Abstract
Let G be a simple algebraic group over \(\mathbb{C} \)and let P be a parabolic subgroup of G with commutative unipotent radical N + ≠ {1}. To each non-open P-orbit C on Lie(N +) we construct an action of Lie(G) on the coordinate algebra \(\mathbb{C}\left[ {\bar C} \right] \) of the closure \(\bar C \) of C. This is a natural generalization of Goncharov’s result dealing with the case where C is the orbit of the highest root vector. It is also shown that \(\mathbb{C}\left[ {\overline C } \right] \) is an irreducible highest weight module.
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References
Beilinson, A. and Bernstein, J. (1981) Localisation de g-modules, C. R. Acad. Sci. Paris 292, 15–18.
Bernstein, J., Gelfand, I. and Gelfand, S. (1971) Structures of representations generated by vectors of highest weight, Funct. Anal. App. 5, 1–8.
Bourbaki, N. (1968) Groupes et algèbres de Lie, Ch IV-VI, Hermann, Paris.
De Concini, C., Eisenbud, D. and Procesi, C. (1980) Young diagrams and determinantal varieties, Invent. Math. 56, 129–165.
De Concini, C. and Procesi, C. (1976) A characteristic free approach to representation theory, Adv. Math. 21, 330–354.
Enright, T.J., Howe, R. and Wallach, N.R. (1983) A classification of unitary highest weight modules, in P.C. Trombi (ed.), Representation theory of reductive groups, Birkhäuser, Boston, pp. 97–143.
Enright, T.J. and Joseph, A. (1990) An intrinsic analysis of unitarizable highest weight modules, Math. Ann. 288, 571–594.
Goncharov, A.B. (1982) Construction of the Weil representations of certain simple Lie algebras, Funct. Anal. Appl. 16, 70–71.
Gyoja, A. (1994) Highest weight modules and b-functions of semiinvariants, Publ. RIMS Kyoto Univ. 30, 353–400.
Jakobsen, H.P. (1983) Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52, 385–412.
Johnson, K. (1980) On a ring of invariant polynomials on a hermitian symmetric space, J. Algebra 67, 72–81.
Kashiwara, M. (1989) Representation theory and D-modules on flag varieties, Astérisque 173–174, 55–109.
Levasseur, T. and Stafford, J.T. (1989) Rings of differential operators on classical rings of invariants, Mem. Amer. Math. Soc. 412, 1–117.
Levasseur, T., Smith, S.P. and Stafford, J.T. (1988) The minimal nilpotent orbit, the Joseph ideal, and differential operator s., J. Algebra 116, 480–501.
Mathieu, O. and Papadopoulo, G. (1997) A combinatorial character formula for some highest weight modules, preprint.
Richardson, R., Röhrle, G. and Steinberg, R. (1992) Parabolic subgroups with abelian unipotent radical, Invent. Math. 110, 649–671.
Sakane, Y. and Takeuchi, M. (1981) On defining equations of symmetric submanifolds in complex projective spaces, J. Math. Soc. Japan 33, 267–279.
Schmid, W. (1969) Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9, 61–80.
Takeuchi, M. (1973) Polynomial representations associated with symmetric bounded domains, Osaka J. Math. 10, 441–475.
Tanisaki, T. (1997) Hypergeometric systems and Radon transforms for Hermitian symmetric spaces, preprint.
Weyl, H. (1946) The classical groups, Princeton University Press, Princeton.
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Tanisaki, T. (1998). Highest Weight Modules Associated to Parabolic Subgroups with Commutative Unipotent Radicals. In: Carter, R.W., Saxl, J. (eds) Algebraic Groups and their Representations. NATO ASI Series, vol 517. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5308-9_5
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DOI: https://doi.org/10.1007/978-94-011-5308-9_5
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