@article{ASENS_1990_4_23_2_311_0, author = {Matumoto, Hisayosi}, title = {$C^{-\infty }${-Whittaker} vectors for complex semisimple {Lie} groups, wave front sets, and {Goldie} rank polynomial representations}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {311--367}, publisher = {Elsevier}, volume = {Ser. 4, 23}, number = {2}, year = {1990}, doi = {10.24033/asens.1605}, mrnumber = {91i:22020}, zbl = {0760.22017}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.1605/} }
TY - JOUR AU - Matumoto, Hisayosi TI - $C^{-\infty }$-Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations JO - Annales scientifiques de l'École Normale Supérieure PY - 1990 SP - 311 EP - 367 VL - 23 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.24033/asens.1605/ DO - 10.24033/asens.1605 LA - en ID - ASENS_1990_4_23_2_311_0 ER -
%0 Journal Article %A Matumoto, Hisayosi %T $C^{-\infty }$-Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations %J Annales scientifiques de l'École Normale Supérieure %D 1990 %P 311-367 %V 23 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.24033/asens.1605/ %R 10.24033/asens.1605 %G en %F ASENS_1990_4_23_2_311_0
Matumoto, Hisayosi. $C^{-\infty }$-Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 23 (1990) no. 2, pp. 311-367. doi : 10.24033/asens.1605. http://www.numdam.org/articles/10.24033/asens.1605/
[BV1] The Local Structure of Characters (J. Funct. Anal., vol. 37, 1980, pp. 27-55). | MR | Zbl
and ,[BV2] Primitive Ideals and Orbital Integrals in Complex Classical Groups (Math. Ann. Vol. 259, 1982, pp. 153-199). | MR | Zbl
and ,[BV3] Primitive Ideals and Orbital Integrals in Complex exceptional Groups (J. Algebra, Vol. 80, 1983, pp. 350-382). | MR | Zbl
and ,[BV4] Unipotent Representations of Complex Semisimple Lie Groups (Ann, of Math., Vol. 121, 1985, pp. 41-110). | Zbl
and ,[BV5] Weyl Group Representations and nilpotent Orbits, in : P. C. TROMBI, editor, "Representation Theory of Reductive Groups" (Progress in Mathematics, Vol. 40, pp. 21-33, Birkhäuser, Boston-Basel-Stuttgart, 1983). | MR | Zbl
and ,[BeG] Tensor Product of Finite and Infinite Dimensional Representations of Semisimple Lie Algebras (Compos. Math., 41, 1980, pp. 245-285). | Numdam | MR | Zbl
and ,[BeyL] Some Numerical Results on the Characters of Exceptional Weyl Groups (Math. Soc. Camb. Phil. Soc., Vol. 84, 1978, pp. 417-426). | MR | Zbl
and ,[BW] "Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups." (Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1980). | MR | Zbl
and ,[BoBr] Differential Operators on Homogeneous Spaces III (Invent. Math., Vol. 80, 1985, pp. 1-68). | MR | Zbl
and ,[BoBrM] Nilpotent Orbits, Primitive Ideals, and Characteristic Classes, preprint, Max-Planck-Institut für Mathematik, Bonn, January, 1988. | Zbl
, and ,[BoKr] Uber primitive ideale in der Einhüllenden einer halbeinfachen Lie-Algebra (Invent. Math., Vol. 39, 1977, pp. 1-53). | EuDML | Zbl
and ,[BoM1] Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes (C.R. Acad. Sci. Paris, T. 292, 1981, pp. 707-710). | MR | Zbl
and ,[BoM2] Partial Resolutions of Nilpotent Varieties (Astérisque, Vol. 101-102, 1983, pp. 23-74). | Numdam | MR | Zbl
and ,[Ca1] A letter to Harish-Chandra, November 30, 1982.
,[Ca2] Jacquet Modules for Real Reductive groups, in : Proceedings of the International Congress of Mathematics, Helsinki, 1978, pp. 557-563. | MR | Zbl
,[Ca3] Canonical Extensions of Harish-Chandra Modules to Representations of G, preprint. | Zbl
,[CD] Sur les représentations induites des groupes semi-simples complexes (Compos. Math., Vol. 34, 1977, pp. 307-336). | EuDML | Numdam | MR | Zbl
and ,[D1] Représentations irréductibles des groupes semi-simples complexes, pp. 26-88 in : Analyse Harmonique sur les Groupes de Lie (Lect. Notes Math., No. 497, Springer-Verlag, Berlin-Heidelberg-New York, 1975). | MR | Zbl
,[D2] Sur la classifications des idéaux primitifs dans l'algèbre de Lie semi-simple (Ann. Math., Vol. 1977, pp. 107-120). | MR | Zbl
,[D3] Polynômes de Vogan pour SL(n, C), pp. 64-76 in : Non-Commutative Harmonic Analysis (Lecture Notes in Mathematics, No. 728, Springer-Verlag, Berlin-Heidelberg-New York, 1979). | MR | Zbl
,[GJ] Oh the Bernstein-Gelfand-Gelfand Resolution and the Duflo sum Formula (Compos. Math., Vol. 43, 1981, pp. 107-131). | EuDML | Numdam | MR | Zbl
and ,[GG1] Categories of Group Representations and the Problem of Classifying Irreducible Representations (Soviet Math. Dokl., Vol. 3, 1962, pp. 1382-1385).
and ,[GG2] Construction of Irreducible Representations of Simple Algebraic Groups Over a Finite field (Soviet Math. Dokl., Vol. 3, 1962, pp. 1646-1649). | MR | Zbl
and ,[Gi] G-modules, Springer's Representations and bivariant Chern Classes (Adv. Math., Vol. 61, 1986, pp. 1-48). | MR | Zbl
,[Go] Horospherical Functions on Symmetric Spaces (Canadian Matehatical Society Conference Proceedings Vol. 1, 1981, pp. 125-133). | Zbl
,[GW] Whittaker Vectors and Conical Vectors (J. Funct. Anal., Vol. 39, 1980, pp. 199-279). | MR | Zbl
and ,[GorM] "Stratified Morse Theory", Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo, 1980.
and ,[Ha1] Whittaker models for semisimple Lie groups (Jpn J. Math., Vol. 5, 1979, pp. 349-401). | MR | Zbl
,[Ha2] Whittaker Models for Representations with Highest Weights, in : Lectures on Harmonic analysis on Lie Groups and Related Topics (Strasbourg, 1979, pp. 45-50, Lect. Math., Vol. 14, Kinokuniya Book Store, Tokyo, 1982). | Zbl
,[Ha3] Whittaker Functions on Semisimple Lie Groups (Hiroshima Math. J., Vol. 13, 1982, pp. 259-293). | MR | Zbl
,[He] A Duality for Symmetric Spaces with Applications to Group Representations (Adv. Math., Vol. 5, 1970, pp. 1-154). | MR | Zbl
,[HS] "A Course in Homological Algebra", GTM No. 4, Springer-Verlag, Berlin-Heidelberg-New York, 1970. | Zbl
and ,[Hi1] Resolution of Singularities of an algebraic geometry over a field of Characteristic 0 (Ann. Math., Vol. 79, 1964, pp. 109-326). | MR | Zbl
,[Hi2] Subanalytic sets, in : "Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. AKIZUKI", pp. 453-493, Kinokuniya Book Store, Tokyo, 1973. | MR | Zbl
,[Hör] Fourier Integral Operators I (Acta Math., Vol. 127, 1971, pp. 79-183). | MR | Zbl
,[Hot1] On Springer's Representations (J. Fac. Sci. Univ. Tokyo, IA Vol. 28, 1982, pp. 863-876). | MR | Zbl
,[Hot2] On Joseph's Construction of Weyl Group Representations (Tohoky Math. J., Vol. 36, 1984, pp. 49-74). | MR | Zbl
,[Hot3] A local formula for Springer's Representation (Adv. Stud. Pure Math., Vol. 6, 1985, pp. 127-138). | MR | Zbl
,[HotK] The Invariant Holonomic System on a Semisimple Lie algebra (Invent. Math., Vol. 75, 1984, pp. 327-358). | EuDML | MR | Zbl
and ,[How] Wave Front Sets of Representations of Lie Groups, in "Automorphic Forms, Representation Theory, and Arithmetic", Bombay, 1981. | MR | Zbl
,[Ja] Fonction de Whittaker associées aux groupes de Chevalley (Bull. Soc. Math. France, Vol. 95, 1967, pp. 243-309). | EuDML | Numdam | MR | Zbl
,[JL] "Automorphic Form on GL (2)" (Lect. Notes Math., No. 114, Springer-Verlag, Berlin-Heidelberg-New York). | MR | Zbl
and ,[Jo1] On the Annihilators of the Simple Subquotients of the Principal series (Ann. Scient. Éc. Norm. Sup. (4), Vol. 10, 1977, pp. 419-440). | EuDML | Numdam | MR | Zbl
,[Jo2] Dixmier's problem for Verma and Principal Series Submodules (J. London Math. Soc., (2), Vol. 20, 1979, pp. 193-204). | MR | Zbl
,[Jo3] W-module Structures in the Primitive Spectrum of the Enveloping Algebra of a Semisimple Lie Algebra, pp. 116-135 in : Non-Commutative Harmonic Analysis (Lect. Notes Math., No. 728, Springer-Verlag, Berlin-Heidelberg-New York, 1979). | MR | Zbl
,[Jo4] Towards the Jantzen conjecture I (Compos. Math., Vol. 40, 1980, pp. 35-67). | EuDML | Numdam | MR | Zbl
,[Jo5] Towards the Jantzen conjecture II (Compos. Math., Vol. 40, 1980, pp. 69-78). | EuDML | Numdam | MR | Zbl
,[Jo6] Kostant's Problem, Goldie rank and Gelfand-Kirillov Conjecture (Invent. Math., Vol. 56, 1980, pp. 191-213). | EuDML | MR | Zbl
,[Jo7] Goldie Rank in the Enveloping Algebra of a Semisimple Lie Algebra I (J. Algebra, Vol. 65, 1980, pp. 269-283). | MR | Zbl
,[Jo8] Goldie Rank in the Enveloping Algebra of a Semisimple Lei Algebra II (J. Algebra, Vol. 65, 1980, pp. 284-306). | MR | Zbl
,[Jo9] Goldie rank in the enveloping algebra of a semisimple Lie algebra III (J. Algebra, Vol. 73, 1981, pp. 295-326). | MR | Zbl
,[Jo10] Towards the Jantzen conjecture III (Compos. Math., Vol. 41, 1981, pp. 23-30). | EuDML | Numdam | MR | Zbl
,[Jo11] On the Classification of Primitive Ideals in the enveloping Algebra of a samisimple Lie Algebra, pp. 30-76 in : Lect. Notes Math., No. 1024, Springer-Verlag, Berlin-Heidelberg-New York, 1983. | MR | Zbl
,[Jo12] On the Cyclicity of vectors Associated with Duflo Involutions, pp. 145-188 in : Non-Commutative Harmonic Analysis (Lect. Notes Math., No. 1243, Springer-Verlag, Berlin-Heidelberg-New York, 1986). | Zbl
,[Kas] The Riemann-Hilbert Problem for Holonomic Systems, Publ. R.I.M.S., Kyoto Univ., Vol. 20, 1984, pp. 319-365. | MR | Zbl
,[KV1] Functions on the Shilov boundary of the Generalized half plane, in : Non-Commutative Harmonic Analysis (Lecture Notes in Mathematics, No. 728, Springer-Verlag, Berlin-Heidelberg-New York, 1979). | MR | Zbl
and ,[KV2] K-types and the Singular Spectrum, in : Non-Commutative Harmonic Analysis (Lect. Notes Math., No. 728, Springer-Verlag, Berlin-Heidelberg-New York, 1979). | MR | Zbl
and ,[KT] The Characteristic cycles of Holomomic Systems on a Flag Manifold-related to the Weyl Group Algebra (Invent. Math., Vol. 77, 1984, pp. 185-198). | EuDML | MR | Zbl
and ,[Kaw1] Generalized Gelfand-Graev Representations and Ennola Duality, in “Algebraic Groups and Related Topics” (Adv. Stud. Pure Math., Vol. 6, Kinokuniya Book Store and North-Holland, 1985, pp. 175-206). | MR | Zbl
,[Kaw2] Generalized Gelfand-Graev Representations of Exceptional Simple Algebraic Groups over a Finite field I (Invent. Math., Vol. 84, 1986, pp. 575-616). | EuDML | MR | Zbl
,[Kaw3] Shintani lifting and Generalized Gelfand-Graev Representations (Proc. Symp. Pure Math., Vol. 47, 1987, pp. 147-163). | MR | Zbl
,[KL1] Representations of Coxeter Groups and Hecke Algebras (Invent. Math. Vol. 53, 1979, pp. 165-184). | EuDML | MR | Zbl
and ,[KL2] A topological Approach to Springer's Representations (Adv. Math., Vol. 38, 1980, pp. 222-228). | MR | Zbl
and ,[Ki1] The Primitive Ideals assoated to Harish-Chandra Modules and Certain harmonic Polynomials. (Thesis, M.I.T., 1979).
,[Ki2] The Character Polynomial of the Annihilator of an Irreducible Harish-Chandra Module (Am. J. Math., Vol. 103, 1981, pp. 1195-1240). | MR | Zbl
,[Kn] “Representation Theory of Semisimple Groups, An Overview Based on Examples”, Princeton Mathematical series 36, Princeton University Press, Lawrenceville, New Jersey, 1986. | MR | Zbl
,[Ko1] Lie algebra Cohomology and the Generalized Borel-Weil Theorem (Ann. Math., Vol. 48, 1978, pp. 101-184). | MR
,[Ko2] On Whittaker Vectors and Representation Theory (Invent. Math., Vol. 48, 1978, pp. 101-184). | EuDML | MR | Zbl
,[Le] Generalized Verma Modules, the Cartan-Helgason Theorem, and the Harish-Chandra Homomorphism (J. Algebra, Vol. 49, 1977, pp. 470-495). | MR | Zbl
,[LeW] Finite- and Infinite-dimensional Representations of Linear Semisimple Groups (Trans. Am. Math. Soc., Vol. 184, 1973, pp. 223-246). | MR | Zbl
and ,[Lo] Sur le problème de la division (Studia Math.), 8, 1959, pp. 87-136). | EuDML | MR | Zbl
,[Lu1] On a Theorem of Benson and Curtis (J. Algebra Vol. 71, 1981, pp. 490-498). | MR | Zbl
,[Lu2] A Class of Irreducible Representations of a Weyl Group (Proc. Kon. Nederl. Akad., series A, Vol. 82, 1979, pp. 323-335). | MR | Zbl
,[Lu3] A class of Irreducible Representations of a Weyl Group II (Proc. Kon. Nederl. Akad., series A, Vol. 85, 1982, pp. 219-226). | MR | Zbl
,[Lu4] “Characters of Reductive Groups over a Finite field” (Annals of Mathematics Studies, No. 107, Princeton University Press, Princeton, New Jersey, 1984). | MR | Zbl
,[Lu5] Cells in Affine Weyl Groups, in : Advanced Studies in Pure Math., Vol. 6, Kinokuniya Book Store and North-Holland, 1985. | MR | Zbl
,[Lu6] Cells in Affine Weyl Groups II (J. Algebra, Vol. 109, 1987, pp. 536-548). | MR | Zbl
,[Lu7] Leading Coefficients of Character Values of Hecke Algebras (Proc. Symp. Pure Math., Vol. 47, 1987, pp. 235-262). | MR | Zbl
,[LuN] Canonical Left Cells in Affine Weyl groups (Adv. Math., Vol. 72, 1988, pp. 284-288). | MR | Zbl
and ,[Ly] Generalized Whittaker Vectors and Representation Theory (Thesis, M.I.T., 1979).
,[Mac] Some Irreducible Representations of Weyl Groups (Bull. London Math. Soc., Vol. 4, 1972, pp. 148-150). | MR | Zbl
,[Mar] Distributions et valeurs au bord des fonctions holomorphes, œuvre de André Martineau, Paris, C.N.R.S., 1977, pp. 439-582.
,[Mat1] Boundary Value Problems for Whittaker Functions on Real Split Semisimple Lie Groups (Duke Math. J., Vol. 53, 1986, pp. 635-676). | MR | Zbl
,[Mat2] Whittaker Vectors and Associated Varieties (Invent. Math., Vol. 89, 1987, pp. 219-224). | EuDML | MR | Zbl
,[Mat3] Cohomological Hardy Space for SU (2,2),, to appear in : Adv. Stud. Pure Math., Vol. 14, Kinokuniya Book Store and North-Holland. | MR | Zbl
,[Mat4] Whittaker Vectors and the Goofman-Wallach Operators, to appear in Acta Math. | Zbl
,[Mat5] Whittaker Vectors and Whittaker Functions for real Semisimple Lie Groups (Theisis, M.I.T., 1988).
,[Mœ1] Modèles de Whittaker et idéaux primitifs complètement premiers dans les algèbres enveloppantes (C. R. Acad. Sci. Paris, Vol. 303, 1986, pp. 845-848). | MR | Zbl
,[Mœ2] Modèles de Whittaker et idéaux primitifs complètement premiers dans les algèbres enveloppantes I, preprint, 1986. | Zbl
,[Mœ3] Modèles de Whittaker et idéaux primitifs complètement premiers dans les algèbres enveloppantes des algèbres de Lie semisimple complexes II, preprint 1986. | Zbl
,[MW] Modèles de Whittaker dégénérés pour des groupes p-adiques (Math. Z., Vol. 196, 1987, pp. 427-452). | EuDML | MR | Zbl
and ,[OW] On Polarizations of Certain Homogeneous Spaces (Hiroshima Math., J., Vol. 2, 1972, pp. 445-482). | MR | Zbl
and ,[R] Modèle de Whittaker et caractères de representations, in : Non-Commutative Harmonic Analysis (Lecture Notes in Pure Mathematics, No. 466, pp. 151-171, Springer-Verlag, Berlin-Heidelberg-New York, 1981). | MR | Zbl
,[Sc] Intégrales d'entrelacement et fonctions de Whittaker (Bull. Soc. Math. France, Vol. 99, 1971, pp. 3-72). | EuDML | Numdam | MR | Zbl
,[Sch] “Théorie des distributions” Hermann, Paris, 1950. | Zbl
,[Sh] The Multiplicity one Theorem for GL(n) (Ann. Math., Vol. 100, 1974, pp. 171-193). | MR | Zbl
,[Sp1] Trigonometric sums, Green Functions of finite Groups and Representations of Weyl Groups (Invent. Math., Vol. 36, 1976, pp. 173-207). | EuDML | MR | Zbl
,[Sp2] A construction of Representations of Weyl Groups (Invent. Math., Vol. 44, 1978, pp. 279-293). | EuDML | MR | Zbl
,[T] “Introduction to Pseudodifferential and Fourier Integral Operators”, Vol. 1, Pseudodifferential Operators, Plenum Press, New York and London, 1980. | Zbl
,[Vo1] Gelfand-Kirillov Dimension for Harish-Chandra Modules (Invent. Math., Vol. 48, 1978, pp. 75-98). | MR | Zbl
,[Vo2] Ordering of the Primitive Spectrum of a Semisimple Lie Algebra (Math. Ann., Vol. 248, pp. 195-203). | MR | Zbl
,[Vo3] “Representations of Real Reductive Lie Groups” (Progress in Mathematics, Birkhäuser, 1982). | Zbl
,[Vo4] The Orbit Method and Primitive Ideals for Semisimple Lie Algebras (Canadian Mathematical Society Conference Proceedings, Vol. 5 “Lie Algebras and Related Topics”, 1986, pp. 381-316). | MR | Zbl
,[Vo5] Irreduicible Characters of Semisimple Lei Groups IV, Character-multiplicity duality (Duke Math. J., Vol. 49, 1982, pp. 943-1073). | Zbl
,[W1] Asymptotic Expansions of Generalized Matrix Entries of Representations of Real Reductive Groups, in : Lie group Representations I (Lecture Notes Pure Math., No. 1024, pp. 287-369, Springer-Verlag, Berlin-Heidelberg-New York, 1983). | MR | Zbl
,[W2] “Real reductive Groups I”, Academic press, 1987. | Zbl
,[W3] Lie Algebra Cohomology and Holomorphic Continuation of Generalized Jacquet Integrals, to appear in : Adv. Stud. Pure Math., Vol. 14, Kinokuniya Book Store and North-Holland. | MR | Zbl
,[War] “Harmonic Analysis on Semi-Simple Lie Groups I”, Die Drundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 188, Springer-Verlag, Berlin-Heidelberg-New York, 1972. | MR | Zbl
,[Y1] On Whittaker Vectors for Generalized Gelfand-Graev Representations of Semisimple Lie Groups (J. Math. Kyoto Univ., Vol. 26, 1986, pp. 263-298). | MR | Zbl
,[Y2] Multiplicity one Theorems for Generalized Gelfand-Graev Representations of Semi-simple Lie Groups and Whittaker Models for the Discrete Series, priprint, 1987.
,Cited by Sources: