Some results in the theory of genuine representations of the metaplectic double cover of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="italic">GSp</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>over p-adic fields

Szpruch, Dani

doi:10.1016/j.jalgebra.2013.05.001

Cited by 7 publications

(13 citation statements)

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“…It can be easily shown that the irreducibility results for principal series representations of the metaplectic double cover of GSp 2n (F ) given in [23] and [24] also agree with the Knapp-Stein Dimension Theorem . The results in [23] and [24] are also unconditional since these are based on [22]. In the case at hand the Knapp-Stein Dimension Theorem is reduced to the following statement: This proposition is contained in [15], where the Knapp-Stein Dimension Theorem is proven for unitary parabolic induction from P to G and from P to G where G is a reductive group defined over F , P is a maximal parabolic subgroup of G, G is a central extension of G by µ n and P is the preimage of P in G. Proof.…”

Section: An Irreducibility Resultmentioning

confidence: 59%

Section: Letmentioning

confidence: 59%

“…This result was proved unconditionally in [9]. It can be easily shown that the irreducibility results for principal series representations of the metaplectic double cover of GSp 2n (F ) given in [23] and [24] also agree with the Knapp-Stein Dimension Theorem . The results in [23] and [24] are also unconditional since these are based on [22].…”

Section: Letmentioning

confidence: 63%

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Plancherel measures for coverings of p-adic SL2(F)

Goldberg

Szpruch

2016

Int. J. Number Theory

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In these notes we compute the Plancherel measures associated with genuine principal series representations of n-fold covers of p−adic SL 2 . Along the way we also compute a higher dimensional metaplectic analog of Shahidi local coefficients. Our method involves new functional equations utilizing the Tate γ-factor and a metaplectic counterpart. As an application we prove an irreducibility theorem.

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Section: An Irreducibility Resultmentioning

confidence: 59%

Section: Letmentioning

confidence: 59%

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Plancherel measures for coverings of p-adic SL2(F)

Goldberg

Szpruch

2016

Int. J. Number Theory

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“…Our argument in this paper is based on the relation between representation theory of GSp 2n (F ) and Sp 2n (F ) given in [21] along with the work of Bump, Friedberg and Hoffstein on the Spherical Whittaker function for Sp 2n (F ). It is the uniqueness of Whittaker model for Sp 2n (F ) which is ultimately responsible for our main result.…”

Section: Introductionmentioning

confidence: 99%

“…This condition is satisfied by any Levi subgroup of GSp 2n (F ), including GSp 2n (F ) itself, provided that n is odd and it is satisfied by T ′ 2n (F ), regardless of the parity of n. Let π be a genuine irreducible smooth admissible representation of M ′ (F ). It was shown in [21] that the restriction of π to M + (F ) is a multiplicity free direct sum of [F * : F * 2 ] summands. This is equivalent to the fact that M /M + (F ) acts freely and transitively on the set of irreducible M + (F ) modules that appears in π.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Genuine Spherical Whittaker Functions on

Szpruch¹

2015

Can. j. math.

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Abstract. Let F be a p-adic field of odd residual characteristic. Let GSp 2n (F) and Sp 2n (F) be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F. Let σ be a genuine, possibly reducible, unramified principal series representation of GSp 2n (F). In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to σ. For odd n, and generically for even n, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of Sp 2n (F). If n is odd then each element in the set has an equivariant property that generalizes the uniqueness result of Gelbart, Howe and Piatetski-Shapiro proven in [?]. Using this symmetric set, we construct a family of reducible genuine unramified principal series representations which have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for n even.

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Local Coefficients and Gamma Factors for Principal Series of Covering Groups

Gao¹,

Shahidi²,

Szpruch³

2023

Memoirs of the AMS

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We consider an n n -fold Brylinski–Deligne cover of a reductive group over a p p -adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of G L 2 GL_2 to S L 2 SL_2 . In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.

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Some results in the theory of genuine representations of the metaplectic double cover ofGSp2n(F)over p-adic fields

Cited by 7 publications

References 20 publications

Plancherel measures for coverings of p-adic SL2(F)

Plancherel measures for coverings of p-adic SL2(F)

Symmetric Genuine Spherical Whittaker Functions on

Local Coefficients and Gamma Factors for Principal Series of Covering Groups

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