Spectral structure and eigenmatrices of self-affine measures with $p$-digits in ${\mathbb{R}}^{\mathbf{n}}$

For a prime number $p>2$, let $\mathbf{0}\in D\subset {\mathbb{Z}}^n$ be a $p$-element digit set satisfying $\mathcal{Z}({\widehat{\delta }}_D)={\cup }_{j=1}^{p-1}(\frac{j}{p}\mathit{a}+{\mathbb{Z}}^n)$ for some $\mathit{a}\in \{(i_1,\dots ,i_n{)}^t:i_k\in [1,p-1]\cap \mathbb{Z},1\le k\le n\}$, where $\mathcal{Z}({\widehat{\delta }}_D)$ is the zero set of the Fourier transform of ${\delta }_D$. Let $Q$ be an integer expansive diagonal matrix in ${\mathbb{R}}^n$, the self-affine measure ${\mu }_{Q,D}$ is defined by $${\mu }_{Q,D}(\cdot )=\frac{1}{\mathrm{\#}D}\sum _{d\in D}{\mu }_{Q,D}(Q(\cdot )-d)\mathrm{.}$$ In this paper, we first provide sufficient condition for a maximal orthogonal family $E_{\mathrm{\Lambda }}=\{e^{-2\pi i⟨\lambda ,x⟩}:\lambda \in \mathrm{\Lambda }\subset {\mathbb{R}}^n\}$ to be an orthogonal basis of $L^2({\mu }_{Q,D})$ when $Q=p\mathrm{diag}[q,\dots ,q]$ with $q\ge 1$. Then we obtain necessary and sufficient conditions for the integer matrix $R$ such that $E_{\mathrm{\Lambda }}$ and $E_{R\mathrm{\Lambda }}$ are both orthogonal basis of $L^2({\mu }_{Q,D})$. Furthermore, for $Q=p\mathrm{diag}[l_1,\dots ,l_n]$ with $\mathrm{\mid }{l}_1\dots l_n\mathrm{\mid }>1$, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of ${\mu }_{Q,D}$.