By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on Sato-Tate theorem calculated the symmetry type of many families of automorphic L-functions of varying weight or level, with the assumption that the family of automorphic representations involved are cohomological at the $\infty$-place.



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> What is the expected answer for the symmetry type of Maass forms and their symmetric powers? (level and weight aspect)
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I am aware of the paper by Alpoge-Miller that Maass forms on $GL(2)/\mathbb{Q}$ of a fixed level and increasing Laplacian eigenvalue should be an orthogonal family. Is there anything known about their symmetric powers and in particular, do we expect them to behave like modular forms? (where it is known that say in fixed level and changing weight, (fixed) even symmetric power of modular forms of weight $k$ form a symplectic family, whereas odd symmetric power of modular forms would form an orthogonal family; cf Duenez-Miller, arXiv:math/0607688)



Symmetry type of non-cohomological automorphic forms
You should read the following preprint of Sarnak, Shin, and Templier:


<http://arxiv.org/abs/1401.5507>


In particular, they study "nice" families $\mathfrak{F}$ of automorphic representations. They assume RH and that there exists $A < \infty$, $\delta < 1$, such that uniformly in $n \geq 1$,
\[\sum\_{\pi \in \mathfrak{F}(x)} \lambda\_{\pi}(n) = t\_{\mathfrak{F}}(n) |\mathfrak{F}(x)| + O(n^A |\mathfrak{F}(x)|^{\delta}),\]
where $\mathfrak{F}(x)$ denotes the set of $\pi \in \mathfrak{F}$ of analytic conductor at most $x$. Then they define invariants
\[i\_1(\mathfrak{F}) = \lim\_{x \to \infty} \frac{1}{x} \sum\_{p \leq x} |t\_{\mathfrak{F}}(p)|^2 \log p,\]
\[i\_2(\mathfrak{F}) = \lim\_{x \to \infty} \frac{1}{x} \sum\_{p \leq x} t\_{\mathfrak{F}}(p)^2 \log p,\]
\[i\_3(\mathfrak{F}) = \lim\_{x \to \infty} \frac{1}{x} \sum\_{p \leq x} t\_{\mathfrak{F}}(p^2) \log p.\]


Then $i\_1(\mathfrak{F}) \geq 1$ with equality iff almost every $\pi$ is cuspidal. We have $0 \leq i\_2(\mathfrak{F}) \leq 1$, with $i\_2(\mathfrak{F}) = 1$ iff almost every $\pi$ is selfdual and $i\_2(\mathfrak{F}) = 0$ iff almost every $\pi$ is not selfdual (i.e. unitary); in this case, $i\_3(\mathfrak{F}) = 0$. We have $-1 \leq i\_3(\mathfrak{F}) \leq 1$, with $i\_2(\mathfrak{F}) = 1$ iff almost every $\pi$ is orthogonal and $i\_3(\mathfrak{F}) = -1$ iff almost every $\pi$ is symplectic.


Here we say that a single automorphic representation $\pi$ is unitary if it is not selfdual, while if it is self-dual, then the completed Rankin-Selberg $L$-function $\Lambda(s,\pi \times \widetilde{\pi})$ factorises as $\Lambda(s,\mathrm{sym}^2 \pi) \Lambda(s,\wedge^2 \pi)$; if the former has a pole at $s = 1$, then $\pi$ is said to be orthogonal, while $\pi$ is symplectic if the latter has a pole at $s = 1$.


So the way to determine the symmetry type of a family of automorphic representations is to use the trace formula to work out what $t\_{\mathfrak{F}}(n)$ ought to look like, then use this to calculate $i\_1(\mathfrak{F}), i\_2(\mathfrak{F}), i\_3(\mathfrak{F})$, with this telling you the symmetry type of the family.


