##
**Weyl quantization for principal series.**
*(English)*
Zbl 1134.22010

Let \(G\) be a connected Lie group. Let \(\pi\) be a unitary irreducible representation of \(G\) on a Hilbert space \(H\). Suppose that the representation \(\pi\) is associated with a coadjoint orbit \(\mathcal O\) of \(G\) by the Kirillov-Kostant method of orbits. A notion of adapted Weyl correspondence was introduced in [B. Cahen, Lett. Math. Phys. 36, No. 1, 65–75 (1996; Zbl 0843.22020); see also B. Cahen, C. R. Acad. Sci. Paris, Sér. I, Math. 325, No. 7, 803–806 (1997; Zbl 0883.22016)]. Roughly speaking, an adapted Weyl correspondence \(W\) on the orbit \(\mathcal O\) is a one-to-one linear correspondence between a class of functions on the orbit \(\mathcal O\) (called symbols) and a class of operators on \(H\) such that the functions \({\widetilde X}\) (\(X \in {\mathfrak g}\)) are symbols and for each \(v\) in a dense subspace of \(H\) and each \(X \in {\mathfrak g}\) we have \(W(i\widetilde X)\,v=d\pi (X)v\). A more precise definition of an adapted Weyl correspondence inspired by the work of Mark Gotay [M. Gotay, Obstructions to Quantization, in: Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo), J. Nonlinear Science Editors, Springer New York, 171–216 (2000; Zbl 1041.53507)] was proposed in [B. Cahen, Differ. Geom. Appl. 25, No. 2, 177–190 (2007; Zbl 1117.81087)].

Our original motivation for constructing adapted Weyl correspondences was to build covariant star-products on coadjoint orbits [B. Cahen, Lett. Math. Phys. 36, No. 1, 65–75 (1996; Zbl 0843.22020)]. A more recent motivation is that adapted Weyl correspondences can be used to study contractions of representations of Lie groups in the setting of the Kirillov-Kostant method of orbits. The basic idea is then to interpret contraction results on the symbols of the representation operators [P. Cotton and A. H. Dooley, J. Lie Theory 7, No. 2, 147–164 (1997; Zbl 0882.22015); B. Cahen, J. Lie Theory 11, No. 2, 257–272 (2001; Zbl 0973.22009)].

In [B. Cahen, Lett. Math. Phys. 36, No. 1, 65–75 (1996; Zbl 0843.22020)], adapted Weyl correspondences on the coadjoint orbits associated to the principal series representations of a connected semisimple non-compact Lie group were constructed by combining the Berezin calculus and a symbolic calculus on the cotangent bundle of a nilpotent Lie group. In [B. Cahen, Differ. Geom. Appl. 25, No. 2, 177–190 (2007; Zbl 1117.81087)], we have considered the case when \(G\) is the semidirect product \(V\rtimes K\) where \(K\) is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space \(V\) and \(\mathcal O\) is a coadjoint orbit of \(G\) associated by the method of orbits to a unitary irreducible representation \(\pi\) of \(G\). Under the assumption that the corresponding little group \(K_0\) is a maximal compact subgroup of \(K\), we have shown that the orbit \(\mathcal O\) is symplectomorphic to the symplectic product \({\mathbb R}^{2n}\times {\mathcal O}' \) where \(n=\dim (K)-\dim (K_0)\) and \({\mathcal O}' \) is a coadjoint orbit of \(K_0\). Thus we have obtained an adapted Weyl correspondence on \(\mathcal O\) by combining the usual Weyl correspondence on \({\mathbb R}^{2n}\) and the Berezin calculus on \({\mathcal O}' \).

In the paper under review, we revisit the case when \(G\) is a connected semisimple non-compact Lie group and \(\mathcal O\) is a coadjoint orbit of \(G\) associated to a principal series representation \(\pi\) of \(G\). We fix an Iwasawa decomposition \(G=KAN\) of \(G\) and we denote by \(M\) the centralizer of \(A\) in \(K\). We use the dequantization procedure introduced in [B. Cahen, Differ. Geom. Appl. 25, 177–190 (2007; Zbl 1117.81087)] in order to obtain an adapted Weyl correspondence on \(\mathcal O\) using only the usual Weyl correspondence and the Berezin calculus. In particular, we obtain an explicit symplectomorphism from a symplectic product \({\mathbb R}^{2n}\times {\mathcal O}' \) onto a dense open set of \(\mathcal O\) where \({\mathcal O}'\) is a coadjoint orbit of \(M\).

Our original motivation for constructing adapted Weyl correspondences was to build covariant star-products on coadjoint orbits [B. Cahen, Lett. Math. Phys. 36, No. 1, 65–75 (1996; Zbl 0843.22020)]. A more recent motivation is that adapted Weyl correspondences can be used to study contractions of representations of Lie groups in the setting of the Kirillov-Kostant method of orbits. The basic idea is then to interpret contraction results on the symbols of the representation operators [P. Cotton and A. H. Dooley, J. Lie Theory 7, No. 2, 147–164 (1997; Zbl 0882.22015); B. Cahen, J. Lie Theory 11, No. 2, 257–272 (2001; Zbl 0973.22009)].

In [B. Cahen, Lett. Math. Phys. 36, No. 1, 65–75 (1996; Zbl 0843.22020)], adapted Weyl correspondences on the coadjoint orbits associated to the principal series representations of a connected semisimple non-compact Lie group were constructed by combining the Berezin calculus and a symbolic calculus on the cotangent bundle of a nilpotent Lie group. In [B. Cahen, Differ. Geom. Appl. 25, No. 2, 177–190 (2007; Zbl 1117.81087)], we have considered the case when \(G\) is the semidirect product \(V\rtimes K\) where \(K\) is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space \(V\) and \(\mathcal O\) is a coadjoint orbit of \(G\) associated by the method of orbits to a unitary irreducible representation \(\pi\) of \(G\). Under the assumption that the corresponding little group \(K_0\) is a maximal compact subgroup of \(K\), we have shown that the orbit \(\mathcal O\) is symplectomorphic to the symplectic product \({\mathbb R}^{2n}\times {\mathcal O}' \) where \(n=\dim (K)-\dim (K_0)\) and \({\mathcal O}' \) is a coadjoint orbit of \(K_0\). Thus we have obtained an adapted Weyl correspondence on \(\mathcal O\) by combining the usual Weyl correspondence on \({\mathbb R}^{2n}\) and the Berezin calculus on \({\mathcal O}' \).

In the paper under review, we revisit the case when \(G\) is a connected semisimple non-compact Lie group and \(\mathcal O\) is a coadjoint orbit of \(G\) associated to a principal series representation \(\pi\) of \(G\). We fix an Iwasawa decomposition \(G=KAN\) of \(G\) and we denote by \(M\) the centralizer of \(A\) in \(K\). We use the dequantization procedure introduced in [B. Cahen, Differ. Geom. Appl. 25, 177–190 (2007; Zbl 1117.81087)] in order to obtain an adapted Weyl correspondence on \(\mathcal O\) using only the usual Weyl correspondence and the Berezin calculus. In particular, we obtain an explicit symplectomorphism from a symplectic product \({\mathbb R}^{2n}\times {\mathcal O}' \) onto a dense open set of \(\mathcal O\) where \({\mathcal O}'\) is a coadjoint orbit of \(M\).

Reviewer: Benjamin Cahen (Metz)

### MSC:

22E46 | Semisimple Lie groups and their representations |

81S10 | Geometry and quantization, symplectic methods |