Finiteness of ℚ-Fano Compactifications of Semisimple Groups with Kähler–Einstein Metrics

Let $G$ be a complex semisimple group. In this note, we give a method to classify $\mathbb Q$-Fano compactifications of $G$. We will prove that there are only finitely many $\mathbb Q$-Fano $G$-compactifications that admit (singular) Kähler–Einstein metrics. As an application, this improves a former result in [ 19].

On the holomorph of finite semisimple groups

Given a finite nonabelian semisimple group 𝐺, we describe those groups that have the same holomorph as 𝐺, that is, those regular subgroups N\simeq G of S(G), the group of permutations on the set 𝐺, such that N_{S(G)}(N)=N_{S(G)}(\rho(G)), where 𝜌 is the right regular representation of 𝐺.

Lower bounds for Maass forms on semisimple groups

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.