Coming from a number-theoretic background, I certainly care about $L$-functions and in particular automorphic ones. For automorphic forms on $SL\_2(\mathbb{Z}) \backslash SL\_2(\mathbb{R})$, $L$-function can be interpreted as the Mellin transform of Fourier expansion at the cusp.


If we look at automorphic forms on $SL\_(n,\mathbb{Z}) \backslash SL(n,\mathbb{R})$, we still have a Fourier expansion in $n-1$ variables since it is periodic with respect to the super-diagonal unipotent group. This gives us Fourier coefficients $A(m\_1,\cdots,m\_{n-1})$. In forming the $L$-functions for $SL(n,\mathbb{R})$, we just look at all the Fourier coefficients $A(m,1,\cdots,1)$ and define
$$L(s) = \sum\_{m=1}^{\infty} A(m,1,\cdots,1)m^{-s}$$
But the most natural thing to do (comparing to the $SL\_2(\mathbb{R})$ case) would be to form the multiple Dirichlet series instead
$$\sum\_{m\_1,\cdots,m\_{n-1}} \frac{A(m\_1,\cdots,m\_{n-1})}{m\_1^{s\_1} \cdots m\_{n-1}^{s\_{n-1}}}$$
It certainly seems that people care "less" about this multiple Dirichlet series: in the whole Langlands business we always take about L-parameters and stuff, which seems to imply that we track only the information of $L$-function but not the whole multiple Dirichlet series.


So here are my questions,


* Is there any conceptual reason why we care more about $L$-functions rather than multiple Dirichlet series?
* Of course, Fourier expansion is available when there is a cusp. For cocompact arithmetic quotients of $GL\_n(\mathbb{R})$ (or other reductive groups in general), can one similarly define a multiple Dirichlet series that should incorporate the data of $L$-function?


Thank you.



Do we care about multiple zeta functions?
It may help clarify things to work out a specific example, although the OP may know this. In case $n=3$, the double Dirichlet series evaluates as
$$ \sum\_{m,n=1}^{\infty} \frac{A\_F(m,n)}{m^{w} n^s} = \frac{L(\overline{F},w) L(F, s)}{\zeta(s+w)}.$$
Here $\overline{F}$ is the contragredient of $F$.
This is known as Bump's double Dirichlet series, and this is worked out in Section 6.6 of Goldfeld's book, *Automorphic forms and $L$-functions for the group $GL(n,\mathbb{R})$*. There are references there for generalizations.


It seems clear (to me) that the $GL\_3$ automorphic $L$-function $L(F,s)$ (and its contragredient, and $\zeta$) are the fundamental objects, but also that there are many interesting Dirichlet series that one can construct from an automorphic form.


