Abstract
In this note, we describe proofs of certain conjectures on functorial, geometric constructions of representations of a reductive Lie group G R . Our methods have applications beyond the conjectures themselves: unified proofs of the basic properties of the maximal and minimal globalizations of Harish-Chandra modules, and a criterion which insures that the solutions of a G R -invariant system of linear differential equations constitute a representation of finite length.
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References
A. Beilinson and J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris 292 (1981), 15–18.
A. Beilinson and J. Bernstein, A generalization of Casselman’s submodule theorem in: Representation Theory of Reductive Groups, Progress in Mathematics, vol. 40, Birkhäuser, Boston 1983, 35–52.
A. Beilinson and J. Bernstein, A proof of Jantzen’s conjecture, Advances in Soviet Math. 16 (1993), 1–50.
A. Beilinson, J. Bernstein, and D. Deligne, Faisceaux pervers, Astérisque 100 (1982), 5–171.
J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578 (1994), Springer.
W. Borho, J.-L. Brylinski, and R. MacPherson, Nilpotent Orbits, Primitive Ideals, and Characteristic Classes, Progress in Mathematics, vol. 78, Birkhäuser, Boston, 1989.
W. Casselman, Jacquet modules for real reductive groups, in: Proc. of the International Congress of Mathematicians, Helsinki, 1980, 557–563.
W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G, Can. Jour. Math. 41 (1989), 385–438.
V. Ginzburg, g-modules, Springer’s representations and bivariant Chern classes, Advances in Math. 61 (1986), 1–48.
R. Goodman and N. Wallach, Whittaker vectors and conical vectors, Jour. Fund. Anal. 39 (1980), 199–279.
Harish-Chandra, Representations of semisimple Lie groups I, Trans. Amer. Math. Soc. 75 (1953), 185–243.
Harish-Chandra, The characters of semisimple Lie groups, Trans. Amer. Math. Soc. 83 (1956), 98–163.
Harish-Chandra, Discrete series for semisimple Lie groups I, Acta Math. 113 (1965), 241–318.
S. Helgason, A duality for symmetric spaces with applications to group representations I, Advances in Math. 5 (1970), 1–154
S. Helgason, A duality for symmetric spaces with applications to group representations II, Advances in Math. 22 (1976), 187–219.
H. Hecht, D. Miličić, W. Schmid, and J. Wolf, Localization and standard modules for real semisimple Lie groups I: The duality theorem, Invent. Math. 90 (1987), 297–332.
H. Hecht and W. Schmid, On the asymptotics of Harish-Chandra modules, Jour. Reine und Angewandte Math. 343 (1983), 169–183.
H. Hecht and J. Taylor, Analytic localization of group representations, Advances in Math. 79 (1990), 139–212.
M. Kashiwara Character, character cycle, fixed point theorem, and group representations, in: Advanced Studies in Pure Math. 14 (1988), 369–378.
M. Kashiwara, Open problems in group representation theory, Proceedings of Taniguchi symposium held in 1986, RIMS preprint 569, Kyoto University, 1987.
M. Kashiwara, Representation theory and D-modules on flag manifolds, Astérisque 173–174 (1989), 55–109.
M. Kashiwara, D-modules and representation theory of Lie groups, RIMS preprint 940, Kyoto University, 1993, to appear in Ann. de l’Institut Fourier.
M. Kashiwara and T. Monteiro-Fernandes, Involutivité des variétés microcaractéristiques, Bull. Soc. Math. France 114 (1986), 393–402.
M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima, and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Annals of Math. 107 (1978), 1–39.
M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
G. Laumon, Sur la catégorie dérivée des D-modules filtrés, in: Algebraic Geometry, Proceedings, 1982, Lecture Notes in Mathematics 1016 (1983), Springer, 151–237.
T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307–320.
H. Matumoto, Whittaker vectors and the Goodman-Wallach operators, Acta Math. 161 (1988), 183–241.
I. Mirković, T. Uzawa, and K. Vilonen, Matsuki correspondence for sheaves, Inventiones Math. 109 (1992), 231–245.
S. J. Prichepionok, A natural topology for linear representations of semisimple Lie algebras, Soviet Math. Doklady 17 (1976), 1564–1566.
W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, thesis, Berkeley 1967. Reprinted in: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Mathematical Surveys and Monographs 31 (1989), Amer. Math. Soc., 223–286.
W. Schmid, Boundary value problems for group invariant differential equations, in: Élie Cartan et les Mathématiques d’Aujourd’hui, Astérisque numéro hors séries (1985), 311–322.
H. Sumihiro, Equivariant completion, Jour. Math. Kyoto Univ. 14 (1974), 1–28.
W. Schmid and K. Vilonen, Character, fixed points and Osborne’s conjecture, Contemp. Math. 145 (1993), 287–303.
W. Schmid and K. Vilonen, Characters, characteristic cycles, and nllpotent orbits, in: Geometry, Topology, and Physics, International Press, Boston, 1994.
W. Schmid and J. Wolf, Geometric quantization and derived functor modules for semisimple Lie groups, Jour. Funct. Anal. 90 (1990), 48–112.
L. Schwartz, Sur le théorème du graphe fermé, Compt. Rend. Acad. Sci. 263 (1966), 602–605.
F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967.
D. Vogan, Gelfand-Kirillov dimension for Harish-Chandra modules, Inventiones Math. 48 (1978), 75–98.
N.R. Wallach, Asymptotic expansion of generalized matrix entries of real reductive groups, in: Lie Group Representations I, Lecture Notes in Mathematics 1024 (1983), Springer, 287–369.
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Kashiwara, M., Schmid, W. (1994). Quasi-Equivariant D-Modules, Equivariant Derived Category, and Representations of Reductive Lie Groups. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics, vol 123. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0261-5_16
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