Sub-principal homomorphisms in positive characteristic

 Let G be a reductive group over an algebraically closed field of characteristic p, and let uG be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ:SL _2/k →G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of φ generalizes the construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, φ is obtained by reduction modulo 𝔭 from a homomorphism of group schemes over a valuation ring 𝒜 in a number field. This permits us to show moreover that the weight spaces of a maximal torus of φ(SL _2/k ) on Lie(G) are ``the same as in characteristic 0'; the existence of a φ with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000). Received: 1 February 2002; in final form: 17 June 2002 / Published online: 1 April 2003 The author was supported in part by a grant from the National Science Foundation.     

Reality properties of conjugacy classes in<Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

LetG be an algebraic group over a fieldk. We callg εG(k) real ifg is conjugate tog ⁻¹ inG(k). In this paper we study reality for groups of typeG ₂ over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG ₂ over special fields and construct examples to show that reality fails for semisimple elements inG ₂ over ℚ and ℚ_p. We show that semisimple elements are real forG ₂ overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG ₂ overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.     

Proof of Springer’s hypothesis

LetG be a reductive group over a finite fieldk of a characteristicp. Π:G _k → AutU is an irreducible representation ofG in “a general position”. Springer formulated a conjecture about values of the character of Π on unipotent elements. This conjecture is proved in the article.     

Central Pairs, Galois Theory and Automorphic Forms

A central pair over a field k of characteristic 0 consists of a finite Abelian group which is equipped with a central 2-cocycle with values in the multiplicative group k ^* of k. In this paper we use specific central pairs to construct a class of projective representations of the absolute Galois group G _k of k and if k is a number field we investigate the liftings of these projective representations to linear representations of G _k . In particular we relate these linear representations to automorphic representations. It turns out that some of these automorphic representations correspond to certain indefinite modular forms already constructed by E. Hecke.     

On the Relation between 2 and ∞ in Galois Cohomology of Number Fields

We remove the assumption p 2 or k is totally imaginary from several well-known theorems on Galois groups with restricted ramification of number fields. For example, we show that the Galois group of the maximal extension of a number field k which is unramified outside 2 has finite cohomological 2-dimension (also if k has real places).     

On simple groups and simple singularities

We give a proof of a characteristicp version of Brieskorn’s theorem, namely, that ifG is a simply connected simple algebraic group of typeA, D orE over an algebraically closed fieldk whose characteristic is very good forG, then the categorical quotient morphismG→G//G _ad yields, when restricted to a general slice through a pointP in the subregular unipotent orbit inG, a miniversal deformation of the rational double point overk of the same type asG.     

The Bar Construction and Affine Stacks

We discuss the equivalence of two constructions of a unipotent group scheme attached to a differential graded algebra over a ?-algebra. The first construction is using the bar resolution and the second is using Toen's theory of affine stacks. We use this to establish the equivalence of two approaches to the comparison theorem in p-adic Hodge theory for the unipotent fundamental group of varieties defined over p-adic fields.     

Varieties for cohomology with twisted coefficients

Let G be a finite group and k a field of characteristic p > 0. In this paper we consider the support variety for the cohomology module Ext _kG ^* (M, N) where M and N are kG-modules. It is the subvariety of the maximal ideal spectrum of H*(G, k) of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus. For other blocks a new nucleus is defined and a similar theorem is proven. However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples. Partially supported by grants from NSF and EPSRC     

On the crystalline cohomology of Deligne–Lusztig varieties

Let $X\to Y^0$ be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic $p>0$. Let Y be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cohomology of X, in particular provides a natural $W[F]$-lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, $Y^0$ and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if $\mathbb{G}$ is a connected reductive algebraic group defined over k and $\mathbb{L}$ a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant $W[F]$-lattice in the cohomology (ℓ-adic or rigid) of the corresponding Deligne–Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups.     

The geometry of k-transvection groups

Let k be a (commutative) field and G a group, then a conjugacy class of Abelian subgroups of G is called a class of k-transvection subgroups in G if and only if it generates G and any two elements of the class either commute or are full unipotent subgroups of the group they generate and which is isomorphic to (P)SL₂(k).In this paper we study the geometry of k-transvection groups. Given a class of k-transvection groups Σ, we consider a partial linear space whose points are the elements of Σ, and whose lines correspond to the groups generated by two noncommuting elements from Σ. We derive several properties of this partial linear space. These properties are used to give a characterization of the geometries of k-transvection groups and provide a classification of groups generated by k-transvection subgroups.     

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group G, a parabolic subgroup P _J , and its unipotent radical U _J , we determine the ring structure of the cohomology ring H^?((U _J )₁, k). We also obtain new results on computing H^?((P _J )₁, L(??)) as an L _J -module where L(??) is a simple G-module with highest weight ?? in the closure of the bottom p-alcove. Finally, we provide generalizations of all our results to small quantum groups at a root of unity.     

Levi decompositions of a linear algebraic group

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G.     

Permutative k-exponential epigroups

We prove that any permutative k-exponential epigroup is a normal band of unipotent epigroups. Moreover, we describe the least regular congruence on such semigroups. Finally, we give an example of a permutative epigroup which is not a band of unipotent epigroups.     

Second cohomology groups for Frobenius kernels and related structures

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k.     

Exponential epigroups

We show that an exponential epigroup is a band of unipotent exponential epigroups (in which the set of idempotents forms a subsemigroup). Also, we investigate the structure of unipotent exponential epigroups. In addition, a characterization of the least normal band of groups congruence is given for an arbitrary medial epigroup. It is also shown that k-exponential eventually regular semigroups are necessarily epigroups.     

On the groups SL2(ℤ[x]) and SL2(k[x,y])

This paper studies free quotients of the groups SL₂(ℤ[x]) and SL₂(k[x, y]),k a finite field. These quotients give information about the relation of the above groups to their subgroups generated by elementary or unipotent elements.     

Uniform boundedness of <Emphasis Type="Italic">p</Emphasis>-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E _s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.     

Generic Representation Theory of the Unipotent Upper Triangular Groups

It is generally believed (and for the most part it is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this article we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups U_n), and under a certain hypothesis relating the characteristic p to both n and the dimension d of a representation (specifically, p ≥ max(n, 2d)), Lie theory is completely sufficient to determine the representation theories of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their “generic” representation theory in characteristic p.     

Large Abelian Unipotent Subgroups of Finite Chevalley Groups

We compute maximal orders of unipotent Abelian subgroups, estimate p-ranks, and describe the structure of Thompson subgroups of maximal unipotent subgroups of finite exceptional groups of Lie type.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.