Generalised Whittaker models as instances of relative Langlands duality.(English)Zbl 07979660
Summary: The recent proposal by Ben-Zvi, Sakellaridis and Venkatesh of a duality in the relative Langlands program, leads, via the process of quantization of Hamiltonian varieties, to a duality theory of branching problems. This often unexpectedly relates twoa priori unrelated branching problems. We examine how the generalised Whittaker (or Gelfand-Graev) models serve as the prototypical example for such branching problems. We give a characterization, for the orthogonal and symplectic groups, of the generalised Whittaker models possibly contained in this duality theory. We then exhibit an infinite family of examples of this duality, which, provably at the local level via the theta correspondence, satisfy the conjectural expectations of duality.
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E57 | Geometric Langlands program: representation-theoretic aspects |
Cite
References:
| [1] | Bakić, P., Theta lifts of generic representations for dual pairs \(( \operatorname{Sp}_{2 n}, \operatorname{O}(V))\), Manuscr. Math., 165, 291-338, 2021 ·Zbl 1503.22010 |
| [2] | Bakić, P.; Hanzer, M., Theta correspondence and Arthur packets: on the Adams conjecture, arXiv preprint |
| [3] | Barbasch, D.; Vogan, D. A., Unipotent representations of complex semisimple groups, Ann. Math., 121, 1, 41-110, 1985 ·Zbl 0582.22007 |
| [4] | Ben-Zvi, D.; Sakellaridis, Y.; Venkatesh, A., Relative Langlands duality, preprint. Available at |
| [5] | Chriss, N.; Ginzburg, V., Representation Theory and Complex Geometry, Modern Birkhäuser Classics, 2010, Birkhäuser: Birkhäuser Boston ·Zbl 1185.22001 |
| [6] | Collingwood, D. H.; McGovern, W. M., Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, 1992 |
| [7] | Crooks, P.; Röser, M., The log symplectic geometry of Poisson slices, J. Symplectic Geom., 20, 1, 135-190, 2022 ·Zbl 1503.53148 |
| [8] | Daszkiewicz, A.; Kraskiewicz, W.; Przebinda, T., Nilpotent orbits and complex dual pairs, J. Algebra, 190, 2, 518-539, 1997 ·Zbl 0870.22007 |
| [9] | Finkelberg, M.; Ukraintsev, I., Hyperspherical equivariant slices and basic classical Lie superalgebras, arXiv preprint |
| [10] | Gan, W. T., Explicit constructions of automorphic forms: theta correspondence and automorphic descent, arXiv preprint |
| [11] | Gan, W. T., Periods and theta correspondence, (Representations of Reductive Groups. Representations of Reductive Groups, Proceedings of Symposia in Pure Mathematics, vol. 101, 2019), 113-132 ·Zbl 1481.11046 |
| [12] | Ginzburg, D., On spin L-functions for orthogonal groups, Duke Math. J., 77, 3, 753-798, 1995 ·Zbl 0832.11020 |
| [13] | Gan, W. L.; Ginzburg, V., Quantization of Slodowy slices, Int. Math. Res. Not., 5, 243-255, 2002 ·Zbl 0989.17014 |
| [14] | Gan, W. T.; Gomez, R., A conjecture of Sakellaridis-Venkatesh on the unitary spectrum of spherical varieties, (Symmetry: Representation Theory and Its Applications. Symmetry: Representation Theory and Its Applications, Progr. Math., vol. 257, 2014, Birkhäuser/Springer: Birkhäuser/Springer New York), 185-226 ·Zbl 1321.22021 |
| [15] | Gan, W. T.; Gurevich, N., Nontempered a-packets of \(G_2\): liftings from \(\widetilde{\operatorname{SL}_2} \), Am. J. Math., 128, 1105-1185, 2006 ·Zbl 1109.22013 |
| [16] | Gan, W. T.; Gross, B. H.; Prasad, D., Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque, 346, 111-170, 2012 ·Zbl 1279.22023 |
| [17] | Guralnick, R. M.; Lawther, R., Generic Stabilizers in Actions of Simple Algebraic Groups, Memoirs of the American Mathematical Society, vol. 300, No. 1502, 2024 ·Zbl 1552.20004 |
| [18] | Ginzburg, D.; Rallis, S.; Soudry, D., Periods, poles of L-functions and symplectic-orthogonal theta lifts, J. Reine Angew. Math., 487, 85-114, 1997 ·Zbl 0928.11025 |
| [19] | Gan, W. T.; Savin, G., Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, Compos. Math., 148, 1655-1694, 2012 ·Zbl 1325.11046 |
| [20] | Gan, W. T.; Savin, G., An exceptional Siegel-Weil formula and poles of the Spin L-function of \(\operatorname{PGSp}_6\), Compos. Math., 156, 6, 1231-1261, 2020 ·Zbl 1476.11076 |
| [21] | Gan, W. T.; Savin, G., Howe duality and dichotomy for exceptional theta correspondences, Invent. Math., 232, 1-78, 2023 ·Zbl 1516.11050 |
| [22] | Guillemin, V.; Sternberg, S., Geometric quantization and multiplicities of group representations, Invent. Math., 67, 3, 515-538, 1982 ·Zbl 0503.58018 |
| [23] | Guillemin, V.; Sternberg, S., Multiplicity-free spaces, J. Differ. Geom., 19, 1, 31-56, 1984 ·Zbl 0548.58017 |
| [24] | Guillemin, V.; Sternberg, S., The Frobenius reciprocity theorem from the symplectic point of view, (Non-linear Partial Differential Operators and Quantization Procedures. Non-linear Partial Differential Operators and Quantization Procedures, Lecture Notes in Mathematics, vol. 1037, 1983, Springer: Springer Berlin, Heidelberg) ·Zbl 0523.58035 |
| [25] | Gan, W. T.; Wan, X., Relative character identities and theta correspondence, (Relative Trace Formulas, Simons Symposia, 2018), 101-186 ·Zbl 1485.11095 |
| [26] | Gaiotto, D.; Witten, E., Supersymmetric boundary conditions in \(N = 4\) super Yang-Mills theory, J. Stat. Phys., 135, 789-855, 2009 ·Zbl 1178.81180 |
| [27] | Gomez, R.; Zhu, C. B., Local theta lifting of generalized Whittaker models associated to nilpotent orbits, Geom. Funct. Anal., 24, 796-853, 2014 ·Zbl 1404.22033 |
| [28] | Hazeltine, A.; Liu, B.; Lo, C. H., On the intersection of local Arthur packets for classical groups, arXiv preprint |
| [29] | Knop, F., On the set of orbits for a Borel subgroup, Comment. Math. Helv., 70, 285-309, 1995 ·Zbl 0828.22016 |
| [30] | Knop, F., Classification of multiplicity free symplectic representations, J. Algebra, 301, 531-553, 2006 ·Zbl 1109.13005 |
| [31] | Knop, F.; Van Steirteghem, B., Classification of smooth affine spherical varieties, Transform. Groups, 11, 495-516, 2006 ·Zbl 1120.14042 |
| [32] | Kapustin, A.; Witten, E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys., 1, 1-236, 2007 ·Zbl 1128.22013 |
| [33] | Liu, Y., Refined Gan-Gross-Prasad conjecture for Bessel periods, J. Reine Angew. Math., 717, 133-194, 2016 ·Zbl 1404.11065 |
| [34] | Li, W. W., The Weil Representation and Its Character, 2008, Mathematisch Instituut: Mathematisch Instituut Leiden, Available at |
| [35] | Prasad, D., Weil representation, Howe duality, and the theta correspondence, 2007, Available at |
| [36] | Pantev, T.; Toen, B.; Vaquie, M.; Vezzosi, G., Shifted symplectic structures, Publ. Math. IHÉS, 117, 271-328, 2013 ·Zbl 1328.14027 |
| [37] | Rumelhart, K. E., Minimal representations of exceptional p-adic groups, Represent. Theory, 1, 133-181, 1997 ·Zbl 0889.22009 |
| [38] | Ratiu, T. S.; Ziegler, F., Symplectic induction, prequantum induction, and prequantum multiplicities, Commun. Contemp. Math., 24, 4, Article 2150057 pp., 2022 ·Zbl 1493.53108 |
| [39] | Sakellaridis, Y., Plancherel decomposition of Howe duality and Euler factorization of automorphic functionals, (Representation Theory, Number Theory, and Invariant Theory. Representation Theory, Number Theory, and Invariant Theory, Progr. Math., vol. 323, 2017, Birkhäuser/Springer: Birkhäuser/Springer Cham), 545-585 ·Zbl 1425.11077 |
| [40] | Sakellaridis, Y.; Venkatesh, A., Periods and harmonic analysis on spherical varieties, Astérisque, 396, 2017, viii+360 pp ·Zbl 1479.22016 |
| [41] | Wan, C.; Zhang, L., Periods of automorphic forms associated to strongly tempered spherical varieties, arXiv preprint |
| [42] | Xue, H., Refined global Gan-Gross-Prasad conjecture for Fourier-Jacobi periods on symplectic groups, Compos. Math., 153, 1, 68-131, 2017 ·Zbl 1436.11055 |
| [43] | Zhu, C. B., Local theta correspondence and nilpotent invariants, (Representations of Reductive Groups. Representations of Reductive Groups, Proceedings of Symposia in Pure Mathematics, vol. 101, 2019), 427-450 ·Zbl 1471.22011 |
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.
