On the intersection of local Arthur packets for classical groups and applications (2201.10539v3)

Abstract: 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.