Fourier coefficients attached to small automorphic representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="normal">SL</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

Ahlén, Olof; Gustafsson, Henrik P. A.; Kleinschmidt, Axel; Liu, Baiying; Persson, Daniel

doi:10.1016/j.jnt.2018.03.022

Cited by 6 publications

(7 citation statements)

References 40 publications

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“…Over F = R, the minimal orbit in U (2, 1) is Rdistinguished but not admissible. In general, the R-distinguished orbits for semi-simple groups are classified in [51, (under the name compact orbits), and comparing this classification with the classification of admissible orbits given in [50,Theorem 3] for classical groups, and [47,48] for exceptional groups, we see that for the groups (6.3) SU(p, q)(with p, q 1), EII, EV, EVI, EVIII, EIX there exist R-distinguished non-admissible orbits (1) . On the other hand, for other real simple groups, all R-distinguished orbits are admissible.…”

Section: Proof Of Theorem 13mentioning

confidence: 92%

“…Then the first quasicritical value of t is t = 4/3. We have S 4/3 = diag(1, −1, 5, 3, 4 1 3 , 2 1 3 ). Then E 14 ∈ g…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…Also, for several exceptional groups, some split and some non-split, there are special non-admissible orbits and admissible nonspecial orbits (see [47,48]). (1) For SU(m, n), the R-distinguished orbits are the ones described by partitions in which all rows of the same size have also the same signs, while admissible orbits are described in Theorem 6.10 (ii) below.…”

Section: Admissible Quasi-admissible and Special Orbitsmentioning

confidence: 99%

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Whittaker supports for representations of reductive groups

Gomez¹,

Gourevitch²,

Sahi³

2021

Annales De l'Institut Fourier

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Section: Proof Of Theorem 13mentioning

confidence: 92%

“…Then the first quasicritical value of t is t = 4/3. We have S 4/3 = diag(1, −1, 5, 3, 4 1 3 , 2 1 3 ). Then E 14 ∈ g…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Section: Admissible Quasi-admissible and Special Orbitsmentioning

confidence: 99%

See 1 more Smart Citation

Whittaker supports for representations of reductive groups

Gomez¹,

Gourevitch²,

Sahi³

2021

Annales De l'Institut Fourier

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“…Definition 2.4. 1 We define a partial order on nilpotent orbits in g * = g * (K) to be the transitive closure of the following relation R:…”

Section: Order On Nilpotent Orbits and Whittaker Supportmentioning

confidence: 99%

A reduction principle for Fourier coefficients of automorphic forms

et al. 2021

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We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

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“…In our work in progress [GGKPS] we aim to express any minimal next-to-minimal automorphic form for these groups through its Whittaker-Fourier coefficients, i.e. period integrals over the nilradical of a Borel subgroup of G against a character of this subgroup, following [MS12,AGKLP]. This is important since the latter integral is Eulerian.…”

mentioning

confidence: 99%

Generalized and degenerate Whittaker quotients and Fourier coefficients

Gourevitch

Sahi²

2019

Representations of Reductive Groups

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The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In order to encompass other representations, one attaches a degenerate (or a generalized) Whittaker model WO, or a Fourier coefficient in the global case, to any nilpotent orbit.In this note we survey some classical and some recent work in this direction -for Archimedean, p-adic and global fields. The main results concern the existence of models. For a representation π, call the set of maximal orbits O with WO that includes π the Whittaker support of π. The two main questions discussed in this note are:(1) What kind of orbits can appear in the Whittaker support of a representation?(2) How does the Whittaker support of a given representation π relate to other invariants of π, such as its wave-front set?

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Fourier coefficients attached to small automorphic representations of SLn(A)

Cited by 6 publications

References 40 publications

Whittaker supports for representations of reductive groups

Whittaker supports for representations of reductive groups

A reduction principle for Fourier coefficients of automorphic forms

Generalized and degenerate Whittaker quotients and Fourier coefficients

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