Differential operators and highest weight representations

Davidson, Mark; Enright, Thomas J.; Stanke, Ronald J.

doi:10.1090/memo/0455

Cited by 39 publications

(46 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also wish to mention that many particular examples of conformally invariant systems have appeared in the literature, without their necessarily being identified as such. One notable class of examples may be found in the work of Davidson, Enright, and Stanke [7].…”

Section: §1 Introductionmentioning

confidence: 99%

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Barchini¹,

Kable²,

Zierau³

2008

Publ. Res. Inst. Math. Sci.

156

View full text Add to dashboard Cite

Several systems of partial differential operators are associated to each complex simple Lie algebra of rank greater than one. Each system is conformally invariant under the given algebra. The systems so constructed yield explicit reducibility results for a family of scalar generalized Verma modules attached to the Heisenberg parabolic subalgebra of the given Lie algebra. Points of reducibility for such families lie in the union of several arithmetic progressions, possibly overlapping. For classical algebras, enough systems are constructed to account for the first point of reducibility in each progression. The relationship between these results and a conjecture of Akihiko Gyoja is explored. §1. Introduction To describe our results, it is first necessary to set the scene. In the body of the paper we shall work most of the time in an exclusively algebraic framework, but it will be useful here to take a more inclusive viewpoint, mixing the analytic and the algebraic. We shall first attempt to explain the significance of our results and place them in context. To some extent, these remarks may be taken as an introduction to a broader investigation of which this work and [2] are the first fruits. Then we shall draw a map to aid the reader in navigating on the admittedly lengthy journey through the proofs.

show abstract

Section: §1 Introductionmentioning

confidence: 99%

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Barchini¹,

Kable²,

Zierau³

2008

Publ. Res. Inst. Math. Sci.

156

View full text Add to dashboard Cite

show abstract

“…If G = O * (2n), this assumption is very minor and it amounts to excluding a (tiny) subset of unitary lowest weight modules with their levels of reduction all being 1. See the paper of Davidson, Enright and Stanke ( §7, [4]) for a precise statement. We remark in passing that this corrects an assertion made (also in passing) in the classification article [6].…”

Section: Introductionmentioning

confidence: 99%

Theta lifting of unitary lowest weight modules and their associated cycles

Nishiyama¹,

Zhu²

2004

Duke Math. J.

View full text Add to dashboard Cite

Abstract. We consider a reductive dual pair (G, G ) in the stable range with G the smaller member and of Hermitian symmetric type. We study the theta lifting of (holomorphic) nilpotent K C -orbits in relation to the theta lifting of unitary lowest weight representations of G . We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under the theta lifting. We also develop a theory for the lifting of covariants arising from double fibrations by affine quotient maps.

show abstract

“…Moreover, the results cover not only the cohomology of p + but also of its appropriate subalgebras. The resulting operators are to be identified not as easily as the Yamabe operator, however there is a close connection of unitarizable highest weight modules and k-invariant differential operators presented in [5]. Consulting the weights of appropriate zeroth and first homology modules, there doesn't appear to be a case which would yield higher GJMS operators as first BGG operators.…”

Section: Resultsmentioning

confidence: 99%

Yamabe operator via BGG sequences

Tuček¹

2012

Arch. Math. (Brno)

View full text Add to dashboard Cite

Differential operators and highest weight representations

Cited by 39 publications

References 0 publications

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Theta lifting of unitary lowest weight modules and their associated cycles

Yamabe operator via BGG sequences

Contact Info

Product

Resources

About