The equivariant spectral function of an invariant elliptic operator. L -bounds, caustics, and concentration of eigenfunctions

Ramacher, Pablo

doi:10.1016/j.matpur.2017.10.005

Cited by 6 publications

(5 citation statements)

References 26 publications

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“…By assumption, the cospheres S * x M are strictly convex, so that for small |t|, the hypersurfaces R,t ι,x will be strictly convex, too. Applying [42,Lemma 3.5] to the integrals I ι (μ, R, t, x, y) with κ ι (x)−κ ι (y) κ ι (x)−κ ι (y) , ω as phase function and ν := μ κ ι (x) − κ ι (y) as asymptotic parameter yields for anyÑ ∈ N the expansion…”

Section: The Spectral Function and Convex Bounds For Eigenfunctionsmentioning

confidence: 99%

“…Suppose further that P commutes with the family of differential operators generated by the action of K , so that the eigenfunctions φ j can be chosen to be compatible with the Peter-Weyl decomposition of L 2 (M) into σ -isotypic components L 2 σ (M), where σ ∈ K . It was then shown in [42,43] that the equivariant convex bound…”

Section: Introductionmentioning

confidence: 99%

“…), it was shown in[42, Eq. (2.8)] that by approximating U (t) in (3.3) by the Fourier integral operatorŪ (t) one obtains a description for the kernel of s μ • σ as the double oscillatory integral…”

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confidence: 99%

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Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Ramacher

Wakatsuki

2020

Math. Z.

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Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

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Section: The Spectral Function and Convex Bounds For Eigenfunctionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Ramacher

Wakatsuki

2020

Math. Z.

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“…Remark 3. Less precise results but in a more general setting were obtained by Ramacher [Ram18] using operator theoretical methods. Combined with an argument of Marshall [Mar14], these were applied by Ramacher-Wakatsuki [RW21] to the sup-norm problem with K-types.…”

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confidence: 99%

Beyond the spherical sup-norm problem

Blomer¹,

Harcos²,

Maga³

et al. 2021

Preprint

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We solve the sup-norm problem for non-spherical Maaß forms on an arithmetic quotient of G = SL2(C) with maximal compact K = SU2(C) when the dimension of the associated K-type gets large. Our results cover the case of vector-valued Maaß forms as well as all the individual scalar-valued Maaß forms of the Wigner basis. They establish the first subconvex bounds for the sup-norm problem in the K-aspect in a non-abelian situation and yield sub-Weyl exponents in some cases. On the way, we develop theory of independent interest for the group G, including localization estimates for generalized spherical functions of high K-type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions.

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“…Note that the additional assumption made in [41, Section 3.2] and[40] that K acts effectively on M is unnecessary.…”

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confidence: 99%

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

Ramacher¹,

Wakatsuki²

2020

Preprint

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In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. In particular, we treat not only spherical, but also non-spherical Hecke-Maass forms, including remainder estimates. Our asymptotic formulas are available for arbitrary simple connected algebraic groups over number fields with cocompact arithmetic subgroups. In preceding studies on distributions of Hecke eigenvalues the trace formula played an important role. Since the latter is currently not available for our families, we rely on Fourier integral operator methods instead. Contents 1. Introduction 1 2. Asymptotics for Hecke eigenvalues and low-lying zeros for PGL(n, R) 2 3. Spectral asymptotics for kernels of Hecke operators 6 4. Non-equivariant asymptotics for Hecke eigenvalues and Sato-Tate equidistribution 13 5. Equivariant asymptotics for Hecke eigenvalues and Sato-Tate equidistribution 19 6. Examples 27 References 28

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The equivariant spectral function of an invariant elliptic operator. L -bounds, caustics, and concentration of eigenfunctions

Cited by 6 publications

References 26 publications

Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Beyond the spherical sup-norm problem

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

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