Arthur-packet equality conjecture for tempered L-packets

Let π=π(λ,s,Oλ,s,L)\pi=\pi(\lambda,s,{{\mathcal O}}^\vee_{\lambda,s},{{\mathcal L}}^\vee) be an irreducible representation in the tempered LL-packet Π(φ)=Π(λ,s,Oλ,s)\Pi(\varphi)=\Pi(\lambda,s,{{\mathcal O}}^\vee_{\lambda,s}), and set Oφ=GOλ,s{\mathbb O}^\vee_\varphi=G^\vee\cdot{{\mathcal O}}^\vee_{\lambda,s}. Arthur-packet equality conjecture. Every constituent satisfies

kˉWF(AZ(π))d(Oφ),{}^{\bar {\mathsf k}}\mathsf{WF}({\mathsf{AZ}}(\pi))\leq d^\vee({\mathbb O}^\vee_\varphi),

and equality is achieved for some π\pi in the LL-packet. This is the tempered-packet reformulation of the preceding Arthur-packet expectation and follows from the broader conjectural picture if the relevant Arthur packets are defined; its general validity remains open.

Sources & referencesView supporting material

Primary source

Dan Ciubotaru and Ju-Lee Kim, “The wavefront set: bounds for the Langlands parameter”, arXiv:2403.14261 (2025).

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