Bertram Kostant 1928-2017

I was sorry to just hear via a comment here about the recent death of Bert Kostant, at the age of 88. MIT has a story about him here.

Kostant was a major figure in the field of representation theory, and perhaps the leading one during the second half of the twentieth century among those with a serious interest in the relations between representation theory and quantum theory. These relations have for a long time now been a deep source of fascination to me, and Kostant’s work has had a great impact on how I think about the subject.

I’ll just list here some of his major papers that I’ve spent significant amounts of time with, characterized by a few major themes:

Borel-Weil-Bott, Lie algebra cohomology, BRST and Dirac cohomology

• Lie algebra cohomology and the generalized Borel-Weil theorem. This paper has had a huge influence. Some notes from my graduate course discussing this and Borel-Weil-Bott are here.
• Symplectic reduction, BRS cohomology and infinite-dimensional Clifford algebras (with Shlomo Sternberg). If you really want to understand the mathematics underlying the BRST method, this is the place to start.
• Dirac cohomology for the Dirac operator. For the context of this and a lot more about the subject, see the book by Huang and Pandzic.

Quantization of the dual of a Lie algebra, W-algebras

The dual of a Lie algebra is a Poisson manifold, and you can ask what happens when you quantize this. For semisimple Lie algebra, reduction with respect to the nilradical is an idea that Kostant pursued, with two examples the following two papers. Applied to loop groups, this is a central idea of the geometric Langlands program. The theory of W-algebras is also an outgrowth of this.

• On Whittaker vectors and representation theory.
• Quantization and representation theory, in the volume Representation theory of Lie groups.

Geometric quantization theory and co-adjoint orbits

Starting around 1970 Kostant did a great deal of work developing the theory of “geometric quantization” and the idea of quantizing co-adjoint orbits to get representations (other figures to mention in this context are Kirillov and Souriau). Some of his papers on this are:

• Orbits, symplectic structures and representation theory.
• Orbits and quantization theory. 1970 ICM talk.
• Quantization and unitary representations. In Lecture notes in Mathematics 170.

All of the three general themes above are closely intertwined, and the relations between them indicate that there is still a lot more to be understood about how quantum theory and representation theory are related, with Kostant’s work undoubtedly playing a large role in developments to come.