Non-tempered automorphic representations of inner forms of $Sp(4)$ (Automorphic representations, automorphic $L$-functions and arithmetic)

For a reductive group $G$ defined over a number field $k$ , an unitary representations of $G(A_{k})$ on the space of $L^{2}$ -automorphic forms $L^{2}(G(k)\backslash G(A_{k}))$ is defined by the right regular action.As for the irreducible decomposition of its discrete spectrumArthur also conjectured the multiplicity of $\pi\in\Pi_{\psi}$ in the associated subspace for $\psi$ .To describe the multiplicity, we need the information about global and local S-group for $\psi$ , and pairings between S-groups and A-packets.In this note, we treat the case that $G$ is a non-split inner form of $Sp(4)$ .$(Sp(4)$ is the isometry group of 4-dimensional symplectic space.)I give an evaluation of the multiplicities of non-tempered irreducible automorphic representations which appear in the residual spectrum, or are CAP representations (Theorem 4.1, Theorem 5.1 and Proposition 6.2).Here a cuspidal representation $\pi$ is said to be of CAP if for any cusp form $\phi$ in $\pi$ which is K-finite where $K$ is a maximal compact subgroup of $G(A_{k})$ , there exists an element $\phi'$ of an irreducible component of the residual spectrum such that $\phi$ and $\phi'$ share the same absolute values of Hecke eigenvalues at almost all places of $k$ .According to Arthur's conjecture, any irreducible non-tempered automorphic representation of $G(A_{k})$ appears in A-packet for some A-parameter $\psi$ of DAP type.Here an A-parameter $\psi$ : $\mathcal{L}_{k}\cross SL(2, \mathbb{C})arrow LG$ where $\mathcal{L}_{k}$ is the hypothetical Langlands group of $k$ and $LG$ is the L-group of $G$ is said to be of DAP type if $\psi$ is elliptic and the restriction to $SL(2, \mathbb{C})$ of $\psi$ is non-trivial.This implies that irreducible non-tempered automorphic representations should be exhausted by the irreducible components of the residual spectrum and CAP representations.From the evaluation of the multiplicity for irreducible non-tempered automorphic rep- resentation of $G(A_{k})$ , we can guess a explicit description of multiplicity of these repre- sentations (Expectation 8.1).Our interest is whether this description coincides with the Arthur's conjectural multiplicity.More precisely, the problem is whether there are pair- ings between S-groups and A-packets such that the description coincides with the Arthur's multiplicity defined by these pairings.Our main result is that such pairings exist (Section