Kazhdan-Lusztig Polynomials Research Articles

Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories

Let $X$ be an irreducible algebraic variety defined over a field $k$, let $\mathcal {R}$ be a sheaf of (noncommutative) noetherian $k$-algebras on $X$ containing the sheaf of regular functions $\mathcal {O}$ and let $R$ be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of $R$ obtained from the local sections of $\mathcal {R}$) there is an equivalence between the category of $R$-modules and the category of sheaves of $\mathcal {R}$-modules which are quasicoherent as $\mathcal {O}$-modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where $R$ is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and $\mathcal {R}$ is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.

On the Kazhdan-Lusztig polynomials for affine Weyl groups

In 191, Kazhdan and Lusztig associated to a Coxeter group W certain polynomials P,,, (y, w E W). In the case where W is a Weyl group, these polynomials (or their values at 1, P,,,,(l)) give multiplicities of irreducible constituents in the Jordan-Holder series of Verma modules for a complex semisimple Lie algebra corresponding to W (see [ 3,4]). Also, in the case where W is an affine Weyl group, it has been conjectured by Lusztig [lo] that these polynomials or their values P,,,(l) enter the character formula of irreducible rational representations for a semisimple algebraic group (corresponding to W) over an algebraically closed field of prime characteristic. The purpose of this paper is to give some formula (4.2, see also 4.10) concerning the Kazhdan-Lusztig polynomials P,,, for affine Weyl groups which appear in this Lusztig conjecture; namely P,,, for y, w “dominant.” (This is a generalization of the q-analogue of Kostant’s weight multiplicity formula in [8].) As a corollary of the formula, we shall show that the Lusztig conjecture above is consistent with the Steinberg tensor product theorem [ 15 1. To be more precise, let P be the weight lattice of a root system R (II) R ‘, a positive root system). The Weyl group W of R acts on P. Hence we can define the semidirect product of W by P, p= W D( P. The element of @ corresponding to A E P is denoted by t,. The group @ contains an afline Weyl group W, = W D( Q as a normal subgroup, where Q is the root lattice. Although # is not a Coxeter group in general, we can define the Bruhat order >, the length function I: p+ La, and the Kazhdan-Lusztig polynomials P,., (y, w E I-V) as natural extensions of those for W,. The main result of this paper (under the specialization q --+ 1) is stated as follows (Corollary 4.10):

Unipotent Representations of Complex Semisimple Groups

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

Projective modules in the category 𝒪_{𝒮}: Loewy series

Let g \mathfrak {g} be a complex, semisimple Lie algebra with a parabolic subalgebra p S {\mathfrak {p}_S} . The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in O S {\mathcal {O}_S} are studied with primary emphasis on the case in which p S {\mathfrak {p}_S} is a Borel subalgebra and O S {\mathcal {O}_S} is the category O \mathcal {O} . An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in O \mathcal {O} , assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan’s statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in O S {\mathcal {O}_S} . Under the stronger assumption of Jantzen’s conjecture, the radical and socle series are computed for self-dual projectives in O \mathcal {O} . An analogous result is formulated for self-dual projectives in O S {\mathcal {O}_S} and proved in certain cases.

Harish-Chandra modules with the unique embedding property

Let G G be a connected semisimple real matrix group. In view of Casselman’s subrepresentation theorem, every irreducible admissible representation of G G may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.

Harish-Chandra Modules with the Unique Embedding Property

Let $G$ be a connected semisimple real matrix group. In view of Casselman’s subrepresentation theorem, every irreducible admissible representation of $G$ may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.

Highest Weight Modules Over Graded Lie Algebras: Resolutions, Filtrations and Character Formulas

In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups. The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen’s character sum formula for a quotient of two Verma modules (one of the main results of this article). Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).

Primitive ideals and orbital integrals in complex exceptional groups

In [2], two related problems were studied for complex classical groups: determination of the primitive spectrum of the enveloping algebra (as a set), and Fourier inversion of unipotent orbital integrals. The first of these was solved completely (extending work of Joseph and others in SL(n, C) and small groups). Although our techniques presumably solve the second as well, up to determination of normalizing constants, we carried out the calculations only for special unipotent classes (see [ 181). We had, of course, tried to apply the same methods to the exceptional groups, but they seem to be inadequate. Since that work was done, however, Brylinski and Kashiwara and Beilinson and Bernstein have established the Kazhdan-Lusztig conjecture, giving character formulas for irreducible highest weight modules [3, 71. Because of a conjecture of Joseph proved in [24], this determines in principle the primitive ideals with a fixed regular integral infinitesimal character: they are in one-to-one correspondence with what Kazhdan and Lusztig call left cells in the Weyl group. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated, requiring one to compute roughly 1 I%‘]* polynomials of degrees on the order of half the number of positive roots. This is very unsatisfactory. However, it

Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas

In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups. The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen’s character sum formula for a quotient of two Verma modules (one of the main results of this article). Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).

Goldie rank in the enveloping algebra of a semisimple Lie algebra, I

Let g be a complex semisimple Lie algebra,U (g) the enveloping algebra ofg, and PrimU (g) the set of primitive ideals ofU (g). In this two-part series the Goldie ranks of the primitive quotients {U(g)/J :J ∈ PrimU(g)} are computed. Lethbe a Cartan subalgebra for g and h* the dual of h. After Duflo [4] there exists a surjection λ ↦ J(λ) of h* onto PrimU(g). In the first part it is shown thath* can be written as a disjoint union of infinite subsets with the following property. On each subset Λ there exists a polynomialpΛ such that for allλ ∈ Λ, pΛ(λ) is the Goldie rank ofU(g). In Part II the computation of thepΛ is reduced to a knowledge of the multiplicities of the simple factors of the Verma modules.