The theory of theta functions can be interpreted as automorphic representations on metaplectic groups (2-fold covering groups of $\mathrm{Sp}_{2}$, or $\mathrm{GL}_2$), and there's also a notion of $n$-fold covering groups which are studied by Brylinski-Deligne, Kubota, Weissman, and many people. Patterson and Bump-Hoffstein constructed cubic analogue of theta functions, i.e. automorphic forms on 3-fold covering groups of $\mathrm{GL}(2)$ and $\mathrm{GL}(3)$.
Since $n$-fold coverings of $G$ are extensions $1 \to \mu_n \to \tilde{G} \to G \to 1$, it seems possible to replace $\mu_n$ with other abelian groups. However, I never saw such an example in the context of automorphic forms and representations. My first thought was that half-integral weight Hilbert modular forms, whose weight is a tuple of half-integers, might be able to interpreted as automorphic forms on such covering groups. However, I realized that they aren't – Hilbert modular forms can be interpreted as automorphic forms on $\mathrm{GL}_{2}$ over real quadratic fields, so that half-integral weight Hilbert modular forms (seem to) correspond to automorphic forms on 2-fold covering groups of $\mathrm{GL}_{2}$ over a real quadratic field.
So my question is: is there any non-trivial and non-cyclic covering groups of a (well-known) reductive group? If there is, what are the corresponding automorphic forms and representations on it?
Best Answer
Let $\tilde{G}$ be a central extension of $G$ by a finite group $A$. Let $V$ be a complex representation of $G$. Then $V$ decomposes as a direct sum over all characters $\chi:A\to \mathbb{C}^\times$. $$ V=\bigoplus_{\chi} V_\chi, $$ where $$ V_\chi = \{v\in V \mid av=\chi(a)v \ \ \ \forall\ a\in A\}.$$
The action of $\tilde{G}$ on any $V_\chi$ factors through a central extension of $G$ by a finite cyclic group. So from this point of view, we don't lose any phenomena when restricting our attention to central extensions by cyclic groups.