Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra

Levasseur, Thierry

doi:10.1090/S0002-9947-01-02851-3

Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra
HTML articles powered by AMS MathViewer

by Thierry Levasseur PDF

Trans. Amer. Math. Soc. 353 (2001), 4189-4202 Request permission

Abstract:

Let $\mathfrak {g}$ be a semisimple complex Lie algebra with adjoint group $G$ and $\mathcal {D}(\mathfrak {g})$ be the algebra of differential operators with polynomial coefficients on $\mathfrak {g}$. If $\mathfrak {g}_0$ is a real form of $\mathfrak {g}$, we give the decomposition of the semisimple $\mathcal {D}(\mathfrak {g})^G$-module of invariant distributions on $\mathfrak {g}_0$ supported on the nilpotent cone.

References

James Arthur, The $L^2$-Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), no. 2, 257–290. MR 1001841, DOI 10.1007/BF01389042
Dan Barbasch and David A. Vogan Jr., The local structure of characters, J. Functional Analysis 37 (1980), no. 1, 27–55. MR 576644, DOI 10.1016/0022-1236(80)90026-9
Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex classical groups, Math. Ann. 259 (1982), no. 2, 153–199. MR 656661, DOI 10.1007/BF01457308
Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), no. 2, 350–382. MR 691809, DOI 10.1016/0021-8693(83)90006-6
Abderrazak Bouaziz, Intégrales orbitales sur les algèbres de Lie réductives, Invent. Math. 115 (1994), no. 1, 163–207 (French). MR 1248081, DOI 10.1007/BF01231757
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Victor Ginsburg, Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 5, 249–252 (French, with English summary). MR 693785
R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), no. 2, 327–358. MR 732550, DOI 10.1007/BF01388568
T. Levasseur, Equivariant $\mathrm {D}$-modules attached to nilpotent orbits in a semisimple Lie algebra, to appear in Transformation Groups, (1998).
T. Levasseur and J. T. Stafford, Invariant differential operators on the tangent space of some symmetric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1711–1741. MR 1738064
T. Levasseur and J. T. Stafford, The kernel of an homomorphism of Harish-Chandra, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 3, 385–397. MR 1386924
T. Levasseur and J. T. Stafford, Semi-simplicity of invariant holonomic systems on a reductive Lie algebra, Amer. J. Math. 119 (1997), no. 5, 1095–1117. MR 1473070
R. Rao, Orbital integrals on reductive groups, Ann. Math., 96 (1972), 505-510.
W. Rossmann, Nilpotent orbital integrals in a real semisimple Lie algebra and representations of Weyl groups, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 263–287. MR 1103593
W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
V. S. Varadarajan, Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer-Verlag, Berlin-New York, 1977. MR 0473111
Nolan R. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), no. 4, 779–816. MR 1212243, DOI 10.1090/S0894-0347-1993-1212243-2