Deligne—Lusztig induction and almost characters

One of the few tools one has when studying the characters of a finite group $G$ is given by induction $\mathrm{Ind}_H^G$ from a subgroup $H \leqslant G$. If $G$ is a finite reductive group and $L \leqslant G$ is a Levi subgroup then, in 1976, Deligne and Lusztig have defined a geometric analogue of this construction, which is typically referred to as Deligne—Lusztig induction $R_L^G$. A fundamental problem is to understand the decomposition of $R_L^G(\chi)$ into its irreducible constituents for any irreducible character $\chi$ of $L$. If $L$ is a torus then this problem was completely settled by Lusztig in the mid to late 80s. Using Shintani descent Asai solved this problem when $\chi$ is a unipotent character of $L$. Assuming $G$ comes from an algebraic group with connected centre then Shoji obtained generalisations of Asai’s results for arbitrary characters. In this talk we describe a new approach to this problem which relies on the validity of the Mackey formula

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