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 𝜌 : 𝐆 → 𝐆 𝐋 ⁡ ( 𝑉 ) be an irreducible ℚ -rational representation of a connected semisimple ℚ -algebraic group G on a complex vector space V , { 𝑎 𝑡 } 𝑡 ∈ ℝ a one-parameter subgroup in a ℚ -split torus in G and 𝜓 : ℝ + → ℝ + a positive function on ℝ + . We define a subset 𝑆 𝜌 ⁡ ( 𝜓 ) of ψ -Diophantine elements in 𝐆 ⁡ ( ℝ ) in terms of the representation ρ and { 𝑎 𝑡 } 𝑡 ∈ ℝ , and prove formulas for the Hausdorff dimension of the complement of 𝑆 𝜌 ⁡ ( 𝜓 ) . We also discuss the connections of our results to Diophantine approximation on flag varieties and rational approximation to linear subspaces in Grassmann varieties.