Modules with bounded weight multiplicities for simple Lie algebras

Benkart, Georgia; Britten, D. J.; Lemire, Francine

doi:10.1007/pl00004314

Cited by 47 publications

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References 11 publications

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“…In the first step Britten and Lemire classified all simple cuspidal modules of degree 1 (see [4]) where deg(M) = sup λ∈h * {dim(M λ )}. Britten and Lemire, and later Benkart, Britten and Lemire, have classified all simple modules of degree 1 when g is of type A or C (see [1]). These modules will play a key role in our Theorem 3.2 below.…”

Section: Cuspidal Modulesmentioning

confidence: 98%

A generalization of the category O of Bernstein–Gelfand–Gelfand

Tomasini

2010

Comptes Rendus. Mathématique

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Section: Cuspidal Modulesmentioning

confidence: 98%

A generalization of the category O of Bernstein–Gelfand–Gelfand

Tomasini

2010

Comptes Rendus. Mathématique

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“…Such modules have been classified by Benkart, Britten and Lemire in [4]. They will be the main object of investigation of this paper.…”

Section: Remark 22 Note That We Require Finite Dimensional Weight Smentioning

confidence: 99%

Unitary representations of the universal cover of SU(1,1) and tensor products

Tomasini

Ørsted

2014

Kyoto J. Math.

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Abstract. In this paper we initiate a study of the relation between weight modules for simple Lie algebras and unitary representations of the corresponding simply-connected Lie groups. In particular we consider in detail from this point of view the universal covering group of SU (1, 1), including new results on the discrete part of tensor products of irreducible representations. As a consequence of these results, we show that the set of smooth vectors of the tensor product intersects trivially some of the representations in the discrete spectrum. IntroductionThe category of weight modules for simple Lie algebras, and in particular those of degree one, has been much studied in recent years. From the point of view of the unitary dual of the corresponding simply-connected Lie group, it is a natural question to find those degree one modules that integrate to unitary representations; they should form a small but interesting class of unitary representations with small Gelfand-Kirillov dimension. In this paper we treat in detail the case of sl(2, C), in effect giving a new proof of the classification due to Pukansky of the unitary dual of the universal covering of SU(1, 1); furthermore we apply this to studying in detail tensor products of such representations, obtaing new results about the discrete spectrum in such tensor products, and about the possible relation between the smooth vectors of the tensor product and the representations in the discrete spectrum. The methods are developed so as to apply to higher rank cases, where similar results are expected to hold.Let us now review the main results of this paper. Let G denote the uiversalThe center of G is generated by exp(2iπH). Let now ρ denote an irreducible unitary representation of G. From Schur's lemma we conclude that ρ(exp(2iπH)) = e −2iπτ 0 I. Therefore,ρ(exp(itH)) := e iτ 0 t ρ(exp(itH)) is a unitary representation of R, with period 2π, and hence is completely reducible. As a consequence, H possesses a complete system of eigenelements. In other word, the corresponding representation of the complexified Lie algebra sl(2, C) is a weight module (see definition 2.1). We shall review the basics of weight module in section 2. In section 3, we classify the unitarisable weight modules for su(1, 1). Recall that a unitarisable module is a module defined on a Hilbert space which is the differential of a unitary module for the universal covering group G. Using the explicit action of sl(2, C), we recover the classification due to Pukansky of the unitary dual of G, which falls into 3 series: the principal series π ǫ,it (0 < ǫ ≤ 1, t ∈ R), the complementary series π c σ,τ (0 < σ, τ < 1), the (continuation of the) discrete series π ± λ (λ > 0), and the extra trivial representation. In section 4 and 5, we study a tensor product V of the form: In section 5, we also give an explicit generator for all submodules in the discrete spectrum of V . As a consequence we prove proposition 5.19. A particular case of this proposition in the above setting is the following Proposition 5.19...

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“…Benkart, Britten, and Lemire [1] proved that infinite-dimensional weight modules, whose weight-subspace dimensions are bounded, exist only for special linear Lie algebras and symplectic Lie algebras. Britten and Lemire [5] classified the weight modules of finitedimensional weight subspaces for symplectic Lie algebras.…”

Section: Introductionmentioning

confidence: 98%

A New Functor fromE₆-mod toE₇-mod

2015

Communications in Algebra

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We find a new representation of the simple Lie algebra of type E 7 on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor from E 6 -Mod to E 7 -Mod. A condition for the functor to map a finite-dimensional irreducible E 6 -module to an infinite-dimensional irreducible E 7 -module is obtained. Our general framework also gives a direct polynomial extension from irreducible E 6 -modules to irreducible E 7 -modules, which can be used to derive Gel'fand-Zetlin bases for E 7 from those for E 6 that can be obtained from those for D 5 according to our earlier work.

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Modules with bounded weight multiplicities for simple Lie algebras

Cited by 47 publications

References 11 publications

A generalization of the category O of Bernstein–Gelfand–Gelfand

A generalization of the category O of Bernstein–Gelfand–Gelfand

Unitary representations of the universal cover of SU(1,1) and tensor products

A New Functor fromE₆-mod toE₇-mod

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Modules with bounded weight multiplicities for simple Lie algebras

Cited by 47 publications

References 11 publications

A generalization of the category O of Bernstein–Gelfand–Gelfand

A generalization of the category O of Bernstein–Gelfand–Gelfand

Unitary representations of the universal cover of SU(1,1) and tensor products

A New Functor fromE6-mod toE7-mod

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A New Functor fromE₆-mod toE₇-mod