Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups
Compositio Mathematica, Volume 49 (1983) no. 3, pp. 309-398.
@article{CM_1983__49_3_309_0,
     author = {Duistermaat, J. J. and Kolk, J. A. C. and Varadarajan, V. S.},
     title = {Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple {Lie} groups},
     journal = {Compositio Mathematica},
     pages = {309--398},
     publisher = {Martinus Nijhoff Publishers},
     volume = {49},
     number = {3},
     year = {1983},
     zbl = {0524.43008},
     mrnumber = {707179},
     language = {en},
     url = {http://www.numdam.org/item/CM_1983__49_3_309_0/}
}
TY  - JOUR
AU  - Duistermaat, J. J.
AU  - Kolk, J. A. C.
AU  - Varadarajan, V. S.
TI  - Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups
JO  - Compositio Mathematica
PY  - 1983
DA  - 1983///
SP  - 309
EP  - 398
VL  - 49
IS  - 3
PB  - Martinus Nijhoff Publishers
UR  - http://www.numdam.org/item/CM_1983__49_3_309_0/
UR  - https://zbmath.org/?q=an%3A0524.43008
UR  - https://www.ams.org/mathscinet-getitem?mr=707179
LA  - en
ID  - CM_1983__49_3_309_0
ER  - 
%0 Journal Article
%A Duistermaat, J. J.
%A Kolk, J. A. C.
%A Varadarajan, V. S.
%T Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups
%J Compositio Mathematica
%D 1983
%P 309-398
%V 49
%N 3
%I Martinus Nijhoff Publishers
%G en
%F CM_1983__49_3_309_0
Duistermaat, J. J.; Kolk, J. A. C.; Varadarajan, V. S. Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. Compositio Mathematica, Volume 49 (1983) no. 3, pp. 309-398. http://www.numdam.org/item/CM_1983__49_3_309_0/

[1] T.F. Banchoff: Computer animation and the geometry of surfaces in 3- and 4-space. Proc. Int. Congr. of Math., Helsinki, 1978, pp. 1005-1013. Helsinki: Academia Scientiarum Fennica 1980. | MR | Zbl

[2] F.A. Berezin and I.M. Gel'Fand: Some remarks on the theory of spherical functions on symmetrical Riemannian manifolds. Trudy Moskov. Mat. Obšč 5 (1956 ) 311-351.English transl.: Amer. Math. Soc. Translations (2) 21 (1962) 193 -238. | MR | Zbl

[3] I.N. Bernstein, I.M. Gel'Fand and S.I. Gel'Fand: Schubert cells and cohomology of the spaces G/P. Uspekhi Mat. Nauk. 28 (1973) 3-26.English transl.: Russ. Math. Surveys 28 (1973) 1-26. | MR | Zbl

[4] A. Borel: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. 57 (1953) 115-207. | MR | Zbl

[5] A. Borel: Kählerian coset spaces of semi-simple Lie groups. Proc. Nat. Acad. Sci., U.S.A. 40 (1954) 1147-1151. | MR | Zbl

[6] A. Borel: Linear Algebraic Groups. New York: Benjamin 1969. | MR | Zbl

[7] A. Borel and J. Tits: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27 (1965) 55-151. | Numdam | MR | Zbl

[8] A. Borel and J. Tits: Compléments à l'article "Groupes réductifs". Inst. Hautes Études Sci. Publ. Math. 41 (1972) 253-276. | Numdam | MR | Zbl

[9] R. Bott: On torsion in Lie groups. Proc. Nat. Acad. Sci., U.S.A. 40 (1954) 586-588. | MR | Zbl

[10] R. Bott: On the iteration of closed geodesics and the Sturm intersection theory. Comm. Pure Appl. Math. 9 (1956) 171-206. | MR | Zbl

[11] R. Bott and H. Samelson: Applications of the theory of Morse to symmetric spaces. Amer. J. Math. 80 (1958) 964-1029. | MR | Zbl

[12] T.E. Cecil and P.J. Ryan: Tight spherical embeddings. In: Global Differential Geometry and Global Analysis. Berlin, 1979, pp. 94-104.Lecture Notes in Mathematics 838. Berlin: Springer-Verlag 1981. | MR | Zbl

[13] J. Chazarain: Formule de Poisson pour les variétés riemanniennes. Invent. Math. 24 (1974) 65-82. | MR | Zbl

[14] L. Cohn: Analytic Theory of the Harish-Chandra C-function. Lecture Notes in Mathematics 429. Berlin: Springer-Verlag 1974. | MR | Zbl

[15] Y. Colin De Verdière: Spectre du Laplacien et longueurs des géodésiques périodiques, II. Comp. Math. 27 (1973) 159-184. | Numdam | MR | Zbl

[16] M. Demazure: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7 (1974) 53-88. | Numdam | MR | Zbl

[17] V.V. Deodhar: On Bruhat ordering and weight-lattice ordering for a Weyl group. Nederl. Akad. Wetensch. Proc. Ser. A81 = Indag. Math. 40 (1978) 423-435. | MR | Zbl

[18] J. Dieudonné: Foundations of Modem Analysis. New York: Academic Press 1960. | MR | Zbl

[19] J.J. Duistermaat: Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974) 207-281. | MR | Zbl

[20] J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan: Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math. 52 (1979) 27-93. | MR | Zbl

[21] C. Ehresmann: Sur la topologie de certaines variétés algébriques réelles. J. Math. Pures Appl. (9) 16 (1937) 69-100. | JFM | Zbl

[22] A. Erdélyi: Asymptotic Expansions. New York: Dover 1956. | MR | Zbl

[23] R. Gangolli: On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups. Ann. of Math. 93 (1971) 150-165. | MR | Zbl

[24] I.M. Gelfand and M.A. Neumark: Unitäre Darstellungen der klassischen Gruppen. Berlin: Akademie-Verlag 1957. | MR | Zbl

[25] S.G. Gindikin: Unitary representations of groups of automorphisms of Riemann symmetric spaces of null curvature. Funkcional Anal. i Priložen 1: 1 (1967) 32-37.English transl.: Functional Anal. Appl. 1 (1967) 28-32. | MR | Zbl

[26] V. Guillemin and A. Pollack: Differential Topology. Englewood Cliffs: Prentice-Hall 1974. | MR | Zbl

[27] H.C. Hansen: On cycles in flag manifolds. Math. Scand. 33 (1973) 269-274. | MR | Zbl

[28] Harish-Chandra: Spherical functions on a semisimple Lie group, I. Amer. J. Math. 80 (1958) 241-310. | MR | Zbl

[29] Harish-Chandra: Spherical functions on a semisimple Lie group, II. Amer. J. Math. 80 (1958) 553-613. | MR | Zbl

[30] Harish-Chandra: Some results on differential equations and their applications. Proc. Nat. Acad. Sci., U.S.A. 45 (1959) 1763-1764. | MR | Zbl

[31] Harish-Chandra: Harmonic analysis on real reductive groups, I. The theory of the constant term. J. of Functional Analysis 19 (1975) 104-204. | MR | Zbl

[32] G.J. Heckman: Projections of Orbits and Asymptotic Behaviour of Multiplicities for Compact Lie Groups. Thesis, Rijksuniversiteit Leiden (1980). | Zbl

[33] S. Helgason: Differential Geometry and Symmetric Spaces. New York: Academic Press 1962. | MR | Zbl

[34] R. Herb: Discrete series characters and Fourier inversion on semisimple real Lie groups. Preprint, University of Maryland (1980). To appear T.A.M.S. | MR | Zbl

[35] R. Hermann: Geometric aspects of potential theory in symmetric spaces. III. Math. Ann. 153 (1964) 384-394. | MR | Zbl

[36] L. Hörmander: Fourier integral operators, I. Acta Math. 127 (1971) 79-183. | MR | Zbl

[37] G.A. Hunt: A theorem of Élie Cartan. Proc. Amer. Math. Soc. 7 (1956) 307-308. | MR | Zbl

[38] A.A. Kirillov: Unitary representations of nilpotent Lie groups. Uspekhi Mat. Nauk. 17 (1962) 57-110.English transl.: Russ. Math. Surveys 17 (1962) 53-104. | MR | Zbl

[39] S.L. Kleiman: Problem 15. Rigorous foundation of Schubert's enumerative calculus. In: Mathematical Developments arising from Hilbert Problems. Proc. Sympos. Pure Math., vol. 28, pp. 445-482. Providence: Amer. Math. Soc. 1976. | MR | Zbl

[40] B. Kostant: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74 (1961) 329-387. | MR | Zbl

[41] B. Kostant: Lie algebra cohomology and generalized Schubert cells. Ann. of Math. 77 (1963) 72-144. | MR | Zbl

[42] B. Kostant: On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. École Norm. Sup. (4) 6 (1973) 413-455. | Numdam | MR | Zbl

[43] N.H. Kuiper: Minimal total absolute curvature for immersions. Invent. Math. 10 (1970) 209-238. | MR | Zbl

[44] S. Lefschetz: Intersections and transformations of complexes and manifolds. Trans. Amer. Math. Soc. 28 (1926) 1-49. | JFM | MR

[45] S. Łojasiewicz: Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964) 449-474. | Numdam | Zbl

[46] J. Milnor: Morse Theory. Ann. of Math. Studies N° 51. Princeton: Princeton University Press 1963. | Zbl

[47] J. Milnor: Lectures on the h-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow. Princeton: Princeton University Press 1965. | MR | Zbl

[48] D. Mumford: Introduction to Algebraic Geometry, preliminary version of Chapters I-III. Cambridge: Harvard Univ. Math. Dept. 1966.

[49] R.W. Richardson, Jr.: Conjugacy classes in Lie algebras and algebraic groups. Ann. of Math. 86 (1967) 1-15. | MR | Zbl

[50] P.J. Sally, JR and G. Warner: The Fourier transform on semisimple Lie groups of real rank one. Acta Math. 131 (1973) 1-26. | MR | Zbl

[51] H. Schubert: Kalkül der Abzählenden Geometrie. Leipzig: Teubner 1879. | JFM

[52] J.T. Schwartz: Differential Geometry and Topology. New York: Gordon and Breach 1968. | Zbl

[53] Séminaire C. Chevalley: Classification des Groupes de Lie Algébriques, vols. 1 en 2. Paris: Secrétariat Mathématique 1958. | Zbl

[54] J.-P. Serre: Représentations linéaires et espaces homogènes kählériennes des groupes de Lie compacts. Séminaire Bourbaki, 1953/1954. Exposé 100. Paris: Secrétariat Mathématique. | Numdam | Zbl

[55] T.A. Springer: The unipotent variety of a semisimple group. In: Algebraic Geometry. Papers presented at the Bombay Colloquium, 1968, pp. 373-391. London: Oxford University Press 1969. | MR | Zbl

[56] R. Steinberg: Conjugacy Classes in Algebraic Groups. Lecture Notes in Mathematics 366. Berlin: Springer-Verlag 1974. | MR | Zbl

[57] D. Sullivan: Combinatorial invariants of analytic spaces. In: Proceedings Liverpool Singularities - Symposium I, 1970, pp. 165-168. Lecture Notes in Mathematics 192. Berlin: Springer-Verlag 1971. | MR | Zbl

[58] M. Takeuchi: Cell decompositions and Morse equalities on certain symmetric spaces. J. Fac. of Sci. Univ. of Tokyo 12 (1965) 81-192. | MR | Zbl

[59] M. Takeuchi and S. Kobayashi: Minimal imbeddings of R-spaces. J. Differential Geometry 2 (1968) 203-215. | MR | Zbl

[60] R. Thom: Sur une partition en cellules associée à une fonction sur une variété. C.R. Acad. Sci. Paris Sér. A-B 228 (1949) 973-975. | MR | Zbl

[61] J. Tits: Sur les R-espaces. C.R. Acad. Sci. Paris Sér. A-B 239 (1954) 850-852. | MR | Zbl

[62] J. Tits: Sur certaines classes d'espaces homogènes de groupes de Lie. Acad. Roy. Belg. Cl. Sci. Mem. Coll. 29 (1955), no. 3. | MR | Zbl

[63] J. Tits: Espaces homogènes complexes compacts. Comment. Math. Helv. 37 (1962) 111-120. | MR | Zbl

[64] V.S. Varadarajan: Lie groups, Lie Algebras, and Their Representations. Englewood Cliffs: Prentice-Hall 1974. | MR | Zbl

[65] V.S. Varadarajan: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Mathematics 576. Berlin: Springer-Verlag 1977. | MR | Zbl

[66] B.L. Van Der Waerden: Topologische Begründung des Kalküls der abzählenden Geometrie. Math. Ann. 102 (1930) 337-362. | JFM | MR

[67] B.L. Van Der Waerden: Einführung in die Algebraische Geometrie. Berlin: Springer 1939. | JFM | Zbl