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Hiraga--Ichino--Ikeda formal degree conjecture

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Assume the local Langlands correspondence for a split connected reductive group G\mathbf{G} over a non-Archimedean local field FF of characteristic zero. Let Π(G)\Pi(\mathbf{G}) be the irreducible smooth representations of G(F)\mathbf{G}(F), let ϕ\phi be an LL-parameter for G\mathbf{G}, and let Sϕ\mathcal{S}_{\phi} be its associated finite component group. Let Sϕ♮\mathcal{S}_{\phi}^{\natural} be the variant used in the conjecture, and let ⟨−,π⟩\langle-,\pi\rangle denote the irreducible character of Sϕ\mathcal{S}_{\phi} attached to π\pi in the LL-packet. For a non-trivial additive character ψF\psi_F, let γ(0,Ad⁡∘ϕ,ψF)\gamma(0,\operatorname{Ad}\circ\phi,\psi_F) be the gamma factor of the adjoint representation, where Ad⁡\operatorname{Ad} is the action of the Langlands dual group on Lie⁡(G^)/Lie⁡(Z(G^))\operatorname{Lie}(\hat{\mathbf{G}})/\operatorname{Lie}(Z(\hat{\mathbf{G}})).

Hiraga--Ichino--Ikeda's formal degree conjecture. Suppose that π∈Π(G)\pi\in\Pi(\mathbf{G}) is a discrete series representation whose LL-parameter is ϕ\phi. Then

deg⁡(π)=⟨1,π⟩∣Sϕ♮∣⋅∣γ(0,Ad⁡∘ϕ,ψF)∣.\deg(\pi)=\frac{\langle1,\pi\rangle}{|\mathcal{S}_{\phi}^{\natural}|}\cdot\left|\gamma(0,\operatorname{Ad}\circ\phi,\psi_F)\right|.

The conjecture predicts a precise relation between formal degrees, LL-packets, component groups, and adjoint gamma factors. It is known in a number of important cases but is not established uniformly for all split connected reductive groups in the stated generality.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Hiraga–Ichino–Ikeda formal degree conjecture

    Let GG be a reductive group over a nonarchimedean local field FF, let φ\varphi be a discrete Langlands parameter, and let π∈Πφ2(G,F)\pi \in \Pi_{\varphi}^2(G,F) correspond to an irreducible representation ρπ\rho_{\pi} of the component group associated with φ\varphi. Write γ(φ)=γ(φ,V,0)\gamma(\varphi)=\gamma(\varphi,V,0) for the gamma factor attached to V=Lie⁡(G^)/Lie⁡(Z(G^)ΓF)V=\operatorname{Lie}(\hat G)/\operatorname{Lie}(Z(\hat G)^{\Gamma_F}), and let μZ\G\mu_{Z\backslash G} be the chosen Haar measure on Z\GZ\backslash G. Hiraga–Ichino–Ikeda formal degree conjecture. For every such φ\varphi and π\pi,

    d(\pi, \mu_{Z\backslash G})=\frac{\dim \rho_{\pi}}{\\#\mathcal{S}_{\varphi}^\sharp}\lvert\gamma(\varphi)\rvert.

    This predicts that formal degrees are governed by adjoint gamma factors and the dimensions of the representations parametrizing local Langlands packets. It is known in a number of cases, but the general statement depends on the local Langlands correspondence and remains unresolved in full generality.

    source: Anantha Krishna B, “The formal degree conjecture for groups over local function fields”, arXiv:2605.26031 (2026).

References

Primary source

Guy Henniart and Masao Oi, “On Swan exponents of symmetric and exterior square Galois representations”, arXiv:2307.15248 (2023).

automorphic-representationsformal-degreeslocal-langlandslocal-langlands-correspondencerepresentation-theory
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