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Local Coefficients and Gamma Factors for Principal Series of Covering Groups
About this Title
Fan Gao, Freydoon Shahidi and Dani Szpruch
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 283, Number 1399
ISBNs: 978-1-4704-5681-8 (print); 978-1-4704-7400-3 (online)
DOI: https://doi.org/10.1090/memo/1399
Published electronically: January 19, 2023
Keywords: Covering groups,
Whittaker functionals,
local coefficients matrix,
scattering matrix,
gamma factor,
Plancherel measure
Table of Contents
Chapters
- 1. Introduction
- 2. Covering groups
- 3. Local coefficients matrix and scattering matrix
- 4. Plancherel measure and gamma factor
- 5. Exceptional point and the Casselman–Shalika formula
- 6. The fundamental group $\pi _1(\overline {G}_{Q,n}^\vee )$ and periodicity of $b_{W,n}$
- 7. Two Poincaré series
- 8. Principal series of $\overline {GL}_2$ and $\overline {SL}_2$
- 9. Local coefficients matrix for $\overline {SL}_2$
- 10. Local coefficients matrices for $\overline {GL}_2$ and invariants via restrictions
Abstract
We consider an $n$-fold Brylinski–Deligne cover of a reductive group over a $p$-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of $GL_2$ to $SL_2$. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.[\resetbiblist{9999999}]*labels=alphabetic
- Mahdi Asgari and Freydoon Shahidi, Generic transfer for general spin groups, Duke Math. J. 132 (2006), no. 1, 137–190. MR 2219256, DOI 10.1215/S0012-7094-06-13214-3
- Mahdi Asgari and Freydoon Shahidi, Generic transfer from $\rm GSp(4)$ to $\rm GL(4)$, Compos. Math. 142 (2006), no. 3, 541–550. MR 2231191, DOI 10.1112/S0010437X06001904
- William D. Banks, Heredity of Whittaker models on the metaplectic group, Pacific J. Math. 185 (1998), no. 1, 89–96. MR 1653196, DOI 10.2140/pjm.1998.185.89
- William D. Banks, Jason Levy, and Mark R. Sepanski, Block-compatible metaplectic cocycles, J. Reine Angew. Math. 507 (1999), 131–163. MR 1670203, DOI 10.1515/crll.1999.011
- I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 425030
- Corinne Blondel, Uniqueness of Whittaker model for some supercuspidal representations of the metaplectic group, Compositio Math. 83 (1992), no. 1, 1–18. MR 1168120
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
- Ben Brubaker, Valentin Buciumas, Daniel Bump, and Nathan Gray, A Yang-Baxter equation for metaplectic ice, Commun. Number Theory Phys. 13 (2019), no. 1, 101–148. MR 3951106, DOI 10.4310/CNTP.2019.v13.n1.a4
- Ben Brubaker, Valentin Buciumas, Daniel Bump, and Solomon Friedberg, Hecke modules from metaplectic ice, Selecta Math. (N.S.) 24 (2018), no. 3, 2523–2570. MR 3816510, DOI 10.1007/s00029-017-0372-0
- Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein, Weyl group multiple Dirichlet series. I, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 91–114. MR 2279932, DOI 10.1090/pspum/075/2279932
- Ben Brubaker, Daniel Bump, and Solomon Friedberg, Weyl group multiple Dirichlet series. II. The stable case, Invent. Math. 165 (2006), no. 2, 325–355. MR 2231959, DOI 10.1007/s00222-005-0496-2
- B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series. III. Eisenstein series and twisted unstable $A_r$, Ann. of Math. (2) 166 (2007), no. 1, 293–316. MR 2342698, DOI 10.4007/annals.2007.166.293
- Jean-Luc Brylinski and Pierre Deligne, Central extensions of reductive groups by $\mathbf K_2$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 5–85. MR 1896177, DOI 10.1007/s10240-001-8192-2
- Mark Budden, Local coefficient matrices of metaplectic groups, J. Lie Theory 16 (2006), no. 2, 239–249. MR 2197591
- Mark Budden and Geoff Goehle, Local coefficient matrices and the metaplectic correspondence, J. Lie Theory 27 (2017), no. 3, 657–670. MR 3592285
- Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508, DOI 10.1017/CBO9780511609572
- Yuanqing Cai, Solomon Friedberg, David Ginzburg, and Eyal Kaplan, Doubling constructions and tensor product $L$-functions: the linear case, Invent. Math. 217 (2019), no. 3, 985–1068. MR 3989257, DOI 10.1007/s00222-019-00883-4
- Yuanqing Cai, Solomon Friedberg, and Eyal Kaplan, Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations. preprint, available at arXiv:1802.02637.
- W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
- W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
- Gautam Chinta and Paul E. Gunnells, Constructing Weyl group multiple Dirichlet series, J. Amer. Math. Soc. 23 (2010), no. 1, 189–215. MR 2552251, DOI 10.1090/S0894-0347-09-00641-9
- Gautam Chinta and Omer Offen, A metaplectic Casselman-Shalika formula for $\textrm {GL}_r$, Amer. J. Math. 135 (2013), no. 2, 403–441. MR 3038716, DOI 10.1353/ajm.2013.0013
- J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to $\textrm {GL}_N$, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 5–30. MR 1863734, DOI 10.1007/s10240-001-8187-z
- J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163–233. MR 2075885, DOI 10.1007/s10240-004-0020-z
- P. Deligne, Sommes de Gauss cubiques et revêtements de $\textrm {SL}(2)$ [d’après S. J. Patterson], Séminaire Bourbaki (1978/79), Lecture Notes in Math., vol. 770, Springer, Berlin, 1980, pp. Exp. No. 539, pp. 244–277 (French). MR 572428
- Wee Teck Gan and Fan Gao, The Langlands-Weissman program for Brylinski-Deligne extensions, Astérisque 398 (2018), 187–275 (English, with English and French summaries). L-groups and the Langlands program for covering groups. MR 3802419
- 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, Generalized Bump-Hoffstein conjecture for coverings of the general linear groups, J. Algebra 499 (2018), 183–228. MR 3758499, DOI 10.1016/j.jalgebra.2017.12.002
- Fan Gao, Hecke $L$-functions and Fourier coefficients of covering Eisenstein series, Doc. Math. 26 (2021), 465–522. MR 4282793
- 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 and Martin H. Weissman, Whittaker models for depth zero representations of covering groups, Int. Math. Res. Not. IMRN 11 (2019), 3580–3620. MR 3961710, DOI 10.1093/imrn/rnx235
- I. M. Gel′fand and D. A. Kajdan, Representations of the group $\textrm {GL}(n,K)$ where $K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted Press, New York-Toronto, Ont., 1975, pp. 95–118. MR 404534
- Stephen Gelbart and I. I. Piatetski-Shapiro, Distinguished representations and modular forms of half-integral weight, Invent. Math. 59 (1980), no. 2, 145–188. MR 577359, DOI 10.1007/BF01390042
- S. Gelbart and I. Piatetski-Shapiro, On Shimura’s correspondence for modular forms of half-integral weight, Automorphic forms, representation theory and arithmetic (Bombay, 1979) Tata Inst. Fundam. Res. Stud. Math., vol. 10, Springer, Berlin-New York, 1981, pp. 1–39. MR 633657
- 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
- Marcela Hanzer, On the cuspidal support of a generic representation, J. Lie Theory 28 (2018), no. 1, 71–78. MR 3683372
- Bruno Kahn, Le groupe des classes modulo $2$, d’après Conner et Perlis, Seminar on number theory, 1984–1985 (Talence, 1984/1985) Univ. Bordeaux I, Talence, 1985, pp. Exp. No. 26, 29 (French). MR 848382, DOI 10.1038/314026a0
- Bruno Kahn, Sommes de Gauss attachées aux caractères quadratiques: une conjecture de Pierre Conner, Comment. Math. Helv. 62 (1987), no. 4, 532–541 (French). MR 920055, DOI 10.1007/BF02564460
- Eyal Kaplan, Representations distinguished by pairs of exceptional representations and a conjecture of Savin, Int. Math. Res. Not. IMRN 2 (2016), 604–643. MR 3493427, DOI 10.1093/imrn/rnv150
- Eyal Kaplan, Theta distinguished representations, inflation and the symmetric square $L$-function, Math. Z. 283 (2016), no. 3-4, 909–936. MR 3519988, DOI 10.1007/s00209-016-1627-8
- Eyal Kaplan, The double cover of odd general spin groups, small representations, and applications, J. Inst. Math. Jussieu 16 (2017), no. 3, 609–671. MR 3646283, DOI 10.1017/S1474748015000250
- Eyal Kaplan, The characterization of theta-distinguished representations of $\textrm {GL}(n)$, Israel J. Math. 222 (2017), no. 2, 551–598. MR 3722261, DOI 10.1007/s11856-017-1600-1
- Eyal Kaplan, Doubling constructions and tensor product L-functions: coverings of the symplectic group. preprint, available at arXiv:1902.00880.
- Camelia Karimianpour, The Stone-von Neumann Construction in Branching Rules and Minimal Degree Problems. PhD Thesis (2016), University of Ottawa, available at http://mysite.science.uottawa.ca/hsalmasi/report/thesis-camelia.pdf.
- D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142. MR 743816
- Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
- Henry H. Kim and Muthukrishnan Krishnamurthy, Stable base change lift from unitary groups to $\textrm {GL}_n$, IMRP Int. Math. Res. Pap. 1 (2005), 1–52. MR 2149370
- Henry H. Kim and Freydoon Shahidi, Functorial products for $\textrm {GL}_2\times \textrm {GL}_3$ and the symmetric cube for $\textrm {GL}_2$, Ann. of Math. (2) 155 (2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart. MR 1923967, DOI 10.2307/3062134
- Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
- 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
- Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
- Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 579181
- 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
- Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214
- Peter J. McNamara, Metaplectic Whittaker functions and crystal bases, Duke Math. J. 156 (2011), no. 1, 1–31. MR 2746386, DOI 10.1215/00127094-2010-064
- 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
- J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field. MR 3729270, DOI 10.1017/9781316711736
- C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (1987), no. 3, 427–452 (French). MR 913667, DOI 10.1007/BF01200363
- Calvin C. Moore, Group extensions of $p$-adic and adelic linear groups, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 157–222. MR 244258
- Shiv Prakash Patel, A theorem of Mœglin and Waldspurger for covering groups, Pacific J. Math. 273 (2015), no. 1, 225–239. MR 3290452, DOI 10.2140/pjm.2015.273.225
- Shiv Prakash Patel and Dipendra Prasad, Multiplicity formula for restriction of representations of $\widetilde {\textrm {GL}_2}(F)$ to $\widetilde {\textrm {SL}_2}(F)$, Proc. Amer. Math. Soc. 144 (2016), no. 2, 903–908. MR 3430864, DOI 10.1090/proc12721
- Manish Patnaik and Anna Puskás, On Iwahori-Whittaker functions for metaplectic groups, Adv. Math. 313 (2017), 875–914. MR 3649240, DOI 10.1016/j.aim.2017.04.005
- S. J. Patterson, Metaplectic forms and Gauss sums. I, Compositio Math. 62 (1987), no. 3, 343–366. MR 901396
- François Rodier, Whittaker models for admissible representations of reductive $p$-adic split groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Proc. Sympos. Pure Math., Vol. XXVI, Amer. Math. Soc., Providence, RI, 1973, pp. 425–430. MR 354942
- Gordan Savin, On unramified representations of covering groups, J. Reine Angew. Math. 566 (2004), 111–134. MR 2039325, DOI 10.1515/crll.2004.001
- Gordan Savin, A nice central extension of $GL_r$. preprint.
- 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, Functional equation satisfied by certain $L$-functions, Compositio Math. 37 (1978), no. 2, 171–207. MR 498494
- Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
- Freydoon Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973–1007. MR 816396, DOI 10.1215/S0012-7094-85-05252-4
- Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain $L$-functions, Ann. of Math. (2) 127 (1988), no. 3, 547–584. MR 942520, DOI 10.2307/2007005
- Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
- 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
- J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- T. A. Springer, Linear algebraic groups, 2nd ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. MR 2458469
- Robert Steinberg, Générateurs, relations et revêtements de groupes algébriques, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) Librairie Universitaire, Louvain, 1962, pp. 113–127 (French). MR 153677
- Robert Steinberg, Lectures on Chevalley groups, University Lecture Series, vol. 66, American Mathematical Society, Providence, RI, 2016. Notes prepared by John Faulkner and Robert Wilson; Revised and corrected edition of the 1968 original [ MR0466335]; With a foreword by Robert R. Snapp. MR 3616493, DOI 10.1090/ulect/066
- Toshiaki Suzuki, Rankin-Selberg convolutions of generalized theta series, J. Reine Angew. Math. 414 (1991), 149–205. MR 1092629, DOI 10.1515/crll.1991.414.149
- Toshiaki Suzuki, Metaplectic Eisenstein series and the Bump-Hoffstein conjecture, Duke Math. J. 90 (1997), no. 3, 577–630. MR 1480547, DOI 10.1215/S0012-7094-97-09016-5
- Toshiaki Suzuki, Distinguished representations of metaplectic groups, Amer. J. Math. 120 (1998), no. 4, 723–755. MR 1637947
- Toshiaki Suzuki, On the theta series of $p$-th order, J. Reine Angew. Math. 651 (2011), 1–95. MR 2774311, DOI 10.1515/CRELLE.2011.009
- W. Jay Sweet Jr., A computation of the gamma matrix of a family of $p$-adic zeta integrals, J. Number Theory 55 (1995), no. 2, 222–260. MR 1366572, DOI 10.1006/jnth.1995.1139
- Dani Szpruch, Uniqueness of Whittaker model for the metaplectic group, Pacific J. Math. 232 (2007), no. 2, 453–469. MR 2366363, DOI 10.2140/pjm.2007.232.453
- Dani Szpruch, The Langlands-Shahidi method for the metaplectic group and applications (2010). PhD thesis (2010), Tel Aviv University, available at arXiv:1004.3516.
- Dani Szpruch, On the existence of a $p$-adic metaplectic Tate-type $\tilde \gamma$-factor, Ramanujan J. 26 (2011), no. 1, 45–53. MR 2837718, DOI 10.1007/s11139-010-9277-7
- Dani Szpruch, Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method, Israel J. Math. 195 (2013), no. 2, 897–971. MR 3096578, DOI 10.1007/s11856-012-0140-y
- Dani Szpruch, Some results in the theory of genuine representations of the metaplectic double cover of $GSp_{2n}(F)$ over p-adic fields, J. Algebra 388 (2013), 160–193. MR 3061683, DOI 10.1016/j.jalgebra.2013.05.001
- Dani Szpruch, Symmetric genuine spherical Whittaker functions on $\overline {GSp_{2n}(F)}$, Canad. J. Math. 67 (2015), no. 1, 214–240. MR 3292701, DOI 10.4153/CJM-2013-033-5
- Dani Szpruch, A short proof for the relation between Weil indices and $\epsilon$-factors, Comm. Algebra 46 (2018), no. 7, 2846–2851. MR 3780827, DOI 10.1080/00927872.2017.1399407
- 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
- Boaz Tamir, On $L$-functions and intertwining operators for unitary groups, Israel J. Math. 73 (1991), no. 2, 161–188. MR 1135210, DOI 10.1007/BF02772947
- Shiang Tang, Principal series representations of metaplectic groups. preprint, available at arXiv:1706.05145.
- J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) Academic Press, London, 1967, pp. 305–347. MR 217026
- J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp. 3–26. MR 546607
- J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366
- J.-L. Waldspurger, La formule de Plancherel pour les groupes $p$-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235–333 (French, with French summary). MR 1989693, DOI 10.1017/S1474748003000082
- André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012
- André Weil, Basic number theory, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the second (1973) edition. MR 1344916
- 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
- Martin H. Weissman, Managing metaplectiphobia: covering $p$-adic groups, Harmonic analysis on reductive, $p$-adic groups, Contemp. Math., vol. 543, Amer. Math. Soc., Providence, RI, 2011, pp. 237–277. MR 2798431, DOI 10.1090/conm/543/10738
- Martin H. Weissman, Split metaplectic groups and their L-groups, J. Reine Angew. Math. 696 (2014), 89–141. MR 3276164, DOI 10.1515/crelle-2012-0111
- Martin H. Weissman, Covers of tori over local and global fields, Amer. J. Math. 138 (2016), no. 6, 1533–1573. MR 3595494, DOI 10.1353/ajm.2016.0046
- Martin H. Weissman, L-groups and parameters for covering groups, Astérisque 398 (2018), 33–186 (English, with English and French summaries). L-groups and the Langlands program for covering groups. MR 3802418