Nilpotent orbits, associated cycles and Whittaker models for highest weight representations
Astérisque, no. 273 (2001) , 169 p.
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     title = {Nilpotent orbits, associated cycles and {Whittaker} models for highest weight representations},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {273},
     year = {2001},
     mrnumber = {1845713},
     zbl = {0968.22001},
     language = {en},
     url = {http://www.numdam.org/item/AST_2001__273__R1_0/}
}
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Nishiyama, Kyo; Ochiai, Hiroyuki; Taniguchi, Kenji; Yamashita, Hiroshi; Kato, Shohei. Nilpotent orbits, associated cycles and Whittaker models for highest weight representations. Astérisque, no. 273 (2001), 169 p. http://numdam.org/item/AST_2001__273__R1_0/

[NOT] Kyo Nishiyama, Hiroyuki Ochiai and Kenji Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case -. In this volume. | Zbl

[Y] Hiroshi Yamashita, Cayley transform and generalized Whittaker models for irreducible highest weight modules. In this volume. | Zbl

[KO] Shohei Kato and Hiroyuki Ochiai, The degrees of orbits of the multiplicity-free actions. In this volume. | Zbl

[1] D. Barbasch and D. A. Vogan, Jr., The local structure of characters, J. Funct. Anal. 37 (1980), no. 1, 27-55. | MR | Zbl | DOI

[2] M. G. Davidson, T. J. Enright and R. J. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. 94 (1991), no. 455, iv+102 pp. | MR | Zbl

[3] T. Enright, R. Howe and N. Wallach, A classification of unitary highest weight modules, in Representation theory of reductive groups (Park City, Utah, 1982), 97-143, | MR | Zbl

T. Enright, R. Howe and N. Wallach, A classification of unitary highest weight modules Progr. Math., 40, Birkhäuser, Boston, Boston, Mass., 1983. | MR | Zbl

[4] W. Fulton, Intersection theory, Springer, Berlin, 1984. | MR | Zbl | DOI

[5] Akihiko Gyoja and Hiroshi Yamashita, Associated variety, Kostant-Sekiguchi correspondence, and locally free U ( n ) -action on Harish-Chandra modules. J. Math. Soc. Japan 51 (1999), no. 1, 129-149. | MR | Zbl | DOI

[6] J. Harris, Algebraic geometry, A first course. Corrected reprint of the 1992 original, Springer, New York, 1995. | MR

[7] V. G. Kac, Some remarks on nilpotent orbits. J. Algebra, 64 (1980), 190-213. | MR | Zbl | DOI

[8] K. W. J. Kadell, The Selberg-Jack symmetric functions. Adv. Math., 130 (1997), 33-102. | MR | Zbl | DOI

[9] Noriaki Kawanaka, Generalized Gel'fand-Graev representations and Ennola duality, in Algebraic groups and related topics (Kyoto/Nagoya, 1983), 175-206 Adv. Stud. Pure Math., 6, North-Holland, Amsterdam-New York, 1985. | MR | Zbl | DOI

[10] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1-47. | MR | Zbl | EuDML | DOI

[11] Hisayosi Matumoto, Whittaker vectors and associated varieties. Invent. Math. 89 (1987), no. 1, 219-224. | MR | Zbl | EuDML | DOI

[12] Hisayosi Matumoto, C - -Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations, Ann. Sci. Éc. Norm. Sup. 23 (1990), 311-367. | MR | Zbl | EuDML | Numdam | DOI

[13] K. Nishiyama and H. Ochiai, Bernstein degree of singular unitary highest weight representations of the metaplectic group. Proc. Japan Acad., 75, Ser. A (1999), 9-11. | MR | Zbl | DOI

[14] W. Rossmann, Picard-Lefschetz theory and characters of a semisimple Lie group, Invent. Math. 121 (1995), 579-611. | MR | Zbl | EuDML | DOI

[15] F. Sato, On the stability of branching coefficients of rational representations of reductive groups, Comment. Math. Univ. St. Paul. 42 (1993), no. 2, 189-207. | MR | Zbl

[16] W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive groups, to appear in Annals of Math. | MR | Zbl | EuDML

[17] M. Tagawa, Generalized Whittaker models for unitarizable highest weight representations (in Japanese), Master Thesis, Kyoto University, 1998.

[18] D. Vogan, Singular unitary representations, in Non commutative harmonic analysis and Lie groups, LNM 880 (1981), pp. 506 - 535. | MR | Zbl | DOI

[19] D. A. Vogan, Jr., Associated varieties and unipotent representations, in Harmonic analysis on reductive groups (Brunswick, ME, 1989), 315-388, Progr. Math. 101, Birkhäuser, Boston, Boston, MA, 1991. | MR | Zbl | DOI

[20] D. A. Vogan, Jr., The method of coadjoint orbits for real reductive groups, in Representation Theory of Lie Groups (eds. J. Adams and D. Vogan), 177-238, IAS/Park City Mathematics Series 8, AMS, 2000. | MR | Zbl | DOI

[1] D. Barbasch and D. A. Vogan, Jr., The local structure of characters, J. Funct. Anal. 37 (1980), no. 1, 27-55. | MR | Zbl | DOI

[2] W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), no. 3, 437-476. | MR | Zbl | EuDML | DOI

[3] W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), no. 1, 1-68. | MR | Zbl | EuDML | DOI

[4] R. Brylinski and B. Kostant, Minimal representations of E 6 , E 7 and E 8 and the generalized Capelli identity, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), no. 7, 2469-2472. | MR | Zbl | DOI

[5] R. Brylinski and B. Kostant, Minimal representations, geometric quantization, and unitarity, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), no. 13, 6026-6029. | MR | Zbl | DOI

[6] R. Brylinski and B. Kostant, Lagrangian models of minimal representations of E 6 , E 7 and E 8 , in Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993), 13-63, Progr. Math., 131, Birkhäuser, Boston, Boston, MA, 1995. | MR | Zbl

[7] J.-T. Chang, Characteristic cycles of holomorphic discrete series, Trans. Amer. Math. Soc. 334 (1992), no. 1, 213-227. | MR | Zbl | DOI

[8] J.-T. Chang, Characteristic cycles of discrete series for -rank one groups, Trans. Amer. Math. Soc. 341 (1994), 603-622. | MR | Zbl

[9] M. G. Davidson, T. J. Enright and R. J. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. 94 (1991), no. 455, iv+102 pp. | MR | Zbl

[10] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, GTM 150, Springer, New York, 1995. | MR | Zbl

[11] T. Enright, R. Howe and N. Wallach, A classification of unitary highest weight modules, in Representation theory of reductive groups (Park City, Utah, 1982), 97-143, Progr. Math., 40, Birkhäuser, Boston, Boston, Mass., 1983. | MR | Zbl | DOI

[12] W. Fulton, Intersection theory, Springer, Berlin, 1984. | MR | Zbl | DOI

[13] W. Fulton, Young tableaux, With applications to representation theory and geometry. Cambridge Univ. Press, Cambridge, 1997. | MR | Zbl

[14] S. Gelbart, Holomorphic discrete series for the real symplectic group, Invent. Math. 19 (1973), 49-58. | MR | Zbl | EuDML | DOI

[15] Giovanni Zeno Giambelli, Sulle varietà rappresentate coll'annulare determinanti minori contenuti in un determinante simmetrico generico di forme, Atti R. Accad. Sci. Torino 41 (1906), 102 - 125. | JFM

[16] J. Harris, Algebraic geometry, A first course. Corrected reprint of the 1992 original, Springer, New York, 1995. | MR

[17] J. Harris and L. W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71-84. | MR | Zbl | DOI

[18] T. Hayata, Whittaker functions of generalized principal series on S U ( 2 , 2 ) , J. Math. Kyoto Univ. 37 (1997), no. 3, 531-546. | MR | Zbl | DOI

[19] T. Hayata and T. Oda, An explicit integral representation of Whittaker functions for the representations of the discrete series -the case of S U ( 2 , 2 ) , J. Math. Kyoto Univ. 37 (1997), no. 3, 519-530. | MR | Zbl | DOI

[20] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. | MR | Zbl

[21] R. Hotta, Rings and Fields, I., Iwanami Lecture Series, The Foundation of Modern Mathematics 15, Iwanami Shoten 1997.

[22] R. Howe, Reciprocity laws in the theory of dual pairs, in Representation theory of reductive groups (Park City, Utah, 1982), 159-175, Progr. Math., 40, Birkhäuser, Boston, Boston, Mass., 1983. | MR | Zbl | DOI

[23] R. Howe, Perspectives on invariant theory : Schur duality, multiplicity-free actions and beyond, in The Schur lectures (1992) (Tel Aviv), 1-182, Bar-Han Univ., Ramat Gan, 1995. | MR | Zbl

[24] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539-570. | MR | Zbl | DOI

R. Howe, Erratum to : "Remarks on classical invariant theory", Trans. Amer. Math. Soc. 318 (1990), no. 2, 823. | MR | Zbl

[25] R. Howe, Dual pairs in physics : harmonic oscillators, photons, electrons, and singletons, in Applications of group theory in physics and mathematical physics (Chicago, 1982), 179-207, Amer. Math. Soc, Providence, R.I., 1985. | MR | Zbl

[26] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565-619. | MR | Zbl | EuDML | DOI

[27] A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. Ecole Norm. Sup. (4) 9 (1976), no. 1, 1-29. | MR | Zbl | EuDML | Numdam | DOI

[28] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1-47. | MR | Zbl | EuDML | DOI

[29] A. A. Kirillov, Characters of unitary representations of Lie groups, Funkcional. Anal, i Priložen 2 (1968), no. 2 40-55. | MR | Zbl

[30] B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101-184. | MR | Zbl | EuDML | DOI

[31] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. | MR | Zbl | DOI

[32] T. E. Lynch, Generalized Whittaker vectors and representation theory, Thesis, M.I.T., 1979. | MR

[33] H. Matsumura, Commutative ring theory, Translated from the Japanese by M. Reid, Cambridge Univ. Press, Cambridge, 1986. | MR | Zbl

[34] H. Matumoto, Whittaker vectors and associated varieties, Invent. math. 89 (1987), 219 - 224. | MR | Zbl | EuDML | DOI

[35] H. Matumoto, C - -Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations, Ann. Sci. Ecole Norm. Sup. (4) 23 (1990), no. 2, 311-367. | MR | Zbl | EuDML | Numdam | DOI

[36] H. Matumoto, Whittaker vectors and the Goodman-Wallach operators, Acta Math. 161 (1988), no. 3-4, 183-241. | MR | Zbl | DOI

[37] K. Nishiyama and H. Ochiai, Bernstein degree of singular unitary highest weight representations of the metaplectic group. Proc. Japan Acad., 75, Ser. A (1999), 9 - 11. | MR | Zbl | DOI

[38] T. Oda, An explicit integral representation of Whittaker functions on S p ( 2 ; 𝐑 ) for the large discrete series representations, Tôhoku Math. J. (2) 46 (1994), no. 2, 261-279. | MR | Zbl | DOI

[39] V. L. Popov and E. B. Vinberg, Invariant theory. in Algebraic geometry. IV, A translation of Algebraic geometry. 4 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, Translation edited by A. N. Parshin and I. R. Shafarevich, Encyclopaedia of Mathematical Sciences, 55. Springer, Berlin, 1994. | Zbl | MR | DOI

[40] S. Rallis and G. Schiffmann, Weil representation. I. Intertwining distributions and discrete spectrum, Mem. Amer. Math. Soc. 25 (1980), no. 231, iii+203 pp. | MR | Zbl

[41] W. Rossmann, Kirillov's character formula for reductive Lie groups, Invent. Math. 48 (1978), no. 3, 207-220. | MR | Zbl | EuDML | DOI

[42] F. Sato, On the stability of branching coefficients of rational representations of reductive groups, Comment. Math. Univ. St. Paul. 42 (1993), no. 2, 189-207. | MR | Zbl

[43] W. Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47-144. | MR | Zbl | EuDML | DOI

[44] W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive groups, to appear in Annals of Math. | MR | Zbl | EuDML

[45] T. A. Springer, Invariant theory, Lecture Notes in Mathematics, Vol. 585. Lecture Notes in Math., 585, Springer, Berlin, 1977. | MR | Zbl

[46] K. Taniguchi, Discrete series Whittaker functions of S U ( n , 1 ) and Spin ( 2 n , 1 ) , J. Math. Sci. Univ. Tokyo 3 (1996), no. 2, 331-377. | MR | Zbl

[47] P. Torasso, Sur le caractère de la représentation de Shale-Weil de Mp ( n , 𝐑 ) et Sp ( n , 𝐂 ) , Math. Ann. 252 (1980), no. 1, 53-86. | MR | Zbl | EuDML | DOI

[48] E. B. Vinberg and V. L. Popov, On a class of quasihomogeneous affine varieties. Math. USSR Izvestja, 6 (1972), 743 - 758. | Zbl | DOI

[49] D. A. Vogan, Jr., Gel'fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75-98. | MR | Zbl | EuDML | DOI

[50] D. Vogan, Singular unitary representations. In Non commutative harmonic analysis and Lie groups, LNM 880 (1981), pp. 506 - 535. | MR | Zbl | DOI

[51] D. A. Vogan, Jr., Associated varieties and unipotent representations, in Harmonic analysis on reductive groups (Brunswick, ME, 1989), 315-388, Progr. Math. 101, Birkhäuser, Boston, Boston, MA, 1991. | MR | Zbl | DOI

[52] D. A. Vogan, Jr., The Method of Coadjoint Orbits for Real Reductive Groups, in Representation Theory of Lie Groups, 177-238, IAS/Park City math. ser. 8, Amer. Math. Soc, (2000). | MR | Zbl | DOI

[53] H. Yamashita, Embeddings of discrete series into induced representations of semisimple Lie groups. II. Generalized Whittaker models for S U ( 2 , 2 ) , J. Math. Kyoto Univ. 31 (1991), no. 2, 543-571. | MR | Zbl | DOI

[54] H. Yamashita, private communication (1999). (See Yamashita's article in this volume.)

[1] L. Barchini, Szegö kernels associated with Zuckerman modules, J. Funct. Anal., 131 (1995), 145-182. | MR | Zbl | DOI

[2] M. G. Davidson, T. J. Enright and R. J. Stanke, Covariant differential operators, Math. Ann., 288 (1990), 731-739. | MR | Zbl | EuDML | DOI

[3] M. G. Davidson, T. J. Enright and R. J. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. No. 455, American Mathematical Society, Providence, R.I., 1991. | MR | Zbl

[4] M. G. Davidson and T. J. Stanke, Szegö maps and highest weight representations, Pacific J. Math., 158 (1993), 67-91. | MR | Zbl | DOI

[5] T. J. Enright and A. Joseph, An intrinsic analysis of unitarizable highest weight modules, Math. Ann., 288 (1990), 571-594. | MR | Zbl | EuDML | DOI

[6] T. J. Enright and R. Parthasarathy, A proof of a conjecture of Kashiwaxa and Vergne. in "Noncommutative harmonic analysis and Lie groups (Marseille, 1980)", Lecture Notes in Math., 880, Springer, Berlin-New York, 1981, pp. 74-90. | MR | Zbl | DOI

[7] T. J. Enright, R. Howe, and N. R. Wallach, A classification of unitary highest weight modules, in "Representation theory of reductive groups (Park City, Utah, 1982 ; P.C.Trombi ed.)", Progress in Math., Vol. 40, Birkhäuser, 1983, pp.97-143. | MR | Zbl

[8] A. Gyoja and H. Yamashita, Associated variety, Kostant-Sekiguchi correspondence, and locally free U ( n ) -action on Harish-Chandra modules, J. Math. Soc. Japan, 51 (1999), 129-149. | MR | Zbl | DOI

[9] M. Harris and H. P. Jakobsen, Singular holomorphic representations and singular modular forms, Math. Ann., 259 (1982), 227-244. | MR | Zbl | EuDML | DOI

[10] R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series, Invent. Math., 26 (1974), 133-178. | MR | Zbl | EuDML | DOI

[11] M. Kashiwara and W. Schmid, Quasi-equivariant 𝒟 -modules, equivariant derived category, and representations of reductive Lie groups, in "Lie theory and geometry (J.L.Brylinski et al. eds.)", Birkhäuser, 1994, pp.457-488. | MR | Zbl

[12] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math., 44 (1978), 1-47. | MR | Zbl | EuDML | DOI

[13] S. Kato and H. Ochiai, The degrees of orbits of the multiplicity actions, in this volume. | Zbl

[14] N. Kawanaka, Generalized Gelfand-Graev representation and Ennola duality, in "Algebraic groups and related topics", Advanced Studies in Pure Math., 6 (1985), 175-206. | MR | Zbl | DOI

[15] N. Kawanaka, Generalized Gelfand-Graev representations of exceptional simple algebraic groups over a finite field. I, Invent. Math., 84 (1986), 575-616. | MR | Zbl | EuDML | DOI

[16] A. W. Knapp, Lie groups beyond an introduction. Progress in Mathematics Vol. 140, Birkhäuser, Boston-Besel-Stuttgart, 1996. | MR | Zbl

[17] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math., 85 (1963), 327-404. | MR | Zbl | DOI

[18] B. Kostant, On Whittaker vectors and representation theory, Invent. Math., 48 (1978), 101-184. | MR | Zbl | EuDML | DOI

[19] K. Koike and I. Terada, Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank, Adv. Math., 79 (1990), 104-135. | MR | Zbl | DOI

[20] H. P. Jakobsen, An intrinsic classification of the unitarizable highest weight modules as well as their associated varieties, Compositio Math. 101 (1996), 313-352. | MR | Zbl | EuDML | Numdam

[21] A. Joseph, Annihilators and associated varieties of unitary highest weight modules, Ann. Sci. Éc. Norm. Sup., 25 (1992), 1-45. | MR | Zbl | EuDML | Numdam | DOI

[22] H. Matumoto, Whittaker vectors and associated varieties, Invent. Math., 89 (1987), 219-224. | MR | Zbl | EuDML | DOI

[23] H. Matumoto, Whittaker vectors and the Goodman-Wallach operators, Acta Math., 161 (1988), 183-241. | MR | Zbl | DOI

[24] H. Matumoto, C - -Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations, Ann. Sci. Éc. Norm. Sup., 23 (1990), 311-367. | MR | Zbl | EuDML | Numdam | DOI

[25] H. Matumoto, C - -Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets. Compositio Math. 82 (1992), 189-244. | MR | Zbl | EuDML | Numdam

[26] C. Moeglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes p -adiques, Math. Z. 196 (1987), 427-452. | MR | Zbl | EuDML | DOI

[27] K. Nishiyama, H. Ochiai and K. Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case, in this volume. | Zbl

[28] W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, Dissertation, University of California, Berkeley, 1967 ; reprinted in "Representation theory and harmonic analysis on semisimple Lie groups (P.L Sally and D.A Vogan eds.)", Mathematical Surveys and Monograph Vol. 31, Amer. Math. Soc, 1989, pp.223- 286. | MR | Zbl

[29] W. Schmid, Boundary value problems for group invariant differential equations, in "Elie Cartan et les Mathématiques d'Aujourd'hui", Astérisque, Numéro hors-série, 1985, pp.311-321. | MR | Numdam | Zbl

[30] R. S. Strichartz, The explicit Fourier decomposition of L 2 ( S O ( n ) / S O ( n - m ) ) , Canad. J. Math., 27 (1975), 294-310. | MR | Zbl

[31] M. Tagawa, Generalized Whittaker models for unitarizable highest weight representations (in Japanese), Master Thesis, Kyoto University, 1998.

[32] M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math., 136 (1976), 1-59. | MR | Zbl | DOI

[33] D. A. Vogan, Associated varieties and unipotent representations, in "Harmonic Analysis on Reductive Groups (W.Barker and P.Sally eds.)," Progress in Math., Vol. 101, Birkhäuser, 1991, pp.315-388. | MR | Zbl | DOI

[34] N. R. Wallach, The analytic continuation of the discrete series. II, Trans. Amer. Math. Soc, 251 (1979), 19-37. | MR | Zbl | DOI

[35] N. R. Wallach, Real reductive groups I, Pure and Applied Mathematics Vol. 132, Academic Press, San Diego-London, 1988. | MR | Zbl

[36] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, Berlin-Heidelberg-New York, 1972. | MR | Zbl

[37] H. Weyl, The classical groups. Their invariants and representations, Eighth printing, Princeton University Press, Princeton, NJ, 1973. | Zbl

[38] H. W. Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, J. Funct. Anal., 129 (1995), 428-454. | MR | Zbl | DOI

[39] H. Yamashita, On Whittaker vectors for generalized Gelfand-Graev representations of semisimple Lie groups, J. Math. Kyoto Univ., 26 (1986), 263-298. | MR | Zbl | DOI

[40] H. Yamashita, Finite multiplicity theorems for induced representations of semisimple Lie groups II : Applications to generalized Gelfand-Graev representations, J. Math. Kyoto. Univ., 28 (1988), 383-444. | MR | Zbl | DOI

[41] H. Yamashita, Multiplicity one theorems for generalized Gelfand-Graev representations of semisimple Lie groups and Whittaker models for the discrete series, Advanced Studies in Pure Math. 14 (1988), 31-121. | MR | Zbl | DOI

[42] H. Yamashita, Embeddings of discrete series into induced representations of semisimple Lie groups I,: General theory and the case of S U ( 2 , 2 ) , Japan. J. Math., 16 (1990), 31-95; | MR | Zbl | DOI

H. Yamashita, Embeddings of discrete series into induced representations of semisimple Lie groups II : generalized Whittaker models for S U ( 2 , 2 ) , J. Math. Kyoto Univ., 31 (1991), 543-571. | MR | Zbl | DOI

[43] H. Yamashita, Criteria for the finiteness of restriction of U ( 𝔤 ) -modules to subalgebras and applications to Harish-Chandra modules: a study in relation to the associated varieties, J. Funct. Anal., 121 (1994), 296-329. | MR | Zbl | DOI

[44] H. Yamashita, Description of the associated varieties for the discrete series representations of a semisimple Lie group: An elementary proof by means of differential operators of gradient type, Comment. Math. Univ. St. Paul., 47 (1998), 35-52. | MR | Zbl

[45] T. Yoshinaga and H. Yamashita, The embeddings of discrete series into principal series for an exceptional real simple Lie group of type G 2 , J. Math. Kyoto Univ., 36 (1996), 557-595. | MR | Zbl | DOI

[1] T. J. Enright, R. Howe and N. R. Wallach, A classification of unitary highest weight modules, Progress in Math. 40 (1983) 97-143, Birkhäuser. | MR | Zbl

[2] T. J. Enright and A. Joseph, An intrinsic analysis of unitarizable highest weight modules, Math. Ann. 288 (1990) 571-594. | MR | Zbl | EuDML | DOI

[3] J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford, 1994. | MR | Zbl

[4] W. Fulton, Intersection theory, Springer, Berlin, 1984. | MR | Zbl | DOI

[5] S. Helgason, Differential geometry, Lie groups, and symmetric spaces Academic Press, New York, San Francisco, London. (1978). | MR | Zbl

[6] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991) 565-619. | MR | Zbl | EuDML | DOI

[7] H. P. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, Jour. Funct. Anal. 52 (1983) 385-412. | MR | Zbl | DOI

[8] K. D. Johnson, On a ring of invariant polynomials on a hermitian symmetric space, J. Algebra, 67 (1980) 72-81. | MR | Zbl | DOI

[9] A. Joseph, Annihilators and associated varieties of unitary highest weight modules, Ann. scient. École Normal Superior, (1992) 1-45. | MR | Zbl | EuDML | Numdam

[10] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980) 190-213. | MR | Zbl | DOI

[11] K. W. J. Kadell, The Selberg-Jack symmetric functions, Adv. Math. 130 (1997) 33-102. | MR | Zbl | DOI

[12] S. Kato, The Gelfand-Kirillov dimension and Bernstein degree of a unitary highest weight module, Master thesis, Kyushu University (2000) in Japanese.

[13] A. Knapp, Representation Theory of Semisimple Groups, An Overview Based on Examples, Princeton Univ. Press, 1986 | MR | Zbl | DOI

[14] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 93 (1971) 753-809. | MR | Zbl | DOI

[15] J. S. Li, The correspondences of infinitesimal characters for reductive dual pairs in semisimple Lie groups, Duke Math. J. 97 (1999) 347-377. | MR | Zbl | DOI

[16] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford, 1995 | MR | Zbl

[17] C. C. Moore, Compactification of symmetric spaces, II, Amer. J. Math. 86 (1964), 358-378. | MR | Zbl | DOI

[18] K. Nishiyama and H. Ochiai, Bernstein degree of singular unitary highest weight representations of the metaplectic group, Proc. Japan Acad. Ser. A, 75 (1999) 9-11. | MR | Zbl | DOI

[19] K. Nishiyama, H. Ochiai and K. Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case -, in this volume. | Zbl

[20] H. Rubenthaler, Les paires duales dans les algèbres de Lie r'eductives, Astérisque 219 (1994), 1-121. | MR | Zbl

[21] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970) 61-80. | MR | Zbl | EuDML | DOI

[22] D. Vogan, Associated varieties and unipotent representations, Progress in Math. 101 (1991) 315-388. | MR | Zbl

[23] N. R. Wallach, The analytic continuation of the discrete series, I, II, Trans. Amer. Math. Soc. 251 (1979) 1-17, 19-37. | MR | Zbl

[24] H. Yamashita, Cayley transform and generalized Whittaker models for irreducible highest weight modules, in this volume. | Zbl

[Intro] Introduction to this volume.

[NOT] Kyo Nishiyama, Hiroyuki Ochiai and Kenji Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case -. In this volume. | Zbl

[Y] Hiroshi Yamashita, Cayley transform and generalized Whittaker models for irreducible highest weight modules. In this volume. | Zbl

[1] S. S. Gelbart, A theory of Stiefel harmonics. Trans. AMS 192 (1974), 29 - 50. | MR | Zbl | DOI

[2] A. Gyoja and H. Yamashita, Associated variety, Kostant-Sekiguchi correspondence, and locally free U ( n ) -action on Harish-Chandra modules. J. Math. Soc. Japan 51 (1999), no. 1, 129-149. | MR | Zbl | DOI

[3] Jian-Shu Li, Singular unitary representations of classical groups. Invent. Math. 97 (1989), no. 2, 237-255. | MR | Zbl | EuDML | DOI

[4] H. Matumoto, C - -Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets. Compositio Math. 82 (1992), 189-244. | MR | Zbl | EuDML | Numdam

[5] F. Sato, On the stability of branching coefficients of rational representations of reductive groups, Comment. Math. Univ. St. Paul. 42 (1993), no. 2, 189-207. | MR | Zbl

[6] W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive groups. To appear in Annals of Math. | MR | Zbl | EuDML

[7] P. Trapa, Annihilators, associated varieties, and the theta correspondence, preprint, November 1999.