Lusztig Conjecture Research Articles

On the decomposition matrices of the quantized Schur algebra

We prove the decomposition conjecture of Leclerc and Thibon for the Schur algebra. We also give a new approach to the Lusztig conjecture for the dimension of the simple U(sl_k)-modules at roots of unity via canonical bases of the Hall algebra.

Upper and Lower Bounds for Kazhdan–Lusztig Polynomials

We give upper and lower bounds for the Kazhdan–Lusztig polynomials of any Coxeter groupW. IfWis finite we prove that, for any k≥0, thekth coefficient of the Kazhdan–Lusztig polynomial of two elementsu,vofWis bounded from above by a polynomial (which depends only onk) inl(v) −l(u). In particular, this implies the validity of Lascoux–Schutzenberger's conjecture for all sufficiently long intervals, and gives supporting evidence in favour of the Dyer–Lusztig conjecture.

D‐Affinity and Toric Varieties

1991 Mathematics Subject Classification 32C38.

The Kazhdan–Lusztig conjecture for 𝒲 algebras

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of 𝒲 algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, which constructs the 𝒲 algebras from affine Kac–Moody (KM) algebras and in a similar fashion 𝒲 modules from KM modules. Assuming certain properties of this functor, the 𝒲 characters are subsequently derived from the Kazhdan–Lusztig conjecture for KM algebras. The result can be formulated in terms of a double coset of the Weyl group of the KM algebra: the Hasse diagrams give the embedding diagrams of the Verma modules and the Kazhdan–Lusztig polynomials give the multiplicities in the characters.

On the computation of unipotent characters of finite classical groups

We determine the constants appearing in the Lusztig conjecture in the case of unipotent characters of classical groupsG subject to the condition that the characteristic is odd and the center ofG is connected. This enables us to compute the character values of unipotent characters of such finite classical groups.

Proof of the Kazhdan–Lusztig Conjecture for Kac–Moody Algebras (The CharacterschLωρ−ρ)

Proof of the Kazhdan–Lusztig Conjecture for Kac–Moody Algebras (The CharacterschLωρ−ρ)

KOSZUL ALGEBRAS AND THE FROBENIUS AUTOMORPHISM

LET %’ be a highest weight category [CPSl] with finitely many simple objects and such that every object has finite length. Then Ce = mod S for a finite dimensional quasi-hereditary algebra S. In a series of papers, [CPS4,5,6,7], we have studied various Kazhdan-Lusztig theories for %. This concept arises naturally in, for example, the modular representation theory of semisimple algebraic groups. As shown in [CPS4; § 41, the existence of a Kazhdan-Lusztig theory in this case is equivalent to the validity of the Lusztig conjecture (and leads to a considerable simplification of that conjecture). Also, the theory permits a combinatorial calculation of the Yoneda extension groups for irreducible representations with regular high weight in the Jantzen region. More generally, as shown in [CPS7; 5 21, if (e has a Kazhdan-Lusztig theory, its homological dual is also a highest weight category. When the algebra S is (positively) graded, the associated category ?& of finitely generated, graded S-modules has the structure of a graded highest weight category. (For a discussion of this notion, see [CPS7; $ 11.) By analogy with the ungraded theory, one can consider the notion of a graded Kazhdan-Lusztig theory. This concept has been developed in detail in [CPS4,7]. The property of having a graded Kazhdan-Lusztig theory is closely related to another property of graded algebras: a positively graded algebra S is called Koszul provided each irreducible S-module L, when regarded as a graded S-module concentrated in degree 0, has a projective resolution * * * --+ P-‘-+ PO+ L + 0 in %& in which each P-” is generated by its term in grade n. (see [P], [BGS].) A main result in [CPS’I; (3.9)] establishes that if %& has a graded KazhdanLusztig theory, the associated algebra S is Koszul. The converse also holds in the presence of an (ungraded) Kazhdan-Lusztig theory on (e. We begin in Q 1 by formulating a new criterion guaranteeing that a (finite dimensional) k-algebra S is Koszul. Our result states essentially that, given an algebra S and an automorphism (7, if the Ext-algebra associated to S can be given its natural graded structure by means of the

Dimensions of the Simple Restricted Modules for the Restricted Contact Lie Algebra

According to [2], the finite-dimensional restricted simple Lie algebras over an algebraically closed field of characteristic p > 7 are either classical or of Cartan type. Lusztig has a conjecture (see [8, p. 294]) which provides a formula for the dimensions of the simple modules for the classical algebras. (In this paper, modules are assumed to be left, finitedimensional and restricted.) Anderson, Jantzen, and Soergel [1] have recently shown that for p A 0 the Lusztig conjecture follows from an analogous conjecture (also due to Lusztig) for certain quantum groups. They have also pointed out that this latter conjecture is implied by recent work of Kazhdan and Lusztig and a paper by Casian, although Casian’s paper has not yet convinced all of its readers. The simple modules for the Cartan-type algebras fall into two classes. There is a large class (see [11] and [5]) consisting of modules that are induced from simple modules for the homogeneous component of degree zero extended trivially to positive components (constructed as “mixed products” in [11]). The degree-zero homogeneous components of Cartan-type algebras are either classical or nearly so. Therefore, the Lusztig conjecture can be used to get a formula for the dimensions of the modules in this class. The small class consists of modules corresponding to “exceptional” weights. In [11] the dimensions of these modules were determined for three of the four Cartan-type algebras, namely, the Witt, special and hamiltonian algebras. (There the exceptional weights turned out to be fundamental dominant weights.) In this paper, we obtain the dimensions (assuming p is not too small) in the remaining case of the contact algebra. In the process, we determine the contragredients of the simple modules.

Socle series of cohomology groups of line bundles on G / B

The G-module structure of the cohomology groups of line bundles over the flag variety G/B for a semisimple algebraic group G is studied. It is proved that, generically, their socle filtrations satisfy the same sum formula as Andersen's filtrations if Lusztig's conjecture is true. Another purpose of this paper is to calculate the socle series of the H 0's for the p-regular weights close to a chamber wall for the group of type G 2, because the multiplicity is not free in this case.

The joint relations and the set [formula omitted]1 in certain crystallographic groups

For any Coxeter group W, D. Kazhdan and G. Lusztig (1979, Invent. Math. 53, 165–184) introduced some joint relations among the elements of W by which certain representations of W and its associated Hecke algebra H are constructed. Recently, in a letter to Lusztig, T. A. Springer obtained some interesting results concerning the connection between some non-zero structural coefficients of H and the joint relations in W when W is a certain crystallographic group. He investigated some properties of a subset D 1 of W (Springer denoted D 1 by F in his letter) which plays an important role for the joint relations in W. In the present paper, we shall develop some results of Springer. We shall deduce further joint relations in W from some non-zero structural coefficients of H . In particular, we shall make some detailed discussion for the set D 1. This includes the approach to the proof of a conjecture of Lusztig concerning the characterization of the set D 1, the connection between the finiteness of D 1 and some joint relations in W, the explicit description for some elements of D 1, and the connectedness of the set D 0(Ω) ∪ D (2) 1(Ω) (see 3.2 and 5.2 for the notations).

Extensions of simple modules for finite Chevalley groups

Let G be an almost simple algebraic group over an algebraically closed field k of positive characteristic p. We assume that G is defined and split over the prime field 5,. For it 3 1 we denote by G(n) the finite group consisting of the points of G over the field with p” elements. Our first result states that for a certain class of simple modules for G(n), the only extensions that can occur are those which are restrictions of G-extensions, see Theorem 2.8. The full story about G-extensions is not known yet except for a few rather small groups. However, much more is known about extensions for the algebraic group than about those for the finite group. For instance, the linkage principle [l] gives an upper bound for which G-extensions may occur. Moreover, a proof of the Lusztig conjecture on the irreducible characters for G also implies strong statements about G-extensions (see [3, Proposition 2.81). It is an open question whether all extensions between simple G(n)-modules can be found via the extension theory for G. We prove that in generic cases this is in fact so (see Theorem 3.2). Moreover, for the groups X(3, p) and Sp(4, p) we are able to determine completely all extensions between simple modules from the corresponding (known) results for the algebraic groups X.(3, k) and Sp(4, k). Our results extend previous results by Cline, Parshall and Scott [S, 91 (who consider extensions of the trivial module by a “minimal” module) and Smith [lS] (who considers extensions between certain “minimal” modules). The method we use is a generalization of the one used by Humphreys [ 131 (who proves the existence of certain “selfextensions” for SP(4, P)). It should be pointed out that our methods do not work for p small. On the other hand, everything we do generalizes easily to the twisted Chevalley groups (see Theorem 3.4). We demonstrate this by working out all extensions for SU(3, p) (see Section 4.3).

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):

Extensions of Modules for Algebraic Groups

Let G be a connected semi-simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p. In this paper we show how one can obtain the extensions of any two simple G-modules if one knows the socles of all the G-modules ExtG, (L, L ') where G1 denotes the kernel of the Frobenius endomorphism on G and L as well as L' are simple G1-modules. Alternatively, we prove how all such G-extensions can be found once one knows the extensions of any two simple G-modules belonging to a specified (finite) set of simple modules with small highest weight. Such information is available in certain low rank cases. It should also be pointed out that it is possible from Lusztig's conjecture on irreducible characters [11] to derive the dimensions of Ext (L1, L2) when L1 and L2 are simple with such small highest weights. To obtain our results on Exth we need to know that there exists no G1-extension of a simple module by itself. This turns out to be somewhat

On Ext1 for semisimple groups and infinitesimal subgroups

Let G be an affine, connected, semisimple, simply connected algebraic group over an algebraically closed field k of non-zero characteristic p and, for a positive integer n, let Gn be the nth infinitesimal subgroup (the kernel of the nth power of the Frobenius morphism). The purpose of this note is to point out that, for certain rational G-modules V, the natural homomorphismis the zero map, for m ≽ 0. For such modules V, taking m = 0, we see from the 5-term exact sequence of Lyndon-Hochschild-Serre that the inflation-restriction sequenceis exact and thus the dimension of H1(G, V), when finite, may be calculated by adding the dimensions of the outer terms. The practical value of this result seems to lie in the special case n = 1 and V = L(λ) ⊗ L(µ), the tensor product of simple rational G-modules. In this case H1(G/G1) VG1 is H1(G, L(λ′) ⊗ L(¯′)) for λ′, ¯′ ‘smaller’ than λ µ and the above exact sequence gives an inductive procedure for computing Ext1G between simple rational (G-modules once the (G-module structure of Ext1G1 (equi valently Ext1u1 where% is the restricted enveloping algebra of the Lie algebra of G), for simple G1-modules, is known. Using the rather complicated structure of Weyl modules Cline, in (4), calculated Ext1G between simple modules for G = SL2(k) (this process was reversed in (5)). The present note has been used by S. EIB. Yehia, (17), in finding the corresponding result for SL3(k). We believe these Ext groups to be of increasing importance in view of their connection with Lusztig's conjecture, (14), on the characters of the simple rational G-modules and the related conjecture, 7–2 of (l), of Andersen.

On the structure of Weyl modules

Let M x (resp. Vx) denote the irreducible module (resp. the Weyl module) with highest weight Z of a connected reductive algebraic group G over k, with k an algebraically closed field of characteristic p >0 . In [3] we proved that a necessary condition for M~ to be a composition factor of V X is that 2 is strongly linked to )6 We proved more generally that the same is true for all cohomology groups of the line bundles on G/B, B a Borel subgroup, induced by )~ and its conjugates under the Weyl group W. Even though Very little is known about the higher cohomology groups of line bundles on G/B (in particular it is not known exactly when they are indecomposable as G-modules) the existence of certain canonical maps between them enables us to deduce some properties of their composition factors and of the structure of Weyl modules. In this paper we shall explore the method from [3] somewhat further. First we obtain an estimate on the set of weights in HI(D, i>0, ;g a character o r b (as in [3] we write Hi00 short for Hi(G/B, L00)). We prove that if )~ is dominant then Z is the unique highest weight in H~(W)(w.z) for all w~W, and it occurs with multiplicity 1. Here ~(w) denotes the length of w and so in characteristic zero, Ht('~)(w.z) is the unique non-vanishing cohomology group [7]. Next we prove that when 2 and X are dominant and 2 is maximal among the weights strongly linked to ;g then M~ occurs exactly once as a composition factor of V z. For p-regular weights this result was proved by Jantzen [9, Satz 10] (see also Carter's and Lusztig's results for groups of type A, [5, Theorem 4.1]). The above result on the highest weight in Ht(W)(w.x) gives a canonical homomorphism Ht{W)(w.z)-~H~ In this way we obtain for each reduced expression for w0, the longest element in W, a filtration of H~ (and of V.~). Using these filtrations we are able to prove the conjecture of Carter and Lusztig [5, p. 239] about intertwining homomorphisms between certain Weyl modules in most cases. We conclude the paper with a general formula for the character of the image of intertwining homomorphisms in the direction of a simple root. As illustrations we give details in the 2 simplest non-trivial cases: type A 2 and B2, Z in the bottom p<alcove.