nLab representative function

Definition

Let either

• $G$ be a Hausdorff topological group and $F$ either the field of real or complex numbers or

• $G$ be a linear algebraic group over an infinite field $F$ .

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

In the (Hausdorff) 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 $f:G\to F$ is continuous or respectively regular then it is a representative function iff

• all left translates $L_g f : h\mapsto f(h g)$ where $g\in G$ form a finite dimensional $F$-vector space

what is true iff

• all right translates $R_g f : h\mapsto f(g h)$ where $g\in G$ form a finite dimensional $F$-vector space

The set of all representative functions on $G$ is a Hopf $F$-algebra.

Hochschild1971 If $G$ is an arbitrary monoid with multiplication $m:G\times G\to G$ then $m$ induces a map $m^*:Fun(G,k)\to Fun(G\times G,k)$, $m^*(f):f\mapsto f\circ m$. We say that $f\in Fun(G,k)$ is representative if $m^*(f)$ is in the image of the canonical map $Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k)$. Equivalently, $f$ is representative if the span of all functions $g\cdot f : h\mapsto f(h\cdot g)$ is finite dimensional. It follows then that $m^*(f)$ is in fact in (the image of) $R(G)\otimes R(G)$ where $R(G)$ is the space of all representative functions on $G$.

Relations to other spaces of functions on a group

Let $G$ be an algebraic subgroup of $GL(d,k)$. The Hopf algebra of regular (“coordinate”) functions $\mathcal{O}(G)$ is a finitely generated subHopf algebra of the Hopf algebra of continuous representative functions on $G$, CartierHopf.

Peter-Weyl theorem says that the representative functions on a compact topological group form a dense subspace of the space of all continuous functions.

Literature

Related entries include Tannaka duality, locally compact topological group, Hopf algebra, harmonic analysis, representation theory, linear algebraic group, distribution on an affine algebraic group

• 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

• G. Hochschild, Introduction to algebraic group schemes, 1971