The Tate conjecture over finite fields for projective schemes related to Coxeter orbits

Abstract

Let $G$ be a simple algebraic group, defined over a finite field $\gF_q$, with Frobenius map $F$. Let $X^{\bul}_f$ be the Hansen-Demazure-Deligne-Lusztig compactification of the Deligne-Lusztig variety $X_f$ of $G$ associated with a Coxeter element in the Weyl group $W_G$ of $G$, and let $X^{\bul}_{f,0}$ be the $\gF_q \delta$-structure on $X^{\bul}_f$ over the finite extension $\gF_q \delta$ of $\gF_q$ determined by $F^{\delta} : X^{\bul}_f \to X^{\bul}_f$, where $\delta$ is the smallest positive integer such that $F^{\delta}$ is the identity map on $W_G$. We shall give an affirmative answer to the Tate conjecture over finite fields for algebraic cycles on $X^{\bul}_{f,0}$ and related projective schemes.