Annales Mathématiques
Blaise Pascal
no. 2
Constant term in Harish-Chandra’s limit formula
Mladen Božičević1
1 Department of Geotechnical Engineering University of Zagreb Hallerova aleja 7 42000 Varaždin Croatia
Annales mathématiques Blaise Pascal, Volume 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
Keywords: Flag variety, equivariant sheaf, characteristic cycle, coadjoint orbit, Liouville measure
Mladen Božičević 1

1 Department of Geotechnical Engineering University of Zagreb Hallerova aleja 7 42000 Varaždin Croatia 
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Mladen Božičević. Constant term in Harish-Chandra’s  limit formula. Annales mathématiques Blaise Pascal, Volume 15 (2008) no. 2, pp. 153-168. doi : 10.5802/ambp.245. https://ambp.centre-mersenne.org/articles/10.5802/ambp.245/

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

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

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

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

[7] T. Matsuki 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] W. Rossmann; A. Connes; M. Duflo; A. Joseph; R. Rentschler 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] W. Rossmann 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] W. Schmid; W. Barker; P. Sally 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] W. Schmid; K. Vilonen Characteristic cycles of constructible sheaves, Invent. Math., Volume 124 (1996), pp. 451-502 | DOI | MR | Zbl

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

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

[14] M. Vergne 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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