Expanding Cone and Applications to Homogeneous Dynamics

Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

Lifting of Outer Actions of Groups II

In this paper we continue to study the lifting of outer actions of groups and introduce the concept of equivalence of two lifting homomorphisms, maximal lifting homomorphisms and maximal split extensions. Some criteria are obtained.

On a Certain Residual Spectrum of Sp8

Let G = Sp2n be the symplectic group defined over a number field F. Let 𝔸 be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of G(𝔸) acting on the Hilbert space L2 (G(F) \ G(𝔸)). Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: , where (M, π) is a Levi subgroup with a cuspidal automorphic representation π taken modulo conjugacy. (Here we normalize π so that the action of the maximal split torus in the center of G at the archimedean places is trivial.) and is a space of residues of Eisenstein series associated to (M, π). In this paper, we will completely determine the space , when M ≃ GL2 × GL2. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than GLn.