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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

##### MSC:
 20G05 Representation theory for linear algebraic groups