A shrinking target problem in homogeneous spaces of semisimple algebraic groups

Abstract

In this paper, we study a shrinking target problem with target at infinity in a homogeneous space of a semisimple algebraic group from the representation-theoretic point of view. Let $\rho:\mathbf G\to\mathbf{GL}(V)$ be an irreducible $\mathbb Q$-rational representation of a connected semisimple $\mathbb Q$-algebraic group $\mathbf G$ on a complex vector space $V$, $\{a_t\}_{t\in\mathbb R}$ a one-parameter subgroup in a $\mathbb Q$-split torus in $\mathbf G$ and $\psi:\mathbb R_+\to\mathbb R_+$ a positive function on $\mathbb R_+$. We define a subset $S_\rho(\psi)$ of $\psi$-Diophantine elements in $\mathbf G(\mathbb R)$ in terms of the representation $\rho$ and $\{a_t\}_{t\in\mathbb R}$, and prove formulas for the Hausdorff dimension of the complement of $S_{\rho}(\psi)$. We also discuss the connections of our results to Diophantine approximation on flag varieties and rational approximation to linear subspaces in Grassmann varieties.