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

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.

Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρηlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

Boundary Cohomology

This chapter discusses some relevant details of the cohomology of the boundary of the Borel–Serre compactification of the locally symmetric space SGKf. It first illustrates a spectral sequence converging to boundary cohomology. The chapter then turns to the cohomology of ∂PSG to better understand the cohomology of the boundary. Finally, the chapter describes the contribution of the discrete but noncuspidal spectrum to cohomology. It formulates the consequences of the description of the discrete spectrum in Mœglin–Waldspurger for the square integrable cohomology. In a sense, the chapter makes their results more explicit. It works at a transcendental level: the coefficient systems are ℂ-vector spaces.

The Cohomology of GLn

This chapter addresses the cohomology of GLn. It first discusses the adèlic locally symmetric space. Next, the chapter turns to the highest weight modules 𝓜λ‎ and the sheaves 𝓜̃λ‎. From there, the chapter illustrates the cohomology of the sheaves 𝓜̃λ‎. A fundamental problem at the heart of this monograph is to understand the arithmetic information contained in the sheaf-theoretically defined cohomology groups H ⦁(SGKf,𝓜̃λ‎,E). Finally, the chapter briefly discusses how to refine many of the foregoing considerations to talk about integral sheaves and their cohomology and why this is interesting. This aspect is not so relevant for the results in this book, but it will become relevant when applying certain refinements of the results to arithmetic questions.

Higher rank rigidity for Berwald spaces

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded non-positive flag curvature with rank at least two whose universal cover is irreducible is a locally symmetric space or a locally Minkowski space.

Hyperbolic rank rigidity for manifolds of -pinched negative curvature

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift

Given a measureon a locally symmetric spaceobtained as a weak-* limit of probability measures associated with eigenfunctions of the ring of invariant differential operators, we construct a measureon the homogeneous spaceX= Γ\Gthat liftsand is invariant by a connected subgroupA1⊂Aof positive dimension, whereG=NAKis an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, thenis also the limit of measures associated with Hecke eigenfunctions on X. This generalizes results of the author with A.Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

Teichmüller space of negatively curved metrics on Gromov–Thurston manifolds is not contractible

In this paper we prove that for all n = 4k - 2, k ≥ 2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that [Formula: see text] is nontrivial. [Formula: see text] denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity. Gromov–Thurston branched cover manifolds provide examples of negatively curved manifolds that do not have the homotopy type of a locally symmetric space. These manifolds will be used in this paper to prove the above stated result.

Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

The squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.