# 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

• 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

Last revised on August 22, 2023 at 17:17:08. See the history of this page for a list of all contributions to it.