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Lusztig's algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable., 25 pages, the final version, to appear in Nagoya Math. J

Let $G = GL_N$ over an algebraically closed field of odd characteristic, and $\theta$ an involutive automorphism on $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1} \}$ is regarded as a symmetric space $G/G^{\theta}$. Let $G^{\iota\theta}_{uni}$ be the set of unipotent elements in $G^{\iota\theta}$. $H$ acts on $G^{\iota\theta}_{uni}$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G^{\iota\theta}_{uni}$ and irreducible representations of various symmetric groups., Comment: 76 pages. Final version, to appear in Tokyo J. of Mathematics

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $\lambda, \mu$ of $r$-partitions and a sign $+, -$. It is expected that there exists a close connection between those Kostka functions and the intersection cohomology associated to the enhanced variety $X$ of level $r$. In this paper, we study combinatorial properties of Kostka functions by making use of the geometry of $X$. In particular, we show that if $\mu$ is of the form $\mu = (-,\dots, -, \xi)$ and $\lambda$ is arbitrary, $K^-_{\lambda, \mu}(t)$ has a Lascoux-Sch\"utzenberger type combinatorial description., Comment: 21 pages

Let V be a 2n dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V) and H = Sp_{2n} be the symplectic group contained in G. We call X = G/H \times V the exotic symmetric space, since its "unipotent" part is isomorphic to Kato's exotic nilcone. Let W be the Weyl group of type C_n. Kato established the Springer correspondence between irreducible characters of W and H-orbits in the exotic nilcone, based on the Ginzburg theory of affine Hecke algebras. In this paper, we develpoe a theory of character sheaves on X, and give an alternate proof for the Springer correspondence based on the theroy of character sheaves., 56 pages, the proof of Proposition 4.8 revised, and typos corrected. The results unchanged

In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula., 24 pages

Let G be a simple algebraic group over C with the Weyl group W. For a unipotent element u of G, let B_u be the variety of Borel subgroups of G containing u. Let L be a Levi subgroup of a parabolic subgroup of G with the Weyl subgroup W_L. Assume that u is in L and let B_u^L be the corresponding variety for L. The induction theorem of Springer representations describes the W-module structure of H*(B_u) in terms of the W_L-module structure of H*(B_u^L). In this paper, we prove a certain refinement of the induction theorem by considering the action of a cyclic group of order e on H^*(B_u). As a corollary, we obtain a description of the values of Green functions at e-th root of unity., 18 pages. In this revised version, some errors are corrected. In particular some restrictions are added to the main results

Hall-Littlewood functions and Green functions associated to complex reflection groups $W = G(r, 1, n)$ were constructed in [S1] by means of symbols, which are a generalization of partitions. In this paper, we consider such functions in the case where the symbols are of very special type, called "limit symbols". The situation becomes simple, and is close to the case of symmetric groups when the symbols tends to the "limit". In the case where $W$ is a Weyl group of type $B_n$, we give a closed formula for Hall-Littlewood functions, and verify some of the conjectures stated in [S1] for the case of Green functions attached to limit symbols.

Let U q ( gl m ⊕ p ) be the quantized universal enveloping algebra of gl m ⊕ p . Let θ be the automorphism of U q ( gl m ⊕ p ) which is defined on generators by E i ↦ E i − m , F i ↦ F i − m , K j ↦ K j − m for any i ∈ Z / p m Z ∖ { 0 ¯ , m ¯ , … , ( p − 1 ) m ¯ } and any j ∈ Z / p m Z . Let H ( p , p , n ) be the Hecke algebra of type G ( p , p , n ) with parameters q , e , where e is a primitive pth root of unity. In this paper we establish a Schur–Weyl reciprocity between H ( p , p , n ) and a twisted tensor product of U q ( gl m ⊕ p ) and the group algebra for 〈 θ 〉 (a cyclic group of order p) by using the results in [J. Hu, J. Algebra 274 (2004) 446–490].

Let H n, r be the Ariki–Koike algebra associated to the complex reflection group Wn, r = G(r, 1, n). In this paper, we give a new presentation of H n, r by making use of the Schur–Weyl reciprocity for H n, r established by M. Sakamoto and T. Shoji (1999, J. Algebra, 221, 293–314). This allows us to construct various non-parabolic subalgebras of H n, r. We construct all the irreducible representations of H n, r as induced modules from such subalgebras. We show the existence of a partition of unity in H n, r, which is specialized to a partition of unity in the group algebra C Wn, r. Then we prove a Frobenius formula for the characters of H n, r, which is an analogy of the Frobenius formula proved by A. Ram (1991, Invent. Math.106, 461–488) for the Iwahori–Hecke algebra of type A.