Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups

Adler, Jeffrey D.; DeBacker, Stephen

doi:10.1515/crll.2004.080

Cited by 32 publications

(47 citation statements)

References 23 publications

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“…Hence ker 1 (E, Z ( I γ )) is trivial. This and the inflationrestriction sequence show that ker 1 (F, Z ( I γ )) is the subset of locally trivial elements in H 1 ( E/F , Z ( I γ )), where we have written E/F for Gal(E/F). In particular,…”

Section: An Element Defined By a Semisimple γ ∈ G(f)mentioning

confidence: 80%

“…For the main term the estimate (12.21) shows that the contribution of the higher moments is negligible. It remains the two terms M (1) and M (2) as defined in (12.22).…”

Section: Resultsmentioning

confidence: 99%

“…The authors do not know how to do it without invoking the (conjectural) global L-parameter for π . The automorphic Sato-Tate conjecture is known in the limited cases of (the restriction of scalars of) GL 1 and GL 2 [6,7]. In an ideal world the conjecture should be closely related to Langlands functoriality.…”

Section: Sato-tate Theorem For Familiesmentioning

confidence: 99%

“…To our knowledge the Katz-Sarnak heuristics for such families are not treated in the literature, except in the case of G = G L (1) where harmonic families coincide with the geometric families treated by Katz-Sarnak (e.g. Dirichlet L-series with quadratic character are the geometric families of hyperelliptic curves in [59, §10]).…”

Section: Criterion 12 Let R : L G → Gl(d C) Be a Continuous L-homommentioning

confidence: 99%

“…We shall use this observation below when adding the contribution of D v (F k , ) which will then produce produce the integral ∞ −∞ (y)dy = (0). (1) and M (2) Recall that by assumption…”

Section: Sum Over Primesmentioning

confidence: 99%

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Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

2015

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We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let L G = G Gal(F/F) be the associated L-group and r : L G → GL(d, C) a continuous homomorphism which is irreducible and does not factor through Gal(F/F). The families under consideration consist of discrete automorphic representations of G(A F ) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1):75-102, 1997). As an application we study the distribution of the lowlying zeros of the associated family of L-functions L(s, π, r ), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r ∨ then the symmetry type is unitary. Otherwise there is a bilinear form on C d which realizes the isomorphism between r and r ∨ . If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

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Section: An Element Defined By a Semisimple γ ∈ G(f)mentioning

confidence: 80%

“…For the main term the estimate (12.21) shows that the contribution of the higher moments is negligible. It remains the two terms M (1) and M (2) as defined in (12.22).…”

Section: Resultsmentioning

confidence: 99%

Section: Sato-tate Theorem For Familiesmentioning

confidence: 99%

Section: Criterion 12 Let R : L G → Gl(d C) Be a Continuous L-homommentioning

confidence: 99%

Section: Sum Over Primesmentioning

confidence: 99%

See 3 more Smart Citations

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

2015

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Stable Distributions Supported on the Nilpotent Cone for the Group G 2

DeBacker

Kazhdan

Progress in Mathematics

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Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions

DeBacker

Kazhdan

2011

Transformation Groups

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The goal of Murnaghan-Kirillov theory is to associate to an irreducible smooth representation of a reductive p-adic group a family of regular semisimple orbital integrals in the Lie algebra with the following property: the character of π is given, on a well determined set, by an explicit combination of the Fourier transforms of these orbital integrals. Subject to certain restrictions, we adapt arguments of Waldspurger to show that, for depth-zero irreducible smooth supercuspidal representations, this problem may be reduced to a similar one for distributions associated to Lusztig functions.

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Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups

Cited by 32 publications

References 23 publications

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

Stable Distributions Supported on the Nilpotent Cone for the Group G 2

Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions

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