Derived geometric Satake for $\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911840564412416 |
|---|---|
| author | Devalapurkar, Sanath K. |
| author_facet | Devalapurkar, Sanath K. |
| contents | In this note, we study the local relative geometric Langlands conjecture of Ben-Zvi--Sakellaridis--Venkatesh for the spherical subgroup $\mathrm{PGL}_2^\mathrm{diag}$ of the triple product $\mathrm{PGL}_2^{\times 3}$ (and also for the spherical subgroup $\mathrm{G}_2$ of $\mathrm{SO}_8/μ_2$), whose corresponding Langlands dual $\mathrm{SL}_2^{\times 3}$-variety can be identified with the symplectic vector space $(\mathbf{A}^2)^{\otimes 3} \cong \mathbf{A}^8$ of $2\times 2 \times 2$-cubes. Our analysis relies on a construction of Bhargava relating $2 \times 2 \times 2$-cubes to Gauss composition on quadratic forms, arising here as the moment map for the Hamiltonian $\mathrm{SL}_2^{\times 3}$-action on $(\mathbf{A}^2)^{\otimes 3}$, and the Cayley hyperdeterminant as studied by Gelfand-Kapranov-Zelevinsky. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09853 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derived geometric Satake for $\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}$ Devalapurkar, Sanath K. Representation Theory Algebraic Geometry Algebraic Topology In this note, we study the local relative geometric Langlands conjecture of Ben-Zvi--Sakellaridis--Venkatesh for the spherical subgroup $\mathrm{PGL}_2^\mathrm{diag}$ of the triple product $\mathrm{PGL}_2^{\times 3}$ (and also for the spherical subgroup $\mathrm{G}_2$ of $\mathrm{SO}_8/μ_2$), whose corresponding Langlands dual $\mathrm{SL}_2^{\times 3}$-variety can be identified with the symplectic vector space $(\mathbf{A}^2)^{\otimes 3} \cong \mathbf{A}^8$ of $2\times 2 \times 2$-cubes. Our analysis relies on a construction of Bhargava relating $2 \times 2 \times 2$-cubes to Gauss composition on quadratic forms, arising here as the moment map for the Hamiltonian $\mathrm{SL}_2^{\times 3}$-action on $(\mathbf{A}^2)^{\otimes 3}$, and the Cayley hyperdeterminant as studied by Gelfand-Kapranov-Zelevinsky. |
| title | Derived geometric Satake for $\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}$ |
| topic | Representation Theory Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2404.09853 |