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

Abstract

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.

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

“…They also derived corresponding subconvex L ∞ -bounds based on the presence of an additional family of commuting operators given by the Hecke algebra [16,20]. In a forthcoming article [24] we shall extend their results to compact arithmetic quotients of semisimple algebraic groups relying on the asymptotic description of the integrals (1.9) given in Theorems 3.3 and 3.4. For a general overview on eigenfunctions on Riemannian manifolds, we refer to the survey articles [38,37].…”

Addendum to 'The equivariant spectral function of an invariant elliptic operator'

“…For their proof, a careful examination of the remainder in the stationary phase expansion of the relevant spectral kernels is necessary. These bounds are crucial for deriving hybrid subconvex bounds for Hecke-Maass forms on compact arithimetic quotients of semisimple Lie groups in the eigenvalue and isotypic aspect [6].…”

Beyond the spherical sup-norm problem

“…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. For compact arithmetic quotients of SL 2 (C), and for φ ∈ V as before, [RW21, Th.…”