Baby Verma Modules Research Articles

A note on the Loewy lengths of baby Verma modules for modular Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and g=Lie(G). We study the representations of the reductive Lie algebra g with p-character χ of standard Levi-form in this note. We obtain the similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. [16, II.Lem.7.20]). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.

The First Cohomology of o s p ( 1,2 ) with Coefficients in Baby Verma Modules and Simple Modules

Over a field of characteristic [Formula: see text], the first cohomology of the orthogonal symplectic Lie superalgebra [Formula: see text] with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of [Formula: see text]. As a byproduct, the first cohomology of [Formula: see text] with coefficients in the restricted enveloping algebra (under the adjoint action) is not trivial.

On Donkin’s tilting module conjecture I: lowering the prime

In this paper the authors provide a complete answer to Donkin’s Tilting Module Conjecture for all rank 2 2 semisimple algebraic groups and SL 4 ⁡ ( k ) \operatorname {SL}_{4}(k) where k k is an algebraically closed field of characteristic p > 0 p>0 . In the process, new techniques are introduced involving the existence of ( p , r ) (p,r) -filtrations, Lusztig’s character formula, and the G r G_{r} T-radical series for baby Verma modules.

Explicit calculations in an infinitesimal singular block of SLn

AbstractLet $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$, there exists a singular block in the category of $G_1$-modules, which contains precisely $n+1$ irreducible modules. We are interested in the ‘lift’ of this block to the category of $G_1T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.

A simple character formula

In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than 2h-1, where h is the Coxeter number of h. This provides a replacement for Lusztig's conjecture, valid under a reasonable assumption on the characteristic.

A criterion for irreducibility of parabolic baby Verma modules of reductive Lie algebras

Let G be a connected, reductive algebraic group over an algebraically closed field k of prime characteristic p and g=Lie(G). In this paper, we study representations of g with a p-character χ of standard Levi form. When g is of type An,Bn,Cn or Dn, a sufficient condition for the irreducibility of standard parabolic baby Verma g-modules is obtained. This partially answers a question raised by Friedlander and Parshall (1990) in [2]. Moreover, as an application, in the special case that g is of type An or Bn, and χ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen (1999) in [6].

Jantzen filtration and strong linkage principle for modular Lie superalgebras

Abstract In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and obtain a sum formula in the corresponding Grothendieck groups. Consequently, we formulate a strong linkage principle for such categories.

Representations of 𝔰𝔩3 in characteristic 3

Let [Formula: see text] be the special linear Lie algebra [Formula: see text] of rank 2 over an algebraically closed field [Formula: see text] of characteristic 3. In this paper, we classify all irreducible representations of [Formula: see text], which completes the classification of the irreducible representations of [Formula: see text] over an algebraically closed field of arbitrary characteristic. Moreover, the multiplicities of baby Verma modules in projective modules and simple modules in baby Verma modules are given. Thus we get the character formulae for simple modules and the Cartan invariants of [Formula: see text].

On Ext-transfer for Reductive Lie Algebras

Let G be a connected reductive algebraic group over an algebraically closed field of prime characteristic p and 𝔤 be the Lie algebra of G. In this paper, we study the representations of 𝔤 when p-character has standard Levi form. An Ext-transfer from the Ext-groups of induced 𝔤-modules to its Levi subalgebras is obtained. Furthermore, we reduce the computation of the multiplicities of simple factors in baby Verma modules over 𝔤 to its Levi subalgebras.

On representations of restricted Lie superalgebras

Simple modules for restricted Lie superalgebras are studied. The indecom-posability of baby Kac modules and baby Verma modules is proved in some situation. In particular, for the classical Lie superalgebra of type A(n|0), the baby Verma modules Z χ (λ) are proved to be simple for any regular nilpotent p-character χ and typical weight λ. Moreover, we obtain the dimension formulas for projective covers of simple modules with p-characters of standard Levi form.

Inverse limits in representations of a restricted Lie algebra

Let (g, (p)) be a restricted Lie algebra over an algebraically closed field of characteristic p> 0. Then the inverse limits of "higher" reduced enveloping algebras {uχs (g) | s ∈ N} with χ running over g ∗ make representations of g split into different "blocks". In this paper, we study such an infinite- dimensional algebra Aχ(g) := lim − Uχs (g )f or a givenχ ∈ g ∗ . A module category equivalence is built between subcategories of U (g)-mod and Aχ(g)-mod. In the case of reductive Lie algebras, (quasi) generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.

Complexity, periodicity and one-parameter subgroups

Using the variety of infinitesimal one-parameter subgroups introduced in [55, 56] by Suslin-Friedlander-Bendel, we define a numerical invariant for representations of an infinitesimal group scheme G. For an indecomposable G-module M of complexity 1, this number, which may also interpreted as the height of a “vertex” UM ⊆ G, is related to the period of M . In the context of the Frobenius category of GrT -modules associated to a smooth reductive group G and a maximal torus T ⊆ G, our methods give control over the behavior of the Heller operator of such modules, as well as precise values for the periodicity of their restrictions to Gr. Applications include the structure of stable Auslander-Reiten components of GrT -modules as well as the distribution of baby Verma modules.

Representations of Lie superalgebras in prime characteristic II: The queer series

The modular representation theory of the queer Lie superalgebra q ( n ) over characteristic p > 2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p -characters χ and a criterion for the semisimplicity of the corresponding reduced enveloping algebras U χ ( q ( n ) ) . A ( 2 p ) -power divisibility of dimensions of q ( n ) -modules with nilpotent p -characters is established. The representation theory of q ( 2 ) is treated in detail. We formulate a Morita super-equivalence conjecture for q ( n ) with general p -characters which is verified for n = 2 .

Lusztig's conjecture as a moment graph problem

We show that Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic is equivalent to the generic multiplicity conjecture, which gives a formula for the Jordan-H"older multiplicities of baby Verma modules over the corresponding Lie algebra. Then we give a short overview of a recent proof of the latter conjecture for almost all base fields via the theory of sheaves on moment graphs.

Support varieties, AR-components, and good filtrations

Let G be a reductive group, defined over the Galois field \({\mathbb{F}_p}\) with p being good for G. Using support varieties and covering techniques based on G r T-modules, we determine the position of simple modules and baby Verma modules within the stable Auslander–Reiten quiver Γ s (G r ) of the rth Frobenius kernel of G. In particular, we show that the almost split sequences terminating in these modules usually have an indecomposable middle term. Concerning support varieties, we introduce a reduction technique leading to isomorphisms$$\mathcal{V}_{G_r}(Z_r(\lambda)) \cong \mathcal{V}_{G_{r-d}}(Z_{r-d}(\mu))$$for baby Verma modules of certain highest weights \({\lambda, \mu \in X(T)}\), which are related by the notion of depth.

BABY VERMA MODULES FOR RATIONAL CHEREDNIK ALGEBRAS

This paper introduces baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t = 0 (the so-called rational Cherednik algebras at parameter t = 0, and presents their most basic properties. Baby Verma modules are then used to answer several problems posed by Etingof and Ginzburg, and to give an elementary proof of a theorem of Finkelberg and Ginzburg. 2000 Mathematics Subject Classification 16Rxx, 16S38, 05E10.

Filtrations on G 1 T -Modules

Let G be an almost simple algebraic group defined over Fp for some prime p. Denote by G1 the first Frobenius kernel in G and let T be a maximal torus. In this paper we study certain Jantzen type filtrations on various modules in the representation theory of G1T. We have such filtrations on the baby Verma modules Zλ, where λ is a character of T. They are obtained via a certain deformation of the natural homomorphism from Zλ into its contravariant dual Zλτ. Using the same deformation we construct for each projective G1T-module Q a filtration of the vector space . We then prove that this filtration may also be described in terms of the above-mentioned homomorphism Z(λ) → Z(λ)τ and this leads us to a sum formula for our filtrations. When Q is indecomposable with highest weight in the bottom alcove (with respect to some special point) we are able to compute the filtrations on Fλ(Q) explicitly for all λ. This is then the starting point of an induction which proceeds via wall crossings to higher alcoves. If our filtrations behave as expected under such wall crossings then we obtain a precise relation betweenthe dimensions of the layers in the filtrations of Fλ(Q) for an arbitrary indecomposable projective Q and the coefficients in the corresponding Kazhdan–Lusztig polynomials. We conclude the paper by proving that the above results in the G1T theory have some analogues in the representation theory of G (where, however, we have to work with representations of bounded highest weights) and the corresponding theory for quantum groups at roots of unity. These results extend previous work by the first author. 2000 Mathematics Subject Classification: 20G05, 20G10, 17B37.