On the cubic Shimura lift to $PGL(3)$: Hecke correspondences
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arXiv
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| Format: | Preprint |
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2025
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| author | Friedberg, Solomon Offen, Omer |
| author_facet | Friedberg, Solomon Offen, Omer |
| contents | In this paper we establish a new Fundamental Lemma for Hecke correspondences. Let $F$ be a local field containing the cube roots of unity. We exhibit an algebra isomorphism of the spherical Hecke algebra of $PGL_3(F)$ and the spherical Hecke algebra of anti-genuine functions on the cubic cover $G'$ of $SL_3(F)$. Then we show that there is a matching (up to a specific transfer factor) of distributions on the two groups for all functions that correspond under this isomorphism. On $PGL_3(F)$ the distributions are relative distributions attached to a period involving the minimal representation on $SO_8$, while on $G'$ they are metaplectic Kuznetsov distributions. This Fundamental Lemma is a key step towards establishing a relative trace formula that would give a new global Shimura lift from genuine automorphic representations on the triple cover of $SL_3$ to automorphic representations on $PGL_3$, and also characterize the image of the lift by means of a period. It extends the matching for the unit elements of the Hecke algebras established by the authors in prior work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06963 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the cubic Shimura lift to $PGL(3)$: Hecke correspondences Friedberg, Solomon Offen, Omer Number Theory Representation Theory Primary 11F70, Secondary 11F25, 11F27, 11F67, 11F72, 22E50, 22E55 In this paper we establish a new Fundamental Lemma for Hecke correspondences. Let $F$ be a local field containing the cube roots of unity. We exhibit an algebra isomorphism of the spherical Hecke algebra of $PGL_3(F)$ and the spherical Hecke algebra of anti-genuine functions on the cubic cover $G'$ of $SL_3(F)$. Then we show that there is a matching (up to a specific transfer factor) of distributions on the two groups for all functions that correspond under this isomorphism. On $PGL_3(F)$ the distributions are relative distributions attached to a period involving the minimal representation on $SO_8$, while on $G'$ they are metaplectic Kuznetsov distributions. This Fundamental Lemma is a key step towards establishing a relative trace formula that would give a new global Shimura lift from genuine automorphic representations on the triple cover of $SL_3$ to automorphic representations on $PGL_3$, and also characterize the image of the lift by means of a period. It extends the matching for the unit elements of the Hecke algebras established by the authors in prior work. |
| title | On the cubic Shimura lift to $PGL(3)$: Hecke correspondences |
| topic | Number Theory Representation Theory Primary 11F70, Secondary 11F25, 11F27, 11F67, 11F72, 22E50, 22E55 |
| url | https://arxiv.org/abs/2507.06963 |