Abstract
Let H be a semisimple algebraic group, K a maximal compact subgroup of G:=H({{mathbb {R}}}), and Gamma subset H({{mathbb {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 to corresponding bounds on the arithmetic quotient Gamma backslash 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.
Highlights
Let M be a closed1 Riemannian manifold M of dimension d and P0 : C∞(M) → L2(M) an elliptic classical pseudodifferential operator on M of degree m, where C∞(M) denotes the space of smooth functions on M and L2(M) the space of square-integrable functions on M
The goal of this paper is to extend the spherical subconvex bounds (1.5) and (1.6) to non-spherical situations, that is, to non-trivial K -types in the Peter–Weyl decomposition of L2( \G) for Hecke–Maass forms of rank 1 and a large class of compact arithmetic quotients \G, sharpening the bounds (1.1) and (1.2) in case that the eigenfunctions φ j are Hecke–Maass forms
As our first main result, we extend the bound (1.5) to automorphic forms on G of arbitrary
Summary
Let M be a closed Riemannian manifold M of dimension d and P0 : C∞(M) → L2(M) an elliptic classical pseudodifferential operator on M of degree m, where C∞(M) denotes the space of smooth functions on M and L2(M) the space of square-integrable functions on M. With the identification K S1 [0, 2π ), any K -type σl ∈ K can be realized as the character σl (θ ) = eilθ , θ ∈ [0, 2π ), l ∈ Z, and we denote by Lσ2l,χ ( \G) the σl -isotypic component of Lχ2 ( \G) It is shown in Theorem 5.5 that for any orthonormal basis φ j j≥0 of L2( χ \G) consisting of Hecke–Maass forms (of rank 1) with Beltrami–Laplace eigenvalues. 3 we give a description of the asymptotic behaviour of spectral function of an elliptic operator by means of Fourier integral operators, and explain how convex bounds can be deduced from this in equivariant and non-equivariant situations Based on these results, we derive spectral asymptotics for kernels of Hecke operators in Sect. Throughout the paper, N := {0, 1, 2, 3, . . .} will denote the set of natural numbers, while N∗ := {1, 2, 3, . . .}
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