Let \(G\) be a finite group and \(\langle F\rangle\) a cyclic group acting on \(G\). Let \(j\) denote the injective map \(g\mapsto gF\) from the set \(G\) into the set \(G\rtimes\langle F\rangle\). One calls \(F\)-class functions on \(G\) the functions of type \(f\circ j\) where \(f\) is a class function on \(G\rtimes\langle F\rangle\). The Shintani descent for a connected algebraic reductive group \(\mathbf G\) defined over an algebraic closure \(\overline\mathbb{F}_q\) of a finite field \(\mathbb{F}_q\) of \(q\) elements, equipped with two rational structures given by Frobenius endomorphisms \(F\) and \(F'\), provides an isometry from the space of \(F'\)-class functions on \({\mathbf G}^F\) (the finite group of the \(F\)-invariant points of \(\mathbf G\)) and the space of \(F\)-class functions on \({\mathbf G}^{F'}\). Two cases are especially interesting: when \(F'\) is a power of \(F\), and when \(F\) and \(F'\) are equal up to an automorphism of \(\mathbf G\). In the paper under review the author reduces the first case to the second one by using restriction of scalars. Restriction of scalars is a functor which is defined by using a non-connected algebraic group which is a wreath product of a finite number of copies of the group \(\mathbf G\) by a finite cyclic group which permutes circularly these copies.
Let \(i\) denote the mapping which is induced on the functions on a group by passing to the inverse. In the first part, a result of commutation of the Shintani descent \(\text{Sh}\) (resp. \(i\circ\text{Sh}\)) with Deligne-Lusztig induction (resp. Deligne-Lusztig restriction) is proved by using results of F. Digne, J. Michel [Ann. Sci. Ec. Norm. Supér., IV. Sér. 27, No. 3, 345-406 (1994; Zbl 0846.20040)].
Let \(m\) be a positive integer. Assuming that the characteristic of \(\mathbb{F}_q\) is good for \(\mathbf G\) (in order to have connected centralizers of unipotent elements) and that it does not divide \(m\) (in order that the quasi-central automorphism \(\tau\) of the group \({\mathbf G}^m\) defined by \(\tau(g_1,g_2,\ldots,g_{m-1},g_m):=(g_2,g_3,\ldots,g_m,g_1)\) is semisimple), it is proved that the Deligne-Lusztig virtual characters are their own Shintani descendent from \(F^m\) to \(F\). The last section extends this to the Shintani descent from \(F^r\) to \(F^s\). These are strong generalizations of previous results by A. Gyoja [Osaka J. Math. 16, 1-30 (1979; Zbl 0416.20032)].