On Whittaker vectors and representation theory

Kostant, Bertram

doi:10.1007/bf01390249

Cited by 494 publications

(443 citation statements)

References 13 publications

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“…Let Z ∩ (e + g g g nw ) red denote the intersection Z ∩ (e +g g g nw ) as an algebraic variety with reduced scheme structure. The following result is a double analogue of [K3,Theorem 1.2]. Note that even in the classical case the proof given below is simpler than the argument in [K3].…”

Section: Some Cohomology and Generating Functionsmentioning

confidence: 85%

Principal nilpotent pairs in a semisimple Lie algebra 1

Ginzburg¹

2000

Inventiones Mathematicae

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Abstract. This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula.In a different direction, each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of sl n , the conjugacy classes of principal nilpotent pairs and the irreducible representations of the Symmetric group, S n , are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple S n -modules in terms of Young's symmetrisers.First results towards a complete classification of all principal nilpotent pairs in a simple Lie algebra are presented at the end of this paper in an Appendix, written by A. Elashvili and D. Panyushev.

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Section: Some Cohomology and Generating Functionsmentioning

confidence: 85%

Principal nilpotent pairs in a semisimple Lie algebra 1

Ginzburg¹

2000

Inventiones Mathematicae

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“…To state the theorem, let Λ 0 := ∞ and Λ n+2 := −∞. As is explained in § 1, write |v| : 5) and each K-type occurs in π i,j with multiplicity one.…”

Section: 2mentioning

confidence: 99%

“…We call these the standard Whittaker (g, K)-modules. Note that these are not the "standard Whittaker module" defined in [5]. It is easy to show that these are K-admissible and then have finite length (Corollary 2.4).…”

Section: Introductionmentioning

confidence: 99%

On the composition series of the standard Whittaker $(\mathfrak {g},K)$-modules

Taniguchi

2012

Trans. Amer. Math. Soc.

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Abstract. For a real reductive linear Lie group G, the space of Whittaker functions is the representation space induced from a non-degenerate unitary character of the Iwasawa nilpotent subgroup. Defined are the standard Whittaker (g, K)-modules, which are K-admissible submodules of the space of Whittaker functions. We first determine the structures of them when the infinitesimal characters characterizing them are generic. As an example of the integral case, we determine the composition series of the standard Whittaker (g, K)-module when G is the group U (n, 1) and the infinitesimal character is regular integral.

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“…It is well known ( [5]) that for almost all ν, the dimension of Wh(ν) is equal to the order of the Weyl group W ∼ = S n of G. Hashizume ([3]) constructed the basis function M n (ν; g) of Wh(ν), which will be explained below. Shalika ([7]) proved the uniqueness of the element in Wh(ν) which has the moderate growth property and contributes to the Fourier expansions of automorphic forms.…”

Section: Definition 1 -A Smooth Function F On G Is Called Class One mentioning

confidence: 99%

A remark on Whittaker functions on SL$(n,{\Bbb R})$

Ishii¹

2005

Annales De L’institut Fourier

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On Whittaker vectors and representation theory

Cited by 494 publications

References 13 publications

Principal nilpotent pairs in a semisimple Lie algebra 1

Principal nilpotent pairs in a semisimple Lie algebra 1

On the composition series of the standard Whittaker $(\mathfrak {g},K)$-modules

A remark on Whittaker functions on SL$(n,{\Bbb R})$

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