Moment maps and Riemannian symmetric pairs

We study Hamiltonian actions of a compact Lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold. The fixed-point set of the involution on the manifold is a Lagrangian submanifold. We investigate its image under the moment map and conclude that the intersection with the Weyl chamber is an easily described subpolytope of the Kirwan polytope. Of special interest is the integral K?hler case, where much stronger results hold. In particular, we obtain convexity theorems for closures of orbits of the noncompact dual group (in the sense of the theory of symmetric pairs). In the abelian case these results were obtained earlier by Duistermaat. We derive explicit inequalities for polytopes associated with real flag varieties. Received: 8 February 1999 / in revised form: 25 October 1999 / Published online: 8 May 2000     

Torsion-free groups and modules with the involution property

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of p-adic integers.     

Dualité dans l’espace des caractères virtuels tempérés d’un groupe réductif <Emphasis Type="Italic">p</Emphasis>-adique

Résumé We define an involution on the set of tempered virtual characters of a connected reductive p-adic group. This involution commutes with character of parabolic induction and with truncation. It also preserves the irreducible characters up to sign and the elliptic inner product.     

On the hermitian picard group of a congruence order

In this note methods are developed which permit us to concretely calculate the hermitian Picard group of a congruence order A over a discrete valuation ring, with respect to an involution α of A. In particular, this allows us to exhibit some explicit examples of algebras A whose hermitian Picard group strongly depends upon the chosen involution on A.     

Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.     

On groups of exponent 36

Locally finiteness is proved for a group of exponent 36 containing an involution and no elements of order 6.     

The main involutions of the metaplectic group

We determine the set of automorphisms of a metaplectic group which lift the main involution of the general linear group over an infinite field. Some basic properties of these automorphisms are also established.

     

Involutory decomposition of a group and twisted subsets with few involutions

A subset K of some group C is called twisted if 1 ∈ K and x, y ∈ K implies that xy ^?1 x belongs to K. We use the concept of twisted subset to investigate and generalize the concept of involutory decomposition of a group. A group is said to admit involutory decomposition if it contains some involution such that the group is the product of the centralizer of the involution and the set of elements inverted by the involution. We study the twisted subsets with at most one involution. We prove that if a twisted subset has no involutions at all then it generates a subgroup of odd order.     

The group Sp<Subscript>10</Subscript>(ℤ) is (2,3)-generated

It is proved that the group Sp₁₀(ℤ) is generated by an involution and an element of order 3.     

D'Alembert's functional equations on metabelian groups

Summary. We extend d'Alembert's classical functional equation by replacing the domain of definition Bbb R {Bbb R} of the solutions by a metabelian group G and simultaneously replacing the group involution by an arbitrary involution of G. We find all complex valued solutions. In particular we show that the continuous solutions have the same form as in the abelian case if G is connected.     

Symmetric Elements Under Oriented Involutions in Group Rings

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ? of the group G, an oriented invoultion ^σ? of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.     

Zero cycles on some involution varieties

The group of zero cycles on involution varieties corresponding to central simple algebras of index at most two with an orthogonal involution is computed. This computation generalizes the Karpenko-Swan theorem on zero cycles on projective nonsingular quadrics. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 93–105.     

The order of a group of even order

We will give an estimation of the order of a group of even order by the order of the centralizer of an involution using the classification of finite simple groups.

     

A local–global principle for algebras with involution and hermitian forms

 Weakly hyperbolic involutions are introduced and a proof is given of the following local–global principle: a central simple algebra with involution of any kind is weakly hyperbolic if and only if its signature is zero for all orderings of the ground field. Also, the order of a weakly hyperbolic algebra with involution is a power of two, this being a direct consequence of a result of Scharlau. As a corollary an analogue of Pfister's local–global principle is obtained for the Witt group of hermitian forms over an algebra with involution. Received: 29 October 2001; in final form: 9 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 16K20, 11E39     

Pierce sheaf for semirings with involution

We introduce the concept of Pierce sheaf for semirings with involution, an analog of Pierce sheaf for rings. We construct maximal spectrum, Pierce congruence, Pierce sheaf of semirings with involution, Pierce stalk of semiring with involution. We prove main theorem on the isomorphism of semiring with involution and semiring with involution of global sections of Pierce sheaf.     

Characterization of the linear groups PSL(2,11) and PSL (2,13)

We prove the nonsimplicity of a finite group containing an involution τ such that the quotient group C(τ)/{τ} the Frobenius group with an additional factor of odd prime order acting transitively on the nonunit elements of the kernel. Based on this we obtain a characterization of the linear groups PSL(2,11) and PSL(2,13).     

Nilpotency of group ring units symmetric with respect to an involution

Let F be an infinite field of characteristic different from 2. Let G be a torsion group having an involution ∗, and consider the units of the group ring FG that are symmetric with respect to the induced involution. We classify the groups G such that these symmetric units satisfy a nilpotency identity (x₁,…,x_n)=1.     

Star-group identities and groups of units

Analogous to *-identities in rings with involution we define *-identities in groups. Suppose that G is a torsion group with involution * and that F is an infinite field with char F ≠ 2. Extend * linearly to FG. We prove that the unit group U{\mathcal{U}} of FG satisfies a *-identity if and only if the symmetric elements U⁺{\mathcal{U}^+} satisfy a group identity.     

On (2, 3)-Generation of Low-Dimensional Symplectic Groups Over the Integers

We prove that the group Sp₈(?) is generated by an involution and an element of order 3, while Sp₆(?) is not.