Invariant differential operators on the tangent space of some symmetric spaces
Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1711-1741.

Soient 𝔤 une algèbre de Lie semi-simple et ϑ une involution de 𝔤. Si 𝔨= Ker (ϑ-I) et 𝔭= Ker (ϑ+I), nous étudions les opérateurs différentiels, 𝒟(𝔭) K , sur 𝔭 qui sont invariants sous l’action du groupe adjoint K de 𝔨. Soit τ:𝔨 Der 𝒪(𝔭) la différentielle de cette action. Nous démontrons que, pour une classe d’espaces symétriques (𝔤,𝔨) considérée par Sekiguchi, on a d𝒟(𝔭):d𝒪 ( 𝔭 ) K =0=𝒟(𝔭)τ(𝔨). Une conséquence immédiate de cette égalité est le résultat suivant de Sekiguchi : Soient (𝔤 0 ,𝔨 0 ) une forme réelle de l’un de ces espaces symétriques (𝔤,𝔨), et T une distribution K 0 -invariante sur 𝔭 0 à support dans l’ensemble des éléments singuliers; alors, T=0. Dans le cas diagonal (𝔤,𝔨)=(𝔤 𝔤 ,𝔤 ) ce résultat bien connu est dû à Harish-Chandra.

Let 𝔤 be a complex, semisimple Lie algebra, with an involutive automorphism ϑ and set 𝔨= Ker (ϑ-I), 𝔭= Ker (ϑ+I). We consider the differential operators, 𝒟(𝔭) K , on 𝔭 that are invariant under the action of the adjoint group K of 𝔨. Write τ:𝔨 Der 𝒪(𝔭) for the differential of this action. Then we prove, for the class of symmetric pairs (𝔤,𝔨) considered by Sekiguchi, that d𝒟(𝔭):d𝒪 ( 𝔭 ) K =0=𝒟(𝔭)τ(𝔨). An immediate consequence of this equality is the following result of Sekiguchi: Let (𝔤 0 ,𝔨 0 ) be a real form of one of these symmetric pairs (𝔤,𝔨), and suppose that T is a K 0 -invariant eigendistribution on 𝔭 0 that is supported on the singular set. Then, T=0. In the diagonal case (𝔤,𝔨)=(𝔤 𝔤 ,𝔤 ) this is a well-known result due to Harish-Chandra.

@article{AIF_1999__49_6_1711_0,
     author = {Levasseur, Thierry and Stafford, J. Toby},
     title = {Invariant differential operators on the tangent space of some symmetric spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1711--1741},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     doi = {10.5802/aif.1736},
     mrnumber = {2001b:16025},
     zbl = {0943.22015},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1736/}
}
TY  - JOUR
AU  - Levasseur, Thierry
AU  - Stafford, J. Toby
TI  - Invariant differential operators on the tangent space of some symmetric spaces
JO  - Annales de l'Institut Fourier
PY  - 1999
SP  - 1711
EP  - 1741
VL  - 49
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1736/
DO  - 10.5802/aif.1736
LA  - en
ID  - AIF_1999__49_6_1711_0
ER  - 
%0 Journal Article
%A Levasseur, Thierry
%A Stafford, J. Toby
%T Invariant differential operators on the tangent space of some symmetric spaces
%J Annales de l'Institut Fourier
%D 1999
%P 1711-1741
%V 49
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1736/
%R 10.5802/aif.1736
%G en
%F AIF_1999__49_6_1711_0
Levasseur, Thierry; Stafford, J. Toby. Invariant differential operators on the tangent space of some symmetric spaces. Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1711-1741. doi : 10.5802/aif.1736. http://www.numdam.org/articles/10.5802/aif.1736/

[1] M.F. Atiyah, Characters of semi-simple Lie groups, (Lectures given in Oxford), Mathematical Institute, Oxford, 1976.

[2] D. Barbasch and D.A. Vogan, The Local Structure of Characters, J. Funct. Anal., 37 (1980), 27-55. | MR | Zbl

[3] A. Borel et al., Algebraic D-modules, Academic Press, Boston, 1987. | MR | Zbl

[4] W. Borho and H. Kraft, Über die Gelfand-Kirillov Dimension, Math. Annalen, 220 (1976), 1-24. | MR | Zbl

[5] J. Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc., 288 (1985), 125-137. | MR | Zbl

[6] O. Gabber, The integrability of the characteristic variety, Amer. J. Math, 103 (1981), 445-468. | MR | Zbl

[7] Harish-Chandra, Invariant distributions on Lie algebras, Amer. J. Math., 86 (1964), 271-309. | MR | Zbl

[8] Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math., 86 (1964), 534-564. | MR | Zbl

[9] Harish-Chandra, Invariant eigendistributions on a semisimple Lie algebra, Inst. Hautes Études Sci. Publ. Math., 27 (1965), 5-54. | Numdam | MR | Zbl

[10] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. | Zbl

[11] S. Helgason, Groups and Geometric Analysis, Academic Press, 1984.

[12] R. Hotta, Introduction to D-modules, (Lectures at the Inst. Math. Sci., Madras), Math. Institute, Tohoku University, Sendai, 1986.

[13] R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math., 75 (1984), 327-358. | MR | Zbl

[14] A. Joseph, Quantum groups and their primitive ideals, Springer-Verlag, Berlin-New York, 1995. | MR | Zbl

[15] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971), 753-809. | MR | Zbl

[16] A. Kowata, Spherical hyperfunctions on the tangent space of symmetric spaces, Hiroshima Math. J., 21 (1991), 401-418. | MR | Zbl

[17] T. Levasseur and J.T. Stafford, Invariant differential operators and an homomorphism of Harish-Chandra, J. Amer. Math. Soc., 8 (1995), 365-372. | MR | Zbl

[18] T. Levasseur and J.T. Stafford, The kernel of an homomorphism of Harish-Chandra, Ann. Scient. Éc. Norm. Sup., 29 (1996), 385-397. | Numdam | MR | Zbl

[19] T. Levasseur and J.T. Stafford, Semi-simplicity of invariant holonomic systems on a reductive Lie algebra, Amer. J. Math., 119 (1997), 1095-1117. | MR | Zbl

[20] T. Levasseur and R. Ushirobira, Adjoint vector fields on the tangent space of semisimple symmetric spaces, J. of Lie Theory, to appear. | Zbl

[21] M. Lorenz, Gelfand-Kirillov dimension and Poincaré series, Cuadernos de Algebra 7, Universidad de Granada, 1988. | Zbl

[22] J.C. Mcconnell and J.C. Robson, Noncommutative Noetherian Rings, John Wiley, Chichester, 1987. | MR | Zbl

[23] J.C. Mcconnell and J.T. Stafford, Gelfand-Kirillov dimension and associated graded modules, J. Algebra, 125 (1989), 197-214. | MR | Zbl

[24] J.S. Milne, Étale Cohomology, Princeton University Press, 1980. | MR | Zbl

[25] M. Noumi, Regular Holonomic Systems and their Minimal Extensions I, in "Group Representations and Systems of Differential Equations", Advanced Studies in Pure Mathematics, 4 (1984), 209-221. | MR | Zbl

[26] H. Ochiai, Invariant functions on the tangent space of a rank one semisimple symmetric space, J. Fac. Sci. Univ. Tokyo, 39 (1992), 17-31. | MR | Zbl

[27] D.I. Panyushev, The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties, Compositio Math., 94 (1994), 181-199. | Numdam | MR | Zbl

[28] V.L. Popov and E.B. Vinberg, Invariant Theory, in "Algebraic Geometry IV", (Eds: A.N. Parshin and I.R. Shafarevich), Springer-Verlag, Berlin, Heidelberg, New York, 1991.

[29] R.W. Richardson, Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke J. Math., 57 (1988), 1-35. | MR | Zbl

[30] G.W. Schwarz, Differential operators on quotients of simple groups, J. Algebra, 169 (1994), 248-273. | MR | Zbl

[31] G.W. Schwarz, Lifting differential operators from orbit spaces, Ann. Sci. École Norm. Sup., 28 (1995), 253-306. | Numdam | MR | Zbl

[32] J. Sekiguchi, The Nilpotent Subvariety of the Vector Space Associated to a Symmetric Pair, Publ. RIMS, Kyoto Univ., 20 (1984), 155-212. | MR | Zbl

[33] J. Sekiguchi, Invariant Spherical Hyperfunctions on the Tangent Space of a Symmetric Space, in "Algebraic Groups and Related Topics", Advanced Studies in Pure Mathematics, 6 (1985), 83-126. | MR | Zbl

[34] P. Slodowy, Simple singularities and Simple Algebraic Groups, Lecture Notes in Mathematics 815, Springer-Verlag, Berlin-New York, 1980. | MR | Zbl

[35] M. Van Den Bergh, Some rings of differential operators for Sl2-invariants are simple, J. Pure and Applied Algebra, 107 (1996), 309-335. | MR | Zbl

[36] V.S. Varadarajan, Harmonic Analysis on Real Reductive Groups, Part I, Lecture Notes in Mathematics 576, Springer-Verlag, Berlin-New York, 1977. | MR | Zbl

[37] E.B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR Izvestija, 10 (1976), 463-495. | MR | Zbl

[38] N. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc., 6 (1993), 779-816. | MR | Zbl

[39] H. Weyl, The Classical Groups, Princeton University Press, Princeton, 1939. | JFM

Cité par Sources :