Characters, bi-modules and representations in Lie group harmonic analysis
نویسنده
چکیده
This paper is a personal look at some issues in the representation theory of Lie groups having to do with the role of commutative hypergroups, bi-modules, and the construction of representations. We begin by considering Frobenius’ original approach to the character theory of a finite group and extending it to the Lie group setting, and then introduce bi-modules as objects intermediate between characters and representations in the theory. A simplified way of understanding the formalism of geometric quantization, at least for compact Lie groups, is presented, which leads to a canonical bi-module of functions on an integral coadjoint orbit. Some meta-mathematical issues relating to the construction of representations are considered.
منابع مشابه
Monomial Irreducible sln-Modules
In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.
متن کاملRepresentations of Double Coset Lie Hypergroups
We study the double cosets of a Lie group by a compact Lie subgroup. We show that a Weil formula holds for double coset Lie hypergroups and show that certain representations of the Lie group lift to representations of the double coset Lie hypergroup. We characterize smooth (analytic) vectors of these lifted representations.
متن کاملInfinite-dimensional Lie Superalgebras and Hook Schur Functions
Making use of a Howe duality involving the infinite-dimensional Lie superalgebra ĝl∞|∞ and the finite-dimensional group GLl of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of ĝl∞|∞ in terms of hook Schur functions. We use the reduction procedure of ĝl∞|∞ to ĝln|n to derive a character formula for a certain class of level 1 highest weight ir...
متن کاملModular Invariance of Characters of Vertex Operator Algebras
In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual ac...
متن کاملAffinization of Category O for Quantum Groups
Let g be a simple Lie algebra. We consider the category Ô of those modules over the affine quantum group Uq(ĝ) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category Ô. In particular, we develop the theory of q-characters a...
متن کامل