Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

By W. Gan, B. Gross, Dipendra Prasad. 2009

492 citations recorded by the source corpus.

We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2237fd17db5cca1e7f1f0d43e2b0b94b3a0b5e26

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In Algebra and Number Theory

1. E. Viehmann
2. James Newton
3. Michael Rapoport
4. J. Thorne
5. Wei Zhang
6. Xuhua He
7. C. Skinner
8. Peter Scholze

In Discrete Mathematics and Combinatorics

1. E. Viehmann
2. Michael Rapoport
3. J. Thorne
4. Wei Zhang
5. James Newton
6. Xuhua He
7. Chong-Quan Zhang
8. Thomas J. Haines

In Mathematical Physics

1. E. Viehmann
2. Wei Zhang
3. Michael Rapoport
4. Xuhua He
5. James Newton
6. J. Thorne
7. Thomas J. Haines
8. R. Kottwitz

Citations in the corpus

• On arithmetic fundamental lemmas (2012, background): “A conjectural generalization of the Gross–Zagier formula to higher-dimensional Shimura varieties has been proposed, for instance, in recent preprints of Gan–Gross–Prasad [ 3 ]a nd Zhang [32].”
• Symmetric power functoriality for holomorphic modular forms, II (2020, background): “∨ i and for each jwe have σj∼=σ c j . For each kwe have τk6∼=τ wc,∨ k . •For each i, ρi is conjugate self-dual of sign (−1)n−1 and for each j, σj is conjugate self-dual of sign (−1)n, in the sense of [GGP12]. Then an extension ϕF+ v exists if and only if each integer mjis even. If the extension ϕF+ ”
• Towards a theory of local Shimura varieties (2014, methodology): “here we assume that [b] is basic. 7.1. The Kottwitz conjecture. To state the conjecture, we will have to make use of some aspects of a conjectural Langlands correspondence. A Langlands parameter (cf. [19], §8) for a group Gover F, ϕ: WF −→L G= Gˆ ⋊WF , is called discrete, if the connected component ”