nLab
representative function

Definition

Let either

  • GG be a Hausdorff topological group and FF either the field of real or complex numbers or

  • GG be a linear algebraic group over an infinite field FF .

Then an FF-valued representative function on GG is FF-valued function which arises in the form tρt\circ\rho where ρ:GEndV\rho:G\to End V is a representation of GG on a finite dimensional FF-vector space and t:EndVFt:End V\to F a linear functional.

In the topological group case the representative function is automatically continuous and in the algebraic case it is automatically a regular function.

On the other hand if we assume that a function f:GFf:G\to F is continuous or respectively regular then it is a representative function iff

  • all left translates L gf:hf(hg)L_g f : h\mapsto f(h g) where gGg\in G form a finite dimensional FF-vector space

what is true iff

  • all right translates R gf:hf(gh)R_g f : h\mapsto f(g h) where gGg\in G form a finite dimensional FF-vector space

The set of all representative functions on GG is a Hopf FF-algebra.

Literature

Related entries include Tannaka duality, locally compact topological group, Hopf algebra, harmonic analysis, representation theory

  • G. Hochschild, G. D. Mostow, Representations and representative functions of Lie groups, Annals of Mathematics, Second ser. 66 (1957) 495–542 jstor MR0098796

  • Pierre Cartier, A primer of Hopf algebras, Frontiers in number theory, physics, and geometry II, 537–615, Springer 2007.; preprint IH'ES 2006/40 pdf

  • J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

  • A. Derighetti Representative functions on topological groups, Comm. Math. Helv. 44:1 (1969) 476–483

Created on September 2, 2017 at 15:40:13. See the history of this page for a list of all contributions to it.