The Multiplicity One Theorem for GL n

Shalika, Joseph

doi:10.2307/1971071

Cited by 359 publications

(261 citation statements)

References 3 publications

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“…When G = GL n , every cuspidal representation is globally generic [57] and therefore our conjecture agrees with the general belief of the validity of the Ramanujan conjecture for GL n .…”

Section: Ramanujan Conjecture For Quasisplit Groupssupporting

confidence: 88%

Infinite Dimensional Groups and Automorphic L-Functions

Shahidi¹

2005

Pure and Applied Mathematics Quarterly

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“…When G = GL n , every cuspidal representation is globally generic [57] and therefore our conjecture agrees with the general belief of the validity of the Ramanujan conjecture for GL n .…”

Section: Ramanujan Conjecture For Quasisplit Groupssupporting

confidence: 88%

Infinite Dimensional Groups and Automorphic L-Functions

Shahidi¹

2005

Pure and Applied Mathematics Quarterly

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“…Indeed, since the spherical Hecke algebra of compactly supported K -biinvariant functions is commutative, K is a Gelfand subgroup of G(F), so the functional L z is determined (up to scalar) by the fact that z (π(k)φ) = z (φ) for all k ∈ K . For the Whittaker functional, the corresponding uniqueness result was obtained by Gelfand and Graev, by Piatetski-Shapiro and by Shalika [1974]. The Whittaker integral, like the intertwining integral, is convergent for z in an open set, and has analytic continuation to all z.…”

Section: G(f) Hmentioning

confidence: 73%

Unique functionals and representations of Hecke algebras

Brubaker¹,

Bump

Friedberg

2012

Pacific J. Math.

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To the memory of Jonathan David Rogawski Rogawski (1985) used the affine Hecke algebra to model the intertwining operators of unramified principal series representations of p-adic groups. On the other hand, a representation of this Hecke algebra in which the standard generators act by Demazure-Lusztig operators was introduced by Lusztig (1989) and applied by Kazhdan and Lusztig (1987) to prove the Deligne-Langlands conjecture. These operators appear in various other contexts. Ion (2006) used them to express matrix coefficients of principal series representations in terms of nonsymmetric Macdonald polynomials, while Brubaker, Bump and Licata (2011) found essentially the same operators underlying recursive relationships for Whittaker functions. Here we explain the role of unique functionals and Hecke algebras in these contexts and revisit the results of Ion from the point of view of Brubaker et al.

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“…In the 1970s Piatetski-Shapiro and Shalika [131], [169] independently developed their "Whittaker" expansions on GL(n) in order to generalize the expansion (7.18) of Jacquet-Langlands. The Whittaker function on GL(n, A Q ) is given by the integral (7.27)…”

Section: Jacquet-piatetski-shapiro-shalika (1979)mentioning

confidence: 99%

Riemann’s zeta function and beyond

Gelbart

Miller

2003

Bull. Amer. Math. Soc.

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Abstract. In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of L-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of L-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of L-functions.

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The Multiplicity One Theorem for GL n

Cited by 359 publications

References 3 publications

Infinite Dimensional Groups and Automorphic L-Functions

Infinite Dimensional Groups and Automorphic L-Functions

Unique functionals and representations of Hecke algebras

Riemann’s zeta function and beyond

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