On the intersection of local Arthur packets for classical groups and applications

In this paper, for symplectic and split odd special orthogonal groups, we develop an account of theory on the intersection problem of local Arthur packets. Specifically, following Atobe's reformulation on M{\oe}glin's construction of local Arthur packets, we give a complete set of operators on the construction data, based on which, we provide algorithms and Sage codes to determine whether a given representation is of Arthur type. Furthermore, for any representation $\pi$ of Arthur type, we give a precise formula for the set $$ \Psi(\pi)=\{ \text{local Arthur parameter }\psi \ | \ \text{the local Arthur packet } \Pi_{\psi} \text{ contains } \pi\}.$$ Our results have many applications, including the precise counting of tempered representations in any local Arthur packet, specifying and characterizing "the" local Arthur parameter in $\Psi(\pi)$ for $\pi$, especially when $\pi$ belongs to several local Arthur packets but does not belong to any local $L$-packet of Arthur type.