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

Abstract

Abstract Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> , and $$\Gamma \subset H({{\mathbb {Q}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Γ</mml:mi><mml:mo>⊂</mml:mo><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Γ</mml:mi><mml:mo>\</mml:mo><mml:mi>G</mml:mi><mml:mo>/</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:math> to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Γ</mml:mi><mml:mo>\</mml:mo><mml:mi>G</mml:mi></mml:mrow></mml:math> 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.

Locations

• Mathematische Zeitschrift - View - PDF