the same eigenvalues on every exponential function summand of an orbit function with eigenvalue $-4\pi\langle \lambda,\lambda\rangle$.

### Orthogonality

For any two complex squared integrable functions $\phi(x)$ and $\psi(x)$ defined on the fundamental region $F$, we define a continuous scalar product $$\begin{gathered}
\label{def_cont_scalar_product}
\l\phi(x),\psi(x)\r:=\int\limits_{{F}}\phi(x)\overline{\psi(x)}{\rm d}x.\end{gathered}$$ Here, integration is carried out with respect to the Euclidean measure, the bar means complex conjugation and $x\in {F}$, where ${F}$ is the fundamental region of either $W$ or $W^e$ (note that the fundamental region of $W^e$ is $F^e=F\cup r_i F$, where $r_i\in W$).

Any pair of orbit functions from the same family is orthogonal on the corresponding fundamental region with respect to the scalar product (\[def\_cont\_scalar\_product\]), namely $$\begin{gathered}
\label{cont_orthog c funcs}
\l C_{\lambda}(x),C_{\lambda'}(x)\r=|W_\lambda|\cdot|F|\cdot\delta_{\lambda\lambda'},
\\\label{cont_orthog s funcs}
\l S_{\lambda}(x),S_{\lambda'}(x)\r=|W|\cdot|F|\cdot\delta_{\lambda\lambda'},
\\\label{cont_orthog e funcs}
\l E_{\lambda}(x),E_{\lambda'}(x)\r=|W^e_{\lambda}|\cdot|F^e|\cdot\delta_{\lambda\lambda'},\end{gathered}$$ where $\delta_{\lambda\lambda'}$ is the Kronecker delta, $|W|$ is the order of the Weyl group, $|W_{\lambda}|$ and $|W^e_{\lambda}|$ are the sizes of the Weyl group orbits (the number of distinct points in the orbit), and $|F|$ and $|F^e|$ are volumes of fundamental regions. The volume $|F|$ was calculated in [@HrivnakPatera2009].

Proof of the relations (\[cont\_orthog c funcs\],\[cont\_orthog s funcs\],\[cont\_orthog e funcs\]) follows from the orthogonality of the usual exponential functions and from the 