Modular Deligne-Lusztig theory

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I study the representation theory of a certain class of finite groups : the finite reductive groups. Since the work of Steinberg, one knows how to construct such a group as the (possibly twisted) rational points of an algebraic reductive group defined over the algebraic closure of a finite field. For example, the group GL(n,q) is obtained by restricting the coefficients of any non-singular matrix to the finite field with q elements. From that geometrical point of view, two directions have been investigated : - how to deduce algebraic and structural properties from the theory of reductive groups over an algebraically closed field; - how to produce representations coming from actions of the finite group on a family of algebraic varieties. This last point has been initiated by Deligne and Lusztig and intensively studied in a series of papers by Lusztig.There are different notions of representations, depending on the coefficient ring. Any representation can be viewed as a lattice or a vector space, together with a linear action of the group. The advantage of the construction of Deligne and Lusztig is that there is a canonical way to linearize the geometric action of the group, with respect to a large family of rings, including the ring of l-adic integers Z_l, where l is a prime number different form the characteristic of the finite field. I am interested in the modular representation theory in non-defining characteristic, that is, the case where this ring is a finite extension of Z_l. In some way, it is the most sophisticated case, since the others are obtained by scalar extensions or reduction mod l. With the pioneer work of Bonnaf and Rouquier in this direction, there are now good evidences that the Deligne-Lusztig varieties should encode many parts of this theory, including blocks, projective modules, decomposition matrices. The main aspect of my work during this Fellowship will be to extract and decode this information. I will be particularly interested in Brou's conjecture and in the modular analogue of Lusztig's character sheaves.