Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters
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| Format: | Preprint |
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2025
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| author | Balodis, Kristaps Cunningham, Clifton Fiori, Andrew Zhang, Qing |
| author_facet | Balodis, Kristaps Cunningham, Clifton Fiori, Andrew Zhang, Qing |
| contents | In this paper we prove the forward direction of the conjecture of Gross-Prasad that an L-packet $Π_ϕ(G)$ contains a generic representation if and only if $L(s, ϕ, \Ad)$ is regular at $s=1$, by assuming the local Langlands correspondence and the $p$-adic Kazhdan-Lusztig hypothesis. We then prove an analogous statement for ABV-packets, which together with Vogan's conjecture on ABV-packets implies that if Arthur's conjectures for $G$ are known, then one direction of Shahidi's enhanced genericity conjecture holds: If an Arthur packet $Π_ψ(G)$ contains a generic representation, then $ϕ_ψ$ is tempered. In the case where the infinitesimal parameters in question are all unramified, we obtain converses to the above statements. We also offer some speculation about the relationship between Arthur type representations, and singularities in varieties of Langlands parameters defined by Vogan. Finally, we recover some facts about central characters using the results above. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters Balodis, Kristaps Cunningham, Clifton Fiori, Andrew Zhang, Qing Representation Theory 11F70, 32S60 In this paper we prove the forward direction of the conjecture of Gross-Prasad that an L-packet $Π_ϕ(G)$ contains a generic representation if and only if $L(s, ϕ, \Ad)$ is regular at $s=1$, by assuming the local Langlands correspondence and the $p$-adic Kazhdan-Lusztig hypothesis. We then prove an analogous statement for ABV-packets, which together with Vogan's conjecture on ABV-packets implies that if Arthur's conjectures for $G$ are known, then one direction of Shahidi's enhanced genericity conjecture holds: If an Arthur packet $Π_ψ(G)$ contains a generic representation, then $ϕ_ψ$ is tempered. In the case where the infinitesimal parameters in question are all unramified, we obtain converses to the above statements. We also offer some speculation about the relationship between Arthur type representations, and singularities in varieties of Langlands parameters defined by Vogan. Finally, we recover some facts about central characters using the results above. |
| title | Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters |
| topic | Representation Theory 11F70, 32S60 |
| url | https://arxiv.org/abs/2504.04163 |