# Bertram Kostant

Bertram Kostant
Bertram Kostant at a workshop on “Enveloping Algebras and Geometric Representation Theory” in Oberwolfach, 2009
BornMay 24, 1928
DiedFebruary 2, 2017 (aged 88)
NationalityAmerican
Alma materPurdue University
University of Chicago (PhD)
Known forKostant partition function
AwardsWigner Medal (2016)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
University of California, Berkeley
ThesisRepresentations of a Lie algebra and its enveloping algebra on a Hilbert space
Doctoral studentsArlie Petters
James Lepowsky
Stephen Rallis
James Harris Simons
Moss Sweedler
David Vogan
Birgit Speh

Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician.[1]

## Early life and education

Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945.[2] He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D. from the University of Chicago in 1954, under the direction of Irving Segal, where he wrote a dissertation on representations of Lie groups.

## Career in mathematics

After time at the Institute for Advanced Study, Princeton University, and the University of California, Berkeley, he joined the faculty at the Massachusetts Institute of Technology, where he remained until his retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. He has given several lectures on the Lie group E8.[3] He has been one of the principal developers of the theory of geometric quantization. His introduction of the theory of prequantization has led to the theory of quantum Toda lattices. The Kostant partition function is named after him. With Gerhard Hochschild and Alex F. T. W. Rosenberg, he is one of the namesakes of the Hochschild–Kostant–Rosenberg theorem which describes the Hochschild homology of some algebras.[4]

His students include James Harris Simons, James Lepowsky, Moss Sweedler, David Vogan, and Birgit Speh. At present he has more than 100 mathematical descendants.

## Awards and honors

Kostant's honors include election to the National Academy of Sciences in 1978. In 2012 he became a fellow of the American Mathematical Society.[5] He was an Invited Speaker of the ICM with talk Orbits and quantization theory in 1970 at Nice.[6]

## Selected publications

• "Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold". Trans. Amer. Math. Soc. 80: 528–542. 1955. doi:10.1090/S0002-9947-1955-0084825-8.
• "A formula for the multiplicity of a weight". Trans. Amer. Math. Soc. 93: 53–73. 1959. doi:10.1090/S0002-9947-1959-0109192-6.
• "Lie algebra cohomology and the generalized Borel-Weil theorem" (PDF). Annals of Mathematics. 74 (2): 329–387. 1961. doi:10.2307/1970237.
• "Lie group representations on polynomial rings". American Journal of Mathematics. 85 (3): 327–404. 1963. doi:10.2307/2373130.
• "On the existence and irreducibility of certain series of representations". Bulletin of the American Mathematical Society. 75 (4): 627–642. 1969. doi:10.1090/S0002-9904-1969-12235-4.
• "Quantization and unitary representations". In: Lectures in modern analysis and applications III. Lecture Notes in Mathematics 170. 1970. pp. 87–208. doi:10.1007/BFb0079068.
• with Louis Auslander: "Polarization and unitary representations of solvable Lie groups". Inventiones mathematicae. 14: 255–354. 1971. doi:10.1007/BF01389744.
• with Stephen Rallis: "Orbits and representations associated with symmetric spaces". American Journal of Mathematics. 93 (3): 753–809. 1971. doi:10.2307/2373470.
• "On convexity, the Weyl group and the Iwasawa decomposition". Annales Scientifiques de l'École Normale Supérieure. 6 (4): 413–455. 1973. doi:10.24033/asens.1254.
• "Graded manifolds, graded Lie theory, and prequantization". In: Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Math 570. 1977. pp. 177–306. doi:10.1007/Bfb0087788.
• "On Whittaker vectors and representation theory". Inventiones mathematicae. 48 (2): 101–184. 1978. doi:10.1007/BF01390249.
• with David Kazhdan and Shlomo Sternberg: "Hamiltonian group actions and dynamical systems of Calogero type". Communications on Pure and Applied Mathematics. 31 (4): 481–507. 1978. doi:10.1002/cpa.3160310405.
• "The solution to a generalized Toda lattice and representation theory". Advances in Mathematics. 34 (3): 195–338. 1979. doi:10.1016/0001-8708(79)90057-4.
• with Shrawan Kumar: "The nil Hecke ring and cohomology of GP for a Kac-Moody group G". Advances in Mathematics. 62 (3): 187–237. 1986. doi:10.1016/0001-8708(86)90101-5.
• with Shlomo Sternberg: "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras", Annals of Physics, 176 (1): 49–113, 1987, doi:10.1016/0003-4916(87)90178-3
• "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of ${\displaystyle {\mbox{SL}}(2,\mathbb {R} )}$". Represent. Theory. 4: 181–224. 2000. doi:10.1090/S1088-4165-00-00096-0.
• with Gerhard Hochschild and Alex Rosenberg: "Differential forms on regular affine algebras". In: Collected Papers. New York: Springer. 2009. pp. 265–290. doi:10.1007/b94535_14. (Reprinted from Trans. Amer. Math. Soc., vol. 102, no. 3, March 1962, pp. 383–408)
• "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group". In: Collected Papers. 2009. pp. 130–189. doi:10.1007/b94535_11. (Reprinted from the Amer. J. Math., vol. 81, no. 4, Oct. 1959)