# Rankin–Cohen brackets and representations of conformal Lie groups

Michael Pevzner^{[1]}

- [1] University of Reims – FR 3399 CNRS Moulin de la Housse, BP 1037 F-51687, Reims France

Annales mathématiques Blaise Pascal (2012)

- Volume: 19, Issue: 2, page 455-484
- ISSN: 1259-1734

## Access Full Article

top## Abstract

top## How to cite

topPevzner, Michael. "Rankin–Cohen brackets and representations of conformal Lie groups." Annales mathématiques Blaise Pascal 19.2 (2012): 455-484. <http://eudml.org/doc/251123>.

@article{Pevzner2012,

abstract = {This is an extended version of a lecture given by the author at the summer school “Quasimodular forms and applications” held in Besse in June 2010.The main purpose of this work is to present Rankin-Cohen brackets through the theory of unitary representations of conformal Lie groups and explain recent results on their analogues for Lie groups of higher rank. Various identities verified by such covariant bi-differential operators will be explained by the associativity of a non-commutative product induced on the set of holomorphic modular forms by a covariant quantization of the associate para-Hermitian symmetric space.},

affiliation = {University of Reims – FR 3399 CNRS Moulin de la Housse, BP 1037 F-51687, Reims France},

author = {Pevzner, Michael},

journal = {Annales mathématiques Blaise Pascal},

keywords = {Rankin-Cohen brackets; Unitary representations; Conformal groups; Covariant quantization; unitary representations; conformal groups; covariant quantization},

language = {eng},

month = {7},

number = {2},

pages = {455-484},

publisher = {Annales mathématiques Blaise Pascal},

title = {Rankin–Cohen brackets and representations of conformal Lie groups},

url = {http://eudml.org/doc/251123},

volume = {19},

year = {2012},

}

TY - JOUR

AU - Pevzner, Michael

TI - Rankin–Cohen brackets and representations of conformal Lie groups

JO - Annales mathématiques Blaise Pascal

DA - 2012/7//

PB - Annales mathématiques Blaise Pascal

VL - 19

IS - 2

SP - 455

EP - 484

AB - This is an extended version of a lecture given by the author at the summer school “Quasimodular forms and applications” held in Besse in June 2010.The main purpose of this work is to present Rankin-Cohen brackets through the theory of unitary representations of conformal Lie groups and explain recent results on their analogues for Lie groups of higher rank. Various identities verified by such covariant bi-differential operators will be explained by the associativity of a non-commutative product induced on the set of holomorphic modular forms by a covariant quantization of the associate para-Hermitian symmetric space.

LA - eng

KW - Rankin-Cohen brackets; Unitary representations; Conformal groups; Covariant quantization; unitary representations; conformal groups; covariant quantization

UR - http://eudml.org/doc/251123

ER -

## References

top- Katsuma Ban, On Rankin-Cohen-Ibukiyama operators for automorphic forms of several variables, Comment. Math. Univ. St. Pauli 55 (2006), 149-171 Zbl1137.11034MR2294926
- Y. Choie, B. Mourrain, P. Solé, Rankin-Cohen brackets and invariant theory, J. Algebraic Combin. 13 (2001), 5-13 Zbl1039.11024MR1817700
- Henri Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), 271-285 Zbl0311.10030MR382192
- Paula Beazley Cohen, Yuri Manin, Don Zagier, Automorphic pseudodifferential operators, Algebraic aspects of integrable systems 26 (1997), 17-47, Birkhäuser Boston, Boston, MA Zbl1055.11514MR1418868
- Alain Connes, Henri Moscovici, Modular Hecke algebras and their Hopf symmetry, Mosc. Math. J. 4 (2004), 67-109, 310 Zbl1122.11023MR2074984
- Alain Connes, Henri Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (2004), 111-130, 311 Zbl1122.11024MR2074985
- Gerrit van Dijk, Michael Pevzner, Ring structures for holomorphic discrete series and Rankin-Cohen brackets, J. Lie Theory 17 (2007), 283-305 Zbl1123.22009MR2325700
- Wolfgang Eholzer, Tomoyoshi Ibukiyama, Rankin-Cohen type differential operators for Siegel modular forms, Internat. J. Math. 9 (1998), 443-463 Zbl0919.11037MR1635181
- Amine M. El Gradechi, The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators, Adv. Math. 207 (2006), 484-531 Zbl1161.11331MR2271014
- Jacques Faraut, Adam Korányi, Analysis on symmetric cones, (1994), The Clarendon Press Oxford University Press, New York Zbl0841.43002MR1446489
- Mogens Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (1980), 253-311 Zbl0462.22006MR569073
- P. Gordan, Vorlesungen über Invariantentheorie. Herausgegeben von G. Kerschensteiner. Zweiter Band: Binäre Formen. 360 S., (1887), Leipzig. Teubner MR917266
- S. Gundelfinger, Zur Theorie der binären Formen., J. Reine Angew. Math (1887), 413-424
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 80 (1978), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York Zbl0993.53002MR514561
- Roger Howe, Eng-Chye Tan, Nonabelian harmonic analysis, (1992), Springer-Verlag, New York Zbl0768.43001MR1151617
- Soji Kaneyuki, Masato Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8 (1985), 81-98 Zbl0585.53029MR800077
- Toshiyuki Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1998), 100-135 Zbl0937.22008MR1600074
- Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642 Zbl0229.22026MR245725
- G. Ólafsson, B. Ørsted, The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal. 81 (1988), 126-159 Zbl0678.22008MR967894
- Peter J. Olver, Classical invariant theory, 44 (1999), Cambridge University Press, Cambridge Zbl0971.13004MR1694364
- Peter J. Olver, Jan A. Sanders, Transvectants, modular forms, and the Heisenberg algebra, Adv. in Appl. Math. 25 (2000), 252-283 Zbl1041.11026MR1783553
- Toshio Ōshima, Toshihiko Matsuki, A description of discrete series for semisimple symmetric spaces, Group representations and systems of differential equations (Tokyo, 1982) 4 (1984), 331-390, North-Holland, Amsterdam Zbl0577.22012MR810636
- Lizhong Peng, Genkai Zhang, Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal. 210 (2004), 171-192 Zbl1050.22020MR2052118
- M. Pevzner, Analyse conforme sur les algèbres de Jordan, J. Aust. Math. Soc. 73 (2002), 279-299 Zbl1019.17011MR1926074
- Michael Pevzner, Rankin-Cohen brackets and associativity, Lett. Math. Phys. 85 (2008), 195-202 Zbl1167.53075MR2443940
- Joe Repka, Tensor products of holomorphic discrete series representations, Canad. J. Math. 31 (1979), 836-844 Zbl0373.22006MR540911
- Ichirô Satake, Algebraic structures of symmetric domains, 4 (1980), Iwanami Shoten, Tokyo Zbl0483.32017MR591460
- Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970), 61-80 Zbl0219.32013MR259164
- Robert S. Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis 12 (1973), 341-383 Zbl0253.43013MR352884
- André Unterberger, Julianne Unterberger, Algebras of symbols and modular forms, J. Anal. Math. 68 (1996), 121-143 Zbl0857.43015MR1403254
- Don Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 57-75 Zbl0806.11022MR1280058
- Genkai Zhang, Rankin-Cohen brackets, transvectants and covariant differential operators, Math. Z. 264 (2010), 513-519 Zbl1189.32013MR2591818

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.