A canonical approach to multiplicity formulas for induced and restricted representations of nilpotent Lie groups.

*(English)*Zbl 0667.22004Let G be a simply connected nilpotent Lie group, K a connected Lie subgroup and \({\mathfrak g}\), \({\mathfrak k}\) their Lie algebras. Let \(\pi\in \hat G\), \(\sigma\in \hat K\) be irreducible unitary representations. The authors study the decomposition of the induced representation ind(K\(\uparrow G,\sigma)\) and the restricted representation \(\pi |_ K\) into irreducibles. This had already been accomplished in former papers of them as well as independently by Grelaud (see e.g. [the authors and G. Grelaud, Trans. Am. Math. Soc. 304, 549-583 (1987; Zbl 0629.22005)]), however by methods which partially were “ad hoc” and not easily interpretable in terms of group actions. The approach in the present paper is based on new canonical decompositions of ind(K\(\uparrow G,\sigma)\) resp. \(\pi |_ K\), formulated here only for the case of the induced representation: If \(\sigma\in \hat K\) is associated with the coadjoint orbit \({\mathcal O}_{\sigma}\subset {\mathfrak k}^*\) via the Kirillov correspondence, and if P: \({\mathfrak g}^*\to {\mathfrak k}^*\) denotes the canonical projection, then there is a natural measure class [d\(\ell]\) on the variety \(P^{-1}({\mathcal O}_{\sigma})\subset {\mathfrak g}^*\), obtained from the canonical K-invariant measure on \({\mathcal O}_{\sigma}\) and Euclidean measure on the fibers parallel to \({\mathfrak k}^{\perp}\), such that
\[
ind(K\uparrow G,\sigma)\cong \int^{\oplus}_{P^{-1}({\mathcal O}_{\sigma})/Ad^*(K)}\pi_{\ell}d{\dot \ell},
\]
where \(\pi_{\ell}\in \hat G\) is associated with \(\ell \in {\mathfrak g}^*\) and [d\({\dot \ell}]\) is the measure class on the quotient space obtained from the push-forward of \(d\ell.\)

A careful analysis of this decomposition then allows the determination of the multiplicity of any \(\omega\in \hat G\) appearing in the above decomposition, and via this approach it comes as no big surprise that this multiplicity is given by the number of \(Ad^*(K)\) orbits in \({\mathcal O}_{\omega}\cap P^{-1}({\mathcal O}_{\sigma})\), thus yielding a simpler and more transparent proof of this multiplicity formula than in the afore-mentioned paper.

Important tools for the proofs are, besides the usual induction over dimension and Kirillov theory, Pukanszky’s parametrization of all orbits for a unipotent action on a vector space and the theory of semi-algebraic sets.

A careful analysis of this decomposition then allows the determination of the multiplicity of any \(\omega\in \hat G\) appearing in the above decomposition, and via this approach it comes as no big surprise that this multiplicity is given by the number of \(Ad^*(K)\) orbits in \({\mathcal O}_{\omega}\cap P^{-1}({\mathcal O}_{\sigma})\), thus yielding a simpler and more transparent proof of this multiplicity formula than in the afore-mentioned paper.

Important tools for the proofs are, besides the usual induction over dimension and Kirillov theory, Pukanszky’s parametrization of all orbits for a unipotent action on a vector space and the theory of semi-algebraic sets.

Reviewer: D.Müller

##### MSC:

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

22D30 | Induced representations for locally compact groups |

14Pxx | Real algebraic and real-analytic geometry |

##### Keywords:

simply connected nilpotent Lie group; Lie algebras; irreducible unitary representations; induced representation; restricted representation; group actions; decompositions; Kirillov correspondence; multiplicity; Pukanszky’s parametrization; unipotent action; semi-algebraic sets
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\textit{L. Corwin} and \textit{F. P. Greenleaf}, Commun. Pure Appl. Math. 41, No. 8, 1051--1088 (1988; Zbl 0667.22004)

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