Quantification of coadjoint orbits and the theory of contractions (Q2730483)

The purpose of this paper is to establish on a simple but non-trivial example the compatibility (in the sense described below) of two techniques in the representation theory of Lie groups. The first of these techniques is the so called Kirillov-Kostant-Souriau method which tries to describe the unitary dual of a Lie group \(G\) in terms of coadjoint orbits of \(G\) on the dual of its Lie algebra. Roughly speaking, a unitary representation should be obtained from a coadjoint orbit by a ``quantization'' of its symplectic structure. We shall not be concerned here with the difficulty of this program in the case of semisimple Lie groups, since the author considers only orbits and representations for which this ``quantization'' procedure is well understood. Let \(G\) be a connected Lie group with Lie algebra \(\mathfrak g\). Let \(\mathcal O\) be a coadjoint orbit of \(G\) on \(\mathfrak g^*\), and assume that attached to \(\mathcal O\) there is a unitary irreducible representation \(\pi\) of \(G\) in a Hilbert space \(\mathcal H\). For all \(X \in \mathfrak g\), define NEWLINE\[NEWLINE \widetilde X(\xi)=\langle \xi,X \rangle ,\qquad (\xi \in \mathcal O \subset \mathfrak g^*). NEWLINE\]NEWLINE Thus \(\widetilde X\) is a regular function on \(\mathcal O\). Following the terminology of the author, call a one-to-one linear map \(f \mapsto W(f)\) from a certain class of functions on \(\mathcal O\) called symbols to a certain class of operators on \(\mathcal H\), adapted Weyl correspondence such that the functions \(\widetilde X\), \(X \in \mathfrak g\), are symbols and such that there exists a dense subspace \(D\) of \(\mathcal H\) of \(\mathcal C^\infty\)-vectors satisfying NEWLINE\[NEWLINE W(i \widetilde X)\cdot \varphi=d\pi(X)\cdot \varphi \qquad (X\in \mathfrak g) \quad (\varphi \in D). NEWLINE\]NEWLINE This definition is an attempt to generalize directly the usual ``quantization rules'', in particular, if \(W\) is adapted NEWLINE\[NEWLINE [W(\widetilde X),W(\widetilde Y)]=-iW(\{ \widetilde X,\widetilde Y \}),\qquad (X,Y\in \mathfrak g),NEWLINE\]NEWLINE where \(\{ .,.\}\) is the Poisson product associated to the symplectic structure on \(\mathcal O\). The other technique mentioned above is the theory of contractions of semisimple groups introduced by A. H. Dooley and J. W. Rice. The typical situation occurs as follows. Let \(G\) be a connected semisimple Lie group, and let \(K\) be a reductive subgroup of \(G\). Let \(V\) be a \(K\)-stable subspace of \(\mathfrak g\) such that \(\mathfrak g=\mathfrak k \oplus V\), and form the semi-direct product \(G'=V\times K\). For all \(\lambda >0\) consider \(c_\lambda: V\times K \rightarrow G\), NEWLINE\[NEWLINE c_\lambda(v,k)=\exp_G(\lambda v)\cdot k \qquad (v\in V,k\in K). NEWLINE\]NEWLINE Its differential \(dc_\lambda: V\times \mathfrak k \rightarrow \mathfrak g\) is given by NEWLINE\[NEWLINE dc_\lambda(v,A)=\lambda v+ k \qquad (v\in V,A\in \mathfrak k). NEWLINE\]NEWLINE For all \(X,Y \in \mathfrak g\), one has NEWLINE\[NEWLINE \lim_{\lambda \rightarrow 0} dc_\lambda^{-1} ([dc_\lambda(X), dc_\lambda(Y)]_{\mathfrak g}) =[X,Y]_{V\times \mathfrak k}.NEWLINE\]NEWLINE Using this, one can make a connection between harmonic analysis on \(G\) and on \(G'\). In some cases, for instance when \(\mathfrak g=\mathfrak k \bigoplus \mathfrak p\) is a Cartan decomposition, or in the example below, generic unitary representations of \(G'\) are obtained as ``contractions'' of principal series of \(G\). The author studies the case \(G=SO_e(n+1,1)\), \(G'=\mathbb R^{n+1}\times SO_e(n,1)\). He recalls the construction of an adapted Weyl correspondence for some coadjoint orbits of \(G'\) (orbits with real mass), and for coadjoint orbits of \(G\) corresponding to the principal series. He shows that contractions of representations from \(G\) to \(G'\) are obtained via the corresponding notion of contractions of adapted Weyl correspondences.