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

TL;DR

This paper proves that the Fourier coefficients of Maass forms for SL(n,Z) are multiplicative and are eigenvalues of the Hecke algebra, extending known results from special cases to more general coefficients.

Contribution

It establishes the multiplicativity of general Fourier coefficients of Maass forms for SL(n,Z), showing they are eigenvalues of the Hecke algebra under certain coprimality conditions.

Findings

Fourier coefficients are eigenvalues of the Hecke algebra.

General coefficients satisfy multiplicativity relations.

Results extend known multiplicativity from special to general coefficients.

Abstract

The Fourier coefficients of a Maass form $\phi$ for SL$(n,\mathbb Z)$ are complex numbers $A_\phi(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form $A_\phi(m_1,1,\ldots,1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_\phi(m_1,\ldots,m_{n-1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations $$A_\phi\big(m_1m_1',\;m_2m_2', \;\ldots\; m_{n-1}m_{n-1}'\big) = A_\phi\big(m_1,m_2,\ldots,m_{n-1})\cdot A_\phi(m_1',m_2',\ldots,m_{n-1}'\big)$$ provided the products $\prod\limits_{i=1}^{n-1} m_i$ and $\prod\limits_{i=1}^{n-1} m_i'$ are relatively prime to each other.