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Foncteur de Lusztig et fonctions de Green généralisées. (French) Zbl 0541.20026
Let \({\mathcal G}\) be a connected reductive algebraic group defined over \(F_ q\) and let G be the finite group of \(F_ q\)-rational points of \({\mathcal G}\). A regular subgroup \({\mathcal L}\) of \({\mathcal G}\) is an \(F_ q\)- rational Levi subgroup of a parabolic subgroup \({\mathcal P}\), which itself might not be \(F_ q\)-rational. Then the subgroup L of \(F_ q\)-rational points of \({\mathcal L}\) is called a regular subgroup of G. G. Lusztig [Invent. Math. 34, 201-213 (1976; Zbl 0371.20039)] defined a map \(R^ G_ L\) from the Grothendieck group \({\mathcal R}(L)\) of \(\bar Q_{\ell}\)- representations of L into \({\mathcal R}(G)\), which reduces to the Deligne- Lusztig construction \(R^ G_ T\) when \(L=T\) is a torus and which reduces to inflating and inducing from P when \({\mathcal P}\) itself if \(F_ q\)-rational. In this paper the authors state some formal properties of the map \(R^ G_ L\). If \(\chi\) is a character of L, they state a character formula (Theorem 2.2) for the virtual character \(R^ G_ L(\chi)\) of G, which involves a ”Green function” \(Q^ G_ L:G\times L\to \bar Q_{\ell}\) which has its support on \(G_ u\times L_ u\), \(G_ u\) being the set of unipotent elements of G, and is analogous to a formula of P. Deligne and G. Lusztig given in [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] for the maps \(R^ G_ T\). If \(^*R^ G_ L:{\mathcal R}(G)\to {\mathcal R}(L)\) is the adjoint map to \(R^ G_ L\), there is a similar formula for \({}^*R^ G_ L\). Several consequences are given, one of them being a generalization of a ”Curtis-type” formula used by P. Fong and B. Srinivasan [Invent. Math. 69, 109-153 (1982; Zbl 0507.20007)] in finding the blocks of general linear groups. The authors also indicate a simple computation for the dimension of \(R^ G_ L(\chi)\) in terms of dim \(\chi\), assuming the result for the map \(R^ G_ T\) (which has been proved by Deligne- Lusztig in loc. cit.). Various other properties of the map \(R^ G_ L\) are stated, including the following: \(R^ G_ L\) commutes with the ”Shintani map” on the space of class functions of G defined in an earlier paper of the authors [C. R. Acad. Sci., Paris, Sér. A 291, 571-574 and 651-653 (1980; Zbl 0456.20021 and Zbl 0456.20020)] provided the characteristic of \(F_ q\) is ”good”.
Reviewer: B.Srinivasan

20G05 Representation theory for linear algebraic groups