  We study the action of a finite group on the Riemann-Roch space of certain
divisors on a curve. If $G$ is a finite subgroup of the automorphism group of a
projective curve $X$ over an algebraically closed field and $D$ is a divisor on
$X$ left stable by $G$ then we show the irreducible constituents of the natural
representation of $G$ on the Riemann-Roch space $L(D)=L_X(D)$ are of dimension
$\leq d$, where $d$ is the size of the smallest $G$-orbit acting on $X$. We
give an example to show that this is, in general, sharp (i.e., that dimension
$d$ irreducible constituents can occur). Connections with coding theory, in
particular to permutation decoding of AG codes, are discussed in the last
section. Many examples are included.
