Bertram Kostant was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.
Alexandre Aleksandrovich Kirilloff is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups and Lie groups. In particular he introduced the orbit method into representation theory. He is an emeritus professor at the University of Pennsylvania.
In mathematics, the orbit method establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra. The theory was introduced by Kirillov for nilpotent groups and later extended by Bertram Kostant, Louis Auslander, Lajos Pukánszky and others to the case of solvable groups. Roger Howe found a version of the orbit method that applies to p-adic Lie groups. David Vogan proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.
Victor Gershevich (Grigorievich) Kac is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-discovered Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities. He classified the finite-dimensional simple Lie superalgebras, and found the Kac determinant formula for the Virasoro algebra. He is also known for the Kac–Weisfeiler conjectures with Boris Weisfeiler.
The Atlas of Lie Groups and Representations is a mathematical project to solve the problem of the unitary dual for real reductive Lie groups.
David Alexander Vogan Jr. is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups.
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related.
In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One of these describes the irreducible admissible (g, K)-modules, for g a Lie algebra of a reductive Lie group G, with maximal compact subgroup K, in terms of tempered representations of smaller groups. The tempered representations were in turn classified by Anthony Knapp and Gregg Zuckerman. The other version of the Langlands classification divides the irreducible representations into L-packets, and classifies the L-packets in terms of certain homomorphisms of the Weil group of R or C into the Langlands dual group.
Anthony W. Knapp is an American mathematician at the State University of New York, Stony Brook working on representation theory, who classified the tempered representations of a semisimple Lie group.
Jeffrey David Adams (born 1956) is a mathematician at the University of Maryland who works on unitary representations of reductive Lie groups and who led the project Atlas of Lie Groups and Representations that calculated the characters of the representations of E8. The project to calculate the representations of E8 has been compared to the Human Genome Project in scope. Together with Dan Barbasch and David Vogan, he co-authored a monograph on a geometric approach to the Langlands classification and Arthur's conjectures in the real case.
Toshiyuki Kobayashi is a Japanese mathematician known for his original work in the field of Lie theory, and in particular for the theory of discontinuous groups (lattice in Lie groups) and the application of geometric analysis to representation theory. He was a major developer in particular of the theory of discontinuous groups for non-Riemannian homogeneous spaces and the theory of discrete breaking symmetry in unitary representation theory.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by Zuckerman and Jantzen. Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.
In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups.
In the field of mathematics known as representation theory, an L-packet is a collection of irreducible representations of a reductive group over a local field, that are L-indistinguishable, meaning they have the same Langlands parameter, and so have the same L-function and ε-factors. L-packets were introduced by Robert Langlands in, .
In mathematical representation theory, the Eisenstein integral is an integral introduced by Harish-Chandra in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms. Harish-Chandra used Eisenstein integrals to decompose the regular representation of a semisimple Lie group into representations induced from parabolic subgroups. Trombi gave a survey of Harish-Chandra's work on this.
Nolan Russell Wallach is a mathematician known for work in the representation theory of reductive algebraic groups. He is the author of the two-volume treatise Real Reductive Groups.
Birgit Speh is Goldwin Smith Professor of Mathematics at Cornell University. She is known for her work in Lie groups, including Speh representations.
Jing-Song Huang is a Chair Professor in the Department of Mathematics of Hong Kong University of Science and Technology. His research interests are representations of Lie groups and harmonic analysis. After graduating from Peking University, he went to Massachusetts Institute of Technology and received his PhD degree in 1989, under the supervision of David Vogan.
Kari Kaleva Vilonen is a Finnish mathematician, specializing in geometric representation theory. He is currently a professor at the University of Melbourne.
