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G. Lusztig

Publications and source records attributed to G. Lusztig.

At least 127 records · Page 7Linked to original sources
arXiv2009

Study of antiorbital complexes

Let E be a finite dimensional vector space over an algebraic closure of a finite field with a given linear action of a connected linear algebraic group K and let E' be the dual space. A complex of l-adic sheaves on E is said to be orbital if it is a simple perverse sheaf whose support is a single K-orbit. A complex of l-adic sheaves on E is said to be biorbital if it is orbital and if its Deligne Fourier transform is orbital on E'. In this paper we study examples of biorbital complexes arising in the case where E is an eigenspace of a semisimple automorphism of a reductive Lie algebra.

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arXiv2009

From groups to symmetric spaces

In this paper we examine various properties/constructions which are known for reductive groups and we do some experiments to see to what extent they generalize to symmetric spaces.

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arXiv2009

Unipotent elements in small characteristic, IV

We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth subvarieties indexed by the unipotent classes in the corresponding group over complex numbers. We obtain explicit results in type A,C and D.

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arXiv2009

On some partitions of a flag manifold

In an earlier paper by Kazhdan and the author, a map from the set of unipotent classes in a reductive connected group over C to the conjugacy classes in the Weyl group was defined. Here we present some experimental evidence for a possibly alternative definition of that map in terms of the pieces in certain partitions of the flag manifold.

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arXiv2008

Unipotent elements in small characteristic, III

We give a uniform description of the decomposition of the unipotent variety of a classical group in arbitrary characteristic into pieces (considered in a non-uniform way in the earlier parts of this paper).

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arXiv2008

Remarks on Springer's representations

We give an apriori description of a set of irreducible representations of a Weyl group which parametrize the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre concerning the conjugacy class of a power of a unipotent element in a connected reductive group.

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arXiv2008

Character sheaves on disconnected groups, X

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

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arXiv2008

Notes on character sheaves

In the first section we study a functor of Bezrukavnikov, Finkelberg and Ostrik defined on character sheaves; we compute it in a Grothendieck group taking weights into account. In the second section we enlarge the class of character sheaves to a larger class of simple perverse sheaves which behaves well under tensor product (unlike the character sheaves themselves).

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arXiv2007

A survey of total positivity

We survey the history of totally positive matrices and the generalization to Lie groups. We describe a reduction of a bilinear form to a canonical form (generalizing the case of symplectic nondegenerate forms) using ideas from total positivity; we also place this in a Lie group context. We give a short exposition of results of Fock and Goncharov on the study of homomorphisms of the fundamental group of a closed surface into a Lie group.

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arXiv2007

Irreducible representations of finite spin groups

We perform the computations necessary to establish a multiplicity one statement for the irreducible representations of a finite spin group which in turn yields the classification of irreducible representations of finite spin groups. (The multiplicity one statement was previously stated without proof.)

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arXiv2006

Unipotent elements in small characteristic, II

Let G be a special orthogonal group over an algebraically closed field of characteristic exponent p. In this paper we extend certain aspects of the Dynkin-Kostant theory of unipotent elements in G (when p=1) to the general case (including p=2).

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arXiv2006

A class of perverse sheaves on a partial flag manifold

We introduce a class of perverse sheaves on a partial flag manifold of a connected reductive group G defined over a finite field which are equivariant under the action of the group of rational points of G. The definition of this class is similar to the definition of parabolic character sheaves.

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arXiv2006

Graded Lie algebras and intersection cohomology

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum decomposition of LG determined by i. Assume that n is not zero. For any G^i-orbit O in L_nG and any irreducible G^i-equivariant local system L on O we consider the restriction of some cohomology sheaf of the intersection cohomology complex of the closure of O with coefficients in L to another orbit O' contained in the closure of O. For any irreducible G^i-equivariant local system L' on O' we would like to compute the multiplicity of L' in that restriction. We present an algorithm which helps in computing that multiplicity.

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