Constant term in Harish-Chandra’s limit formula
Božičević, Mladen1
1 Department of Geotechnical Engineering University of Zagreb Hallerova aleja 7 42000 Varaždin Croatia
Annales mathématiques Blaise Pascal, Tome 15 (2008) no. 2, pp. 153-168.

Let G ℝ be a real form of a complex semisimple Lie group G. Recall that Rossmann defined a Weyl group action on Lagrangian cycles supported on the conormal bundle of the flag variety of G. We compute the signed average of the Weyl group action on the characteristic cycle of the standard sheaf associated to an open G ℝ -orbit on the flag variety. This result is applied to find the value of the constant term in Harish-Chandra’s limit formula for the delta function at zero.

DOI : 10.5802/ambp.245
Classification : 22E46, 22E30
Mots-clés : Flag variety, equivariant sheaf, characteristic cycle, coadjoint orbit, Liouville measure
Božičević, Mladen 1

1 Department of Geotechnical Engineering University of Zagreb Hallerova aleja 7 42000 Varaždin Croatia 
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     title = {Constant term in {Harish-Chandra{\textquoteright}s}  limit formula},
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Božičević, Mladen. Constant term in Harish-Chandra’s  limit formula. Annales mathématiques Blaise Pascal, Tome 15 (2008) no. 2, pp. 153-168. doi : 10.5802/ambp.245. http://www.numdam.org/articles/10.5802/ambp.245/

[1] Barbasch, D.; Vogan, D.; Trombi, P. Weyl group representations and nilpotent orbits, Representations of Reductive Groups, Progr. Math. 40, Birkhäuser, Boston, 1982, pp. 21-32 | MR | Zbl

[2] Bernstein, J.; Lunts, V. Equivariant sheaves and functors, Lecture Notes in Mathematics 1578, Springer-Verlag, Berlin, 1994 | MR | Zbl

[3] Božičević, M. Limit formulas for groups with one conjugacy class of Cartan subgroups, Ann. Inst. Fourier, Volume 58 (2008), pp. 1213-1232 | DOI | EuDML | Numdam | MR | Zbl

[4] Harish-Chandra Fourier transform on a semisimple Lie algebra II, Amer. J. Math., Volume 79 (1957), pp. 733-760 | DOI | MR | Zbl

[5] Kashiwara, M.; Schapira, P. Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer-Verlag, Berlin, 1990 | MR | Zbl

[6] Libine, M. A localization argument for characters of reductive Lie groups, J. Funct. Anal., Volume 203 (2003), pp. 197-236 | DOI | MR | Zbl

[7] Matsuki, T. The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, Volume 31 (1979), pp. 331-357 | DOI | MR | Zbl

[8] Rossmann, W.; Connes, A.; Duflo, M.; Joseph, A.; Rentschler, R. Nilpotent orbital integrals in a real semisimple Lie algebra and representations of the Weyl groups, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progr. Math. 92, Birkhäuser, Boston, 1990, pp. 263-287 | MR | Zbl

[9] Rossmann, W. Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra, Invent. Math., Volume 121 (1995), pp. 531-578 | DOI | EuDML | MR | Zbl

[10] Schmid, W.; Barker, W.; Sally, P. Construction and classification of irreducible Harish-Chandra modules, Harmonic analysis on reductive groups, Progr. Math. 101, Birkhäuser, Boston, 1991, pp. 235-275 | MR | Zbl

[11] Schmid, W.; Vilonen, K. Characteristic cycles of constructible sheaves, Invent. Math., Volume 124 (1996), pp. 451-502 | DOI | MR | Zbl

[12] Schmid, W.; Vilonen, K. Two geometric character formulas for reductive Lie groups, J. Amer. Math. Soc., Volume 11 (1998), pp. 799-867 | DOI | MR | Zbl

[13] Varadarajan, V.S. Lie groups, Lie algebras, and their representations, Graduate Texts in Math. 102, Springer-Verlag, New York, 1984 | MR | Zbl

[14] Vergne, M. Polynômes de Joseph et représentation de Springer, Ann. Sci. École Norm. Sup. (4), Volume 23 (1990), pp. 543-562 | Numdam | MR | Zbl

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