Duality property in coinduced representations of Lie superalgebras. (Propriété de dualité dans les représentations coinduites de superalgèbres de Lie.) (French) Zbl 0815.17022

Chemla, Sophie

Ann. Inst. Fourier 44, No. 4, 1067-1090 (1994).

Summary: We give a new proof of a duality theorem about coinduced representations proved by M. Duflo [Invent. Math. 67, 385-393 (1982; Zbl 0501.17006)] in the case of finite dimensional Lie algebras and generalized by the author in Math. Ann. 297, No. 2, 371-382 (1993; Zbl 0788.17021). Our proof involves the algebra of differential operators as well as Bernstein’s correspondence between left and right \(D\)-modules. We also give an interpretation of the result in terms of induced representations of Lie supergroups. Our proof holds over a field of characteristic 0.

MSC:

17B70	Graded Lie (super)algebras
17A70	Superalgebras
17B25	Exceptional (super)algebras

Keywords:

duality theorem; coinduced representations; finite-dimensional Lie algebras; algebra of differential operators; \(D\)-modules; induced representations of Lie supergroups

Citations:

Zbl 0501.17006; Zbl 0788.17021

References:

[1]	[Bla] , Induced and produced representations of Lie algebras, Trans. Amer. Math. Soc., 144 (1969), 457-474. · Zbl 0295.17002
[2]	[BB] , , Differential operators on homogeneous spaces I, Invent. Math., 69 (1982), 437-476. · Zbl 0504.22015
[3]	[C] , Cohomologie locale de Grothendieck et représentations induites de superalgèbres de Lie, Mathematische Annalen, 297 (1993), 371-382. · Zbl 0788.17021
[4]	[Du] , Sur les idéaux induits dans les algèbres enveloppantes, Invent. Math., 67 (1982), 385-393. · Zbl 0501.17006
[5]	[Fe] , Global dimension of rings of differential operators, Trans. Moscow. Math. Soc., 1 (1982), 123-147. · Zbl 0484.13020
[6]	[K] , Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Math. n° 570, Springer-Verlag, 1975. · Zbl 0358.53024
[7]	[L] , Introduction to the theory of supermanifolds, Uspeki Mat. Nauk, 35, 1 (1980). · Zbl 0462.58002
[8]	[Le] , Critères d’induction et de coinduction pour certains anneaux d’opérateurs différentiels, J. Algebra, 110, n° 2 (1987), 530-562. · Zbl 0629.16003
[9]	[Ma] , Gauge field theory and complex geometry. A Series of Comprehensive Studies in Mathematics, Springer-Verlag, 1988.
[10]	[Ri] , Differential forms on general commutative algebras, Trans. Amer. Math. Soc., 108 (1963), 195-222. · Zbl 0113.26204
[11]	[Sch] , The theory of Lie superalgebras, Lecture Notes in Math., n° 716, Springer-Verlag, 1979. · Zbl 0407.17001