## Invariant measures on nilpotent orbits associated with holomorphic discrete series

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- by Mladen Božičević
- Represent. Theory
**25**(2021), 732-747 - DOI: https://doi.org/10.1090/ert/580
- Published electronically: August 18, 2021
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## Abstract:

Let $G_\mathbb R$ be a real form of a complex, semisimple Lie group $G$. Assume $G_\mathbb R$ has holomorphic discrete series. Let $\mathcal W$ be a nilpotent coadjoint $G_\mathbb R$-orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on $\mathcal W$ as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the positive chamber defined by the Borel subalgebra associated with holomorphic discrete series.## References

- Joseph Bernstein and Valery Lunts,
*Equivariant sheaves and functors*, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR**1299527**, DOI 10.1007/BFb0073549 - Armand Borel,
*Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts*, Ann. of Math. (2)**57**(1953), 115–207 (French). MR**51508**, DOI 10.2307/1969728 - Armand Borel,
*Linear algebraic groups*, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR**1102012**, DOI 10.1007/978-1-4612-0941-6 - Walter Borho and Robert MacPherson,
*Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 15, 707–710 (French, with English summary). MR**618892** - Mladen Božičević,
*A limit formula for elliptic orbital integrals*, Duke Math. J.**113**(2002), no. 2, 331–353. MR**1909221**, DOI 10.1215/S0012-7094-02-11325-8 - Mladen Božičević,
*Characteristic cycles of standard sheaves associated with open orbits*, Proc. Amer. Math. Soc.**136**(2008), no. 1, 367–371. MR**2350425**, DOI 10.1090/S0002-9939-07-08986-1 - Mladen Božičević,
*Limit formulas for groups with one conjugacy class of Cartan subgroups*, Ann. Inst. Fourier (Grenoble)**58**(2008), no. 4, 1213–1232 (English, with English and French summaries). MR**2427959**, DOI 10.5802/aif.2383 - Mladen Božičević,
*Homology groups of conormal varieties*, Mediterr. J. Math.**4**(2007), no. 4, 407–418. MR**2369235**, DOI 10.1007/s00009-007-0126-x - Jen-Tseh Chang,
*Characteristic cycles of holomorphic discrete series*, Trans. Amer. Math. Soc.**334**(1992), no. 1, 213–227. MR**1087052**, DOI 10.1090/S0002-9947-1992-1087052-3 - B. Kostant and S. Rallis,
*Orbits and representations associated with symmetric spaces*, Amer. J. Math.**93**(1971), 753–809. MR**311837**, DOI 10.2307/2373470 - Masaki Kashiwara and Pierre Schapira,
*Sheaves on manifolds*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR**1074006**, DOI 10.1007/978-3-662-02661-8 - Toshihiko Matsuki,
*The orbits of affine symmetric spaces under the action of minimal parabolic subgroups*, J. Math. Soc. Japan**31**(1979), no. 2, 331–357. MR**527548**, DOI 10.2969/jmsj/03120331 - D. Miličić,
*Localization and representation theory of reductive Lie groups*, Preliminary manuscript, 1993. - I. Mirković, T. Uzawa, and K. Vilonen,
*Matsuki correspondence for sheaves*, Invent. Math.**109**(1992), no. 2, 231–245. MR**1172690**, DOI 10.1007/BF01232026 - 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. Rossmann,
*Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra*, Invent. Math.**121**(1995), no. 3, 531–578. MR**1353308**, DOI 10.1007/BF01884311 - W. Rossmann,
*Picard-Lefschetz theory and characters of a semisimple Lie group*, Invent. Math.**121**(1995), no. 3, 579–611. MR**1353309**, DOI 10.1007/BF01884312 - Wilfried Schmid and Kari Vilonen,
*Characteristic cycles of constructible sheaves*, Invent. Math.**124**(1996), no. 1-3, 451–502. MR**1369425**, DOI 10.1007/s002220050060 - Wilfried Schmid and Kari Vilonen,
*Two geometric character formulas for reductive Lie groups*, J. Amer. Math. Soc.**11**(1998), no. 4, 799–867. MR**1612634**, DOI 10.1090/S0894-0347-98-00275-6 - Wilfried Schmid and Kari Vilonen,
*Characteristic cycles and wave front cycles of representations of reductive Lie groups*, Ann. of Math. (2)**151**(2000), no. 3, 1071–1118. MR**1779564**, DOI 10.2307/121129 - Jir\B{o} Sekiguchi,
*Remarks on real nilpotent orbits of a symmetric pair*, J. Math. Soc. Japan**39**(1987), no. 1, 127–138. MR**867991**, DOI 10.2969/jmsj/03910127 - Hiroshi Yamashita,
*Cayley transform and generalized Whittaker models for irreducible highest weight modules*, Astérisque**273**(2001), 81–137 (English, with English and French summaries). Nilpotent orbits, associated cycles and Whittaker models for highest weight representations. MR**1845715**

## Bibliographic Information

**Mladen Božičević**- Affiliation: Department of Geotechnical Engineering, University of Zagreb, Hallerova 7, 42000 Varaždin, Croatia
- ORCID: 0000-0002-2588-723X
- Email: mladen.bozicevic@gmail.com
- Received by editor(s): January 26, 2021
- Received by editor(s) in revised form: May 25, 2021
- Published electronically: August 18, 2021
- Additional Notes: The author was partially supported by grant no. 4176 of the Croatian Science Foundation
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 732-747 - MSC (2020): Primary 22E46; Secondary 22E30
- DOI: https://doi.org/10.1090/ert/580
- MathSciNet review: 4301562