no. 4
Covariant bi-differential operators on matrix space
Clerc, Jean-Louis1
1 Institut Élie Cartan, Université de Lorraine 54506 Vandœuvre-lès Nancy (France)
Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1427-1455.

A family of bi-differential operators from C ∞ (Mat(m,ℝ)×Mat(m,ℝ)) into C ∞ (Mat(m,ℝ)) which are covariant for the projective action of the group SL(2m,ℝ) on Mat(m,ℝ) is constructed, generalizing both the transvectants and the Rankin–Cohen brackets (case m=1).

On construit une famille d’opérateurs bi-différentiels de C ∞ (Mat(m,ℝ)×Mat(m,ℝ)) dans C ∞ (Mat(m,ℝ)) qui sont covariants pour l’action projective du groupe SL(2m,ℝ) sur Mat(m,ℝ). Dans le cas m=1, cette construction fournit une nouvelle approche des transvectants et des crochets de Rankin–Cohen.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3114
Classification: 22E45, 58J70
Keywords: Covariant differential operators, Knapp–Stein intertwining operators, Zeta functional equation, transvectants, Rankin–Cohen brackets
Mot clés : Opérateurs différentiels covariants, opérateurs d’entrelacement de Knapp–Stein, équation fonctionnelle de Zeta, transvectants, crochets de Rankin–Cohen
Clerc, Jean-Louis 1

1 Institut Élie Cartan, Université de Lorraine 54506 Vandœuvre-lès Nancy (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AIF_2017__67_4_1427_0,
     author = {Clerc, Jean-Louis},
     title = {Covariant bi-differential operators on matrix space},
     journal = {Annales de l'Institut Fourier},
     pages = {1427--1455},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {4},
     year = {2017},
     doi = {10.5802/aif.3114},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3114/}
}
TY  - JOUR
AU  - Clerc, Jean-Louis
TI  - Covariant bi-differential operators on matrix space
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 1427
EP  - 1455
VL  - 67
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3114/
DO  - 10.5802/aif.3114
LA  - en
ID  - AIF_2017__67_4_1427_0
ER  - 
%0 Journal Article
%A Clerc, Jean-Louis
%T Covariant bi-differential operators on matrix space
%J Annales de l'Institut Fourier
%D 2017
%P 1427-1455
%V 67
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3114/
%R 10.5802/aif.3114
%G en
%F AIF_2017__67_4_1427_0
Clerc, Jean-Louis. Covariant bi-differential operators on matrix space. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1427-1455. doi : 10.5802/aif.3114. https://aif.centre-mersenne.org/articles/10.5802/aif.3114/

[1] Barchini, Leticia; Sepanski, Mark R.; Zierau, Roger Positivity of zeta distributions and small unitary representations, The ubiquitous heat kernel (Contemporary Mathematics), Volume 398, American Mathematical Society, 2006, pp. 1-46

[2] Beckmann, Ralf; Clerc, Jean-Louis Singular invariant trilinear forms and covariant (bi)-differential operators under the conformal group, J. Funct. Anal., Volume 262 (2012) no. 10, pp. 4341-4376 | DOI

[3] Ben Saïd, Salem The functional equation of zeta distributions associated with non-Euclidean Jordan algebras., Can. J. Math., Volume 58 (2006) no. 1, pp. 3-22 | DOI

[4] Bopp, Nicole; Rubenthaler, Hubert Local zeta functions attached to the minimal spherical series for a class of symmetric spaces, Mem. Am. Math. Soc., 821, American Mathematical Society, 2005, 233 pages

[5] Clerc, Jean-Louis Singular conformally invariant trilinear forms. II: The higher multiplicity cases (to appear in Transform. Groups)

[6] van Dijk, Gerrit; Pevzner, Michael Ring structures for holomorphic discrete series and Rankin-Cohen brackets, J. Lie Theory, Volume 17 (2007) no. 2, pp. 283-305

[7] El Gradechi, Amine M. The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators, Adv. Math., Volume 207 (2006) no. 2, pp. 484-531 | DOI

[8] Faraut, Jacques; Korányi, Adam Analysis on symmetric cones, Oxford Mathematical Publications, Clarendon Press, 1994, xii+382 pages

[9] Gelbart, Stephen S. Fourier analysis on matrix space, Mem. Am. Math. Soc., 108, American Mathematical Society, 1971, 77 pages

[10] Ibukiyama, Tomoyoshi; Kuzumaki, Takako; Ochiai, Hiroyuki Holonomic systems of Gegenbauer polynomials of matrix arguments related with Siegel modular forms, J. Math. Soc. Japan, Volume 64 (2012) no. 1, pp. 273-316 | DOI

[11] Knapp, Anthony W. Representation theory of semisimple groups, an overview based on examples, Princeton Mathematical Series, 36, Princeton University Press, 1986, xvii+773 pages

[12] Kobayashi, Toshiyuki; Kubo, Toshihisa; Pevzner, Michael Vector-valued covariant differential operators for the Möbius transformation, Lie theory and its applications in physics (Springer Proceedings in Mathematics & Statistics), Volume 111, Springer, 2014, pp. 67-85

[13] Kobayashi, Toshiyuki; Pevzner, Michael Differential symmetry breaking operators. II: Rankin-Cohen operators for symmetric pairs, Sel. Math., Volume 22 (2016) no. 2, pp. 847-911 | DOI

[14] Muller, Iris Décomposition orbitale des espaces préhomogènes réguliers de type parabolique commutatif et application, C. R. Acad. Sci., Paris, Volume 303 (1986), pp. 495-498

[15] Olver, Peter J. Classical Invariant Theory, London Mathematical Society Student Texts, 44, Cambridge University Press, 1999, xxi+280 pages

[16] Olver, Peter J.; Petitot, Michel; Solé, Patrick Generalized Transvectants and Siegel modular forms, Adv. Appl. Math., Volume 38 (2007) no. 3, pp. 404-418 | DOI

[17] Peng, Lizhong; Zhang, Genkai Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal., Volume 210 (2004) no. 1, pp. 171-192 | DOI

[18] Sato, Mikio; Shintani, Takuro On zeta functions associated with prehomogeneous vector spaces, Ann. Math., Volume 100 (1974), pp. 131-170 | DOI

[19] Stein, Elias M. Analysis in matrix spaces and some new representations of SL(N,ℂ), Ann. Math., Volume 86 (1967), pp. 461-490 | DOI

[20] Tate, John Torrence Fourier analysis in number fields and Hecke’s zeta-functions, Princeton University, USA (1950) (Ph. D. Thesis)

[21] Unterberger, André; Unterberger, Julianne Algebras of symbols and modular forms, J. Anal. Math., Volume 68 (1996), pp. 121-143 | DOI

[22] Zagier, Don Modular forms and differential operators, Proc. Indian Acad. Sci., Volume 104 (1994) no. 1, pp. 57-75

[23] Zhang, Genkai Rankin-Cohen brackets, transvectants and covariant differential operators, Math. Z., Volume 264 (2010) no. 3, pp. 513-519 | DOI

Cited by Sources: