The weak orthogonality relation for Fourier coefficients of Maass forms

Let $\phi_j$ be Hecke–Maass forms for $\operatorname{SL}(N,\mathbb{Z})$ with Fourier coefficients $A_j(m_1,\ldots,m_{N-1})$. Under the same assumptions on the test functions $h_T$ and weights $\omega_j(T)$ as in the orthogonality relation, define

$$\omega_j(T)=\frac{h_T(\nu^{(j)})}{L(1,\phi_j,\operatorname{Ad})}.$$

The weak orthogonality relation. For positive integers $m_1,\ldots,m_{N-1}$,

$$\lim_{T\to\infty}\frac{\sum_{j=1}^{\infty}A_j(m_1,\ldots,m_{N-1})\omega_j(T)}{\sum_{j=1}^{\infty}\omega_j(T)}=\begin{cases}1,&m_1=\cdots=m_{N-1}=1,\\0,&\text{otherwise.}\end{cases}$$

This is introduced as a weaker consequence of the full orthogonality relation and is used as a conjectural input for the paper’s distribution results; no resolution is given.