A Study on the zeta functions attached with the irreducible representations of finite reductive groups

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1. Zeta functions and functional equations associated with them for representations of a finite group G were first discussed by T.A.Springer (1971) and I.G.Macdonald (1985). We showed first that these functional equations also hold for the irreducible representations of Hecke algebra H which is an endomorphism algebra of an induced representation with multiplicity free.Then we studied the explicit examples and applications ((1)(2) joint work with C.W.Curtis). The results are as follows :(1) It can be shown that if G is a finite reductive group and if His an endomorphism algebra of Gelfand-Graev representation, the epsilon factor which appears in the functional equation is exactly the Gauss sum studied by Saito-Shinoda. Using this fact we obtained an explicit expression of the Fourier transform of e, which is the identity of Hin the case of general linear groups.(2) We showed that in the case of GL(n,q), values of irreducible representations of H on a standard basis element, which corresponds to a Coxeter element, become generalized Kloosterman sums. This implies that there should exist a close relation a between Davenport-Hasse type equations of Kloosterman sums and the norm maps of Hecke algebras.(3) In case of GL(4,q) we obtained almost all values of irreducibles representations of H on standard basis elements. They can be called as Kloosterman sums of higher degree.2. We also studied this project from relating fields.(1) Gomi and Shinoda generalized a result of J.McKay (1999) on coinvariant algebras of finite linear groups (joint with I.Nakamura).(2) Yokonuma studied discrete sets and associated dynamical systems in view of symmetry with a non-commutative setting. Nakashima and Koga studied from the view of quantum groups ; particularly Nakashima studied geometric crystals on Schubert varieties.(3) Wada, Tsunogai and Tsuzuki respectively studied this project in view from the number theory.