Generalized Bump-Hoffstein conjecture for coverings of the general linear groups. (English) Zbl 1471.11170
The Bump-Hoffstein conjecture concerns the pair \((\widetilde{\mathrm{GL}}^{(n)}_r,\widetilde{\mathrm{GL}}^{(n)}_n)\), with \(r <n\), of Kazhdan-Patterson \(n\)-fold coverings and a pair of representations \((\pi, \Theta)\), where \(\pi\) is generic and \(\Theta\) is a theta representations. This paper investigates the extent to which the conjecture could be generalized to Brylinski-Deligne coverings.
The author defines the notion of a fundamental pair \((\widetilde{\mathrm{GL}}_r,\widetilde{\mathrm{GL}}_R)\), \(r < R\), of \(n\)-fold coverings and introduces Whittaker models for coverings of \(\mathrm{GL}_r\). Suppose that \(\pi\) is a generic unramified representation of \(\widetilde{\mathrm{GL}}_r\) and \(\Theta= \Theta(\widetilde{\mathrm{GL}}_R, \chi)\) the theta representation associated to an exceptional character \(\chi\). Then \(\Theta\) possesses a unique Whittaker model \(W_\Theta\). Let \(W_\pi\) be any Whittaker model of \(\pi\). The author conjectures that, under some additional conditions, the Rankin-Selberg integral of \(\overline{W_\Theta}\) against \(W_\pi\) is equal to a certain \(L\)-function associated to \(\pi\) and \(\chi\) multiplied by \(W_\pi(1) \cdot \overline{W_\Theta}(1).\)
For the pair \((\widetilde{\mathrm{GL}}^{(n)}_r,\widetilde{\mathrm{GL}}^{(n)}_n)\) of Kazhdan-Patterson \(n\)-fold coverings with \(r <n\), this is the Bump-Hoffstein conjecture and it holds by the works of D. Bump and S. Friedberg [Proc. Symp. Pure Math. 66, Part 2, 1–17 (1999; Zbl 0983.11026)] and D. Ginzburg [Compos. Math. 154, No. 4, 671–684 (2018; Zbl 1431.11067)]. The author provides evidence for the generalized Bump-Hoffstein conjecture by proving some special cases, namely the case \(r=2\) and also the case when \(\pi\), in addition to being generic, is also a theta representation.
The author defines the notion of a fundamental pair \((\widetilde{\mathrm{GL}}_r,\widetilde{\mathrm{GL}}_R)\), \(r < R\), of \(n\)-fold coverings and introduces Whittaker models for coverings of \(\mathrm{GL}_r\). Suppose that \(\pi\) is a generic unramified representation of \(\widetilde{\mathrm{GL}}_r\) and \(\Theta= \Theta(\widetilde{\mathrm{GL}}_R, \chi)\) the theta representation associated to an exceptional character \(\chi\). Then \(\Theta\) possesses a unique Whittaker model \(W_\Theta\). Let \(W_\pi\) be any Whittaker model of \(\pi\). The author conjectures that, under some additional conditions, the Rankin-Selberg integral of \(\overline{W_\Theta}\) against \(W_\pi\) is equal to a certain \(L\)-function associated to \(\pi\) and \(\chi\) multiplied by \(W_\pi(1) \cdot \overline{W_\Theta}(1).\)
For the pair \((\widetilde{\mathrm{GL}}^{(n)}_r,\widetilde{\mathrm{GL}}^{(n)}_n)\) of Kazhdan-Patterson \(n\)-fold coverings with \(r <n\), this is the Bump-Hoffstein conjecture and it holds by the works of D. Bump and S. Friedberg [Proc. Symp. Pure Math. 66, Part 2, 1–17 (1999; Zbl 0983.11026)] and D. Ginzburg [Compos. Math. 154, No. 4, 671–684 (2018; Zbl 1431.11067)]. The author provides evidence for the generalized Bump-Hoffstein conjecture by proving some special cases, namely the case \(r=2\) and also the case when \(\pi\), in addition to being generic, is also a theta representation.
Reviewer: Dubravka Ban (Carbondale)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
Keywords:
Brylinski-Deligne covering groups; theta representations; Whittaker functionals; Bump-Hoffstein conjecture; dual groupsReferences:
| [1] | Ban, Dubravka; Jantzen, Chris, The Langlands quotient theorem for finite central extensions of \(p\)-adic groups, Glas. Mat. Ser. III, 48(68), 2, 313-334 (2013), MR3151110 · Zbl 1304.22019 |
| [2] | Bourbaki, Nicolas, Lie Groups and Lie Algebras. Chapters 4-6, Elements of Mathematics (Berlin) (2002), Springer-Verlag: Springer-Verlag Berlin, translated from the 1968 French original by Andrew Pressley. MR1890629 · Zbl 0983.17001 |
| [3] | Brylinski, Jean-Luc; Deligne, Pierre, Central extensions of reductive groups by \(K_2\), Publ. Math. Inst. Hautes Études Sci., 94, 5-85 (2001), MR1896177 · Zbl 1093.20027 |
| [4] | Bump, Daniel; Friedberg, Solomon, Metaplectic generating functions and Shimura integrals, (Automorphic Forms, Automorphic Representations, and Arithmetic. Automorphic Forms, Automorphic Representations, and Arithmetic, Fort Worth, TX, 1996. Automorphic Forms, Automorphic Representations, and Arithmetic. Automorphic Forms, Automorphic Representations, and Arithmetic, Fort Worth, TX, 1996, Proc. Sympos. Pure Math., vol. 66 (1999), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-17, MR1703755 · Zbl 0983.11026 |
| [5] | Bump, Daniel; Furusawa, Masaaki; Ginzburg, David, Non-unique models in the Rankin-Selberg method, J. Reine Angew. Math., 468, 77-111 (1995), MR1361787 · Zbl 0833.11017 |
| [6] | Bump, Daniel; Hoffstein, Jeffrey, On Shimura’s correspondence, Duke Math. J., 55, 3, 661-691 (1987), MR904946 · Zbl 0631.10015 |
| [7] | Bump, Daniel; Hoffstein, Jeffrey, Some conjectured relationships between theta functions and Eisenstein series on the metaplectic group, (Number Theory. Number Theory, New York, 1985/1988. Number Theory. Number Theory, New York, 1985/1988, Lecture Notes in Math., vol. 1383 (1989), Springer: Springer Berlin), 1-11, MR1023915 · Zbl 0677.10021 |
| [8] | Chinta, Gautam; Offen, Omer, A metaplectic Casselman-Shalika formula for \(GL_r\), Amer. J. Math., 135, 2, 403-441 (2013), MR3038716 · Zbl 1294.22012 |
| [9] | Finkelberg, Michael; Lysenko, Sergey, Twisted geometric Satake equivalence, J. Inst. Math. Jussieu, 9, 4, 719-739 (2010), MR2684259 · Zbl 1316.22019 |
| [10] | Gan, Wee Teck; Gao, Fan, The Langlands-Weissman program for Brylinski-Deligne extensions, Astérisque (2016), in press |
| [11] | Gao, Fan, The Langlands-Shahidi L-functions for Brylinski-Deligne extensions, Amer. J. Math. (2016), in press · Zbl 1425.11109 |
| [12] | Gao, Fan, Distinguished theta representations for certain covering groups, Pacific J. Math., 290, 2, 333-379 (2017) · Zbl 1395.11082 |
| [13] | Gelbart, Stephen; Howe, Roger; Piatetski-Shapiro, Ilya, Uniqueness and existence of Whittaker models for the metaplectic group, Israel J. Math., 34, 1-2, 21-37 (1979), (1980). MR571393 · Zbl 0441.22015 |
| [14] | Ginzburg, David, Generating functions on covering groups, preprint, available at · Zbl 1431.11067 |
| [15] | Goetze, Thomas, Euler products associated to metaplectic automorphic forms on the 3-fold cover of \(GSp(4)\), Trans. Amer. Math. Soc., 350, 3, 975-1011 (1998), MR1401521 · Zbl 0895.11019 |
| [16] | Kazhdan, D. A.; Patterson, S. J., Metaplectic forms, Publ. Math. Inst. Hautes Études Sci., 59, 35-142 (1984), MR743816 · Zbl 0559.10026 |
| [17] | McNamara, Peter J., Principal series representations of metaplectic groups over local fields, (Multiple Dirichlet Series, L-Functions and Automorphic Forms. Multiple Dirichlet Series, L-Functions and Automorphic Forms, Progr. Math., vol. 300 (2012), Birkhäuser/Springer: Birkhäuser/Springer New York), 299-327 · Zbl 1283.11081 |
| [18] | McNamara, Peter J., The metaplectic Casselman-Shalika formula, Trans. Amer. Math. Soc., 368, 4, 2913-2937 (2016), MR3449262 · Zbl 1338.22009 |
| [19] | Patterson, S. J., Metaplectic forms and Gauss sums. I, Compos. Math., 62, 3, 343-366 (1987), MR901396 · Zbl 0632.10030 |
| [20] | Piatetski-Shapiro, I.; Rallis, S., A new way to get Euler products, J. Reine Angew. Math., 392, 110-124 (1988), MR965059 · Zbl 0651.10021 |
| [21] | Reich, Ryan Cohen, Twisted geometric Satake equivalence via gerbes on the factorizable Grassmannian, Represent. Theory, 16, 345-449 (2012), MR2956088 · Zbl 1302.22014 |
| [22] | Shahidi, Freydoon, Eisenstein Series and Automorphic \(L\)-Functions, Amer. Math. Soc. Colloq. Publ., vol. 58 (2010), American Mathematical Society: American Mathematical Society Providence, RI, MR2683009 · Zbl 1215.11054 |
| [23] | Gordan Savin, A nice central extension of \(G L_r\); Gordan Savin, A nice central extension of \(G L_r\) |
| [24] | Suzuki, Toshiaki, Rankin-Selberg convolutions of generalized theta series, J. Reine Angew. Math., 414, 149-205 (1991), MR1092629 · Zbl 0733.11017 |
| [25] | Suzuki, Toshiaki, Metaplectic Eisenstein series and the Bump-Hoffstein conjecture, Duke Math. J., 90, 3, 577-630 (1997), MR1480547 · Zbl 0895.11020 |
| [26] | Suzuki, Toshiaki, Distinguished representations of metaplectic groups, Amer. J. Math., 120, 4, 723-755 (1998), MR1637947 · Zbl 0921.11028 |
| [27] | Szpruch, Dani, Uniqueness of Whittaker model for the metaplectic group, Pacific J. Math., 232, 2, 453-469 (2007), MR2366363 · Zbl 1221.11130 |
| [28] | Weissman, Martin H., Metaplectic tori over local fields, Pacific J. Math., 241, 1, 169-200 (2009), MR2485462 · Zbl 1230.11064 |
| [29] | Weissman, Martin H., L-groups and parameters for covering groups, Astérisque (2016), in press, available at · Zbl 1369.14058 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
