Wavefront set conjecture for generic local Langlands parameters

Let FF be a local field of characteristic zero, let GG be a classical group over FF, and let πΠF(G)\pi\in\Pi_F(G) have a generic local LL-parameter. For each symbol \square, let WF(π)smax\operatorname{WF}_{\square}(\pi)^{s-\mathrm{max}} denote the maximal members of the corresponding wavefront set under the FF-stable topological order over NF(g)\mathcal{N}_F(\mathfrak{g})_\circ, and let WF(π)rmax\operatorname{WF}_{\square}(\pi)^{r-\mathrm{max}} denote the maximal members under the FF-rational topological order. Wavefront set conjecture. The identities

WFtr(π)smax=WFari(π)smax=WFwm(π)smax\operatorname{WF}_{\mathrm{tr}}(\pi)^{s-\mathrm{max}}=\operatorname{WF}_{\mathrm{ari}}(\pi)^{s-\mathrm{max}}=\operatorname{WF}_{\mathrm{wm}}(\pi)^{s-\mathrm{max}}

and

WFtr(π)rmax=WFari(π)rmax=WFwm(π)rmax\operatorname{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}=\operatorname{WF}_{\mathrm{ari}}(\pi)^{r-\mathrm{max}}=\operatorname{WF}_{\mathrm{wm}}(\pi)^{r-\mathrm{max}}

should both hold. This conjecture seeks to identify analytic, arithmetic, and generalized-Whittaker descriptions of maximal wavefront data for generic parameters; the paper presents substantial progress toward it for real classical groups.

Sources & referencesView supporting material

Primary source

Dihua Jiang, Dongwen Liu, Zhikang Luo, Jia-Jun Ma and Lei Zhang, “Arithmetic Wavefront Set and Microlocal Structure of Harish-Chandra Character”, arXiv:2606.05735 (2026).

Additional references

2 papers in this index state this conjecture (2022–2026). The statement above is taken from the most recent of them; the others are arXiv:2207.04700.

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