Simple G-modules Research Articles

Contramodules for algebraic groups: Induction and projective covers

In this paper we investigate contramodules for algebraic groups. Namely, we give contramodule analogs to two 20th century results about comodules. Firstly, we show that induction of contramodules over coordinate rings of algebraic groups is exact if and only if the associated quotient variety is affine. Secondly, we give an inverse limit theorem for constructing projective covers of simple G-modules using G-structures of projective covers of simple modules of the first Frobenius kernel, G1. We conclude by showing that the inverse limit theorem is a special case of a more general phenomenon between injective covers in k[G]-Comod and projective covers in k[G]-Contra.

On character formulas for simple and tilting modules

We show that the characters of tilting modules can be used, in a concrete and explicit way, to obtain the simple characters of a connected reductive algebraic group G over an algebraically closed field k of characteristic p, for all p. Thus, once an algorithm for computing the characters of the indecomposable tilting G-modules has been found, an algorithm for the simple characters has been also. An immediate implication is that work by Achar, Makisumi, Riche, and Williamson can be used to obtain the characters for simple G-modules when p>h, the Coxeter number of G.

INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS BY CLASSICAL GROUPS

Let W be a finite-dimensional representation of a reductive algebraic group G. The invariant Hilbert scheme \( \mathcal{H} \) is a moduli space that classifies the G-stable closed subschemes Z of W such that the affine algebra k[Z] is the direct sum of simple G-modules with prescribed multiplicities. In this article, we consider the case where G is a classical group acting on a classical representation W and k[Z] is isomorphic to the regular representation of G as a G-module. We obtain families of examples where \( \mathcal{H} \) is a smooth variety, and thus for which the Hilbert–Chow morphism \( \gamma :\mathcal{H}\to W//G \) is a canonical desingularization of the categorical quotient.

Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type

Let \(\mathbb{F}\) be an algebraically closed field of prime characteristic p > 2, and g be a simple Lie superalgebra of special type or Hamiltonian type over \(\mathbb{F}\). We construct the simple g-modules with non-singular characters of height more than one, and some simple modules with singular characters of height more than five. Furthermore, for the case of special type Lie superalgebras, we also construct a class of simple modules with regular semisimple characters of height one. All those simple modules mentioned above are proved to be reduced Kac modules.

Simple modules of exceptional groups with normal closures of maximal torus orbits

Let G be an exceptional simple algebraic group, and let T be a maximal torus in G. In this paper, for every such G, we find all simple rational G-modules V with the following property: for every vector v ∈ V, the closure of its T-orbit is a normal affine variety. To solve this problem, we use a combinatorial criterion of normality formulated in terms of weights of simple G-modules. This paper continues the works of the second author in which the same problem was solved for classical linear groups.

A Gelfand Model for Weyl Groups of Type D2n

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.

Unbounding Ext

We produce examples in the cohomology of algebraic groups which answer two questions of Parshall and Scott. Specifically, if G=SL2, then we show: (a) dimExtG2(L,L) can be arbitrarily large for a simple module L; and (b) if we define γm=maxLdimHm(G,L) where the maximum is taken over all simple G-modules L, then the sequence {γm} grows exponentially fast with m.

Support varieties for Frobenius kernels of classical groups

Let G be a classical simple algebraic group over an algebraically closed field k of characteristic p>0, and denote by G(r) the r-th Frobenius kernel of G. We show that for p large enough, the support variety of a simple G-module over G(r) can be described in terms of support varieties of simple G-modules over G(1). We use this, together with the computation of the varieties VG(1)(H0(λ)), given by Jantzen (1987) in [8] and by Nakano et al. (2002) in [10], to explicitly compute the support variety of a block of Dist(G(r)).

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x 1, x 2, . . . , x n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x 1, x 2, . . . , x n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.

Actions of the Derived Group of a Maximal Unipotent Subgroup on G-Varieties

Let U be a maximal unipotent subgroup of a connected semisimple group G and U′ the derived group of U. We study actions of U′ on affine G-varieties. First, we consider the algebra of U′ invariants on G/ U. We prove that is a polynomial algebra of Krull dimension 2r, where r = rk(G). A related result is that, for any simple finite-dimensional G-module V, is a cyclic U/ U′-module. Second, we study “symmetries” of Poincare series for U′-invariants on affine conical G-varieties. The results we obtain are very similar to those for the algebras of U-invariants. Third, we obtain a classification of simple G-modules V with polynomial algebras of U′-invariants (for G simple).

On the coordinate ring of spherical conjugacy classes

Let G be a simple algebraic group over an algebraically closed field k of characteristic zero and $${\mathcal{O}}$$ be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring $${k[\mathcal{O}]}$$ of $${\mathcal{O}}$$ into simple G-modules.

Tensor product decompositions and open orbits in multiple flag varieties

For a connected semisimple algebraic group G, we consider some special infinite series of tensor products of simple G-modules whose G-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open G-orbits in multiple flag varieties and address the problem of classifying such series.

Good l-filtrations for q- [formula omitted

Let k be an algebraically closed field of characteristic p, possibly zero, and G = q - GL 3 ( k ) , the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take G to be GL 3 ( k ) . We first determine the extensions between simple G-modules for both G and G 1 , the first Frobenius kernel of G. We then determine the submodule structure of certain induced modules, Z ˆ ( λ ) , for the infinitesimal group G 1 B . We induce this structure to G to obtain a good l-filtration of certain induced modules, ∇ ( λ ) , for G. We also determine the homomorphisms between these induced modules for G.

Le schéma Quot invariant

Given an affine scheme X with an action of a reductive group G and a G-linearized coherent sheaf M, we construct the “invariant Quot scheme” that parametrizes the quotients of M whose space of global sections is a direct sum of simple G-modules with fixed finite multiplicities.Then we determine the invariant Quot scheme in a simple situation, where X is the cone of primitive vectors of a simple G-module and M is the free sheaf on X generated by another simple G-module. This invariant Quot scheme has only one point, that is reduced in most of the cases. The only cases where it is not reduced occur when X is the cone of primitive vectors of a quadratic vector space V of odd dimension, under the action of Spin(V).

A note on the tensor product of restricted simple modules for algebraic groups

Let G be a semisimple simply connected algebraic group over an algebraically closed field of positive characteristic p. Denote by G 1 its first Frobenius kernel. In this note, we determine for which group G the restriction to G 1 of any indecomposable G-summand of the tensor product of any two restricted simple G-modules remains indecomposable.

Casimir Operators and Monodromy Representations of Generalised Braid Groups

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections ∇κ on h with values in any finite-dimensional g-module V and simple poles on the root hyperplanes. The corresponding monodromyre presentation of the braid group Bg of type g is a deformation of the action of (afinite extension of) W on V. The residues of ∇κ are the Casimirs κα of the subalgebra sslα2 ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊂= V under the κα implies that, for generic values of the parameter, the braid group Bg acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g-modules if g = sl3 but that this is not the case if g ≇ sl2,sl3. We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin’s braid group Bn on the zero-weight spaces of all simple Usln-modules for n≥4. Finally, we study their reducibility of the action of the Casimirs on the zero-weight spaces of self-dual g-modules and obtain complete classification results for g = sln and g2.

Complexity of Homogeneous Spaces and Growth of Multiplicities

The complexity of a homogeneous space G/H under a reductive group G is by definition the codimension of general orbits in G/H of a Borel subgroup B\subseteq G. We give a representation-theoretic interpretation of this number as the exponent of growth for multiplicities of simple G-modules in the spaces of sections of homogeneous line bundles on G/H. For this, we show that these multiplicities are bounded from above by the dimensions of certain Demazure modules. This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity.

On the good filtration dimension of Weyl modules for a linear algebraic group

Let G be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible. In this paper we calculate the Weyl filtration di- mension of the induced G-modules,∇(�) and the simple G-modules L(�), fora regular weight. We use this to calculate some Ext groups of the form Ext � ∇(�),�(µ) � , ExtL(�),L(µ) � , and Ext � ∇(�),∇(µ) � , where �,µ are regular and �(µ) is the Weyl module of highest weight µ. We then deduce the projective dimensions and injective dimensions for L(�), ∇(�) and �(�) for � a regular weight in associated generalised Schur algebras. We also deduce the global dimension of the Schur algebras for GLn, S(n,r), when p > n and for S(mp,p) with m an integer.

On Extensions of Simple Modules over Symmetric and Algebraic Groups

Here and below the cohomology of algebraic groups is the rational cohomology (see [13, I.4]). This result is quite useful for calculating some Ext-groups between simple G-modules. For example, it immediately implies that ExtGLλ‘; Lλ‘‘ = 0: We would be interested in having a result similar to (1) for the symmetric group 6n: For a partition λ of n we denote by S the corresponding Specht module, and, if λ is p-regular, we write D for the corresponding irreducible module over F6n (see [11]). It is known that the Specht modules play a role similar to that of Weyl modules for algebraic groups. So we are interested in results connecting Ext6nD;D‘ and Hom6nrad Sλ;Dμ‘. The first main result of the paper is

Complexity for modules over finite Chevalley groups and classical Lie algebras

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,