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Green Functions Associated to Complex Reflection Groups, II
Abstract
Green functions associated to complex reflection groups G(e,1,n) were discussed in the author's previous paper. In this paper, we consider the case of complex reflection groups W=G(e,p,n). Schur functions and Hall–Littlewood functions associated to W are introduced, and Green functions are described as the transition matrix between those two symmetric functions. Furthermore, it is shown that these Green functions are determined by means of Green functions associated to various G(e′,1,n′). Our result involves, as a special case, a combinatorial approach to the Green functions of type Dn.
... In this case the results of [Sho01] are not directly applicable. The Green functions of the Weyl group of type D are described in [Sho02], also using the theory of symmetric functions. For the Weyl group W (D N/2 ), the main idea will be to use [Sho02,Proposition 4.9], which relates the Green functions with certain Green functions of the Weyl group of type B/C and A. ...
... The Green functions of the Weyl group of type D are described in [Sho02], also using the theory of symmetric functions. For the Weyl group W (D N/2 ), the main idea will be to use [Sho02,Proposition 4.9], which relates the Green functions with certain Green functions of the Weyl group of type B/C and A. ...
... Furthermore, we write χ ∼ χ ′ if Λ χ;k and Λ χ ′ ;k are similar. In [Sho01] (for type B) and in [Sho02] (for type D) the following theorem is proved: ...
The generalised Springer correspondence for $\text{SO}(N,\mathbb{C})$ attaches to a pair $(C,E)$, where $C$ is a unipotent class of $\text{SO}(N,\mathbb{C})$ and $E$ is an irreducible $G$-equivariant local system on $C$, an irreducible representation $\rho(C,E)$ of a relative Weyl group of $\text{SO}(N,\mathbb{C})$. We call $C$ the unipotent support of $\rho(C,E)$. Each generalised Springer representation $\rho(C,E)$ appears with multiplicity 1 in the top cohomology of a certain variety. Consider the representation $\bar\rho(C,E)$ obtained by summing over all the cohomology groups of this variety. It is well-known that $C$ is strictly minimal in the closure ordering among the unipotent supports of the irreducbile subrepresentations of $\bar\rho(C,E)$ and that $\rho(C,E)$ appears in $\bar\rho(C,E)$ with multiplicity $1$. Suppose $C$ is parametrised by an orthogonal partition consisting of only odd parts. We prove that there exists a unique $(C^{\text{max}},E^{\text{max}})$ such that $\rho(C^{\text{max}},E^{\text{max}})$ appears with multiplicity $1$ in $\bar\rho(C,E)$ and such that $C^{\text{max}}$ is strictly maximal among the unipotent supports of the irreducible subrepresentations of $\bar \rho(C,E)$. Let $\text{sgn}$ be the sign representation of the relevant relative Weyl group. We also show that there exists a unique $(C^{\text{min}}, E^{\text{min}})$ such that $\rho(C^{\text{min}},E^{\text{min}})$ appears with multiplicity $1$ in $\text{sgn} \otimes \bar\rho(C,E)$ and such that $C^{\text{min}}$ is strictly minimal among the unipotent supports of the irreducible constituents of $\text{sgn} \otimes \bar\rho(C,E)$, and we show that $\rho(C^{\text{min}},E^{\text{min}}) = \text{sgn} \otimes \rho(C^{\text{max}},E^{\text{max}})$. This is a direct analogue to a similar maximality and minimality result for $\text{Sp}(2n,\mathbb{C})$ by Waldspurger.
... It has already been observed by various people that this algorithm is something that lends itself to generalization to complex reflection groups (see [13] [31] [32]). ...
... (see [13, Conjectures 2.5–2.7], as well as [20] [33]). Separately, Shoji [31] [32] has studied the algorithm for imprimitive complex reflection groups by partitioning the characters using combinatorial objects called " symbols " (these are generalizations of the symbols and u-symbols that occur in the representation theory of algebraic groups of classical type). An important feature of the original Lusztig–Shoji algorithm is that its output obeys certain integrality and positivity conditions. ...
Recent work by a number of people has shown that complex reection groups give rise to many representation-theoretic structures
(e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these
structures are actually describing the representation theory of as-yet undescribed objects called spetses, of which reductive algebraic groups ought to be a special case.
In this paper we carry out the Lusztig–Shoji algorithm for calculating Green functions for the dihedral groups. With a suitable
set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl
group case, so we may think of it as describing the geometry of the “unipotent variety” associated to a spets. From this,
we determine the possible “Springer correspondences,” and we show that, as is true for algebraic groups, each special piece
is rationally smooth, as is the full unipotent variety.
... Based on this reciprocity, the second author gave (see [20]) a Frobenius type formula for the characters of H K (n, r). In a more recent paper [21], the second author extended the classical Schur–Weyl reciprocity to the case between the complex reflection group G(r, p, n) (where p | r) and a certain subgroup of GL M . So it is natural to ask whether a q-version of such reciprocity also exists, that is, between the cyclotomic Hecke algebra of type G(r, p, n) and certain " quantum group. ...
Let Uq(glm⊕p) be the quantized universal enveloping algebra of glm⊕p. Let θ be the automorphism of Uq(glm⊕p) which is defined on generators by Ei↦Ei−m, Fi↦Fi−m, Kj↦Kj−m for any i∈Z/pmZ∖{0¯,m¯,…,(p−1)m¯} and any j∈Z/pmZ. Let H(p,p,n) be the Hecke algebra of type G(p,p,n) with parameters q,ε, where ε 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 Uq(glm⊕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].
... The connection to Springer representations of Weyl groups and the representations of Chevalley groups over finite fields has been developed extensively by Lusztig, Shoji and others; a good survey of the current theory is in [Shj1] and the recent papers [Shj2] show how this theory is beginning to extend its reach outside Lie theory into the realm of complex reflection groups. ...
Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Sch\"utzenberger's positive formula for the Kostka-Foulkes poynomials in the type A case.
Let W be the Weyl group of type BCn. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of W which is of type BCn− 1. Then we show that these two restriction formulas are equivalent, and discuss how the results can be used to examine the existence of affine pavings of Springer fibers corresponding to the symplectic Lie algebra in characteristic 2.
In this paper, we consider the set of r-symbols in a full generality. We construct Hall-Littlewood functions and Kostka functions associated to those r-symbols. We also discuss a multi-parameter version of those functions. We show that there exists a general algorithm of computing the multi-parameter Kostka functions. As an application, we show that the generalized Green functions of symplectic groups can be described combinatorially in terms of our (one-parameter) Kostka functions.
To a spetsial complex reflection group, equipped with a root lattice in the
sense of Nebe, we attach a certain finite set playing a role which is analogous
to the role of the set of unipotent classes of an algebraic group. In the case
of imprimitive groups, we give a combinatoric parametrization of it in terms of
Malle-Shoji generalized symbols. This result provides a link between the works
of Shoji on Green functions for complex reflection groups and of Broue, Kim,
Malle, Rouquier, et. al. on the cyclotomic Hecke algebras and their families of
characters.
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A un groupe de reflexions complexe spetsial, muni d'un reseau radiciel au
sens de Nebe, nous associons un certain ensemble fini qui doit jouer un role
analogue a celui de l'ensemble des classes unipotentes d'un groupe algebrique.
Dans le cas des groupes imprimitifs, nous en donnons un parametrage
combinatoire en termes des symboles generalises de Malle et Shoji. Ce resultat
fournit un lien entre les travaux de Shoji sur les fonctions de Green pour les
groupes de reflexions complexes et ceux de Broue, Kim, Malle, Rouquier, et al.
sur les algebres de Hecke cyclotomiques et leurs familles de caracteres.
Let W be the complex reflection group . In the author's previous paper [J. Algebra 245 (2001) 650–694], Hall–Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type Bn, 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.
We interpret the orthogonality relation of Kostka polynomials arising from
complex reflection groups (c.f. [Shoji, Invent. Math. 74 (1983), J. Algebra 245
(2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra.
This leads us to the notion of Kostka system, which can be seen as a
categorical counter-part of Kostka polynomials. Then, we show that every
generalized Springer correspondence [Lusztig, Invent. Math. 75 (1984)] (in good
characteristic) gives rise to a Kostka system. This enables us to see the
top-term generation property of the (twisted) homology of generalized Springer
fibers, and the transition formula of Kostka polynomials between two
generalized Springer correspondences of type $\mathsf{BC}$. The latter provides
an inductive algorithm to compute Kostka polynomials by upgrading
[Ciubotaru-Kato-K, Invent. Math. 178 (2012)] \S 3 to its graded version. In the
appendices, we present purely algebraic proofs that Kostka systems exist for
type $\mathsf{A}$ and asymptotic type $\mathsf{BC}$ cases, and therefore one
can skip geometric sections \S 3--5 to see the key ideas and basic
examples/techniques.
Let G be a reductive algebraic group, with nilpotent cone N and flag variety G/B. We construct an exact functor from perverse sheaves on N to locally constant sheaves on G/B, and we use it to study Ext-groups of simple perverse sheaves on N in terms of the cohomology of G/B. As an application, we give new proofs of some known results on stalks of perverse sheaves on N. Comment: 14 pages
Let = glm1 ⊕ ··· ⊕ glmr be a Levi subalgebra of glm, with m = ∑ri = 1mi, and V the natural representation of the quantum group Uq(). We construct a representation of the Ariki–Koike algebra n, r on the n-fold tensor space of V, commuting with the action of Uq(), and prove the Schur–Weyl reciprocity for the actions of Uq() and n, r on it.
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.
This article is the result of experiments performed using computer programs written in the GAP language. We describe an algorithm which computes a set of rational functions attached to a finite Coxeter group W. Conjecturally, these rational functions should be polynomials, and in the case where W is the Weyl group of a Chevalley group G defined over ${\funnyF}_q$, the values of our polynomials at q should give the number of ${\funnyF}_q$-rational points of Lusztig's special pieces in the unipotent variety of G. The algorithm even works for complex reflection groups. We give a number of examples which show, in particular, that our conjecture is true for all types except possibly $B_n$ and $D_n$.