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.     

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.     

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.     

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     

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.     

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.     

Isotropy of non-nilpotent Riemannian solvable Lie groups

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.     

Spectral Bounds on Orbifold Isotropy

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Petersen, as well as comparison geometry techniques developed by Grove and Petersen.This research was partially supported by NSF grant DMS 0072534.