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The unicity of types conjecture for supercuspidal representations

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Let G=G(F)G=\mathbf{G}(F) be the group of rational points of a connected reductive algebraic group over a non-archimedean local field, and let π\pi be a supercuspidal representation of GG. An archetype is a GG-conjugacy class of typical irreducible representations of a maximal compact subgroup of GG for the inertial support [G,π]G[G,\pi]_G. The unicity of types conjecture. There exists a [G,π]G[G,\pi]_G-archetype, and there exists at most one [G,π]G[G,\pi]_G-archetype defined on each GG-conjugacy class of maximal compact subgroups of GG. This conjecture asserts existence and unicity of maximal-compact types for supercuspidal representations of arbitrary pp-adic groups; the source presents it as a generalization of the unicity of types known for general linear groups.

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  1. The unicity of types conjecture for supercuspidal representations

    Fix a cuspidal datum Σ=(G⃗,x,σ,r⃗,ϕ⃗)\Sigma=(\vec{\mathbf{G}},x,\sigma,\vec{r},\vec{\phi}) of GG, an irreducible supercuspidal representation π\pi of GG, and a [G,π]G[G,\pi]_G-type (J,λ)(J,\lambda), with λ=σ⊗κ\lambda=\sigma\otimes\kappa. Let KK be a maximal compact subgroup of GG and let (K,τ)(K,\tau) be a [G,π]G[G,\pi]_G-type. Unicity of types conjecture. There exists a g∈Gg\in G such that gJ⊂K^gJ\subset K and

    Hom⁡K(τ,Ind⁡gJKλ)≠0.\operatorname{Hom}_K\left(\tau,\operatorname{Ind}_{^gJ}^K\lambda\right)\neq 0.

    This conjecture predicts that every maximal-compact type for the Bernstein component determined by π\pi contains, after conjugation, the explicitly constructed type (J,λ)(J,\lambda) in its compact induction. Its status is not determined by the supplied source information.

    source: Peter Latham and Monica Nevins, “Typical representations via fixed point sets in Bruhat–Tits buildings”, arXiv:1909.05895 (2020).

References

Primary source

Peter Latham, “The unicity of types for depth-zero supercuspidal representations”, arXiv:1606.01691 (2017).

p-adic-groupsrepresentation-theorysupercuspidal-representationstypes
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