Monomial representations of certain exponential Lie groups
type: Item
instance of: scholarly article
title: Monomial representations of certain exponential Lie groups
zbMATH Open document ID: 0675.22002
DOI: 10.1016/0022-1236(89)90033-5
author: Junko Inoue
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let G be a connected Lie group with Lie algebra \({\mathfrak g}\). We note by \(\Delta_ G\) the modular function of G. Let \(\pi\) be an irreducible unitary representation of G. We denote by \({\mathcal H}^{\infty}_{\pi}\) the space of \(C^{\infty}\) vectors for \(\pi\), provided with the usual topology, and by \({\mathcal H}_{\pi}^{-\infty}\) its antidual. Given a closed subgroup H and its unitary character \(\chi\), define the space of semi-invariant generalized vectors for (\(\pi\),H,\(\chi)\); \[ ({\mathcal H}_{\pi}^{-\infty})^{H,\chi \Delta^{1/2}_{H,G}}=\{a\in {\mathcal H}_{\pi}^{-\infty};\quad \pi (h)a=\chi (h)\Delta^{1/2}_{H,G}(h)a,\quad \forall h\in H\}, \] where \(\Delta_{H,G}=\Delta_ H/\Delta_ G\). On the other hand, one constructs a monomial representation \(\tau =ind^ G_ H\chi\) of G, which is decomposed into irreducible constituents \(\tau =\int^{\oplus}_{\hat G}m(\pi)\pi d\mu (\pi)\) with respect to some Borel measure \(\mu\) on the unitary dual \(\hat G\) of G. For multiplicities m(\(\pi)\), it is natural to ask a kind of reciprocity, whether the equalities \(m(\pi)=\dim ({\mathcal H}_{\pi}^{-\infty})^{H,\chi \Delta^{1/2}_{H,G}}\) hold \(\mu\)-almost everywhere. Now we assume G exponential and use the orbit method. Let \(\theta\) : \({\mathfrak g}^*\to \hat G\) be the Kirillov-Bernat mapping. Assume that H is connected and let \({\mathfrak h}\) denote its Lie algebra. Then there exists a linear form f on \({\mathfrak g}\) such that f([\({\mathfrak h},{\mathfrak h}])=0\) and that \(\chi =\chi_ f\), \(\chi_ f(h)=e^{\sqrt{-1}f(\log h)}\) for \(\forall h\in H=\exp {\mathfrak h}\). We denote by \({\mathfrak h}^{\perp}\) the annihilator of \({\mathfrak h}\) in \({\mathfrak g}^*\). In these situations, \(\mu\) is supported on \(\theta (f+{\mathfrak h}^{\perp})\) and m(\(\pi)\) is equal to the number of H-orbits contained in \((f+{\mathfrak h}^{\perp})\cap \theta^{- 1}(\pi)\). The author studies the space (\({\mathcal H}_{\pi}^{- \infty})^{H,\chi_ f\Delta^{1/…
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 4105253
zbMATH Keywords: Frobenius reciprocity
zbMATH Keywords: connected Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: modular function
zbMATH Keywords: irreducible unitary representation
zbMATH Keywords: unitary character
zbMATH Keywords: monomial representation
zbMATH Keywords: irreducible constituents
zbMATH Keywords: unitary dual
zbMATH Keywords: multiplicities
zbMATH Keywords: orbit method
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90033-5
OpenAlex ID: W2077172177
cites work: Q5662168
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description: scientific article; zbMATH DE number 4105253