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
Messaoud, A., and A. Rahali. “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