G. Lusztig

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All published works (414)

Constructible representations and Catalan numbers

2025-05-26

G. Lusztig , Eric Sommers

Parabolic character sheaves and Hecke algebras

2025-04-03

G. Lusztig

We show that certain Iwahori-Hecke algebras with unequal parameters can be realized in the framework of parabolic character sheaves. We show that certain Iwahori-Hecke algebras with unequal parameters can be realized in the framework of parabolic character sheaves.

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On the trace of Coxeter elements

2024-11-27

G. Lusztig

Let W be a Weyl group and let w be a Coxeter elememt of minimal length of W. In the early 1970's I.G.Macdonald stated that the trace of w on … Let W be a Weyl group and let w be a Coxeter elememt of minimal length of W. In the early 1970's I.G.Macdonald stated that the trace of w on an irreducible representation of W is 0,1 or -1. In this paper we give a proof of this statement and of an Iwahori-Hecke algebra version of it.

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Sharpness versus bluntness in affine Weyl groups

2024-11-14

G. Lusztig

In this paper we give an explanation of the bijection between arithmetic and geometric diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on … In this paper we give an explanation of the bijection between arithmetic and geometric diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on algebra.

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Total positivity in the space of maximal tori

2024-11-08

G. Lusztig

Let G be a connected reductive group over the complex numbers with a fixed pinning. We define and study the totally positive part of the set of maximal tori of … Let G be a connected reductive group over the complex numbers with a fixed pinning. We define and study the totally positive part of the set of maximal tori of G.

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Almost special representations of Weyl groups

2024-05-07

G. Lusztig

Let $c$ be the family of irreducible representations of a Weyl group $W$ corresponding to a two-sided cell of $W$. We define a subset $A_c$ of $c$ which contains the … Let $c$ be the family of irreducible representations of a Weyl group $W$ corresponding to a two-sided cell of $W$. We define a subset $A_c$ of $c$ which contains the special representation of $W$ in $c$ and is in canonical bijection with the set of constructible representations of $W$ attached to $c$.

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On the new bases attached to families of Weyl groups

2024-03-26

G. Lusztig

We study the new basis of the (complexified) Grothendieck group of unipotent representations of a split reductive group over a finite field. For exceptional types we use a definition of … We study the new basis of the (complexified) Grothendieck group of unipotent representations of a split reductive group over a finite field. For exceptional types we use a definition of the new basis which differs from the earlier one.

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Constructible representations and Catalan numbers

2024-03-04

G. Lusztig , Eric Sommers

We establish a connection between constructible representations (arising in the study of left cells in Weyl groups) and Catalan numbers. We establish a connection between constructible representations (arising in the study of left cells in Weyl groups) and Catalan numbers.

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Unipotent character sheaves and strata of a reductive group

2023-11-29

G. Lusztig

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive group over an algebraically closed field. We define a surjective map from … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive group over an algebraically closed field. We define a surjective map from the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper S left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CS(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of unipotent character sheaves on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (up to isomorphism) to the set of strata of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. To do this we use the generalized Springer correspondence. We also give a new parametrization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C upper S left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">CS(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of data coming from bad characteristic.

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The quantum group U̇ and flag manifolds over the semifield Z

2023-09-01

G. Lusztig

We give a parametrization of the canonical basis of the modified quantum group corresponding to a root datum in terms of the flag manifold over the semifield Z associated to … We give a parametrization of the canonical basis of the modified quantum group corresponding to a root datum in terms of the flag manifold over the semifield Z associated to the reductive group corresponding to the dual root datum.

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On bases of certain Grothendieck groups

2023-06-01

G. Lusztig

In a previous paper we have defined a second basis of the Grothendieck group of unipotent representations of a split reductive group over a finite field.In this paper we extend … In a previous paper we have defined a second basis of the Grothendieck group of unipotent representations of a split reductive group over a finite field.In this paper we extend this to the case of non-split special orthogonal groups.

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From families in Weyl groups to Springer representations

2023-03-01

G. Lusztig

Let G be a simple reductive group over the complex numbers.Let W be the Weyl group of G.We propose a description of the Springer representations of W associated to various … Let G be a simple reductive group over the complex numbers.Let W be the Weyl group of G.We propose a description of the Springer representations of W associated to various unipotent classes of G by a purely algebraic method involving the families of various reflection subgroups of W and which is suitable to computer calculations.

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On bases of certain Grothendieck groups, II

2023-01-01

G. Lusztig

In previous papers the author introduced a new basis of the Grpthendieck group of unipotent representations of a finite Chevalley group. In type D the definition of this basis was … In previous papers the author introduced a new basis of the Grpthendieck group of unipotent representations of a finite Chevalley group. In type D the definition of this basis was stated without proof. In this paper we provide the missing proof.

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Families of isotropic subspaces of a symplectic Z/2-vector space

2023-01-01

G. Lusztig

For a symplectic vector space V over Z/2 with a given circular basis we give a non-inductive definition of a family of isotropic subspaces of V with remarkable properties. For a symplectic vector space V over Z/2 with a given circular basis we give a non-inductive definition of a family of isotropic subspaces of V with remarkable properties.

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Sharpness versus bluntness in affine Weyl groups

2023-01-01

G. Lusztig

We give an explanation of the bijection between geometric and arithmetic diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on algebra. The second … We give an explanation of the bijection between geometric and arithmetic diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on algebra. The second version contains much additional material compared to the first version.

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Involutions in Weyl groups and nil-Hecke algebras

2022-12-01

G. Lusztig , Jr. Vogan David A.

In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W … In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W .In this paper we show that the specialization of this action at the parameter 0 has a simple description.0.1.Let W be a Coxeter group and let S be the set of simple reflections of W ; we assume that S is finite.Let w → |w| be the length function on W .Let H be the Iwahori-Hecke algebra attached to W . Recall that H is the free Z[u]module with basis {T w ; w ∈ W } (u is an indeterminate) with (associative) multiplication characterized byLet w → w * be an automorphism with square 1 of W preserving S and let I * = {w ∈ W ; w * = w -1 } be the set of "twisted involutions" in W .Let M be the free Z[u]-module with basis {a x ; x ∈ I * }.For any s ∈ S we define a Z[u]-linear map T s : M → M byT s a x = (u 2 -1)a x + u 2 a sxs * if x ∈ I * , sx = xs * , |sx| = |x| -1.

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Total Positivity in Symmetric Spaces

2022-10-04

G. Lusztig

In this paper we extend the theory of total positivity for reductive groups to the case of symmetric spaces. In this paper we extend the theory of total positivity for reductive groups to the case of symmetric spaces.

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A parametrization of unipotent representations

2022-09-01

G. Lusztig

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cell representations of the Weyl … We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cell representations of the Weyl group in the same two-sided cell.We use this map to parametrize the unipotent representations.0.1.Let G be a simple algebraic group defined and split over a finite field F q .Let U be the set of isomorphism classes of irreducible unipotent representations (over C) of the finite group G(F q ).Let W be the Weyl group of G and let Irr(W ) be the set of isomorphism classes of irreducible representations (over C) of W .In [2] a partition of Irr(W ) into families is described and in [4] a partition U = ⊔ c U c of U (with c running over the families of Irr(W )) is introduced.Moreover, in [4, §4] to any family c we have associated a finite group G c and a bijection (a) U c ↔ M (G c ).Here, for any finite group Γ, M (Γ) is the set of Γ-conjugacy classes of pairs (x, ρ) were x ∈ Γ and ρ is an irreducible representation (over C) of the centralizer Z Γ (x) of x in Γ; let C[M (Γ)] (resp.N[M (Γ)]) be the vector space of formal C-linear combinations of elements in M (Γ) and let A Γ : C[M (Γ)] → C[M (Γ)] be the non-abelian Fourier transform of [2] (a linear isomorphism with square 1).Let N[M (Γ)] (resp.R ≥0 [M (Γ)]) be the set of vectors of C[M (Γ)] which are linear combinations with coefficients in N

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Distinguished strata in a reductive group

2022-06-30

G. Lusztig

The set of strata of a reductive group can be viewed as an enlargement of the set of unipotent classes. In this paper the notion of distinguished unipotent class is … The set of strata of a reductive group can be viewed as an enlargement of the set of unipotent classes. In this paper the notion of distinguished unipotent class is extended to this larger set. The strata of a Weyl group are also introduced and studied.

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Adjacency for special representations of a Weyl group

2022-06-01

G. Lusztig

represent an element of ( ǫ represent an element of ( ǫ

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ON THE SATAKE ISOMORPHISM

2022-05-27

G. Lusztig

Abstract In a 1983 paper, the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper, we give new proofs for some results of that paper, … Abstract In a 1983 paper, the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper, we give new proofs for some results of that paper, one based on the theory of J -rings and one based on the known character formula for rational representations of a reductive group in positive, large characteristic. We also give an extension of that formula to disconnected groups.

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Coordinate rings and birational charts

2022-01-05

Sergey Fomin , G. Lusztig

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Total positivity and conjugacy classes

2022-01-01

Xuhua He , G. Lusztig

In this paper, we study the interaction between the totally positive monoid $G_{\ge 0}$ attached to a connected reductive group $G$ with a pinning and the conjugacy classes in $G$. … In this paper, we study the interaction between the totally positive monoid $G_{\ge 0}$ attached to a connected reductive group $G$ with a pinning and the conjugacy classes in $G$. In particular, we study how a conjugacy class meets the various cells of $G_{\ge0}$. We also state a conjectural Jordan decomposition for $G_{\ge0}$ and prove it in some special cases.

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On induction of class functions

2021-05-07

G. Lusztig

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive group defined over a finite field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive group defined over a finite field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Levi subgroup (defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) of a parabolic subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We define a linear map from class functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis bold upper F Subscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(\mathbf {F}_q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to class functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis bold upper F Subscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(\mathbf {F}_q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This map is independent of the choice of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that for large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> this map coincides with the known cohomological induction (whose definition involves a choice of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>).

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Reducing mod 𝑝 complex representations of finite reductive groups

2021-03-02

G. Lusztig

We state a conjecture on the reduction modulo the defining characteristic of a unipotent representation of a finite reductive group. We state a conjecture on the reduction modulo the defining characteristic of a unipotent representation of a finite reductive group.

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Coordinate rings and birational charts

2021-02-06

Sergey Fomin , G. Lusztig

Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., … Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational charts introduced in [L94, L19] in connection with the study of total positivity.

Two partitions of a flag manifold

2021-01-01

G. Lusztig

We point out similarities between two partitions of a flag manifold with pieces indexed by Weyl group elements. One partition is defined using the action of a Frobenius map, the … We point out similarities between two partitions of a flag manifold with pieces indexed by Weyl group elements. One partition is defined using the action of a Frobenius map, the other partition is defined using conjugation by a regular semisimple element.

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Traces on Iwahori-Hecke algebras and counting rational points

2021-01-01

G. Lusztig

Let w be an element of the Weyl group of a reductive group G defined and split over a finite field. We consider the variety of triples (g,B,B') where g … Let w be an element of the Weyl group of a reductive group G defined and split over a finite field. We consider the variety of triples (g,B,B') where g is a unipotent element of G and B, B' are Borel subgroups of G such that B contains g and B',gB'g^{-1} are in relative position w. We show that the number of rational points of this variety can be expressed in terms of a trace on the Iwahori-Hecke algebra. We also show that this variety is smooth, irreducible, if w is elliptic, of minimal length in its conjugacy class.

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Total positivity in symmetric spaces

2021-01-01

G. Lusztig

The theory of total positivity for reductive groups is here extended to the case of symmetric spaces. The theory of total positivity for reductive groups is here extended to the case of symmetric spaces.

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A parametrization of unipotent representations

2021-01-01

G. Lusztig

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl … We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize the unipotent representations.

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Corrigendum to ‘Endoscopy for Hecke categories, character sheaves and representations’

2021-01-01

G. Lusztig , Zhiwei Yun

Abstract We fix an error on a $3$ -cocycle in the original version of the paper ‘Endoscopy for Hecke categories, character sheaves and representations’. We give the corrected statements of … Abstract We fix an error on a $3$ -cocycle in the original version of the paper ‘Endoscopy for Hecke categories, character sheaves and representations’. We give the corrected statements of the main results.

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Coordinate rings and birational charts

2021-01-01

Sergey Fomin , G. Lusztig

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Involutions in Weyl groups and nil-Hecke algebras

2021-01-01

G. Lusztig , Jr Vogan

In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W. … In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W. In this paper we show that the specialization of this action at the parameter 0 has a simple description.

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On the Satake isomorphism

2020-12-01

G. Lusztig

In a 1983 paper the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper we give new proofs for some results of that paper, one … In a 1983 paper the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper we give new proofs for some results of that paper, one based on the theory of J-rings and one based on the known character formula for rational representations of a reductive group in positive, large, characteristic. We also give an extension of that character formula to disconnected groups.

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Positive conjugacy classes in Weyl groups

2020-12-01

G. Lusztig

Let W be a Weyl group.In this paper we introduce the notion of positive conjugacy class of W .This generalizes the notion of elliptic regular conjugacy class in the sense … Let W be a Weyl group.In this paper we introduce the notion of positive conjugacy class of W .This generalizes the notion of elliptic regular conjugacy class in the sense of Springer [9].

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Strata of a disconnected reductive group

2020-10-10

G. Lusztig

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Fourier transform as a triangular matrix

2020-10-03

G. Lusztig

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite dimensional vector space over the field with two elements with a given nondegenerate … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the vector space of complex valued functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper V right-bracket Subscript bold upper Z"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">[V]_{\mathbf {Z}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of integer valued functions. We show that there exists a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-basis of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper V right-bracket Subscript bold upper Z"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">[V]_{\mathbf {Z}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of characteristic functions of certain isotropic subspaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and such that the matrix of the Fourier transform from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to this basis is triangular. We show that this is a special case of a result which holds for any two-sided cell in a Weyl group.

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Springer’s work on unipotent classes and Weyl group representations

2020-09-03

G. Lusztig

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Partial flag manifolds over a semifield

2020-08-26

G. Lusztig

For any semifield <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we define a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> … For any semifield <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we define a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-form of a partial flag manifold of a semisimple group of simply laced type over the complex numbers.

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𝐙/𝐦-graded Lie algebras and perverse sheaves, IV

2020-08-26

G. Lusztig , Zhiwei Yun

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a reductive group over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a reductive group over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assume that the Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:annotation encoding="application/x-tex">\frak g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a given grading <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis German g Subscript German j Baseline German right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">j</mml:mi> </mml:msub> <mml:mo mathvariant="fraktur" stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\frak g_j)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> indexed by a cyclic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Z slash m"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Z}/m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German 0"> <mml:semantics> <mml:msub> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mn mathvariant="fraktur">0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\frak g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a Cartan subalgebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:annotation encoding="application/x-tex">\frak g</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German 0"> <mml:semantics> <mml:msub> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mn mathvariant="fraktur">0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\frak g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts on the variety of nilpotent elements in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mn mathvariant="fraktur">1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\frak g_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with finitely many orbits. We are interested in computing the local intersection cohomology of closures of these orbits with coefficients in irreducible <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant local systems in the case of the principal block. We show that these can be computed by a purely combinatorial algorithm.

Total Positivity in Springer Fibres

2020-06-22

G. Lusztig

Abstract Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally … Abstract Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.

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The Grothendieck group of unipotent representations: A new basis

2020-05-27

G. Lusztig

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis upper F Subscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(F_q)</mml:annotation> </mml:semantics> </mml:math> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis upper F Subscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(F_q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the group of rational points of a simple algebraic group defined and split over a finite field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we define a new basis for the Grothendieck group of unipotent representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis upper F Subscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(F_q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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The flag manifold over the semifield $\bf Z$

2020-03-01

G. Lusztig

Let G be a semisimple group over the complex numbers.We show that the flag manifold B of G has a version B(Z) over the tropical semifield Z on which the … Let G be a semisimple group over the complex numbers.We show that the flag manifold B of G has a version B(Z) over the tropical semifield Z on which the monoid G(Z) attached to G and Z acts naturally.This paper is concerned with the question of defining the flag manifold B(K) over a semifield K with an action of the monoid G(K) so that in the case where K = R >0 we recover B ≥0 with its G ≥0 -action.In [9, 4.9], for any semifield K, a definition of the flag manifold B(K) over K was given (based on ideas of Marsh and Rietsch [10]); but in that definition the lower and upper triangular part of G play an asymmetric role

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Remarks on Affine Springer Fibres

2020-03-01

G. Lusztig

Let h be a regular semisimple element in a complex simple Lie algebra g.Let t be an indeterminate.We consider the "variety" of Iwahori subalgebras of g tensored with the power … Let h be a regular semisimple element in a complex simple Lie algebra g.Let t be an indeterminate.We consider the "variety" of Iwahori subalgebras of g tensored with the power series in t which contain t times h.This variety admits a free action of a free abelian group of rank equal to the rank of g.We describe a fundamental domain for this action.

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Partial flag manifolds over a semifield

2020-01-01

G. Lusztig

For any semifield K we define a K-form of a partial flag manifold of a semisimple group G of simply laced type over the complex numbers. The definition is in … For any semifield K we define a K-form of a partial flag manifold of a semisimple group G of simply laced type over the complex numbers. The definition is in terms of the theory of canonical bases.

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Fourier transform as a triangular matrix

2020-01-01

G. Lusztig

Let V be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let [V] be the vector space of complex valued functions … Let V be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let [V] be the vector space of complex valued functions on V and let [V]_Z be the subgroup of [V] consisting of integer valued functions. We show that there exists a Z-basis of [V]_Z consisting of characteristic functions of certain explicit isotropic subspaces of V such that the matrix of the Fourier transform from [V] to [V] with respect to this basis is triangular. We show that this is a special case of a result which holds for any two-sided cell in a Weyl group.

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On induction of class functions

2020-01-01

G. Lusztig

Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. … Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. We define a linear map from class functions on L(F_q) to class functions on G(F_q). This map is independent of the choice of P. We show that for large q this map coincides with the known cohomological induction (whose definition involves a choice of P).

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Open problems on Iwahori-Hecke algebras

2020-01-01

G. Lusztig

We state four open problems on Iwahori-Hecke algebras. The first one states a relation between some algebras appearing in Solleveld's work and some explicit Hecke algebras appearing in the study … We state four open problems on Iwahori-Hecke algebras. The first one states a relation between some algebras appearing in Solleveld's work and some explicit Hecke algebras appearing in the study of unipotent representations. The second one is a boundedness conjecture. The third one is a conjecture of well-definedness of an asymptotic Hecke algebra. The fourth one concerns positive conjugacy classes in a Weyl group.

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Strata of a disconnected reductive group

2020-01-01

G. Lusztig

Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of … Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of which is a union of $G^0$-conjugacy classes of fixed dimension. In the case where $D=G^0$ this recovers a known partition.

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From families in Weyl groups to Springer representations

2020-01-01

G. Lusztig

Let G be a simple reductive group over the complex numbers. Let W be the Weyl group of G. We propose a description of the Springer representations of W associated … Let G be a simple reductive group over the complex numbers. Let W be the Weyl group of G. We propose a description of the Springer representations of W associated to various unipotent classes of G by a purely algebraic method involving the families of various reflection subgroups of W and which is suitable to computer calculations.

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Common Coauthors

Coauthor	Papers Together
Zhiwei Yun	13
David Kazhdan	12
Xuhua He	6
David A. Vogan	5
Geordie Williamson	4
Ju-Lee Kim	4
Sergey Fomin	3
Pierre Deligne	3
R. W. Carter	2
Dean Alvis	2
Ting Xue	2
Dipendra Prasad	2
Shrawan Kumar	2
Eric Sommers	2
Roger Howe	1
François Labourie	1
Hanspeter Kraft	1
Claudio Procesi	1
明彦行者	1
Nanhua Xi	1
Robert Macpherson	1
Roger W. Carter	1
Michel Broué	1
William M. Goldman	1
Corrado De Concini	1
Alexander Beilinson	1
Jr Vogan	1
John J. Millson	1
G Lehrer	1
Changchang Xi	1
C. De Concini	1
Ivan Cherednik	1
Jr. Vogan David A.	1
M. Gromov	1
Torsten Ekedahl	1
Gunter Malle	1
Jean Michel	1
J. M. Smelt	1
J. A. GREEN	1
N. Spaltenstein	1
Johan L. Dupont	1
Bhama Srinivasan	1
Yavor Markov	1
C. Procesi	1
Richard Schoen	1
Yves Benoist	1
Pierre D. Milman	1
Edward Bierstone	1
W. M. Beynon	1
Gerald W. Schwarz	1

Commonly Cited References

Representations of Coxeter groups and Hecke algebras

1979-06-01

David Kazhdan , G. Lusztig

Intersection cohomology complexes on a reductive group

1984-06-01

G. Lusztig

Representations of Reductive Groups Over Finite Fields

1976-01-01

Pierre Deligne , G. Lusztig

Character sheaves, V

1986-08-01

G. Lusztig

A class of irreducible representations of a Weyl group

1979-01-01

G. Lusztig

Characters of Reductive Groups over a Finite Field. (AM-107)

1984-12-31

G. Lusztig

This book presents a classification of all (complex) irreducible representations of a reductive group with connected centre, over a finite field. To achieve this, the author uses etale intersection cohomology, … This book presents a classification of all (complex) irreducible representations of a reductive group with connected centre, over a finite field. To achieve this, the author uses etale intersection cohomology, and detailed information on representations of Weyl groups.

Character sheaves on disconnected groups, X

2009-04-08

G. Lusztig

We classify the unipotent character sheaves on a fixed connected component of a reductive algebraic group under a mild condition on the characteristic of the ground field. We classify the unipotent character sheaves on a fixed connected component of a reductive algebraic group under a mild condition on the characteristic of the ground field.

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Canonical bases arising from quantized enveloping algebras

1990-01-01

G. Lusztig

0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call … 0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see ?2) bases of PBW type, since for v = 1, they specialize to bases of U+ of the type provided by the Poincare see however ? 12.)

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Trigonometric sums, green functions of finite groups and representations of Weyl groups

1976-12-01

T. A. Springer

Introduction to Quantum Groups

2010-01-01

G. Lusztig

According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping … According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple

Green polynomials and singularities of unipotent classes

1981-11-01

G. Lusztig

Proof of the Deligne-Langlands conjecture for Hecke algebras

1987-02-01

David Kazhdan , G. Lusztig

Classes Unipotentes et Sous-groupes de Borel

1982-01-01

N. Spaltenstein

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Hecke Algebras with Unequal Parameters

2003-02-04

G. Lusztig

These are notes for the Aisenstadt lectures given in may/june 2002 at CRM, Montreal, enlarged and updated in 2014 by taking into account the recent results of Elias and Williamson … These are notes for the Aisenstadt lectures given in may/june 2002 at CRM, Montreal, enlarged and updated in 2014 by taking into account the recent results of Elias and Williamson on Soergel bimodules. The main object is the study of Iwahori-Hecke algebras arising from reductive groups over finite or p-adic fields. We try to extend various results known in the equal parameter case to the case of not necessarily equal parameters.

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From conjugacy classes in the Weyl group to unipotent classes

2011-06-08

G. Lusztig

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UNIPOTENT REPRESENTATIONS OF A FINITE CHEVALLEY GROUP OF TYPE E<sub>8</sub>

1979-01-01

G. Lusztig

Journal Article UNIPOTENT REPRESENTATIONS OF A FINITE CHEVALLEY GROUP OF TYPE E8 Get access GEORGE LUSZTIG GEORGE LUSZTIG Department of Mathematics, Massachusetts Institute of TechnologyCambridge, Massachusetts 02139 USA Search for … Journal Article UNIPOTENT REPRESENTATIONS OF A FINITE CHEVALLEY GROUP OF TYPE E8 Get access GEORGE LUSZTIG GEORGE LUSZTIG Department of Mathematics, Massachusetts Institute of TechnologyCambridge, Massachusetts 02139 USA Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 30, Issue 3, September 1979, Pages 315–338, https://doi.org/10.1093/qmath/30.3.315 Published: 01 September 1979 Article history Received: 06 June 1978 Published: 01 September 1979

Coxeter orbits and eigenspaces of Frobenius

1976-06-01

G. Lusztig

Representations of Finite Chevalley Groups

1978-12-31

G. Lusztig

Introduction Preliminaries The characters $R^G_T(\theta)$ Unipotent representations Some open problems Bibliography. Introduction Preliminaries The characters $R^G_T(\theta)$ Unipotent representations Some open problems Bibliography.

Some examples of square integrable representations of semisimple 𝑝-adic groups

1983-01-01

G. Lusztig

We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer’s reflection representation corresponding to finite Weyl groups, and we show that in many cases … We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer’s reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic group.

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Cuspidal local systems and graded Hecke algebras, I

1988-01-01

G. Lusztig

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Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras

2000-08-10

Meinolf Geck , Götz Pfeiffer

Abstract Finite Coxeter groups and related structures arise naturally in several branches of mathematics, for example, the theory of Lie algebras and algebraic groups. The corresponding Iwahori-Hecke algebras are obtained … Abstract Finite Coxeter groups and related structures arise naturally in several branches of mathematics, for example, the theory of Lie algebras and algebraic groups. The corresponding Iwahori-Hecke algebras are obtained by a certain deformation process. They have applications in the representation theory of groups of Lie type and the theory of knots and links. The aim of this book is to develop the theory of conjugacy classes and irreducible characters, both for finite Coxeter groups and the associated Iwahori-Hecke algebras. The topics range from classical results to more recent developments and are treated in a coherent and self-contained way. This is the first book which develops these subjects both from a theoretical and an algorithmic point of view in a systematic way. All types of finite Coxeter groups are covered.

Cells in Affine Weyl Groups

2018-06-06

G. Lusztig

I. The basis Cw of the Hecke algebra 2. The function a 3. Positivity 4. Left cells and two-sided cells 5. Cells and the function a 6.The case of a … I. The basis Cw of the Hecke algebra 2. The function a 3. Positivity 4. Left cells and two-sided cells 5. Cells and the function a 6.The case of a finite Wey1 group 7.An upper bound for a(w) for w in an affine Wey1 group 8.The subset W(v) of an affine Wey1 group 9. Construction of n-tempered representations 10.Left cells and dihedral subgroups II.Left cells in the affine Wey1 groups A2, B2, G2

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Notes on unipotent classes

1997-01-01

G. Lusztig

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Irreducible representations of finite classical groups

1977-06-01

G. Lusztig

Green Functions and Character Sheaves

1990-03-01

G. Lusztig

purely combinatorial way, using earlier ideas of P. Hall. In the general case, they were introduced [3] in terms of l-adic cohomology, as values of certain virtual representations of R'(O)(T … purely combinatorial way, using earlier ideas of P. Hall. In the general case, they were introduced [3] in terms of l-adic cohomology, as values of certain virtual representations of R'(O)(T is a maximal torus defined over Fq) at unipotent elements; the characters of RG( 0) at arbitrary elements were then expressible in a simple form in terms of Green functions of G and of smaller groups. One drawback of the definition of [3] was that it did not allow one to compute the Green functions except in some simple cases. Another definition of Green functions was proposed by Springer, first in terms of trigonometric sums on the Lie algebra and later in geometrical terms [17]. This definition was applicable only in large characteristic and in that case it was identified with the definition in [3] by Springer [17] and Kazhdan [10]. In [12], I observed that Green functions for GLn can be redefined in terms of intersection cohomology. This led me to a new definition of Green functions

Unipotent Elements in Small Characteristic, II

2008-08-22

G. Lusztig

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Groupes et algèbres de Lie

1971-01-01

Nicolás Bourbaki

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.

Total Positivity in Reductive Groups

1994-01-01

G. Lusztig

An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 … An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 paper [S] and in the 1935 note [GK] of Gantmacher and Krein. (For a recent survey of totally positive matrices, see [A].)

On the finiteness of the number of unipotent classes

1976-10-01

G. Lusztig

Schubert varieties and Poincaré duality

1980-01-01

David Kazhdan , G. Lusztig

On the springer representations of the weyl groups of classical algebraic groups

1979-01-01

Toshiaki Shoji

(1979). On the springer representations of the weyl groups of classical algebraic groups. Communications in Algebra: Vol. 7, No. 16, pp. 1713-1745. (1979). On the springer representations of the weyl groups of classical algebraic groups. Communications in Algebra: Vol. 7, No. 16, pp. 1713-1745.

On the Green polynomials of classical groups

1983-06-01

Toshiaki Shoji

Canonical Bases Arising from Quantized Enveloping Algebras

1990-04-01

G. Lusztig

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Regular elements of semisimple algebraic groups

1965-12-01

Robert Steinberg

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Character Sheaves and Almost Characters of Reductive Groups

1995-03-01

Toshiaki Shoji

On crystal bases of the Q-analogue of universal enveloping algebras

1991-07-01

Masaki Kashiwara

Unipotent characters of the symplectic and odd orthogonal groups over a finite field

1981-06-01

G. Lusztig

Green functions of finite Chevalley groups of type En (n = 6, 7, 8)

1984-06-01

W. M. Beynon , N. Spaltenstein

On quantum groups

1990-06-01

G. Lusztig

Elliptic elements in a Weyl group: a homogeneity property

2012-02-20

G. Lusztig

Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>be a reductive group over an algebraically closed field whose characteristic is not a bad prime for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>. Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>be a reductive group over an algebraically closed field whose characteristic is not a bad prime for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>. Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"><mml:semantics><mml:mi>w</mml:mi><mml:annotation encoding="application/x-tex">w</mml:annotation></mml:semantics></mml:math></inline-formula>be an elliptic element of the Weyl group which has minimum length in its conjugacy class. We show that there exists a unique unipotent class<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>such that the following holds: if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"><mml:semantics><mml:mi>V</mml:mi><mml:annotation encoding="application/x-tex">V</mml:annotation></mml:semantics></mml:math></inline-formula>is the variety of pairs<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis g comma upper B right-parenthesis"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">(g,B)</mml:annotation></mml:semantics></mml:math></inline-formula>where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g element-of upper X"><mml:semantics><mml:mrow><mml:mi>g</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">g\in X</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"><mml:semantics><mml:mi>B</mml:mi><mml:annotation encoding="application/x-tex">B</mml:annotation></mml:semantics></mml:math></inline-formula>is a Borel subgroup such that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B comma g upper B g Superscript negative 1"><mml:semantics><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mi>B</mml:mi><mml:msup><mml:mi>g</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">B,gBg^{-1}</mml:annotation></mml:semantics></mml:math></inline-formula>are in relative position<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"><mml:semantics><mml:mi>w</mml:mi><mml:annotation encoding="application/x-tex">w</mml:annotation></mml:semantics></mml:math></inline-formula>, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"><mml:semantics><mml:mi>V</mml:mi><mml:annotation encoding="application/x-tex">V</mml:annotation></mml:semantics></mml:math></inline-formula>is a homogeneous<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>-space.

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Quantum deformations of certain simple modules over enveloping algebras

1988-08-01

G. Lusztig

Proof of Springer’s hypothesis

1977-12-01

David Kazhdan

A construction of representations of Weyl groups

1978-10-01

T. A. Springer

Fixed point varieties on affine flag manifolds

1988-06-01

David Kazhdan , G. Lusztig

Unipotent classes and special Weyl group representations

2008-05-17

G. Lusztig

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La conjecture de Weil. I

1974-12-01

Pierre Deligne

On some bruhat decomposition and the structure of the hecke rings of p-Adic chevalley groups

1965-12-01

Nagayoshi Iwahori , Hisatake Matsumoto

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Affine Hecke algebras and their graded version

1989-01-01

G. Lusztig

0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s … 0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s satisfy (Ts + l)(Ts v2c(s)) = 0, where C(S) E N depend on s and are subject only to c(s) = c(s') whenever s, s are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the special where c(s) is independent of s and the coroots generate a direct summand. In this special case, the question above has been studied in [1] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out.

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Induce/restrict matrices for exceptional Weyl groups

2005-01-01

Dean Alvis

This manuscript contains tables giving the multiplicities with which irreducible characters of exceptional Weyl groups appear in characters induced from certain reflection subgroups containing maximal parabolic subgroups. This manuscript contains tables giving the multiplicities with which irreducible characters of exceptional Weyl groups appear in characters induced from certain reflection subgroups containing maximal parabolic subgroups.

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On certain varieties attached to a Weyl group element

2010-01-01

G. Lusztig

Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of … Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of G, B is a Borel subgroup of G and B,gBg^{-1} are in relative position w. Then G acts naturally on X. Assume that w has minimal length in its conjugacy class. We show that the set of G-orbits in X has a well defined structure of an affine algebraic variety V. When G is a classical group we show that this variety is an affine space modulo the action of a finite diagonalizable group. In this case we also construct some nontrivial automorphisms of X.

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