Representation theory of algebraic groups, Hecke algebras and canpex refiecion groups

我们就以下三个主题进行了研究。关于有限可约群G(F_Q)的不可约特征标的Lusztig猜想所涉及的标量的确定，以及与计算不可约特征标的算法有关的具有某些良好性质的幂等元的确定。特别地，对于第二个问题，在G(F_Q)=SL_n(F_Q)，Sp_{2n}(F_Q)，SO_{2n+1}(F_Q)，SO{2n}(F_Q)的情况下，我们确定了一类好的幂等元。在经典群的情况下，当特征等于2时，这一结果也成立。利用这一结果，建立了计算特征为偶数的经典群的格林函数的算法，该算法以前存在一定的歧义性。有限约化群的格林函数是Q中的多项式.我们证明了用单位根代替Q得到格林函数值的一个公式.这种公式在…的情况下是已知的其中G(F_Q)=GL_n(F_Q)。在本研究中，我们利用Lusztig的Springer表示的归纳定理，在一般情况下证明了这一点。我们研究了与复反射群G(r，1，n)相关的Hecke代数的Ariki-Koike代数的模表示理论，以及与Ariki-Koike代数相关的分圆Q-Schur代数。我们构造了分圆Q-Schur代数及其商的各种子代数，并通过比较它们的分解数证明了某类分解数的乘积公式；另一方面，利用Yvonne猜想，猜想分圆Q-Schur代数的分解数是由高阶Fock空间的标准基和标准基之间的转换矩阵得到的。在此猜想的基础上，我们证明了Fock空间的一个乘积公式，该公式猜想地对应于原乘积公式Less

We have studied on the following three themes.1. Determination of scalars involved in Lusztig's conjecture concerning irreducible characters of finite reductive groups_G (F_q), and the determination of unipotent elements with certain good properties related to the algorithm of computing irreducible characters. In particular concerning the second problem, in the case where G (F_q) = SL_n(F_q), Sp_{2n}(F_q), SO_{2n+1} (F_q), SO {2n} (F_q) we have determined a class of good unipotent elements. In the case of Classical groups, this results holds also for the case where the characteristic is equal to 2. By this result, an algorithm computing Green functions of classical groups of even characteristic was established, which had involved certain ambiguity before.2. It is known that Green functions of finite reductive groups is a polynomial in q. We have proved a formula for the values of Green functions obtained by substituting a root of unity for q. This type of formula was known in the case … More where G(F_q) = GL_n(F_q) by a combinatorial method. In this study, we have proved it in the general case by making use of induction theorem for Springer representations due to Lusztig.3. We have studied the modular representation theory of Ariki-Koike algebras which are Hecke algebras associated to the complex reflection groups G(r,1,n), and the cyclotomic q-Schur algebras related to the Ariki-Koike algebras. We have constructed various subalgebras of cyclotomic q-Schur algebras and their quotients, and proved a product formulas for certain type of decomposition numbers, by comparing the decomposition numbers of them.On the other hand, thanks to Yvonne's conjecture, it is conjectured that the decomposition numbers of cyclotomic q-Schur algebras are obtained from the transition matrix between standard bases and canonical bases of higher level Fock space. Based on this conjecture", we have proved a product formula for the Fock space which conjecturally corresponds to the original product formula Less