Another proof for the continuity of the Lipsman mapping

  • A. Messaoud
  • A. Rahali Univ. Sfax, Tunisia
Keywords: Lie groups, semidirect product, unitary representations, coadjoint orbits, symplectic induction

Abstract

UDC 515.1

We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \ddagger} /} G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G$. It was pointed out by Lipsman that the correspondence between $\hat{G} $ and ${\mathfrak{g}^{ \ddagger} /} G$ is bijective. Under some assumption on $G$, we give another proof for the continuity of the orbit mapping (Lipsman mapping)
$$\Theta : {\mathfrak{g}^{ \ddagger} /} G - \rightarrow \hat{G} .$$

References

D. Arnal, M. Ben Ammar, M. Selmi, Le problème de la réduction à un sous-groupe dans la quantification par déformation, Ann. Fac. Sci. Toulouse Math. (5), 12, 7 – 27 (1991), http://www.numdam.org/item?id=AFST_1991_5_12_1_7_0

W. Baggett, A description of the topology on the dual spaces of certain locally compact groups, Trans. Amer. Math. Soc., 132, 175 – 215 (1968), https://doi.org/10.2307/1994889

P. Baguis, Semidirect product and the Pukanszky condition, J. Geom. and Phys., 25, 245 – 270 (1998), https://doi.org/10.1016/S0393-0440(97)00028-4

M. Ben Halima, A. Rahali, On the dual topology of a class of Cartan motion groups, J. Lie Theory, 22, 491 – 503 (2012)

M. Elloumi and J. Ludwig, Dual topology of the motion groups $ SO(n) × R^n$, Forum Math., 22, no. 2, 397 – 410 (2008), https://doi.org/10.1515/FORUM.2010.022

J. M. G. Fell, Weak containment and induced representations of groups (II), Trans. Amer. Math. Soc. 110, 424 – 447 (1964), https://doi.org/10.2307/1993690

B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Super., (4), 6, 413 – 455 (1973).

H. Leptin and J. Ludwig, Unitary representation theory of exponential Lie groups, de Gruyter, Berlin (1994), http://www.numdam.org/item?id=ASENS_1973_4_6_4_413_0

R. L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures et Appl., 59, no. 3, 337 – 374 (1980).

A. Rahali, Dual topology of generalized motion groups, Math. Rep., 20(70), no. 3, 233 – 243 (2018).

A. Messaoud, A. Rahali, On the continuity of the Lipsman mapping of semidirect products, Rev. Roum. Math. Pures et Appl., 3(63), 249 – 258 (2018)

Published
15.07.2020
How to Cite
MessaoudA., and RahaliA. “Another Proof for the Continuity of the Lipsman Mapping”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 7, July 2020, pp. 945-51, doi:10.37863/umzh.v72i7.548.
Section
Research articles