Rational G-module Research Articles

Fundamental Theorem of Poisson (A,H)-Hopf modules

Let k be a field, H a Hopf algebra with a bijective antipode and A an H-comodule Poisson algebra. We assume that there is an H-colinear algebra map from H to the Poisson-center of A. We generalize the Fundamental Theorem of (A,H)-Hopf modules to Poisson (A,H)-Hopf modules, and we deduce relative projectivity in the category of Poisson (A,H)-Hopf modules. In many applications, A could be a rational G-module Poisson algebra, where G is an affine algebraic group or the coordinate ring of an affine Poisson variety over C on which acts morphically an algebraic group via automorphisms of Poisson varieties.

Geometric invariants of representations of finite groups

For a finite group τ and a field k of characteristic p dividing the order of τ, we construct a map pτ:Uτ→PYτ of varieties over k with unions of affine spaces as fibers and with (PYτ)/τp-isogenous to ProjH•(τ,k). On Uτ, we construct a “universal p-nilpotent operator” PΘτ which leads to the construction of τ-equivariant coherent sheaves associated to finite dimensional kτ-modules M. These coherent sheaves are algebraic vector bundles on Uτ if M has constant Jordan type. For any finite dimensional kτ-module M, these coherent sheaves are vector bundles when restricted to the complements of the generalized support varieties of M.For a linear algebraic group G, we compare these constructions for the finite group G(Fp) to previous constructions for the infinitesimal group scheme G(r). Our comparison of bundle invariants for a rational G-module M of exponential type <pr upon restriction to G(Fp) and to G(r) is a strengthened form of earlier comparisons of support varieties for M. Our constructions extend to semi-direct products G⋊τ of an infinitesimal group scheme G and a finite group τ, thus to all finite group schemes if k is algebraically closed.

Truncation and the Induction Theorem

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg compares the bounded derived category Db(block(U)) of finite dimensional modules for the principal block of a Lusztig quantum enveloping algebra U at an ℓth root of unity with a special full subcategory Dtriv(B) of the bounded derived category of integrable type 1 modules for a Borel part B⊂U. Specifically, according to this “Induction Theorem” [1, Theorem 3.5.5] the right derived functor of induction IndBU yields an equivalence of categories RIndBU:Dtriv(B)→∼Db(block(U)). Some restrictions on ℓ are required–e.g., ℓ>h, the Coxeter number. It is suggested briefly [1, Remark 3.5.6] that an analog of this equivalence carries over to characteristic p>0 representations of algebraic groups. Indeed, the authors of the present paper have verified, in a separate preprint [6], that there is such an equivalence RIndBG:Dtriv(B)→∼Db(block(G)) relating an analog of Dtriv(B), defined using a Borel subgroup B of a simply connected semisimple algebraic group G, to the bounded derived category of the principal block of finite dimensional rational G-modules. The proof is not without difficulty and supplies new, previously missing details even in the quantum case. The present paper continues the study of the modular case, taking the derived category equivalence as a starting point. The main result here is that, assuming p>2h−2, the equivalence behaves well with respect to certain weight poset “truncations,” making use of a variation by Woodcock [12] on van der Kallen's “excellent order” [10]. This means, in particular, that the equivalence can be reformulated in terms of derived categories of finite dimensional quasi-hereditary algebras. We expect that a similar result holds in the quantum case.In [9, Thm. 2.1] J. Rickard proves a theorem stating existence, in an algebraic groups context, of some derived equivalences with specified corresponding character isometries. In an appendix to this paper, we discuss a similar more recent result [6, Lem. 3.2]. It gives additional information on the behavior of the derived equivalences, giving their action on some natural right-derived induced objects, lifting the given character isometries.

On modules arising from quantum groups at pr-th roots of unity

This paper studies the “reduction mod p” method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group Uζ at a pr-th root of unity. The G-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between ExtGn(V,W) and ExtUζn(V′,W′), when V,W are obtained from V′,W′ by reduction mod p. Since the dimensions of Extn-spaces for Uζ-modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p-th root of unity case and a general pr-th root of unity case are essentially different.

A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup

Let G be a connected, semisimple, simply connected, algebraic group over an algebraically closed field K of positive characteristic. The action of G on itself by conjugation induces on the coordinate algebra K[G], and hence on the coordinate algebra K[Gr] of the infinitesimal subgroup Gr, the structure of a rational G-module. In the recent paper, [3], by Bendel, Nakano, Pillen and Sobaje, the authors ask whether the tensor product of K[Gr] with the rth Steinberg module Str has a good filtration. Here we show that this is the case provided that the characteristic of K is good for G. We do so by relating K[Gr] to a certain quotient of the coordinate algebra K[g] of the Lie algebra g of G and using a Koszul resolution. We note that the argument may also be applied to certain polynomial modules occurring in the context of Doty's Conjecture, [17, 4.2.11].

The Steinberg linkage class for a reductive algebraic group

Let G be a reductive algebraic group over a field of positive characteristic and denote by C(G) the category of rational G-modules. In this note we investigate the subcategory of C(G) consisting of those modules whose composition factors all have highest weights linked to the Steinberg weight. This subcategory is denoted ST and called the Steinberg component. We give an explicit equivalence between ST and C(G) and we derive some consequences. In particular, our result allows us to relate the Frobenius contracting functor to the projection functor from C(G) onto ST .

Filtrations, 1-parameter subgroups, and rational injectivity

We investigate rational G-modules M for a linear algebraic group G over an algebraically closed field k of characteristic p>0 using filtrations by sub-coalgebras of the coordinate algebra k[G] of G. Even in the special case of the additive group Ga, interesting structures and examples are revealed. The “degree” filtration we consider for unipotent algebraic groups leads to a “filtration by exponential degree” applicable to rational G modules for any linear algebraic group G of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational G-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational G-modules. Our investigation leads to the consideration of two new classes of rational G-modules: those that are “mock injective” and those that are “mock trivial”.

On the character of certain tilting modules

Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a (partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.

On the existence of mock injective modules for algebraic groups

Let G be an affine algebraic group scheme over an algebraically closed field k of characteristic p > 0 , and let G r denote the rth Frobenius kernel of G. Motivated by recent work of Friedlander, the authors investigate the class of mock injective G-modules, which are defined to be those rational G-modules that are injective on restriction to G r for all r ⩾ 1 . In this paper, the authors provide necessary and sufficient conditions for the existence of non-injective mock injective G-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.

Invariant forms on irreducible modules of simple algebraic groups

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥0 and let V be an irreducible rational G-module with highest weight λ. When V is self-dual, a basic question to ask is whether V has a non-degenerate G-invariant alternating bilinear form or a non-degenerate G-invariant quadratic form.If p≠2, the answer is well known and easily described in terms of λ. In the case where p=2, we know that if V is self-dual, it always has a non-degenerate G-invariant alternating bilinear form. However, determining when V has a non-degenerate G-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where G is of classical type and λ is a fundamental highest weight ωi, and in the case where G is of type Al and λ=ωr+ωs for 1≤r<s≤l. We also give a solution in some specific cases when G is of exceptional type.As an application of our results, we refine Seitz's 1987 description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If X<Y<SL(V) are simple algebraic groups and V↓X is irreducible, then one of the following holds: (1) V↓Y is not self-dual; (2) both or neither of the modules V↓Y and V↓X have a non-degenerate invariant quadratic form; (3) p=2, X=SO(V), and Y=Sp(V).

Normality and smoothness of simple linear group compactifications

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a projective space of the shape P(End(V)), where V is finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r > 0) is the unique non-adjoint simple group which admits a simple smooth compactification.

Simple modules of classical linear groups with normal closures of maximal torus orbits

Let T be a maximal torus in a classical linear group G. In this paper we find all simple rational G-modules V such that for each vector v ∈ V the closure of the T-orbit of v is a normal affine variety. For every G-module without this property we present a T-orbit with nonnormal closure. To solve this problem, we use a combinatorial criterion of normality which is formulated in terms of the set of weights of a simple G-module. The same problem for G = SL(n) was solved by the author earlier.

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.

On stability of diagonal actions and tensor invariants

For a connected simply connected semisimple algebraic group G we prove the existence of invariant tensors in certain tensor powers of rational G-modules and establish relations between the existence of such invariant tensors and stability of diagonal actions of G on affine algebraic varieties.Bibliography: 12 titles.

Second cohomology for finite groups of Lie type

Let G be a simple, simply-connected algebraic group defined over Fp. Given a power q=pr of p, let G(Fq)⊂G be the subgroup of Fq-rational points. Let L(λ) be the simple rational G-module of highest weight λ. In this paper we establish sufficient criteria for the restriction map in second cohomology H2(G,L(λ))→H2(G(Fq),L(λ)) to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on p and q provided λ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe H2(G(Fq),L(λ)) in terms of rational cohomology for G. We apply our techniques to compute H2(G(Fq),L(λ)) in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.

Restrictions to G(𝔽p) and G(r) of rational G-modules

AbstractWe fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over 𝔽p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(𝔽p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G(𝔽p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(𝔽p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.

Tensoring generalized characters with the Steinberg character

Let G be a reductive connected algebraic group over an algebraic closure of a finite field of characteristic p. Let F be a Frobenius endomorphism on G and write G := G F for the corresponding finite group of Lie type. We consider projective characters of G in characteristic p of the form St ·β, where β is an irreducible Brauer character and St the Steinberg character of G. Let M be a rational G-module affording β on restriction to G. We say that M is G-regular if for every F -stable maximal torus T distinct weight spaces of M are nonisomorphic T -modules. We show that if M is G-regular of dimension d, then the lift of St · β decomposes as a sum of d regular characters of G.

Projective modules for Frobenius kernels and finite Chevalley groups

Let G be a connected reductive algebraic group with a Frobenius morphism F: G → G defined over a finite field F p r . The main result of the paper is to prove that any rational G-module M which is projective when restricted to the Frobenius kernel Gr = Ker(F) is also projective over the split and twisted finite Chevalley groups. In 1987, Parshall conjectured this statement for r = 1 in the split case. The authors verified this in 1999 with the possible exclusion of primes 2 and 3 in non-simply laced cases. The converse of the main result is also discussed for split groups in this paper.

Quasi-hereditary quotients of finite Chevalley groups and Frobenius kernels

Let G be a semisimple connected simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. Denote by Gn its nth Frobenius kernel and by G(pn) its finite subgroup of Fpn-rational points. In this paper we find quotients of the algebra Un = k[Gn]* and of the group algebra kG(pn) whose module category is equivalent to a (highest weight) subcategory of the category of rational G-modules.

Regions in the dominant chamber and nilpotent orbits

Let G be a complex semisimple algebraic group with Lie algebra g . The goal of this note is to show that combining some ideas of Gunnells and Sommers [Math. Res. Lett. 10 (2–3) (2003) 363–373] and Vinberg and Popov [Invariant Theory, in: Algebraic Geometry IV, in: Encyclopaedia Math. Sci., Vol. 55, Springer, Berlin, 1994, pp.123–284] yields a geometric description of the characteristic of a nilpotent G-orbit in an arbitrary (finite-dimensional) rational G-module.