QM Research Seminar: Dirac operators in representation theory

Speaker: Steffen Schmidt (University of Southern Denmark).

Abstract: 
Symmetry serves as a guiding principle in mathematics and physics. The language of symmetry groups and their actionsorganizes a wide range of phenomena. In this setting, Dirac operators form a natural bridge between quantum mechanics, geometry, and representation theory. 

In representation theory, Dirac operators provide conceptual and computational tools for addressing basic problems concerning a (semisimple) Lie group: they furnish explicit constructions, give effective criteria for unitarity, contribute to classification results, and relate the topology of locally symmetric spaces to representation-theoretic data. A foundational instance is Parthasarathy's use of a representation-theoretic Dirac operator in the study of discrete series representations. 

Interest in this circle of ideas intensified in the late 1990s, following Kostant's introduction of the cubic Dirac operator and the subsequent development of \emph{Dirac cohomology}, including Huang–Pandžić’s proofof Vogan's conjecture. 

In this talk, I present a modern construction of (algebraic) Dirac operators and discuss applications of Dirac operators and Dirac cohomology to representations of Lie superalgebras, including localization principles, relations with Duflo–Serganova cohomology, and character formulas.