On local descent to metaplectic groups and local theta correspondence. (English) Zbl 1465.11141
One of the far-reaching conjectures in number theory is the Langlands functoriality conjecture. One way to attack this conjecture is via the profound Ginzburg-Rallis-Soudry descent method with the help of converse theorem, which works well for the functoral lifts from classical groups to general linear groups. Such a method has since been explored intensively and fruitfully globally and locally by D. Ginzburg, S. Rallis, D. Soudry, and their descendents [D. Ginzburg et al., The descent map from automorphic representations of \(\text{GL}(n)\) to classical groups. Hackensack, NJ: World Scientific (2011; Zbl 1233.11056)].
In a preprint not avaible online [“On local descent to inner forms of odd special orthogonal groups”], the author obtains a complete result for a twisted local descent construction for \(\mathrm{SO}_{2n+1}\) from \(\mathrm{GL}_m\) under an assumption, i.e., tower property, first occurrence, supercuspidality and genericity. Given this, in the paper under review, the author studies an analogous twisted version of the local descent construction for metaplectic groups \(\widetilde{\mathrm{Sp}}_{2n}\) from \(\mathrm{GL}_m\) by investigating its relation with its counterpart for \(\mathrm{SO}_{2n+1}\) under Howe’s theta correspondence for the dual pair \((\widetilde{\mathrm{Sp}}_{2n},~\mathrm{SO}_{2n+1})\). By doing so, the author proves all other expected properties under an additioanl assumption on the first occurrence. At the end of the paper, the author investigates the case \(n=1\) in detail unconditionally via a local-global argument.
This is a well-written paper with a detailed background on the local descent construction.
In a preprint not avaible online [“On local descent to inner forms of odd special orthogonal groups”], the author obtains a complete result for a twisted local descent construction for \(\mathrm{SO}_{2n+1}\) from \(\mathrm{GL}_m\) under an assumption, i.e., tower property, first occurrence, supercuspidality and genericity. Given this, in the paper under review, the author studies an analogous twisted version of the local descent construction for metaplectic groups \(\widetilde{\mathrm{Sp}}_{2n}\) from \(\mathrm{GL}_m\) by investigating its relation with its counterpart for \(\mathrm{SO}_{2n+1}\) under Howe’s theta correspondence for the dual pair \((\widetilde{\mathrm{Sp}}_{2n},~\mathrm{SO}_{2n+1})\). By doing so, the author proves all other expected properties under an additioanl assumption on the first occurrence. At the end of the paper, the author investigates the case \(n=1\) in detail unconditionally via a local-global argument.
This is a well-written paper with a detailed background on the local descent construction.
Reviewer: Caihua Luo (Göteborg)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 20G25 | Linear algebraic groups over local fields and their integers |
Citations:
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