×

Uniqueness of linear periods. (English) Zbl 0855.22018

Let \(G= \text{GL} (n, k)\), \(k\) a non-archimedean local field of characteristic 0 and \(H= \text{GL} (p, k) \times \text{GL} ( q, k)\) with \(p+ q= n\), viewed as a subgroup of \(G\). The following result is proved. Let \(\pi\) be an irreducible admissible representation of \(G\). Up to a scalar factor, there is at most one \(H\)-invariant linear form on the space of \(\pi\). This is a consequence of the following theorem. Any distribution on \(G\) which is \(H\)-bi-invariant is invariant under \(g\mapsto g^{-1}\). The proof needs a study of the \(H\)-orbits on the symmetric variety \(G/H\) and eventually the problem is reduced to the study of distributions on the cone of nilpotent elements in the corresponding infinitesimal symmetric space.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] A. Ash and D. Ginzburg : p-adic L-functions for GL(2n) , preprint. · Zbl 0807.11029 · doi:10.1007/BF01231556
[2] A. Ash , D. Ginzburg and S. Rallis : Vanishing of periods of cusp forms over modular symbols , Preprint. · Zbl 0786.11028 · doi:10.1007/BF01445131
[3] J. Bernstein : lectures at the University of Tel Aviv (1989).
[4] D. Bump and S. Friedberg : The exterior Square L-Functions on GL(n) , In: FESTSCHRIFT IN HONOR OF I.I. PIATETSKI-SHAPIRO, Part II , 47-65, Israel Math. Conf. Proc. 3 (1990). · Zbl 0712.11030
[5] W. Casselman and J. Shalika : The unramified principal series of p-adic groups II, The Whittaker function , Compositio Math. 41 (1980), 207-231. · Zbl 0472.22005
[6] Y. Flicker : On distinguished representations , J. reine angew. Math. 418 (1991), 139-172. · Zbl 0725.11026 · doi:10.1515/crll.1991.418.139
[7] S. Friedberg and H. Jacquet : Linear Periods , J. reine angew. Math. 443 (1993), 91-139. · Zbl 0782.11033 · doi:10.1515/crll.1993.443.91
[8] I.M. Gelfand and D. Kajdan : Representations of the group GL(n, k) where K is a local field , in Lie Groups and their Representations , Helstead Press, New York (1971), 95-118. · Zbl 0348.22011
[9] D. Ginzburg , I. Piatetski-Shapiro and S. Rallis : L-functions for O(V) x GL(r) , in preparation. · Zbl 0884.11022
[10] G. Harder , R.P. Langlands and M. Rapoport : Algebraische Zyklen auf Hilbert-Blumenthal-Flachen , J. reine angew. Math. 366 (1986), 53-120. · Zbl 0575.14004
[11] H. Jacquet : [J1] Sur un résultat de Waldspurger II , Comp. Math., 63 (1987), 315-389. · Zbl 0633.10029
[12] On the non vanishing of some L-functions , Proc. Indian Acad. Sci. (Math. Sci.), 97 (1987), 117-155. · Zbl 0659.10031 · doi:10.1007/BF02837819
[13] H. Jacquet , K.F. Lai and S. Rallis : A trace formula for symmetric spaces , Duke Math. Journal., 70 (1993), 305-372. · Zbl 0795.22008 · doi:10.1215/S0012-7094-93-07006-8
[14] H. Jacquet , I.I. Piatetski and J. Shalika : Automorphic forms on GL(3) I , Annals of Math. 109 (1979), 169-212. · Zbl 0401.10037 · doi:10.2307/1971270
[15] H. Jacquet and J. Shalika : On the exterior square L-function , in Automorphic Forms, Shimura Varieties, and L-functions , L. Clozel and J. S. Milne, editors, Perspectives in Mathematics , Vol. 11, Academic Press, 143-225. · Zbl 0695.10025
[16] H. Kraft and C. Procesi : Closures of conjugacy classes of matrices are normal , Inventiones Math. 53 (1979), 227-247. · Zbl 0434.14026 · doi:10.1007/BF01389764
[17] B. Kostant and S. Rallis : Orbits and representations associated with symmetric spaces , Amer. J. Math. 93 (1971), 275-306. · Zbl 0224.22013 · doi:10.2307/2373470
[18] R. Richardson : Orbits, invariants, and representations associated to involutions of reductive groups , Invent. Math. 66 (1982), 287-312. · Zbl 0508.20021 · doi:10.1007/BF01389396
[19] C. Rader and S. Rallis : Spherical Characters on p adic symmetric spaces , to appear in the American Journal of Mathematics. · Zbl 0861.22011 · doi:10.1353/ajm.1996.0003
[20] D. Soudry : A uniqueness theorem for representations of GsO(6) and the strong multiplicity one theorem for generic representations of Gsp(4) , Israel Journal of Mathematics, 58 (1987), 547-584. · Zbl 0642.22003 · doi:10.1007/BF02771692
[21] T. Springer : Some results on Algebraic groups with Involution , in Algebraic Groups and Related Topics, Advanced Studies in Pure Mathematics , 523-543. · Zbl 0628.20036
[22] J.L. Waldspurger : [W1] Sur les coefficients de Fourier des formes modulaires de poids demi-entier , J. Math. Pures Appl. 54 (1985), 375-484. · Zbl 0431.10015
[23] Quelques propriétés arithmétiques des formes modulaires de poids demi-entier , Compositio Math. 54 (1985), 121-171.
[24] Sur les valeurs de certaines fonctions L-automorphes en leur centre de symétrie , Compositio Math. 54 (1985), 173-242. · Zbl 0567.10021
[25] Correspondence de Shimura et Shintani , J. Math. Pures Appl. 59 (1980), 1-133. · Zbl 0412.10019
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.