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Holomorphy of normalized intertwining operators for certain induced representations

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The theory of intertwining operators plays an important role and appears everywhere in the development of the Langlands program. This, in some sense, is a very sophisticated theory, but the basic question of its singularity, in general, is quite unknown, whilst a holomorphicity conjecture of standard intertwining operators for generic standard modules was proposed by Casselman–Shahidi more than 20 years ago. On the other hand, motivated by its deep connection with the longstanding pursuit of constructing automorphic L-functions via the method of integral representations, especially the direct application to the g.c.d. definition of local tensor product L-functions under the framework of Cai–Friedberg–Ginzburg–Kaplan’s generalized doubling method, and the potential application to the ambitious Braverman–Kazhdan/Ngô program which grows out of generalizing Godement–Jacquet’s zeta integral for standard L-functions of general linear groups, we prove the holomorphy of normalized local intertwining operators, normalized in the sense of Casselman–Shahidi, for a family of induced representations of quasi-split classical groups. In particular, Casselman–Shahidi’s holomorphicity conjecture for classical groups is settled here. Our argument is the outcome of an observation of an intrinsically asymmetric property of normalization factors appearing in different reduced decompositions of intertwining operators. Such an approach bears the potential to work in general.

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Acknowledgements

The author would like to thank Eyal Kaplan for his kindness and help. Thanks are also due to the referee for his/her detailed comments. Part of the work was done when the author was a post-doctor at Bar-Ilan University and was supported by the ISRAEL SCIENCE FOUNDATION Grant Number 376/21. The work is also supported by the NATURAL SCIENCE FOUNDATION OF CHINA-Young Scientists Fund Grant Number 12201205 and CUHKSZ Startup Fund UDF01002991.

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    Caihua Luo

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Luo, C. Holomorphy of normalized intertwining operators for certain induced representations. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02744-1

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