Results 1  10
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21
On rationality properties of involutions of reductive groups
 Adv. in Math. 99
, 1993
"... Introduction. Let k be a field of characteristic not two and G a connected linear reductive kgroup. By a kinvolution θ of G, we mean a kautomorphism θ of G of order two. For k = R, C or an algebraically closed field, such involutions have been extensively studied emerging from different interests ..."
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Cited by 35 (19 self)
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Introduction. Let k be a field of characteristic not two and G a connected linear reductive kgroup. By a kinvolution θ of G, we mean a kautomorphism θ of G of order two. For k = R, C or an algebraically closed field, such involutions have been extensively studied emerging from different interests. As manifested in [8, 18, 28], the interactions with
On the set of orbits for a Borel subgroup
 Comment. Math. Helv
, 1995
"... Let X = G/H be a homogeneous variety for a connected complex reductive group G and let B be a Borel subgroup of G. In many situations, it is necessary to study the Borbits in X. An equivalent setting of this problem is to analyze Horbits in the flag variety G/B. The probably best known example is ..."
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Cited by 26 (1 self)
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Let X = G/H be a homogeneous variety for a connected complex reductive group G and let B be a Borel subgroup of G. In many situations, it is necessary to study the Borbits in X. An equivalent setting of this problem is to analyze Horbits in the flag variety G/B. The probably best known example is the Bruhat decomposition of G/B where one
Simulating perverse sheaves in modular representation theory
 Proc. Symposia Pure Math 56
, 1994
"... For some time we have studied the representation theory of semisimple algebraic groups in characteristic p> 0. Of special interest is a celebrated conjecture of Lusztig [Ll] describing the characters of the simple modules when p is not too small relative to the root system. A similar conjecture, by ..."
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Cited by 12 (10 self)
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For some time we have studied the representation theory of semisimple algebraic groups in characteristic p> 0. Of special interest is a celebrated conjecture of Lusztig [Ll] describing the characters of the simple modules when p is not too small relative to the root system. A similar conjecture, by Kazhdan and Lusztig [KLl], for the composition factor multiplicities of Verma modules for semisimple complex Lie algebras, has already been proved [BB] and [BK]. The method of proof was to establish a correspondenceactually an equivalence of categoriesbetween a
Tori invariant under an involutorial automorphism III
"... Abstract. The geometry of the orbits of a minimal parabolic ksubgroup acting on a symmetric kvariety is essential in several area’s, but its main importance is in the study of the representations associated with these symmetric kvarieties (see for example [6], [5], [20] and [31]). Up to an action ..."
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Cited by 8 (6 self)
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Abstract. The geometry of the orbits of a minimal parabolic ksubgroup acting on a symmetric kvariety is essential in several area’s, but its main importance is in the study of the representations associated with these symmetric kvarieties (see for example [6], [5], [20] and [31]). Up to an action of the restricted Weyl group of G, these orbits can be characterized by the Hkconjugacy classes of maximal ksplit tori, which are stable under the kinvolution θ associated with the symmetric kvariety. Here H is a open ksubgroup of the fixed point group of θ. This is the second in a series of papers in which we characterize and classify the Hkconjugacy classes of maximal ksplit tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of non algebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, �adic numbers, finite fields and number fields.
Orbits and invariants associated with a pair of commuting involutions
 Duke Math. J
"... Let σ, θ be commuting involutions of the connected reductive algebraic group G, where σ, θ, and G are defined over a (usually algebraically closed) field k, char k ̸=2. We have fixed point groups H: = G σ and K: = G θ and an action (H ×K)×G → G, where ((h, k), g) ↦ → hgk −1, h ∈ H, k ∈ K, g ∈ G. Let ..."
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Cited by 5 (5 self)
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Let σ, θ be commuting involutions of the connected reductive algebraic group G, where σ, θ, and G are defined over a (usually algebraically closed) field k, char k ̸=2. We have fixed point groups H: = G σ and K: = G θ and an action (H ×K)×G → G, where ((h, k), g) ↦ → hgk −1, h ∈ H, k ∈ K, g ∈ G. Let G /(H × K) denote Spec O(G) H ×K (the categorical quotient). Let A be maximal among subtori S of G such that θ(s) = σ(s) = s −1 for all s ∈ S. There is the associated Weyl group W: = WH ×K(A). We show the following. • The inclusion A → G induces an isomorphism A/W ˜→G /(H × K). In particular, the closed (H × K)orbits are precisely those which intersect A. • The fibers of G → G /(H × K) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. We investigate • the structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the Aweight spaces of g; • the relation of the orbit type stratifications of A/W and G /(H × K). Along the way we simplify some of R. Richardson’s proofs for the symmetric case σ = θ, and at the end we quickly recover results of M. Berger, M. FlenstedJensen, B. Hoogenboom, and T. Matsuki [Ber], [FJ1], [Hoo], [Mat] for the case k = R.
On the classification of kinvolutions
 Adv. Math
"... Abstract. Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a kopen subgroup of the fixed point group of θ and Gk (resp. Hk) the set of krational points of G (resp. H). The variety Gk/Hk is called a symmetric kvar ..."
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Cited by 4 (3 self)
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Abstract. Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a kopen subgroup of the fixed point group of θ and Gk (resp. Hk) the set of krational points of G (resp. H). The variety Gk/Hk is called a symmetric kvariety. These varieties occur in many problems in representation theory, geometry and singularity theory. Over the last few decades the representation theory of these varieties has been extensively studied for k = � and �. Asmostofthe work in these two cases was completed, the study the representation theory over other fields, like local fields and finite fields began. The representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space, like the restricted root systems with Weyl groups, etc. Thus it is essential to study first this structure and the related geometry. In this paper we give a characterization of the isomorphy classes of these symmetric kvarieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers, �adic numbers, finite fields and number fields. 1.
Unipotent Hecke algebras of GLn(Fq
, 2003
"... This paper describes a family of Hecke algebras Hµ = EndG(Ind G U(ψµ)), where U is the subgroup of unipotent uppertriangular matrices of G = GLn(Fq) and ψµ is a linear character of U. The main results combinatorially index a basis of Hµ, provide a large commutative subalgebra of Hµ, and after descr ..."
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Cited by 3 (2 self)
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This paper describes a family of Hecke algebras Hµ = EndG(Ind G U(ψµ)), where U is the subgroup of unipotent uppertriangular matrices of G = GLn(Fq) and ψµ is a linear character of U. The main results combinatorially index a basis of Hµ, provide a large commutative subalgebra of Hµ, and after describing the combinatorics associated with the representation theory of Hµ, generalize the RSK correspondence that is typically found in the representation theory of the symmetric group. 1
Twisted Incidence Algebras and KazhdanLusztigStanley functions
 Adv. Math
, 1997
"... We introduce a new multiplication in the incidence algebra of a partially ordered set, and study the resulting algebra. As an application of the properties of this algebra we obtain a combinatorial formula for the KazhdanLusztigStanley functions of a poset. As special cases this yields new combinat ..."
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Cited by 1 (0 self)
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We introduce a new multiplication in the incidence algebra of a partially ordered set, and study the resulting algebra. As an application of the properties of this algebra we obtain a combinatorial formula for the KazhdanLusztigStanley functions of a poset. As special cases this yields new combinatorial formulas for the parabolic and inverse parabolic KazhdanLusztig polynomials, for the generalized (toric) hvector of an Eulerian poset, and for the LusztigVogan polynomials. 1
Computing Orbits Of Minimal Parabolic kSubgroups Acting On Symmetric kVarieties
"... . The orbits of a minimal parabolic ksubgroup acting on a symmetric kvariety are essential to the study of symmetric kvarieties and their representations. This paper gives an algorithm to compute these orbits and most of the combinatorial structure of the orbit decomposition. There are several w ..."
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Cited by 1 (0 self)
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. The orbits of a minimal parabolic ksubgroup acting on a symmetric kvariety are essential to the study of symmetric kvarieties and their representations. This paper gives an algorithm to compute these orbits and most of the combinatorial structure of the orbit decomposition. There are several ways to describe these orbits, see for example [22, 28, 35]. Fundamental in all these descriptions are the associated twisted involutions in the restricted Weyl group. These describe the combinatorial structure of the orbit decomposition in a similar manner to the special case of orbits of a Borel subgroup acting on a symmetric variety, see [19]. However, the orbit structure in the general case is much more complicated than the special case of orbits of a Borel subgroup. In this paper we first modify the characterization of the orbits of minimal parabolic ksubgroups acting on a symmetric kvarieties given in [22], to illuminate the similarity to the one for orbits of a Borel subgroup acting...
CLOSURES OF KORBITS IN THE FLAG VARIETY FOR U(p, q)
, 905
"... Abstract. We classify the GLp × GLqorbits in the flag variety for GLp+q with rationally smooth closure, showing that they are all either already closed or are pullbacks from orbits with smooth closure in a partial flag variety. 1. ..."
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Cited by 1 (1 self)
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Abstract. We classify the GLp × GLqorbits in the flag variety for GLp+q with rationally smooth closure, showing that they are all either already closed or are pullbacks from orbits with smooth closure in a partial flag variety. 1.