Anthony William Knapp (born 2 December 1941, Morristown, New Jersey) [1] is an American mathematician and professor emeritus at the State University of New York, Stony Brook working in representation theory. For much of his career, Knapp was a professor at Cornell University.
Knapp lived in Baltimore, Maryland, [2] and graduated from the preparatory McDonogh School nearby. [3] He attended Dartmouth College, [4] graduating from there in 1962 and receiving a National Science Foundation Graduate Fellowship to continue his studies. [2] Knapp received his Ph.D. in 1965 from Princeton University under the supervision of Salomon Bochner. [4]
Knapp began his career as a C. L. E. Moore instructor for two years at the Massachusetts Institute of Technology, [4] before gaining a position as an assistant professor at Cornell University in 1967. [5] He was promoted to associate professor there in 1970 and full professor in 1975. [6] Knapp began spending some of his time at SUNY Stony Brook in 1986 and took a full-time position there in 1990. [4]
In a series of papers from 1976 to 1984, Knapp and Gregg Zuckerman gave the classification of tempered representations of semisimple Lie groups. [7] [8] [9] [10]
He won the Leroy P. Steele Prize for Mathematical Exposition in 1997. [11] In 2012 he became a fellow of the American Mathematical Society. [12]
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.
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.
In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.
In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois 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.
Gregg Jay Zuckerman is a mathematician and professor at Yale University working in representation theory. He discovered Zuckerman functors and translation functors, and with Anthony W. Knapp classified the irreducible tempered representations of semisimple Lie groups.
Tonny Albert Springer was a mathematician at Utrecht University who worked on linear algebraic groups, Hecke algebras, complex reflection groups, and who introduced Springer representations and the Springer resolution.
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.
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2, R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
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.
In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of the general linear group GL(V) such that G has an open dense orbit in V. The term prehomogeneous vector space was introduced by Mikio Sato in 1970. These spaces have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified first by Vinberg in his 1960 thesis in the special case when G is simple and later by Sato and Tatsuo Kimura in 1977 in the general case by means of a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of G acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on V which is invariant under the semisimple part of G.
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.
Jens Carsten Jantzen is a German mathematician and professor emeritus at Aarhus University working on representation theory and algebraic groups. He introduced the Jantzen filtration and translation functors.
In mathematical representation theory, a 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 Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagrams they are a different way of classifying simple Lie algebras.
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.
James Edward Humphreys was an American mathematician who worked in algebraic groups, Lie groups, and Lie algebras and applications of these mathematical structures. He is known as the author of several mathematical texts, such as Introduction to Lie Algebras and Representation Theory and Reflection Groups and Coxeter Groups.
In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.