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Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $\sigma$ on ${\mathfrak g}$. Let ${\mathbf B}^{\sigma}$ be the set of $\sigma$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^{\sigma} \simeq \underline{\mathbf B}$ based on his geometric construction of canonical bases. In this paper, we prove (the signed bases version of) this fact, in the case where ${\mathfrak g}$ is finite or affine type, in an elementary way, in the sense that we don't appeal to the geometric theory of canonical bases nor Kashiwara's theory of crystal bases. We also discuss the correspondence between PBW-bases, by using a new type of PBW-bases of ${\mathbf U}_q^-$ obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima., Comment: 43 pages

Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall-Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture., Comment: 22 pages

In our previous paper, Green functions associated to complex reflection groups G(e,1,n) were discussed. It involved a combinatorial approach to the Green functions of classical groups of type B_n or C_n. In this paper, we introduce Green functions associated to complex reflection groups G(e,p,n). We define Schur functions, Hall-Littlewood functions associated to G(e,p,n), and show that our Green functions are obtained as a transition matrix between those two kinds of symmetric functions. As a special case, it gives a combinatorial treatmnet of Green functions of type D_n., Comment: 33pages

Green functions of classical groups are determined by the data from Weyl groups and by certain combinatorial objects called symbols. Generalizing this, we define Green functions associated to complex reflection groups G(e, 1, n) and study their combinatorial properties. We construct Hall–Littlewood functions and Schur functions in our scheme and show that such Green functions are obtained as a transition matrix between those two symmetric functions, as in the case of GLn.

Let g = glm1 ⊕ ··· ⊕ glmr be a Levi subalgebra of glm, with m = ∑ri = 1mi, and V the natural representation of the quantum group Uq(g). We construct a representation of the Ariki–Koike algebra Hn, r on the n-fold tensor space of V, commuting with the action of Uq(g), and prove the Schur–Weyl reciprocity for the actions of Uq(g) and Hn, r on it.

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function l(W) associated to a given generator set S is completely described by the function n(W), introduced in [BM], associated to the root system of W. In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space HW in terms of Demazure operators, and show that HW is isomorphic to the coinvariant algebra SW, which enables us to define a homogeneous basis on SW parametrized by w ϵ W.

In this paper, the Shintani descent of irreducible characters of finite special linear groups is discussed. We parametrize irreducible characters of SL n ( F q ) by making use of (modified) generalized Gelfand–Graev characters. Shintani descent of irreducible characters of SL n ( F q m ) is determined by investigating the Shintani descent of generalized Gelfand–Graev characters. Our results hold for arbitrary p and q .