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

Abstract

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