Whittaker spaces for reducible unitary principal series representations of ̃𝑆𝐿₂(𝐹)

Szpruch, Dani

doi:10.1090/proc/17143

Whittaker spaces for reducible unitary principal series representations of $\widetilde {SL_2(F)}$
HTML articles powered by AMS MathViewer

by Dani Szpruch;

Proc. Amer. Math. Soc. 153 (2025), 1933-1945

DOI: https://doi.org/10.1090/proc/17143

Published electronically: February 20, 2025

HTML | PDF | Request permission

Abstract:

Let $F$ be a $p$-adic field containing the full group of $n^{th}$ roots of 1 and let $\widetilde {{G}}$ be the $n$-fold cover of $SL_2(F)$ constructed by Kubota [On automorphic functions and the reciprocity law in a number field, Kyoto University, Tokyo, 1969]. In this paper we compute the dimension of the space of Whittaker functionals of the two irreducible summands inside a reducible unitary genuine principal series representation of $\widetilde {{G}}$. We also show how these dimensions change when the Whittaker character is modified. As an application we determine the action of the twisted Kazhdan-Patterson $n$-fold cover of $GL_2(F)$ on the two summands. We emphasize that our main results address both ramified and unramified representations and do not rely on the assumption that the cover is tame.

References

Fan Gao, Distinguished theta representations for certain covering groups, Pacific J. Math. 290 (2017), no. 2, 333–379. MR 3681111, DOI 10.2140/pjm.2017.290.333
Fan Gao, The Langlands-Shahidi $L$-functions for Brylinski-Deligne extensions, Amer. J. Math. 140 (2018), no. 1, 83–137. MR 3749191, DOI 10.1353/ajm.2018.0001
Fan Gao, $R$-group and Whittaker space of some genuine representations, J. Inst. Math. Jussieu 22 (2023), no. 1, 213–273. MR 4556932, DOI 10.1017/S1474748021000128
F. Gao, N. Gurevich, and E. Karasiewicz, Genuine whittaker functor and the quantum affine schur–weyl duality, Preprint, arXiv, 2022.
Fan Gao, Freydoon Shahidi, and Dani Szpruch, On the local coefficients matrix for coverings of $\rm SL_2$, Geometry, algebra, number theory, and their information technology applications, Springer Proc. Math. Stat., vol. 251, Springer, Cham, 2018, pp. 207–244. MR 3880389, DOI 10.1007/978-3-319-97379-1_{1}0
Fan Gao, Freydoon Shahidi, and Dani Szpruch, Local coefficients and gamma factors for principal series of covering groups, Mem. Amer. Math. Soc. 283 (2023), no. 1399, v+135. MR 4537303, DOI 10.1090/memo/1399
F. Gao, F. Shahidi, and D. Szpruch, Restrictions, l-parameters, and local coefficients for genuine representations, Preprint, arXiv:2102.08859, (accepted for publication in Mémoires de la Société Mathématique de France, 2023).
David Goldberg and Dani Szpruch, Plancherel measures for coverings of $p$-adic $\text {SL}_2(F)$, Int. J. Number Theory 12 (2016), no. 7, 1907–1936. MR 3544420, DOI 10.1142/S1793042116501189
D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142. MR 743816, DOI 10.1007/BF02698770
C. David Keys and Freydoon Shahidi, Artin $L$-functions and normalization of intertwining operators, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 67–89. MR 944102, DOI 10.24033/asens.1551
Tomio Kubota, On automorphic functions and the reciprocity law in a number field, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 2, Kinokuniya Book Store Co., Ltd., Tokyo, 1969. MR 255490
Caihua Luo, Knapp-Stein dimension theorem for finite central covering groups, Pacific J. Math. 306 (2020), no. 1, 265–280. MR 4109915, DOI 10.2140/pjm.2020.306.265
Peter J. McNamara, Principal series representations of metaplectic groups over local fields, Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., vol. 300, Birkhäuser/Springer, New York, 2012, pp. 299–327. MR 2963537, DOI 10.1007/978-0-8176-8334-4_{1}3
Peter J. McNamara, The metaplectic Casselman-Shalika formula, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2913–2937. MR 3449262, DOI 10.1090/tran/6597
Ralf Schmidt, Some remarks on local newforms for $\rm GL(2)$, J. Ramanujan Math. Soc. 17 (2002), no. 2, 115–147. MR 1913897
Freydoon Shahidi, Eisenstein series and automorphic $L$-functions, American Mathematical Society Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010. MR 2683009, DOI 10.1090/coll/058
Dani Szpruch, On Shahidi local coefficients matrix, Manuscripta Math. 159 (2019), no. 1-2, 117–159. MR 3936136, DOI 10.1007/s00229-018-1052-x
Dani Szpruch, A $p$-adic analog of Hasse-Davenport product relation involving $\epsilon$-factors, Forum Math. 37 (2025), no. 1, 247–267. MR 4846659, DOI 10.1515/forum-2023-0347
Martin H. Weissman, Metaplectic tori over local fields, Pacific J. Math. 241 (2009), no. 1, 169–200. MR 2485462, DOI 10.2140/pjm.2009.241.169
Jiandi Zou, Local metaplectic correspondence and applications, Math. Z. 305 (2023), no. 3, Paper No. 43, 33. MR 4652947, DOI 10.1007/s00209-023-03368-6