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.