Representations of reductive groups distinguished by symmetric subgroups
Tóm tắt
Let G be a complex connected reductive group,
$$G^{\theta }$$
be its fixed point subgroup under a Galois involution
$$\theta $$
and H be an open subgroup of
$$G^{\theta }$$
. We show that any H-distinguished representation
$$\pi $$
satisfies:
By proving the first statement, we give a partial answer to a conjecture by Prasad and Lapid.
Tài liệu tham khảo
Aizenbud, A., Gourevitch, D.: Schwartz functions on Nash manifolds. Int. Math. Res. Notes 2008, 5 (2008)
Aizenbud, A., Gourevitch, D.: Generalized Harish-Chandra descent Gelfand pairs and an Archimedean analog of Jacquet–Rallis’ theorem. Duke Math. J. 149(3), 509–67 (2009)
Aizenbud, A., Gourevitch, D., Minchenko, A.: Holonomicity of spherical characters and applications to multiplicity bounds. Selecta Math. 22, 2325–2345 (2016)
Bernstein, J., Krötz, B.: Smooth Fréchet globalizations of Harish-Chandra modules. Israel J. Math. 199(1), 45–111 (2014)
Delorme, P.: Constant term of smooth \(H\psi \)-spherical functions on a reductive p-adic group. Trans. Am. Math. Soc. 362, 933–955 (2010)
Gelfand, I., Kajdan, D.: Representations of the group \(\rm GL(n,\rm K\rm )\rm \) where \(\rm K\) is a local field, Lie groups and their representations (Proc. Summer School, Bolyai J’ anos Math. Soc., Budapest, 1971). Halsted, New York (1975)
Gourevitch, D., Sahi, S., Sayag, E.: Invariant functionals on the Speh representations. Transform. Groups 20(4), 1023–1042 (2015). MR 3416438
Harish-Chandra “On the theory of the Eisenstein integral”, in Conference on Harmonic Analysis, Lecture Notes in Mathematics 266, pp. 123–149, (1972)
Helminck, A.G., Wang, S.P.: On rationality properties of involutions of reductive groups. Adv. Math. 99, 26–97 (1993)
Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, (1972)
Kemarsky, A.: Gamma factors of distinguished representations of GL \((n, \mathbb{C})\). Pacific J. Math. 278(1), 137–172 (2015). MR 3404670
Knapp, A.W.: Representation theory of semisimple groups, An overview based on examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton (2001)
Knapp, A.W., Zuckerman, G.J.: Classification of irreducible tempered representations of semisimple Lie groups. Proc. Nat. Acad. Sci. USA 73, 2178–2180 (1976)
Kobayashi, T., Oshima, T.: Finite multiplicity theorems for induction and restriction. Adv. Math. 248, 921–944 (2013)
Krotz, B., Schlichtkrull, H.: Multiplicity bounds and the subrepresentation theorem for real spherical spaces. Trans. Am. Math. Soc. 368, 2749–2762 (2016)
Langlands, R. P.: On classification of irreducible representations of real algebraic groups. Unpublished Manuscript (1973)
Lapid, E., Offen, O., Feigon, B.: On representations distinguished by unitary groups. Publ. Math. Inst, Hautes (2012)
Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. 31, 331–357 (1979)
Prasad, D.: A “relative” local Langlands correspondence, preprint, 2012, available at http://www.math.tifr.res.in/~dprasad/relative-L.pdf
Sakellaridis, Y.: A. Venkatesh. Periods and harmonic analysis on spherical varieties. arXiv: 1203.0039
Springer, T.A.: Some results on algebraic groups with involutions, Algebraic groups and related topics. Adv. Stud. Pure Math. 6, 525–543 (1985)
van den Ban, E.: Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces. Nederl. Akad. Wetensch. Indag. Math. 49(3), 225–249 (1987)
Wolf, J.A.: Finiteness of orbit structure for real flag manifolds. Geom. Dedicata 3, 377–384 (1974)
Zhelobenko, D.P., Naimark, M.A.: Description of completely irreducible representations of a semi-simple complex Lie group. 5 (Russian) Dokl. AN SSSR 171, : 25–28. English translation: Soviet Math. Dokl. 7(1966), 1403–1406 (1966)