Springer Proceedings in Mathematics & Statistics

Volume 290 Springer Proceedings in Mathematics & Statistics

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533 Ali Baklouti • Takaaki Nomura Editors

Geometric and Harmonic Analysis on Homogeneous Spaces TJC 2017, Mahdia, Tunisia, December 17–21

123 Editors Ali Baklouti Takaaki Nomura Department of Mathematics, Faculty Faculty of Mathematics of Sciences at Sfax Kyushu University Sfax University Fukuoka, Japan Sfax, Tunisia

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-030-26561-8 ISBN 978-3-030-26562-5 (eBook) https://doi.org/10.1007/978-3-030-26562-5

Mathematics Subject Classification (2010): 22D05, 54B20, 22E40, 22E25, 22E27, 22E45, 33C67, 43A90, 53D05, 53D12, 53C12, 15A04, 17A99, 17D99, 22E46, 32M05, 22E60, 14M17, 81S10

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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface

This book presents a series of important contributions focusing on harmonic analysis and representation theory of Lie groups. All were originally presented at the 5th Tunisian–Japanese conference entitled “Geometric and Harmonic Analysis on Homogeneous Spaces and Applications”, which was held at Mahdia in Tunisia from December 17 to 21, 2017 and was dedicated to the memory of the brilliant Tunisian mathematician Majdi Ben Halima, who passed away at such a young age and contributed so significantly to this series of Tunisian–Japanese conferences. We deeply regret this tragic loss to our community. All of the contributions presented will be of high interest to harmonic analysts worldwide. Examples of the topics covered include: harmonic analysis for four-dimensional real Frobenius Lie algebras; quantization of color Lie bialgebras; self-Chabauty-isolated locally compact groups; Lagrangian submanifolds of stan- dard multisymplectic manifolds; spherical functions for small K-types; and the Poisson characteristic variety of unitary irreducible representations of exponential Lie groups. Two of the contributions are devoted to the scientific life of Majdi Ben Halima, who was passionate about finding new problems and developing bridges between ideas and theories. The contributions selected for publication were subjected to a peer review process by specialists and underwent modifications as required by the referees. They are without exception of a high standard, equivalent to that in the first-class mathematical periodicals. The conference attracted 86 participants, and 29 talks were given by the fore- most researchers from both Tunisia and Japan as well as from several other countries, including France. Lively mathematical discussions were held during and after the talks, and new joint works by some of the participants are currently in progress. The details of the conference are available at the following web page: https://www2.math.kyushu-u.ac.jp/*tnomura/Mahdia/. This series of Tunisian-Japanese conferences started with the conclusion of an agreement in June 2007 regarding academic cooperation between the Faculty of Science of Sfax University in Tunisia and the Faculty of Mathematics at Kyushu University in Japan. The agreement between the faculties was extended for 5 years

v vi Preface in 2012, with new participation of the Institute of Mathematics for Industry, Kyushu University, and then extended once again in 2017 for a further 5 years. Previous conferences in the series were held at Kerkennah Islands, Sousse, Hammamet, and Monastir in 2009, 2011, 2013, and 2015, respectively. The 6th conference is already planned for Djerba in December 2019. All of our conferences are supported by Grants-in-Aid for Scientific Research in Japan (JSPS), the Tunisian Mathematical Society, the Mediterranean Institute for the Mathematical Sciences (MIMS), the Ministry of Higher Education, Scientific Research and Technology in Tunisia, the Faculty of Sciences of Sfax University, and École Doctorale, Sciences Fondamentales, Sfax University. It is our pleasant duty to express our most sincere gratitude to these organizations for continuously supporting our activity. We are always thinking about how to involve younger generations to promote continuation of this academic exchange.

Sfax, Tunisia Ali Baklouti Fukuoka, Japan Takaaki Nomura Majdi Ben Halima a Brilliant Mathematician Turned off at Early Age

It is a tremendous honor to write this tribute to Majdi BEN HALIMA, an intimate Friend, Colleague, Teacher, Scholar, and a Researcher. In whatever role I knew him, from whatever vantage point, he stood apart as someone special! Majdi was committed to research focused on harmonic analysis and representation theory of Lie groups. He was passionate about finding new problems and enabling bridges between ideas and theories. In addition to his devotion to his work and to the improvement of research, he always found time for his colleagues, his friends, and his laboratory companions. Majdi Ben Halima was born on June 24, 1979 in the city of Sfax. He achieved his undergraduate studies in Mathematics at the Faculty of Sciences of Sfax in 2002 with excellent grades. He was amended the first prize for graduating with excel- lence from the Ministry of Higher education, Scientific Research and Technology in Tunisia. He then moved to the University of Paul-Verlaine-Metz in France, where he obtained his Master degree in Mathematics in 2003, after he got a Tunisian Government scholarship for Master and doctoral studies in France. In 2006, he defended his Ph.D. thesis entitled: “Invariant differential operators on homogeneous

vii viii Majdi Ben Halima a Brilliant Mathematician Turned off at Early Age

spaces: Branching rules and applications” under the supervision of Prof. Tilmann Wurzbacher. Right after, he pursued his researches on similar problems, which include: • Branching rules for compact Lie groups and spectra of invariant differential operators • Construction of fuzzy homogeneous spaces and applications • Orbit method for certain classical Lie groups and Lie subgroups. In 2013 Majdi defended his Habilitation thesis at the Faculty of Sciences of Sfax, after having published several important articles. The list of his publications includes: 1. M. Ben Halima, Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians, J. Algebra., 318 (2007), 520– 552. 2. M. Ben Halima, Spectrum of the Hodge Laplacian on complex Grassmannian m þ 2 Gr2ðC Þ; Bull. Sci. Math., 132 (2008), 19–36. 3. M. Ben Halima, Spectrum of twisted Dirac operators on the complex projective space P2q þ 1ðCÞ; Comment. Math. Univ. Carolin., 49 (2008), 437–445. 4. M. Ben Halima and T. Wurzbacher, Fuzzy complex Grassmannians and quantization of line bundles, Abh. Math. Semin. Univ. Hambg., 80 (2010), No. 1, 59–70. 5. M. Ben Halima, Construction of certain fuzzy flag manifolds, Rev. Math. Phys., 5 (2010), 533–548. 6. M. Ben Halima, Generalized Littlewood-Richardson rule and sum of coadjoint orbits of compact Lie groups, Bull. Sci. Math., 135 (2011), 345–352. 7. M. Ben Halima and A. Rahali, On the dual topology of a class of Cartan motion groups, J. Lie Theory., 22 (2012), No. 2, 491–503. 8. M. Ben Halima and A. Rahali, Dual topology of the Heisenberg motion groups, Indian J. Pure. App. Math., 45 (2014), 513–530. 9. M. Ben Halima, Coadjoint orbits of certain motion groups and their coherent states, J. Nonlinear Math. Phys., 20 (2013), 420–430. 10. M. Ben Halima and A. Rahali, Separation of unitary representations of Euclidien motion groups, Not Mat., 35 (2015), 15–22. 11. M. Ben Halima and Massaoud Anis, Corwin-Greenleaf multiplicity function for compact extensions of Rn: Int. J. Math., 26 (2015), No. 10. Majdi was an active member of our Laboratory. He regularly attended our seminars and also participated in several international events. He was an inspiring figure of our department, his serious, deep thinking, and pretty quiet character made of him unanimously a best friend for the whole staff. He very often came to my office and we talked together about teaching, research, family, society problems, and even some intimate issues. For what concerns mathematics, he was the brilliant theorist who made enduring contributions and inspired many researchers to pursue Majdi Ben Halima a Brilliant Mathematician Turned off at Early Age ix his alluring way. We did engage many regular discussions together about some problems related to visible actions on complex solvable homogeneous spaces and many preliminary results were obtained. Following a first suggestion of mine, we co-supervised together the Ph.D. thesis of Ayman Rahali, who is at present an Assistant Professor at the University of Kairouan. The defense ceremony turned on during the third Tunisian-Japanese conference on Geometric and Harmonic Analysis on Homogeneous spaces and Applications TJC3, hosted in Hammamet in 2013. The papers [7, 8, 10] follow from a fruitful collaboration, in which Majdi was the most perseverant engine. He then started the supervision of the Ph.D. thesis of the researcher Anis Massaoud, acting now as “Professeur-Agrégé” at the University of Gafsa and they published together the article [11] above. Majdi passed away on February 04, 2016, before the accomplishment of the project, after he fell sick from a severe disease. As Anis expressed his great enthusiasm, I was offered the immense honor to carry on the rest of this work. The publication [11] was as a starting milestone of a new issue in the theory of branching rules of unitary representations, and left behind many open problems, tackled by Anis in the next chapters of the thesis. We will all have our own personal and proper memories, of the legacy he left in our hearts and our lives, and it is very hard for me today to be up here, imagining his alluring behavior, trying my best to focus on the glad memories Majdi brought to us, rather than the fact that he is no longer here with us today. I do offer my heartfelt respects upon the passing of the talented scientist, to his grieving family, all harmonic analysts worldwide and the scientific community.

Ali Baklouti Sfax, Tunisia [email protected] Homage to Majdi Ben Halima 1979–2016

As I understand, the name Majdi signifies

glory, pride, grandeur, immortality. In retrospective these words are a perfectly fitting description of the life of the shy and polite, young man who entered my office in Metz in France in November 2002. Despite his late arrival to the last year of courses of the Master of Mathematics cycle, he quickly showed his great capacities and wrote an excellent Master thesis on index theory of invariant differential operators under my supervision in 2003. We continued our common path with the work on his Ph.D. on branching rules for linear representations of compact Lie groups with applications to geometric dif- ferential operators. His thesis was brilliant, concise, clear, and full of new insights, and it taught me a lot on the difficult subject of branching rules. His thesis defense took place in June 2009 and he then went back to his beloved hometown Sfax in Tunisia, where he occupied from September 2009 until his untimely death the position of an assistant professor in Mathematics and started a never failing professional career. Let me mention that he always took greatest care of his teaching, as well as of the written and oral presentations of his research. He received his Habilitation degree in Sfax in May 2013 during a ceremony I had the good fortune to be present. But not only his work went on with great success, he also founded a happy family with his wife Afef and their two children Yacine and Emna. I still relive the shock I got when I remember how he told me that he had to cope with a very serious disease. He never gave up fighting for his life, giving us an unforgettable image of strength in the hope to continue to be with his family and based on his faith. Majdi, we will not forget you!

December 2017 Tilmann Wurzbacher

xi Contents

Monomial Representations of Discrete Type of an Exponential Solvable Lie Group ...... 1 Ali Baklouti, Hidenori Fujiwara and Jean Ludwig Self-Chabauty-isolated Locally Compact Groups ...... 57 Hatem Hamrouni and Firas Sadki Quantization of Color Lie Bialgebras ...... 71 Benedikt Hurle and Abdenacer Makhlouf Harmonic Analysis for 4-Dimensional Real Frobenius Lie Algebras ...... 95 Edi Kurniadi and Hideyuki Ishi An Example of Holomorphically Induced Representations of Exponential Solvable Lie Groups ...... 111 Junko Inoue Spherical Functions for Small K-Types ...... 121 Hiroshi Oda and Nobukazu Shimeno A Cartan Decomposition for Non-symmetric Reductive Spherical Pairs of Rank-One Type and Its Application to Visible Actions ...... 169 Atsumu Sasaki Lagrangian Submanifolds of Standard Multisymplectic Manifolds .... 191 Gabriel Sevestre and Tilmann Wurzbacher The Poisson Characteristic Variety of Unitary Irreducible Representations of Exponential Lie Groups ...... 207 Ali Baklouti, Sami Dhieb and Dominique Manchon

xiii About the Editors

Ali Baklouti is Full Professor of Mathematics at the University of Sfax, Tunisia. He received his Ph.D. in Mathematics from the University of Metz (France) in 1995. He was elected as the President of the Tunisian Mathematical Society for two consecutive terms (April 2016–March 2019 and April 2019–March 2022), and he has been the Deputy Director of the Mediterranean Institute of Mathematical Sciences since January 2012. He was also nominated as a member of the “Tunisian Academy of Sciences, Letters and Arts: Beit Elhikma” in December 2016. He has published over 70 papers in peer-reviewed national and international journals and proceedings, as well as a number of book chapters. He is Co-Editor-in-Chief of the Tunisian Journal of Mathematics and an editorial board member for a number of well-known journals. He has delivered several invited talks at national and inter- national conferences.

Takaaki Nomura was a Full Professor of Mathematics at Kyushu University in Japan, and has been a Professor Emeritus since April 2019. He is currently affiliated with Osaka City University, Advanced Mathematical Institute. He has published about 30 original papers in peer-reviewed national and international journals, as well as several survey papers and proceeding articles written both in English and in Japanese. He has written three mathematical books in Japanese, and delivered several invited talks at national and international conferences.

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