Representation theory of algebraic groups, Hecke algebras and complex reflection groups

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(1) I have proved the Luszfig's conjecture concerning the characters of finite special linear group SLn(〓_q) over finite field 〓_q. This conjecture provides as a general algorithm of computing irreducible characters of reductive group G(〓_q) over finite field, which is stated as 「almost characters and characteristic functions of character sheaves coincides with up to scalar multiple」. This conjecture we proved by the head investigator in the case where the center of G is connected. SLn is an example of disconnected enter. We can determine the scalars also, and we obtain a complete algorithm of computing irreducible characters.(2) By the joint work with K.Sorlin, I have determined the deconpositon of the presentation G(〓_q^2) -with G(〓_q^2)/G(〓_q) for G=SLn.