Generic character sheaves on groups over $\kk[\e]/(\e^r)$

Abstract: Let $G$ be a connected reductive group over $\kk$, an algebraic closure of a finite field. For an integer $r\ge 1$ let $G_r=G(\kk[\e]/(\e^r))$ viewed as an algebraic group of dimension $r\dim G$ over $\kk$. We show that the character of the generic principal series representation of $G_r(F_q)$ can be realized by a simple perverse sheaf on $G_r$ provided that $r=2$ or $r=4$ and we give a strategy to prove the same statement for any even $r$. (The case where $r=1$ is already known.)