fact that a given weight $\mu\in P$ belongs to precisely one orbit function.

The families of $C$-, $S$- and $E$-functions are complete on the fundamental domain. The completeness of these systems follows from the completeness of the system of exponential functions; i.e., there does not exist a function $\phi(x)$, such that $\langle\phi(x),\phi(x)\rangle>0$, and at the same time $\langle\phi(x),\psi(x)\rangle=0$ for all functions $\psi(x)$ from the same system.

### Orbit functions of $A_n$ acting in $\R^{n+1}$ {#orb_func_n+1}

Relations (\[lambda\_to\_l\]) allow us to rewrite variables $\lambda$ and $x$ in an orbit function in the $e$-basis. Therefore we can obtain the $C$-, $S$- and $E$- functions acting in $\R^{n+1}$ $$\begin{gathered}
C_\lambda(x) = \sum_{s\in {\rm S}_{n+1}} e^{2\pi  i  (s(\lambda), x)},\\
C_\lambda(x) = \sum_{s\in {\rm S}_{n+1}} ({\rm sgn}\,s)e^{2\pi  i  (s(\lambda), x)},\\
E_\lambda(x) = \sum_{s\in  {\rm Alt}_{n{+}1}}e^{2\pi  i (s(\lambda), x)},\end{gathered}$$ where $(\cdot\,,\cdot)$ is a scalar product in $\R^{n+1}$, ${\rm sgn}\,s$ is the permutation sign, and ${\rm Alt}_{n{+}1}$ is the alternating group acting on an $(n+1)$-tuple of numbers. Note that variables $x$ and $\lambda$ are in the hyperplane $\mathcal{H}$.

Using the identity $\l\lambda,r_ix\r=\l r_i\lambda,x\r$ for the reflection $r_i$, $i=1,\ldots,n$, it can be verified that $$\begin{gathered}
\label{ri_symm_of_C_S}
C_\lambda(r_ix)=C_{r_i\lambda}(x)=C_\lambda(x),\quad
\text{and} \quad
{S}_{r_i\lambda}(x)={S}_\lambda(r_ix)=-{S}_\lambda(x).\end{gathered}$$

Note that it is easy to see for 