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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.

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Equivalent formulations 1

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).

Sources & referencesView supporting material

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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