Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups

J. J. Duistermaat; J. A. C. Kolk; V. S. Varadarajan

Compositio Mathematica (1983)

  • Volume: 49, Issue: 3, page 309-398
  • ISSN: 0010-437X

How to cite

top

Duistermaat, J. J., Kolk, J. A. C., and Varadarajan, V. S.. "Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups." Compositio Mathematica 49.3 (1983): 309-398. <http://eudml.org/doc/89615>.

@article{Duistermaat1983,
author = {Duistermaat, J. J., Kolk, J. A. C., Varadarajan, V. S.},
journal = {Compositio Mathematica},
keywords = {C-function; phase-function; oscillatory integral; asymptotics; critical manifold; Selberg trace formula; special functions},
language = {eng},
number = {3},
pages = {309-398},
publisher = {Martinus Nijhoff Publishers},
title = {Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups},
url = {http://eudml.org/doc/89615},
volume = {49},
year = {1983},
}

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
PB - Martinus Nijhoff Publishers
VL - 49
IS - 3
SP - 309
EP - 398
LA - eng
KW - C-function; phase-function; oscillatory integral; asymptotics; critical manifold; Selberg trace formula; special functions
UR - http://eudml.org/doc/89615
ER -

References

top
  1. [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 Fennica1980. Zbl0443.53002MR562718
  2. [2] F.A. Berezin and I.M. Gel'fand: Some remarks on the theory of spherical functions on symmetrical Riemannian manifolds. Trudy Moskov. Mat. Obšč5 (1956 ) 311-351.English transl.: Amer. Math. Soc. Translations (2) 21 (1962) 193 -238. Zbl0195.42302MR86266
  3. [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. Surveys28 (1973) 1-26. Zbl0289.57024MR429933
  4. [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. Zbl0052.40001MR51508
  5. [5] A. Borel: Kählerian coset spaces of semi-simple Lie groups. Proc. Nat. Acad. Sci., U.S.A. 40 (1954) 1147-1151. Zbl0058.16002MR77878
  6. [6] A. Borel: Linear Algebraic Groups. New York: Benjamin1969. Zbl0186.33201MR251042
  7. [7] A. Borel and J. Tits: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math.27 (1965) 55-151. Zbl0145.17402MR207712
  8. [8] A. Borel and J. Tits: Compléments à l'article "Groupes réductifs". Inst. Hautes Études Sci. Publ. Math.41 (1972) 253-276. Zbl0254.14018MR315007
  9. [9] R. Bott: On torsion in Lie groups. Proc. Nat. Acad. Sci., U.S.A. 40 (1954) 586-588. Zbl0057.02201MR62750
  10. [10] R. Bott: On the iteration of closed geodesics and the Sturm intersection theory. Comm. Pure Appl. Math.9 (1956) 171-206. Zbl0074.17202MR90730
  11. [11] R. Bott and H. Samelson: Applications of the theory of Morse to symmetric spaces. Amer. J. Math.80 (1958) 964-1029. Zbl0101.39702MR105694
  12. [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 Mathematics838. Berlin: Springer-Verlag1981. Zbl0437.53043MR636269
  13. [13] J. Chazarain: Formule de Poisson pour les variétés riemanniennes. Invent. Math.24 (1974) 65-82. Zbl0281.35028MR343320
  14. [14] L. Cohn: Analytic Theory of the Harish-Chandra C-function. Lecture Notes in Mathematics429. Berlin: Springer-Verlag1974. Zbl0342.33026MR422509
  15. [15] Y. Colin De Verdière: Spectre du Laplacien et longueurs des géodésiques périodiques, II. Comp. Math.27 (1973) 159-184. Zbl0281.53036MR319107
  16. [16] M. Demazure: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7 (1974) 53-88. Zbl0312.14009MR354697
  17. [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. Zbl0425.20041MR515601
  18. [18] J. Dieudonné: Foundations of Modem Analysis. New York: Academic Press1960. Zbl0100.04201MR120319
  19. [19] J.J. Duistermaat: Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math.27 (1974) 207-281. Zbl0285.35010MR405513
  20. [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. Zbl0434.58019MR532745
  21. [21] C. Ehresmann: Sur la topologie de certaines variétés algébriques réelles. J. Math. Pures Appl. (9) 16 (1937) 69-100. Zbl0016.07403JFM63.0557.01
  22. [22] A. Erdélyi: Asymptotic Expansions. New York: Dover1956. Zbl0070.29002MR78494
  23. [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. Zbl0232.43007MR289724
  24. [24] I.M. Gelfand and M.A. Neumark: Unitäre Darstellungen der klassischen Gruppen. Berlin: Akademie-Verlag1957. Zbl0077.03405MR85262
  25. [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. Zbl0184.25304MR209405
  26. [26] V. Guillemin and A. Pollack: Differential Topology. Englewood Cliffs: Prentice-Hall1974. Zbl0361.57001MR348781
  27. [27] H.C. Hansen: On cycles in flag manifolds. Math. Scand.33 (1973) 269-274. Zbl0301.14019MR376703
  28. [28] Harish- Chandra: Spherical functions on a semisimple Lie group, I. Amer. J. Math. 80 (1958) 241-310. Zbl0093.12801MR94407
  29. [29] Harish- Chandra: Spherical functions on a semisimple Lie group, II. Amer. J. Math. 80 (1958) 553-613. Zbl0093.12801MR101279
  30. [30] Harish- Chandra: Some results on differential equations and their applications. Proc. Nat. Acad. Sci., U.S.A.45 (1959) 1763-1764. Zbl0161.33803MR175998
  31. [31] Harish- Chandra: Harmonic analysis on real reductive groups, I. The theory of the constant term. J. of Functional Analysis19 (1975) 104-204. Zbl0315.43002MR399356
  32. [32] G.J. Heckman: Projections of Orbits and Asymptotic Behaviour of Multiplicities for Compact Lie Groups. Thesis, Rijksuniversiteit Leiden (1980). Zbl0497.22006
  33. [33] S. Helgason: Differential Geometry and Symmetric Spaces. New York: Academic Press1962. Zbl0111.18101MR145455
  34. [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. Zbl0516.22007MR690050
  35. [35] R. Hermann: Geometric aspects of potential theory in symmetric spaces. III. Math. Ann.153 (1964) 384-394. Zbl0119.09001MR160248
  36. [36] L. Hörmander: Fourier integral operators, I. Acta Math.127 (1971) 79-183. Zbl0212.46601MR388463
  37. [37] G.A. Hunt: A theorem of Élie Cartan. Proc. Amer. Math. Soc.7 (1956) 307-308. Zbl0073.01602MR77075
  38. [38] A.A. Kirillov: Unitary representations of nilpotent Lie groups. Uspekhi Mat. Nauk.17 (1962) 57-110.English transl.: Russ. Math. Surveys17 (1962) 53-104. Zbl0106.25001MR142001
  39. [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. Zbl0336.14001MR429938
  40. [40] B. Kostant: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math.74 (1961) 329-387. Zbl0134.03501MR142696
  41. [41] B. Kostant: Lie algebra cohomology and generalized Schubert cells. Ann. of Math.77 (1963) 72-144. Zbl0134.03503MR142697
  42. [42] B. Kostant: On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. École Norm. Sup. (4) 6 (1973) 413-455. Zbl0293.22019MR364552
  43. [43] N.H. Kuiper: Minimal total absolute curvature for immersions. Invent. Math.10 (1970) 209-238. Zbl0187.18902MR267597
  44. [44] S. Lefschetz: Intersections and transformations of complexes and manifolds. Trans. Amer. Math. Soc.28 (1926) 1-49. Zbl52.0572.02MR1501331JFM52.0572.02
  45. [45] S. Łojasiewicz: Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964) 449-474. Zbl0128.17101
  46. [46] J. Milnor: Morse Theory. Ann. of Math. Studies N° 51. Princeton: Princeton University Press1963. Zbl0108.10401
  47. [47] J. Milnor: Lectures on the h-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow. Princeton: Princeton University Press1965. Zbl0161.20302MR190942
  48. [48] D. Mumford: Introduction to Algebraic Geometry, preliminary version of Chapters I-III. Cambridge: Harvard Univ. Math. Dept. 1966. 
  49. [49] R.W. Richardson, JR.: Conjugacy classes in Lie algebras and algebraic groups. Ann. of Math.86 (1967) 1-15. Zbl0153.04501MR217079
  50. [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. Zbl0305.43007MR450461
  51. [51] H. Schubert: Kalkül der Abzählenden Geometrie. Leipzig: Teubner1879. JFM11.0460.01
  52. [52] J.T. Schwartz: Differential Geometry and Topology. New York: Gordon and Breach1968. Zbl0187.45006
  53. [53] Séminaire C. Chevalley: Classification des Groupes de Lie Algébriques, vols. 1 en 2. Paris: Secrétariat Mathématique1958. Zbl0092.26301
  54. [54] J.-P. Serre: Représentations linéaires et espaces homogènes kählériennes des groupes de Lie compacts. Séminaire Bourbaki, 1953/1954. Exposé 100. Paris: Secrétariat Mathématique. Zbl0121.16203
  55. [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 Press1969. Zbl0195.50803MR263830
  56. [56] R. Steinberg: Conjugacy Classes in Algebraic Groups. Lecture Notes in Mathematics366. Berlin: Springer-Verlag1974. Zbl0281.20037MR352279
  57. [57] D. Sullivan: Combinatorial invariants of analytic spaces. In: Proceedings Liverpool Singularities - Symposium I, 1970, pp. 165-168. Lecture Notes in Mathematics192. Berlin: Springer-Verlag1971. Zbl0227.32005MR278333
  58. [58] M. Takeuchi: Cell decompositions and Morse equalities on certain symmetric spaces. J. Fac. of Sci. Univ. of Tokyo12 (1965) 81-192. Zbl0144.22804MR216517
  59. [59] M. Takeuchi and S. Kobayashi: Minimal imbeddings of R-spaces. J. Differential Geometry2 (1968) 203-215. Zbl0165.24901MR239007
  60. [60] R. Thom: Sur une partition en cellules associée à une fonction sur une variété. C.R. Acad. Sci. Paris Sér. A-B228 (1949) 973-975. Zbl0034.20802MR29160
  61. [61] J. Tits: Sur les R-espaces. C.R. Acad. Sci. Paris Sér. A-B239 (1954) 850-852. Zbl0058.36503MR66395
  62. [62] J. Tits: Sur certaines classes d'espaces homogènes de groupes de Lie. Acad. Roy. Belg. Cl. Sci. Mem. Coll.29 (1955), no. 3. Zbl0067.12301MR76286
  63. [63] J. Tits: Espaces homogènes complexes compacts. Comment. Math. Helv.37 (1962) 111-120. Zbl0108.36302MR154299
  64. [64] V.S. Varadarajan: Lie groups, Lie Algebras, and Their Representations. Englewood Cliffs: Prentice-Hall1974. Zbl0371.22001MR376938
  65. [65] V.S. Varadarajan: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Mathematics576. Berlin: Springer-Verlag1977. Zbl0354.43001MR473111
  66. [66] B.L. Van Der Waerden: Topologische Begründung des Kalküls der abzählenden Geometrie. Math. Ann.102 (1930) 337-362. Zbl55.0992.01MR1512581JFM55.0992.01
  67. [67] B.L. Van Der Waerden: Einführung in die Algebraische Geometrie. Berlin: Springer1939. Zbl0021.25001JFM65.1393.01

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.