Research in Representation Theory and Automorphic Forms

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Abstract Wallach This project will study several related problems in representation theory, non-commutative harmonic analysis and algebraic group theory. In representation theory it involves the construction, study and application of small unitary representations of real reductive groups. In harmonic analysis, it involves dropping the K-finite condition in Paley-Wiener theorems for rapidly decreasing functions on a real reductive group. This analysis will be applied to proving a version of the Casselman-Wallach theorem depending on parameters. This theorem will imply a meromorphic continuation of non-K-finite Eisenstein series. Related to these more analytic problems we will study the algebraic problem of determining graded multiplicities for the irreducible constituents of the action of a reductive algebraic group on an affine cone. The latter work also has applications to the analysis and to the study of measures of entanglement in quantum computing. Representation theory has its roots in nineteenth century invariant theory, early twentieth century quantum mechanics and mid-twentieth century number theory. In this first decade of the twenty first century the theory has returned to its roots. The nineteenth century invariant theory emphasized concrete questions on binary forms with algorithmic solutions. These problems have reemerged and are now being generalized to apply to quantum computation. Early quantum mechanics studied puzzling and weird measurements involving photons, electrons etc. These phenomena led to the Hilbert space approach to quantum mechanics. The philosophical debates of the early quantum mechanics have reemerged as quantum information technology. The Hilbert space approach also gave birth to representation theory, which has as one of its main applications in number theory. The Langlands program has established a goal for the twenty first century to establish a non-commutative class field theory (Wile's proof of Fermat's Last Theorem is actually proof of a special case of the Tanayama-Shimura conjecture which is a special case of the Langlands program). This project is in the interface of all of these exciting directions.