Mundy's Arthur multiplicity conjecture for certain G2 representations

Assume S0=S_0=\emptyset, where S0S_0 is the subset of S(πf)S(\pi_{\mathbf f}) consisting of places pp with πp=Stp\pi_p={\rm St}_p. Let SS(πf)S'\subset S(\pi_{\mathbf f}), let Π\Pi_\infty be the specified quaternionic discrete series representation of G2(R)G_2(\mathbb R), and let Πp+\Pi_p^+ and Πp\Pi_p^- be the local representations defined in the source. Set

Π=ΠpSΠppSΠp+.\Pi=\Pi_\infty\otimes\bigotimes_{p\in S'}\Pi_p^-\otimes\bigotimes_{p\notin S'}'\Pi_p^+.

Mundy's conjecture. The representation Π\Pi occurs in Ldisc2(G2(Q)\G2(A))L^2_{\rm disc}(G_2(\mathbb Q)\backslash G_2(\mathbb A)) with multiplicity zero or one, and it occurs with multiplicity one if and only if

ϵ(12,Sym3(πf))=(1)#S,\epsilon\left(\frac12,\operatorname{Sym}^3(\pi_f)\right)=-(-1)^{\#S'},

equivalently, if and only if #S\#S' is even. This is presented as a special case of an Arthur multiplicity formula. The source subsequently says that, when C=1C=1, the conjecture is true, so its status is solved in that case, but does not establish the unrestricted statement beyond that setting.

Sources & referencesView supporting material

Primary source

Henry H. Kim and Takuya Yamauchi, “On the Fourier expansion of Gan-Gurevich lifts on the exceptional group of type G_2”, arXiv:2411.16953 (2025).

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