Affine algebraic groups with periodic components

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.

We study the group of automorphisms Aut(G) of a given group G from the groups of automorphisms (supposed known) of some of its subgroups; basic results are applied to topological groups and widely developed inside the Borel–Chevalley theory of linear groups; affine algebraic complete groups are studied in detail.

Let G be an affine algebraic group over a field. The representation ring $\mathrm{R}(G)$ has three standard filtrations, defining the same topology on $\mathrm{R}(G)$: augmentation, Chern and Chow, each of which contained in the next one. For split reductive G, motivated by potential applications related to spin groups, we introduce and study one more filtration, containing the previous ones, which we call induced because it is induced by any of the filtrations on the representation ring of a maximal split torus of G. In the case of semisimple simply connected G, this fourth filtration turns out to be equivalent (in the above topological sense) to the previous three. However, for the spin group $G=\operatorname{\mathrm{Spin}}(d)$ over the complex numbers with $d=7,8$, the new filtration is shown to be strictly larger than the others. It is also shown that for $G=\operatorname{\mathrm{Spin}}(d)$ over an arbitrary field and with any $d\geq7$, the Chern and Chow filtrations on $\mathrm{R}(G)$ are not the same, giving new counter-examples to an extension of Atiyah’s conjecture.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V∣H is free. The problem is raised in the recent paper by Kuzucuŏglu and Zalesskiǐ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(Q)-module V of dimension 2 n Π p p ( 2 n ) , where the product is over all primes less than or equal to n+1, such that V is free as a QH-module for every finite subgroup H of GLn(Q). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(Q)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module St p n for G is injective (and projective) on restriction to H for n≫0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

Let G be an affine algebraic group over an algebraically closed field of positive characteristic. Recent work of Hardesty, Nakano, and Sobaje gives necessary and sufficient conditions for the existence of so-called mock injective G -modules, that is, modules which are injective upon restriction to all Frobenius kernels of G . In this article, we give analogous results for contramodules, including showing that the same necessary and sufficient conditions on G guarantee the existence of mock projective contramodules. In order to do this, we first develop contramodule analogs to many well-known (co)module constructions.

An affine algebraic group

In this chapter, we come to those results on affine varieties and algebraic groups which rely in an essential way on the hypothesis that the ground field k is algebraically closed. We will see that, under this hypothesis, the image of a morphism between affine varieties is a relatively ‘thick’ subset, more precisely, it contains an open subset of its closure. This fact will turn out to be a key tool.

Superstability and central extensions of algebraic groups

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose G is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of G has finite extensions which are not residually finite. More precisely, we investigate the group H¯2(Z/n)=limΓ→H2(Γ,Z/n),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2(\\mathbb {Z}/n) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle \\varGamma }\\limits ^{\\textstyle \\rightarrow }} \\end{array} H^2(\\varGamma ,\\mathbb {Z}/n), \\end{aligned}$$\\end{document}where varGamma runs through the arithmetic subgroups of G. Elements of {bar{H}}^2(mathbb {Z}/n) correspond to (equivalence classes of) central extensions of arithmetic groups by mathbb {Z}/n; non-zero elements of {bar{H}}^2(mathbb {Z}/n) correspond to extensions which are not residually finite. We prove that {bar{H}}^2(mathbb {Z}/n) contains infinitely many elements of order n, some of which are invariant for the action of the arithmetic completion {widehat{G(mathbb {Q})}} of G(mathbb {Q}). We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group H¯2(Zl)=limt←H¯2(Z/lt).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2({\\mathbb {Z}_l}) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle t}\\limits ^{\\textstyle \\leftarrow }} \\end{array} {\\bar{H}}^2(\\mathbb {Z}/l^t). \\end{aligned}$$\\end{document}We show that {bar{H}}^2({mathbb {Z}_l})^{widehat{G(mathbb {Q})}} is isomorphic to {mathbb {Z}_l}^c for some positive integer c. When G(mathbb {R}) has no simple components of complex type, we prove that c=b+m, where b is the number of simple components of G(mathbb {R}) and m is the dimension of the centre of a maximal compact subgroup of G(mathbb {R}). In all other cases, we prove upper and lower bounds on c; our lower bound (which we believe is the correct number) is b+m.

ABSTRACTLet σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;σ]∕fK[t;σ] obtained when the twisted polynomial f∈K[t;σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in AutF(K) and n≥m−1, where n is the order of σ and m the degree of f, and obtain partial results for n<m−1. In the case where K∕F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.

Uniquely separable extensions

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

Let G be a semisimple affine algebraic group of inner type over a field F . We write XG for the class of all finite direct products of projective G-homogeneous F varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in XG. More precisely, it is known that the motive of any variety in XG decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group PGLA of automorphisms of a given central simple F algebra A, for any variety in the class XG (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in XG are p-incompressible. If A is a division algebra of degree p for some n ≥ 0, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p;A) of ideals of reduced dimension p for m = 0, 1, . . . , n.

We develop a derived-categorical framework for the categorification of representations arising from algebraic group actions. Our approach unifies localization, descent, and equivariant structures within a single triangulated setting. Central to the construction is the exact localization functor LG : D b G Coh(X) −→ D b Coh([X/G]), defined for any affine algebraic group G acting on a qcqs scheme X. Under standard coherent-descent finiteness hypotheses (summarized in §2.4), the functor LG is the atlasinduced descent equivalence D b G Coh(X) ≃ D b Coh([X/G]); no linear reductivity is required for this equivalence. Linear reductivity is used later only for the exactness of invariants (−) G when comparing equivariant/global sections. Building on this equivalence, we construct a categorified character theory for Ad(G)-equivariant complexes with Weil structure in the constructible ℓ-adic category D b c (GF q , Q ℓ) Ad(G), Weil. We define derived character sheaves and a geometric trace map recovering classical characters via Frobenius traces, providing a unified description of Deligne-Lusztig and Steinberg representations within the derived formalism. Throughout, all proofs are carried out in the triangulated setting; we explicitly separate geometric descent assumptions from those ensuring invariant exactness, and present counterexamples delineating the boundaries of the theory. The framework culminates in conjectural extensions to derived motivic and ∞-categorical contexts, offering a coherent route from equivariant localization to categorified representation theory.