Whittaker vectors and conical vectors

Goodman, Roe; Wallach, Nolan R.

doi:10.1016/0022-1236(80)90013-0

Cited by 69 publications

(53 citation statements)

References 19 publications

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“…[GW80,Mat90,Mat88]) give a partial Archimedean analog of the results of [MW87], by considering non-degenerate ("admissible") characters of the (smaller) nilradicals of bigger parabolic subgroups. However, much less is known in the Archimedean case.…”

Section: 3mentioning

confidence: 99%

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

Gourevitch

Sahi

2012

Sel. Math. New Ser.

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Abstract. In this paper we study irreducible unitary representations of GLn (R) and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker functionals, for both smooth and K-finite vectors, thereby generalizing results of Kostant, Matumoto and others.Our second result relates the annihilator to the sequence of adduced representations, as defined in this setting by one of the authors. Based on those results, we suggest a new notion of rank of a smooth admissible representation of GLn(R), which for unitarizable representations refines Howe's notion of rank.Our third result computes the adduced representations for (almost) all irreducible unitary representations in terms of the Vogan classification.We also indicate briefly the analogous results over complex and p-adic fields.

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Section: 3mentioning

confidence: 99%

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

Gourevitch

Sahi

2012

Sel. Math. New Ser.

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“…is a rational function into U(n )-,,+"• Also n_fl+a and n', have singularity sets contained in that ofñ_)l. The following result is proved in exactly the same way as Corollary 4.3, p. 229 of [7] using the Kac-Casimir operator of g. Using the results of Kac and Kazhdan [14] it is easy to see that the proof of the following result is identical with the proof of Lemma 4.4, p. 231 in [7]. Lemma 3.7.…”

mentioning

confidence: 73%

Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas

Rocha-Caridi¹,

Wallach²

1983

Trans. Amer. Math. Soc.

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Abstract. In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups.The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen's character sum formula for a quotient of two Verma modules (one of the main results of this article).Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).

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“…and, for 12) where the asterisk signifies that the variable of summation, δ mod c ′ O, runs over solutions of…”

Section: Index Of Notationmentioning

confidence: 99%

“…Let ν ∈ C and p ∈ Z. Bruggeman and Motohashi, in Section 6 of [5], employ a method of Goodman and Wallach [12] in constructing a certain family of linear operators (M ν,p ω ) ω∈C , with common domain H(ν, p). In this subsection we detail the salient properties of these operators; we omit the relevant proofs, which may be found in our sources, [5] and [32].…”

Section: 4)mentioning

confidence: 99%

Spectral large sieve inequalities for Hecke congruence subgroups ofSL(2,Z[i])

Watt¹

2014

Journal of Number Theory

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We prove, in respect of an arbitrary Hecke congruence subgroup Γ = Γ 0 (q 0 ) ≤ SL(2, Z[i]), some new upper bounds (or 'spectral large sieve inequalities') for sums involving Fourier coefficients of Γ-automorphic cusp forms on SL(2, C). The Fourier coefficients in question may arise from the Fourier expansion at any given cusp c of Γ (our results are not limited to the case c = ∞). For this reason, our proof is reliant upon an extension, to arbitrary cusps, of the spectral-Kloosterman sum formula for Γ\SL(2, C) obtained by Hristina Lokvenec-Guleska in her doctoral thesis (generalising the sum formulae of Roelof Bruggeman and Yoichi Motohashi for P SL(2, Z[i])\P SL(2, C) in several respects, though not as regards the choice of cusps). A proof of the required extension of the sum formula is given in an appendix.

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Whittaker vectors and conical vectors

Cited by 69 publications

References 19 publications

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas

Spectral large sieve inequalities for Hecke congruence subgroups ofSL(2,Z[i])

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