Related papers: On the additional invariance of the Dirac and Maxw…
We give a superconnection proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid.
This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac-Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the…
Some years ago Ruijsenaars and Schneider initiated the study of mechanical systems exhibiting an action of the Poincare algebra. The systems they discovered were far richer: their models were actually integrable and possessed a natural…
We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of…
Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized…
In [Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants, {\it Trans. Amer. Math. Soc. \bf309} (1988), 581--589] we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general…
We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are…
We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit…
We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to…
In this thesis new objects to the existing set of invariants of Lie algebras are added. These invariant characteristics are capable of describing the nilpotent parametric continuum of Lie algebras. The properties of these invariants, in…
We introduce a slight modification of the usual equivariant $KK$-theory. We use this to give a $KK$-theoretical proof of an equivariant index theorem for Dirac-Schrodinger operators on a non-compact manifold of nowhere positive curvature.…
We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra $\mathcal A\subseteq \mathcal B(\mathscr X)$ which…
For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding $\imath$quantum group. In other words, we provide a differential operator approach to…
Though the irreducible representations of the Poincare' group form the groundwork for the formulation of relativistic quantum theories of a particle, robust classes of such representations are missed in current formulations of these…
In this paper, we define two generalisations of Dirac operators for Drinfeld's Hecke algebra. One generalisation, Parthasarathy operators inherit the notion of the Dirac inequality. The second generalisation, warped Dirac operators are such…
We compute $K$-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $\R^k.$ We discuss the relation between our…
Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of…
Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this talk I want to show how to deform the Fredholm family, in the sense of quantum…
In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order $1$. Namely, assuming that such an…
We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly non-compact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to non-compact…