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Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, thealgebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stableBorel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In thispaper, we completely determine the composition factors of the induced module$Ind_{B}^{G}{tr}=\Bbbk{G}\otimes_{\Bbbk{\bf B}}$ tr (here $\Bbbk{H}$ is thegroup algebra of the group ${H}$, and tr is the trivial $B$-module). Inparticular, we find a new family of infinite dimensional irreducible abstractrepresentations of $G$.

The Permutation Module on Flag Varieties in Cross Characteristic