Isotropy of symplectic involutions

We prove the symplectic analogue of the isotropy theorem for orthogonal involutions. We apply (a modification of) a method due to J.-P. Tignol originally applied to prove the symplectic analogue of the hyperbolicity theorem for orthogonal involutions.     

Hyperbolicity of unitary involutions

We prove the so-called Unitary Hyperbolicity Theorem,a result on hyperbolicity of unitary involutions.The analogous previously known results for the orthogonal and symplectic involutions are formal consequences of the unitary one.While the original proofs in the orthogonal and symplectic cases were based on the incompressibility of generalized Severi-Brauer varieties,the proof in the unitary case is based on the incompressibility of their Weil transfers.     

Eigenvalues of products of unitary matrices and Lagrangian involutions

This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian submanifold of the moduli of unitary representations.     

Pfister involutions

The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.     

Weakly hyperbolic involutions

Pfister’s Local–Global Principle states that a quadratic form over a (formally) real field is weakly hyperbolic (i.e. represents a torsion element in the Witt ring) if and only if its total signature is zero. This result extends naturally to the setting of central simple algebras with involution. The present article provides a new proof of this result and extends it to the case of signatures at preorderings. Furthermore the quantitative relation between nilpotence and torsion is explored for quadratic forms as well as for central simple algebras with involution.     

Isotropy Subgroups of Transformation Groups

In this paper we consider transitive actions of Lie groups on analytic manifolds. We study three cases of analytic manifolds and their corresponding transformation groups. Given a free action on the left, we define left orbit spaces and consider actions on the right by maximal compact subgroups. We show that these actions are transitive and find the corresponding isotropy subgroups. Further, we show that the left orbit spaces are reductive homogeneous spaces. This article thus forms the basis of a forthcoming paper on invariant differential operators on homogeneous manifolds. Partially supported by a Carver Research Initiative Grant.     

Manifolds of linear involutions

The set of bounded linear involutions on a complex Banach space $\text{X}$ is equipped with a Banach manifold structure and an affine connection compatible with its embedding into $\text{B}$($\text{X}$). Geodesic lines are characterized. Moreover, if $\text{X}$ is a Hilbert space and the topology of the self-adjoint part of the manifold is strengthened to be compatible with the Hilbert-Schmidt metric, these geodesics are identified as minimal arcs between pairs of self-adjoint involutions whose straight line distance is less than 2.     

Riordan group involutions

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize the RNA matrix but are of independent interest.     

Partition congruences by involutions

We present a general construction of involutions on integer partitions which enables us to prove a number of modulo 2 partition congruences.     

Clifford algebras of hyperbolic involutions

For a central simple algebra of even degree with hyperbolic orthogonal involution, we describe the canonically induced involution on the even Clifford algebra of . When , and the interesting part of is isomorphic to the canonical involution on an exterior power algebra of B. As a corollary, we get some properties of the involution on the exterior power algebra. Received January 27, 1998; in final form June 7, 2000 / Published online October 30, 2000     

Quotient spaces of anti-holomorphic involutions

Let σ be an anti-holomorphic involution on an almost complex four manifold X, a necessary and sufficient condition is given to determine weather X/σ admits an almost complex structure.     

Orientation-reversing involutions on handlebodies

The observation that the quotient orbifold of an orientation- reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the -fibers of some -bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.

     

Bordism of two commuting involutions

In this paper we obtain conditions for a Whitney sum of three vector bundles over a closed manifold, , to be the fixed data of a -action; these conditions yield the fact that if is the fixed data of a -action, where is the trivial one dimensional bundle, then the same is true for . The results obtained, together with techniques previously developed, are used to obtain, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.