F. P. Greenleaf [in Commun. Pure Appl. Math. 41, 1051- 1088 (1988; Zbl 0667.22004)].) He then considers the solvable case and proves similar theorems in the case where H is normal in G. (The case of induction has previously been done by Grelaud.) Finally, he gives an analysis of the case where H is conormal in G - i.e., where G is a semidirect product of H with a normal N. These last results are not fitted into the form of the earlier ones.

More recently, both the author and H. Fujiwara have announced that the theorems for nilpotent Lie groups described above hold in the exponential solvable case.

Suppose first that G is nilpotent. The reviewer, F. P. Greenleaf, and G. Grelaud gave a solution to the first problem that can be described as follows: let p: \({\mathfrak g}^*\to {\mathfrak h}^*\) be the canonical projection, let \({\mathcal O}_{\nu}\) be the orbit in \({\mathfrak h}^*\) corresponding to \(\nu\), and let \(\mu\) be a finite measure on \(p^{- 1}({\mathcal O}_{\nu})\subseteq {\mathfrak g}^*\) equivalent to Lebesgue measure \((p^{-1}({\mathcal O}_{\nu})\approx {\mathbb{R}}^ k\) for some k). Let \(\mu_*\) be the push-forward of \(\mu\) to \(G^{\wedge}\approx {\mathfrak g}^*/Ad^*(G)\). Then \(Ind^ G_ H\cong \int^{\oplus}_{G^{\wedge}}h_{\pi}\pi d\mu_*(\pi)\), where \(h_{\pi}\) is the number of \(Ad^*(H)\)-orbits in \(p^{-1}({\mathcal O}_{\nu})\cap {\mathcal O}_{\pi}\). (\({\mathcal O}_{\pi}\subseteq {\mathfrak g}^*\) is the orbit corresponding to \(\pi\).) For the second problem, let \(\lambda\) be a finite measure on \({\mathcal O}_{\pi}\) equivalent to the G- invariant measure there, and let \(\lambda_*\) be its push-forward to \(H^{\wedge}\cong {\mathfrak h}^*/Ad^*(H)\) under p. Then \(\pi |_ H\cong \int^{\oplus}_{H^{\wedge}}m_{\nu}\nu d\lambda_*(\nu)\), where \(m_{\nu}\) is the number of \(Ad^*(H)\)-orbits in \(p^{-1}({\mathcal O}_{\nu})\cap {\mathcal O}_{\pi}\). This result was obtained by F. P. Greenleaf and the reviewer.

The author begins by giving a simplified proof for these combined results. He also states the results in a different, but quite natural, form. (An account that is similar in some respects is given by the reviewer and F. P. Greenleaf [in Commun. Pure Appl. Math. 41, 1051- 1088 (1988; Zbl 0667.22004)].) He then considers the solvable case and proves similar theorems in the case where H is normal in G. (The case of induction has previously been done by Grelaud.) Finally, he gives an analysis of the case where H is conormal in G - i.e., where G is a semidirect product of H with a normal N. These last results are not fitted into the form of the earlier ones.

More recently, both the author and H. Fujiwara have announced that the theorems for nilpotent Lie groups described above hold in the exponential solvable case.

Reviewer: L.Corwin

### MSC:

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

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

22E25 | Nilpotent and solvable Lie groups |

22D30 | Induced representations for locally compact groups |

### Keywords:

induced representation; Kirillov theory; connected Lie group; direct integral decompositions; simply connected exponential solvable Lie group; Lie algebra; finite measure; induction; nilpotent Lie groups### Citations:

Zbl 0667.22004
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\textit{R. L. Lipsman}, Trans. Am. Math. Soc. 313, No. 2, 433--473 (1989; Zbl 0683.22009)

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### References:

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