Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb{Z}$)

The Fourier coefficients of a Maass form $\phi$ for SL$(n,\mathbb{Z})$ are complex numbers $A_{\phi }(M)$, where $M=(m_1,m_2,\dots ,m_{n-1})$ and $m_1,m_2,\dots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form $A_{\phi }(m_1,1,\dots ,1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_{\phi }(m_1,\dots ,m_{n-1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations $$A_{\phi }(m_1{m}_1^{\mathrm{\prime }},m_2{m}_2^{\mathrm{\prime }},\dots m_{n-1}{m}_{n-1}^{\mathrm{\prime }})=A_{\phi }(m_1,m_2,\dots ,m_{n-1})\cdot A_{\phi }(m_1^{\mathrm{\prime }},m_2^{\mathrm{\prime }},\dots ,m_{n-1}^{\mathrm{\prime }})$$ provided the products ${\prod }_{i=1}^{n-1}{m}_i$ and ${\prod }_{i=1}^{n-1}{m}_i^{\mathrm{\prime }}$ are relatively prime to each other.