A local twisted trace formula for Whittaker induction of coregular symmetric pairs: the geometric side
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909222786039808 |
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| author | Beuzart-Plessis, Raphaël Wan, Chen |
| author_facet | Beuzart-Plessis, Raphaël Wan, Chen |
| contents | In this paper, we prove the geometric expansion of a local twisted trace formula for the Whittaker induction of any symmetric pairs that are coregular. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur \cite{A91} and Waldspurger \cite{WalFTLtordue}. We also prove a formula for the regular germs of quasi-characters associated to strongly cuspidal functions in terms of certain weighted orbital integrals. As a consequence of our trace formula and the formula for regular germs of quasi-characters, we prove a simple local trace formula of those models for strongly cuspidal test functions which implies a multiplicity formula for these models. We also present various applications of our trace formula and multiplicity formula, including a necessary condition for a discrete L-packet to contain a representation with a unitary Shalika model (resp. a Galois model for classical groups) in terms of the associated Langlands parameter, and we also compute the summation of the corresponding multiplicities for certain discrete L-packets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10845 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A local twisted trace formula for Whittaker induction of coregular symmetric pairs: the geometric side Beuzart-Plessis, Raphaël Wan, Chen Representation Theory Number Theory In this paper, we prove the geometric expansion of a local twisted trace formula for the Whittaker induction of any symmetric pairs that are coregular. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur \cite{A91} and Waldspurger \cite{WalFTLtordue}. We also prove a formula for the regular germs of quasi-characters associated to strongly cuspidal functions in terms of certain weighted orbital integrals. As a consequence of our trace formula and the formula for regular germs of quasi-characters, we prove a simple local trace formula of those models for strongly cuspidal test functions which implies a multiplicity formula for these models. We also present various applications of our trace formula and multiplicity formula, including a necessary condition for a discrete L-packet to contain a representation with a unitary Shalika model (resp. a Galois model for classical groups) in terms of the associated Langlands parameter, and we also compute the summation of the corresponding multiplicities for certain discrete L-packets. |
| title | A local twisted trace formula for Whittaker induction of coregular symmetric pairs: the geometric side |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2312.10845 |