Induced Unipotent Classes

Intersection cohomology complexes on a reductive group.

The term “nilpotent orbits” in the title is an abbreviation for “orbits consisting of nilpotent elements.” We shall consider here such orbits only for the adjoint action of a reductive algebraic group on its Lie algebra.

Nilpotent Orbits in Representation Theory

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Cuspidal local systems and graded Hecke algebras, I

Surface operators in gauge theory are analogous to Wilson and ’t Hooft line operators except that they are supported on a two-dimensional surface rather than a one-dimensional curve In a previous paper, we constructed a certain class of half-BPS surface operators in N = 4 super Yang–Mills theory, and determined how they transform under S-duality Those surface operators depend on a relatively large number of freely adjustable parameters In the present paper, we consider the opposite case of half-BPS surface operators that are “rigid” in the sense that they do not depend on any parameters at all We present some simple constructions of rigid half-BPS surface operators and attempt to determine how they transform under duality This attempt is only partially successful, suggesting that our constructions are not the whole story The partial match suggests interesting connections with quantization We discuss some possible refinements and some string theory constructions which might lead to a more complete picture

background

/pdf/rigid-surface-operators-1lwvrqu27y.pdf

Rigid Surface Operators

algebraic group over R or C. In this paper, we restrict attention to C. We generalize Arthur's definition slightly (or perhaps simply make it more precise). All of the resulting representations, except for a finite set, are then unitarily induced from representations of the same kind on proper parabolic subgroups. We call the finite set remaining special unipotent representations; a precise definition will be given later (Definition 1.17). Our main result (Theorem III of this introduction) is a character formula for any special unipotent representation. Of course such a formula can be deduced from the Kazhdan-Lusztig conjecture (cf. [V3]). The advantages of Theorem III are that it is in closed form, and that it lends itself to verification of some conjectures of Arthur in [A]. So let G be a connected complex semisimple Lie group, and q its Lie algebra. Choose

Induced Unipotent Classes

Citations

Intersection cohomology complexes on a reductive group.

Nilpotent Orbits in Representation Theory

Cuspidal local systems and graded Hecke algebras, I

Rigid Surface Operators

Unipotent representations of complex semisimple groups

Related Papers (5)

Classes unipotentes et sous-groupes de Borel

Nilpotent Orbits In Semisimple Lie Algebra: An Introduction

Intersection cohomology complexes on a reductive group.

Finite Groups of Lie Type: Conjugacy Classes and Complex Characters

Simple singularities and simple algebraic groups