Summary
We study the ring of differential operators \( \mathcal{D} \) (X) on the basic affine space X = G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H*(X \( \mathcal{O} \) X) decomposes as a finite direct sum of nonisomorphic simple \( \mathcal{D} \) (X)-modules, each of which is isomorphic to a twist of \( \mathcal{O} \) (X) by an automorphism of \( \mathcal{D} \) (X).
We also use \( \mathcal{D} \) (X) to study the properties of \( \mathcal{D} \) (Z) for highest weight varieties Z. For example, we prove that Z is \( \mathcal{D} \) -simple in the sense that \( \mathcal{O} \) (Z) is a simple \( \mathcal{D} \) (Z)-module and produce an irreducible G-module of differential operators on Z of degree −1 and specified order.
This paper is dedicated to Tony Joseph on the occasion of his 60thbirthday.
The second author was supported in part by the NSF through the grants DMS-9801148 and DMS-0245320. Part of this work was done while he was visiting the Mittag-Leffler Institute and he would like to thank the Institute for its financial support and hospitality.
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References
A. Astashkevich and R. Brylinski, Exotic differential operators on complex minimal nilpotent orbits, in Advances in Geometry, Prog. Math., Vol. 172, Birkhäuser, Boston, 1999, 19–51.
J. Becker, Higher derivations and integral closure, Amer. J. Math., 100 (1978), 495–521.
J. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Differential operators on the cubic cone, Russian Math. Surveys, 27 (1972), 466–488.
R. Bezrukavnikov, A. Braverman and L. Positselskii, Gluing of abelian categories and differential operators on the basic affine space, J. Inst. Math. Jussieu, 1 (2002), 543–557.
J.-E. Björk, Rings of Differential Operators, North-Holland, Amsterdam, 1979.
N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4,5 et 6, Masson, Paris, 1981.
R. Brylinski and B. Kostant, Minimal representations, geometrical quantization and unitarity, Proc. Natl. Acad. Sci., 91 (1994), 6026–6029.
—, Differential operators on conical Lagrangian manifolds, in Lie Theory and Geometry: in honor of Bertram Kostant on the occasion of his 65th birthday, Prog. Math. Vol. 123, Birkhäuser, Boston, 1994, 65–96.
I. M. Gel’fand and A. A. Kirillov, The structure of the Lie field connected with a split semisimple Lie algebra, Funct. Anal. Appl., 3 (1969), 6–21.
F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, Lecture Notes in Mathematics, Vol. 1673, Springer-Verlag, Berlin Heidelberg, 1997.
R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York Heidelberg Berlin, 1977.
M. Hochster, The Zariski-Lipman conjecture in the graded case, J. Algebra, 47 (1977), 411–424.
A. van den Hombergh and H. de Vries, On the differential operators on the quasi-affine variety G/N, Indag. Math., 40 (1978), 460–466.
Y. Ishibashi, Nakai’s conjecture for invariant subrings, Hiroshima Math. J., 15 (1985), 429–436.
J. C. Jantzen, Representations of Algebraic Groups, Second Ed., Math. Surveys and Monographs Vol. 107, Amer. Math. Soc., Providence, RI, 2003.
A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. éc. Norm. Sup., 9 (1976), 1–29.
—, A surjectivity theorem for rigid highest weight modules, Invent. Math., 92(1988), 567–596.
—, Annihilators and associated varieties of unitary highest weight modules, Ann. Sci. éc. Norm. Sup., 25 (1992), 1–45.
M. Kashiwara, Representation theory and D-modules on flag varieties, Astérisque, 173–174 (1989), 55–109.
G. Kempf, The Grothendieck-Cousin complex of an induced representation, Advances in Mathematics, 29 (1978), 310–396.
H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspekte derMathematik, Vieweg Verlag, 1985.
T. Levasseur, S. P. Smith and J. T. Stafford, The minimal nilpotent orbit, the Joseph ideal, and differential operators, J. Algebra, 116 (1988), 480–501.
T. Levasseur and J. T. Stafford, Rings of Differential Operators on Classical Rings of Invariants, Mem. Amer. Math. Soc. 412, 1989.
—, Invariant differential operators and an homomorphism of Harish-Chandra, J. Amer. Math. Soc., 8 (1995), 365–372.
—, Differential operators on some nilpotent orbits, Representation Theory, 3(1999), 457–473.
M. Lorenz, Gelfand-Kirillov Dimension, Cuadernos de Algebra, No. 7 (Grenada, Spain), 1988.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Grad. Texts in Math., Vol. 30, Amer. Math. Soc., Providence, RI, 2000.
Y. Nakai, On the theory of differentials in commutative rings, J. Math. Soc. Japan, 13(1961), 63–84.
A. Polishchuk, Gluing of perverse sheaves on the basic affine space. With an appendix by R. Bezrukavnikov and the author, Selecta Math. (N.S.), 7 (2001), 83–147.
G. W. Schwarz, Lifting differential operators from orbit spaces, Ann. Sci. Éc. Norm. Sup., 28 (1995), 253–306.
N. N. Shapovalov, On a conjecture of Gel’fand-Kirillov, Funct. Anal. Appl., 7 (1973), 165–6.
—, Structure of the algebra of regular differential operators on a basic affine space, Funct. Anal. Appl., 8 (1974), 37–46.
K. E. Smith, The D-module structure of F-split rings, Math. Research Letters, 2(1995), 377–386.
T. A. Springer, Linear Algebraic Groups, Progress in Math., Vol. 9, Birkhäuser, Boston, MA., 1998.
W. N. Traves, Nakai’s conjecture for varieties smoothed by normalization, Proc. Amer. Math. Soc., 127 (1999), 2245–2248.
E. B. Vinberg and V. L. Popov, On a class of quasi-homogeneous affine varieties, Math. USSR Izvestija, 6 (1972), 743–758.
A. Yekutieli and J. J. Zhang, Dualizing complexes and tilting complexes over simple rings, J. Algebra, 256 (2002), 556–567.
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Levasseur, T., Stafford, J.T. (2006). Differential operators and cohomology groups on the basic affine space. In: Bernstein, J., Hinich, V., Melnikov, A. (eds) Studies in Lie Theory. Progress in Mathematics, vol 243. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4478-4_14
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DOI: https://doi.org/10.1007/0-8176-4478-4_14
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