When Is a Reductive Group Scheme Linear?

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$ . In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$ -valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$ -valued representations.

Lie algebroids arising from simple group schemes

Parabolic Subgroups in a Reductive Group Scheme

Let be a regular local ring containing a field. Let be a reductive group scheme over . We prove that a principal -bundle over is trivial if it is trivial over the field of fractions of . In other words, if is the field of fractions of , then the map of the non-Abelian cohomology pointed sets induced by the inclusion of in has trivial kernel. This result was proved in [1] for regular local rings containing an infinite field.

On the support varieties of tilting modules

Representations of reductive groups over finite local rings of length two

Let $R$ be a semilocal Dedekind domain with fraction field $F$. We show that two hereditary $R$-orders in central simple $F$-algebras which become isomorphic after tensoring with $F$ and with some faithfully flat etale $R$-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for hermitian forms over hereditary $R$-orders with involution. The results can be restated by means of etale cohomology and can be seen as variations of the Grothendieck--Serre conjecture on principal homogeneous bundles of reductive group schemes. Connections with Bruhat--Tits theory are also discussed.

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R U = A U then the conclusion is that A is integral over R.

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

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.

A procedure is given for the construction of a faithful linear representation of the generalized Bondi—Metzner group from each faithful linear representation of the inhomogeneous orthochronous Lorentz group. Unitary representations can be obtained in this way.

on S an , where S denotes the constant local system on S an .Let s S() be a base point, (resp.) be the image of 1 (S(C), s) in GL(V s ) (resp.GL(V s )), and H (resp. H) be its Zariski-closure.By a theorem of Deligne [15], the neutral component of H is a semisimple group.Let (the relative integral monodromy) be the kernel of and U (the relative algebraic monodromy) be the kernel of H H, so that U is a vector group and is a subgroup of U ().In [1] (cf. also [2, 1.4,1.5]), the following result was proven:1.1.1Theorem.Assume that i) A/S has no fixed part, even after pull-back to a finite etale cover of S, ii) .(S) is Zariski-dense in A (i.e. the image of the section is not contained in any proper S-subgroup of A).Then there is a natural isomorphism U () = H 1 (A s , ).

These varieties are conjectured to be abelian varieties up to finite etale coverings. This conjecture is derived from an affirmative answer to the abundance conjecture in minimal model theory. In particular, this is true for n=3.

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup Open image in new window. We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite etale cover Xg = Ω/Γ(g) of X determined by a subgroup Open image in new window depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f : R → Xg* from R into the Satake-Baily-Borel compactification Xg* of Xg, the image f(R) lies in the boundary ∂Xg: = X*g\Xg. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g≥0, we show that there exists a positive number a(n,g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N≥a(n,g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.