Torsionfree sheaves over a nodal curve of arithmetic genus one

We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over ℂ. Let X be a nodal curve of arithmetic genus one defined over ℝ, with exactly one node, such that X does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsionfree sheaves over X of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over X of rank one.     

About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold

Let G be a connected reductive algebraic group defined on an algebraically closed field k of characteristic different from 2. Let B denote the flag variety of G. Let H be a spherical subgroup of G. F. Knop defined an action of the Weyl group W of G on the finite set of the H-orbits in B. Here, we define an invariant, namely the type, separating the orbits of W.     

Points rationnels sur les espaces homogenes et leurs compactifications

Let k be a perfect field and G an algebraic group defined over k. Let X be a homogeneous space of G. We show the following: if there exists a k-variety which is birational to X and which has a smooth k-rational point, then X also has a k-rational point (Theorem 5.7).     

Classification of hermitian forms and semisimple groups over fields of virtual cohomological dimension one

Summary Letk be a perfect field with cdk(i)≤1. It has recently been proved by the author that homogeneous spaces under connected linear groups overk satisfy a Hasse principle with respect to the real closures ofk. Using this result we classify the semisimple algebraic groups overk and, in particular, characterize the anisotropic ones. Similarly we classify the various types of hermitian forms over skew fields overk and exhibit to what extent weak or strong local-global principles hold. In the case wherek is the function field of a smooth projective curveX over ℝ, we also cover the local-global questions vis-à-vis the completions ofk at the points ofX.     

On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F_p. Let q be a power of p and let G(q) be the finite group of F_q-rational points of G. Let F be the Frobenius morphism such that G(q) = G^F. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).     

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL ₃(K),PSp ₄(K) orG ₂(K) for some algebraically closed fieldK. Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we show Theorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field. Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed. We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ₁-categorical polygon have Morley degree 1. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Supported by the Minerva Foundation (Germany). Research Director at the Fund for Scientific Research-Flanders (Belgium).     

Conjugacy classes of non-connected semisimple algebraic groups

Consider a non-connected algebraic group G = G ⋉ Γ with semisimple identity component G and a subgroup of its diagram automorphisms Γ. The identity component G acts on a fixed exterior component Gτ, id ≠ τ ∈ Γ by conjugation. In this paper we will describe the conjugacy classes and the invariant theory of this action. Let T be a τ -stable maximal torus of G and its Weyl group W. Then the quotient space Gτ//G is isomorphic to (T/(1 − τ )(T))/W^τ. Furthermore, exploiting the Jordan decomposition, the reduced fibres of this quotient map are naturally associated bundles over semisimple G-orbits. Similar to Steinberg's connected and simply connected case [22] and under additional assumptions on the fundamental group of G, a global section to this quotient map exists. The material presented here is a synopsis of the Ph.D thesis of the author, cf. [15].     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

On reid’s 3-simplicial matroid theorem

In this paper we prove the following result of Ralph Reid (which was never published nor completely proved). Theorem. Let M be a matroid coordinatizable (representable) over a prime field F. Then there is a 3-simplicial matroid M′ over F which is a series extension of M. The proof we give is different from the original proof of Reid which uses techniques of algebraic topology. Our proof is constructive and uses elementary matrix operations.     

The decomposition of permutation module for infinite Chevalley groups

Let G be a connected reductive group defined over F_q, the finite field with q elements. Let B be a Borel subgroup defined over F_q. In this paper, we completely determine the composition factors of the induced module M(tr) = kG ■tr(where tr is the trivial B-module) for any field k.     

Automorphism groups of fields

We consider pairs (K,G) of an infinite field K or a formally real field K and a group G and want to find extension fields F of K with automorphism group G. If K is formally real then we also want F to be formally real and G must be right orderable. Besides showing the existence of the desired extension fields F, we are mainly interested in the question about the smallest possible size of such fields. From some combinatorial tools, like Shelah’s Black Box, we inherit jumps in cardinalities of K and F respectively. For this reason we apply different methods in constructing fields F: We use a recent theorem on realizations of group rings as endomorphism rings in the category of free modules with distinguished submodules. Fortunately this theorem remains valid without cardinal jumps. In our main result (Theorem 1) we will show that for a large class of fields the desired result holds for extension fields of equal cardinality. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

Homogeneity results for invariant distributions of a reductive p-adic group

Let k denote a complete nonarchimedean local field with finite residue field. Let G be the group of k-rational points of a connected reductive linear algebraic group defined over k. Subject to some conditions, we establish a range of validity for the Harish-Chandra-Howe local expansion for characters of admissible irreducible representations of G. Subject to some restrictions, we also verify two analogues of this result.     

Stability of homogeneous principal bundles with a classical groupas the structure group

Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k and P ⊂ G a parabolic subgroup without any simple factor. Let H be a connected reductive linear algebraic group defined over the field k such that all the simple quotients of H are of classical type. Take any homomorphism π : P → H such that the image of p is not contained in any proper parabolic subgroup of H. Consider the corresponding principal H-bundle E_P(H) = (G × H)/P over G/P. We prove that E_P (H) is strongly stable with respect to any polarization on G/P.     

Spectral norms on valued fields

Let be a perfect valued field, be an algebraic closure of be an extension of to and be the G-spectral norm on Let be an algebraic extension of K and be the completion of L relative to We associate to any element a real number and prove that if for all x in , then and is a zero-dimensional regular ring. We show that and prove that is algebraic over (with some additional conditions on K and L). We give a Galois type correspondence between the set of all closed K-subalgebras of and the subfields of L. We prove that is an algebraic closed and zero-dimensional regular ring. Received: 3 March 1999; in final form: 21 February 2000 / Published online: 4 May 2001     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.     

Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C^*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero. Dedicated to the memory of Peter Slodowy     

Orbits of Real Semisimple Lie Groups on Real Loci of Complex Symmetric Spaces

Let G be a complex semisimple algebraic group and X be a complex symmetric homogeneous G-variety. Assume that both G, X as well as the G-action on X are defined over real numbers.Then G(R) acts on X(R) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of Borel and Ji.     

On presentations of integer polynomial points of simple groups over number fields

Let K be a number field and let A be its ring of integers. Let G be a connected, noncommutative, absolutely almost simple algebraic K-group. If the K-rank of G equals 2, then G(A[t]) is not finitely presented.     

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)