Generalized modular symbols and relative Lie algebra cohomology

Dirac Operators and Lie Algebra Cohomology

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves [V2], [HP1], [K4]. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the correspondin...

Lie Algebra Cohomology and the Representations Ofsemisimple Lie Groups

The Lie algebra cohomology of jets

Let g be a finite-dimensional complex reductive semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible repr...

On modular symbols and the cohomology of Hecke triangle surfaces

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surface...

Operations and Cooperations in Elliptic Cohomology, Part I: Generalized modular forms and the cooperation algebra

This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E``∗E``, while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology. We investigate the structure of the stable operation algebra E``∗E``...