On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910026164076544 |
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| author | Friedberg, Solomon Offen, Omer |
| author_facet | Friedberg, Solomon Offen, Omer |
| contents | The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of $SL_3$ to $PGL_3$, and also characterize the image of the lift. The characterization would be through the nonvanishing of a certain global period involving a function in the space of the automorphic minimal representation $Θ_{SO_8}$ for split $SO_8({\mathbb{A}})$, consistent with a 2001 conjecture of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global distribution on $PGL_3({\mathbb{A}})$ involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of $SL_3({\mathbb{A}})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the Fundamental Lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_01247 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma Friedberg, Solomon Offen, Omer Number Theory Representation Theory 11F70 (Primary) 11F27, 11F67, 11F72, 22E50, 22E55 (Secondary) The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of $SL_3$ to $PGL_3$, and also characterize the image of the lift. The characterization would be through the nonvanishing of a certain global period involving a function in the space of the automorphic minimal representation $Θ_{SO_8}$ for split $SO_8({\mathbb{A}})$, consistent with a 2001 conjecture of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global distribution on $PGL_3({\mathbb{A}})$ involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of $SL_3({\mathbb{A}})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the Fundamental Lemma. |
| title | On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma |
| topic | Number Theory Representation Theory 11F70 (Primary) 11F27, 11F67, 11F72, 22E50, 22E55 (Secondary) |
| url | https://arxiv.org/abs/2202.01247 |