Epsilon Dichotomy for Linear Models: The Archimedean Case

[1] Adams, J. and D. A. Vogan. “Contragredient representations and characterizing the local

Langlands correspondence.” Am. J. Math. 138, no. 3 (2016): 657–82. https://doi.org/10.1353/

ajm.2016.0024.

Downloaded from https://academic.oup.com/imrn/article/2023/20/17853/7188055 by Kyoto University user on 06 November 2023

More generally, let us prove v = Ptriv v for any v ∈ , where v¯ denotes the image of v

17890 M. Suzuki and H. Tamori

[2] Bernstein, J. and B. Krötz. “Smooth Fréchet globalizations of Harish–Chandra modules.” Israel

J. Math. 199, no. 1 (2014): 45–111. https://doi.org/10.1007/s11856-013-0056-1.

[3] Borel, A. and N. R. Wallach. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Mathematical Surveys and Monographs, vol. 67, 2nd ed. Amer.

[4] Broussous, P. and N. Matringe. “Multiplicity one for pairs of Prasad–Takloo-Bighash type.”

Internat. Math. Res. Notices 2021, no. 21 (2021): 16423–47. https://doi.org/10.1093/imrn/

rnz254.

´ “Bruhat filtrations and Whittaker vectors for

[5] Casselman, W., H. Hecht, and D. Miliˇcic.

real groups.” In The Mathematical Legacy of Harish–Chandra (Baltimore, MD, 1998). Proc.

Sympos. Pure Math., vol. 68, 151–90. Amer. Math. Soc., Providence, RI, 2000, https://doi.org/

10.1090/pspum/068/1767896.

[6] Chen, Y. and B. Sun. “Schwartz homologies of representations of almost linear Nash groups.”

J. Funct. Anal. 280, no. 7 (2021): 108817. https://doi.org/10.1016/j.jfa.2020.108817.

[7] Chommaux, M. “Distinction of the Steinberg representation and a conjecture of Prasad and

Takloo-Bighash.” J. Number Theory 202 (2019): 200–19. https://doi.org/10.1016/j.jnt.2019.01.

009.

[8] Fenchel, W. and D. W. Blackett. Convex Cones, Sets and Functions. Princeton University,

Department of Mathematics, Logistics Research Project, 1953.

[9] Gan, W.-T., B. Gross, and D. Prasad. “Symplectic local root numbers, central critical L-values,

and restriction problems in the representation theory of classical groups.” Astérisque 346

(2012): 1–109.

[10] Jacquet, H. “Principal l-functions of the linear group.” In Automorphic Forms, Representations, and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977).

Proc. Sympos. Pure Math., vol. 33.2, part 2, 63–86. Amer. Math. Soc., Providence, RI, 1979,

https://doi.org/10.1090/pspum/033.2/546609.

[11] Knapp, A. W. “Local Langlands correspondence: the Archimedean case.” In Motives (Seattle,

WA, 1991). Proc. Sym. Pure Math., vol. 55.2, 393–410. Amer. Math. Soc., Providence, RI, 1994,

https://doi.org/10.1090/pspum/055.2/1265560.

[12] Knapp, A. W. Representation Theory of Semisimple Groups. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 2001.

[13] Knapp, A. W. and D. A. Vogan. Cohomological Induction and Unitary Representations.

Princeton Mathematical Series, vol. 45. Princeton University Press, Princeton, NJ, 1995,

https://doi.org/10.1515/9781400883936.

[14] Li, N., G. Liu, and J. Yu. “A proof of Casselman’s comparison theorem.” Represent. Theory 25

(2021): 994–1020. https://doi.org/10.1090/ert/591.

[15] Lu, H. “Multiplicity one for the pair (GLn (D), GLn (E)).” Transform. Groups (2022). https://doi.

org/10.1007/s00031-022-09713-z.

[16] Matsuki, T. “The orbits of affine symmetric spaces under the action of minimal parabolic subgroups.” J. Math. Soc. Japan 31, no. 2 (1979): 331–57. https://doi.org/10.2969/jmsj/03120331.

[17] Oshima, T. and J. Sekiguchi.. “The restricted root system of a semisimple symmetric pair.” In

Downloaded from https://academic.oup.com/imrn/article/2023/20/17853/7188055 by Kyoto University user on 06 November 2023

Math. Soc., Providence, RI, 2000, https://doi.org/10.1090/surv/067.

Epsilon Dichotomy for Linear Models 17891

Group Representations and Systems of Differential Equations (Tokyo, 1982). Adv. Stud. Pure

Math., vol. 4, 433–97. North-Holland, Amsterdam, 1984.

[18] Prasad, D. and R. Takloo-Bighash. “Bessel models for GSp(4).” J. Reine Angew. Math. 2011

(2011): 189–243. https://doi.org/10.1515/crelle.2011.045.

(2020): preprint, https://arxiv.org/abs/2005.05615.

[20] Suzuki, M. “Classification of standard modules with linear periods.” J. Number Theory 218

(2021): 302–10. https://doi.org/10.1016/j.jnt.2020.07.005.

[21] Suzuki, M. and H. Xue. “Linear intertwining periods and epsilon dichotomy for linear models.”

Math.Ann. (2023). https://doi.org/10.1007/s00208-023-02615-9.

[22] Tate, J. “Number theoretic background.” In Automorphic Forms, Representations, and LFunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977). Proc. Sympos.

Pure Math., vol. 33, part 2, 3–26. Amer. Math. Soc., Providence, RI, 1979.

[23] Vogan, D. A. Representations of Real Reductive Lie Groups. Progr. Math., vol. 15. Birkhäuser,

Boston, Mass., 1981.

[24] Wallach, N. R. “The asymptotic behavior of holomorphic representations.” Mém. Soc. Math.

Fr. (N.S.) 1 (1964): 291–305. https://doi.org/10.24033/msmf.308.

[25] Warner, G. Harmonic Analysis on Semi-Simple Lie Groups. I. Grundlehren Math. Wiss., vol.

188. Springer, New York–Heidelberg, 1972, https://doi.org/10.1007/978-3-642-50275-0.

[26] Xue, H. “Bessel models for unitary groups and schwartz homology.” (2020): preprint, https://

www.math.arizona.edu/xuehang/lggp_generic_v1.pdf.

[27] Xue, H. “Epsilon dichotomy for linear models.” Algebra Number Theory 15, no. 1 (2021):

173–215. https://doi.org/10.2140/ant.2021.15.173.

[28] Yokota, I. “Exceptional Lie groups.” (2009): preprint, http://arxiv.org/abs/0902.0431.

Downloaded from https://academic.oup.com/imrn/article/2023/20/17853/7188055 by Kyoto University user on 06 November 2023

[19] Sécherre, V. “Représentations cuspidales de glr (d) distinguées par une involution intérieure.”

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