# Talk

Around 1980, Jacquet, Piatetski-Shapiro and Shalika published their paper “Automorphic Forms on $\mathrm{GL}(3)$” and Bump his book “Automorphic Forms on $\mathrm{GL}(3,\mathbb{R})$” in which they founded the theory of automorphic forms on $\mathrm{GL}(3)$ and proved the functional equation for an $L$-function attached to an automorphic representation of $\mathrm{GL}_3(\mathbb{A})$ for the first time.
As in these two works, we will introduce Hecke-Maass cusp forms $\phi\in\pi$ on $\mathrm{GL}(3)$, their first projections $\phi^1$ and their Fourier-Whittaker expansions. To do this, we have to define the corresponding Whittaker functions $W_\phi$, the related dual automorphic forms $\tilde{φ}$ and their dual Whittaker functions $\widetilde{W}_\phi$. We will also have to prove some properties of the Fourier- Whittaker coefficients of Hecke-Maass cusp forms on $\mathrm{GL}(3)$.
In the end of our talk, we will discuss the various functional equations related to the $L$-function on $\mathrm{GL}(3)$ coming from an arbitrary Hecke-Maass cusp form $\phi\in\pi$. This is done without the limitation of $L$-functions coming only from Maass forms as in Bump’s book.