= Deligne-Lusztig induction
{c}
{title2=$R_T^G(\theta)$}
For a <rational maximal torus> $T$, choose a <Borel subgroup> containing it with <unipotent radical> $U$. The alternating compactly supported <étale cohomology> of $\mathcal L_F^{-1}(U)$, in the appropriate $T^F$-isotypic component, defines the <virtual character> of $G^F$ $R_T^G(\theta)$. It is independent of the chosen Borel. When the <Borel subgroup> is rational, this is ordinary induction from its rational points after <inflation of a group representation> from $T^F$.
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