Arthur multiplicity formula

Let $\psi$ be a discrete global Arthur parameter, let $\mathcal S_\psi$ be its finite component group, and let $\epsilon_\psi:\mathcal S_\psi\to\{\pm1\}$ be Arthur's canonical global character. For a member

\pi=\bigotimes_v'\pi_v

of the global $A$ -packet, the local packet pairings give a global character

\langle s,\pi\rangle = \prod_v \langle s_v,\pi_v\rangle .

The Arthur multiplicity formula selects the representations for which this character matches $\epsilon_\psi$ . In a standard multiplicity-free form,

m_\psi(\pi) = \frac{1}{|\mathcal S_\psi|} \sum_{s\in\mathcal S_\psi} \epsilon_\psi(s)\, \langle s,\pi\rangle .

By character orthogonality this is $1$ when the two characters agree and $0$ otherwise.

Meaning

Local packets supply many possible restricted tensor products. The formula is a global reciprocity constraint: local component-group labels must multiply to the distinguished global sign. It therefore explains why a global packet is not simply the Cartesian product of its local packets.

Source of the sign

The character $\epsilon_\psi$ is built from global root numbers of pairs of the cuspidal general-linear-group constituents of $\psi$ . It is not an arbitrary choice and can force a representation out of the discrete spectrum even when all of its local components lie in the prescribed packets.

Scope and variants

The displayed formula is the clean form for the quasi-split classical settings of Arthur's classification. Central quotients, even orthogonal outer automorphisms, non-quasi-split inner forms, and parameters with multiplicity require refinements of the packet and coefficient conventions.

The formula is a theorem for the symplectic and orthogonal groups treated by Arthur and in corresponding established extensions, such as quasi-split unitary groups. It remains part of the conjectural general theory for arbitrary reductive groups.

References

James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Theorem 1.5.2 and Chapter 8, AMS,
AMS.
Chung Pang Mok, “Endoscopic classification of representations of quasi-split unitary groups,” Memoirs of the AMS 235 (2015). AMS.