\((\text O(V\oplus F),\text O(V))\) is a Gelfand pair for any quadratic space \(V\) over a local field \(F\). (English) Zbl 1179.22017
Let \(F\) be an arbitrary local field of characteristic different from \(2\). Consider the standard embedding \(\text O(V) \hookrightarrow\text O(W)\) where \((W:=V\oplus F{\cdot}e,Q)\) is a quadratic space over \(F\) with \(Q(e)=1\). In the paper under review it is proved that \((\text O(V),\text O(W))\) is a Gelfand pair. This means that for every irreducible admissible representation \((\pi,E)\) of \(\text O(W)\) the inequality \(\dim\operatorname{Hom}_{\text O(V)}(E,\mathbb{C})\leq 1\) holds.
The authors use a technique of Gelfand and Kazhdan to deduce this representation theoretic result from considerations about invariant distributions. They show that any distribution on \(\text O(W)\) which is invariant with respect to the standard two-sided action of \(\text O(V)\times\text O(V)\) is also invariant with respect to transposition. The proof of the latter result forms the main part of this paper. When \(F\) is the field of real or complex numbers or a \(p\)-adic field this has previously been obtained by G. van Dijk [Math. Z. 193, 581–593 (1986; Zbl 0613.43009)], S. Aparicio and G. van Dijk [Complex generalized Gelfand pairs (Tambov University Reports, Tambov) (2006)] and E. P. H. Bosman and G. van Dijk [Geom. Dedicata 50, No. 3, 261–282 (1994; Zbl 0829.22015)], respectively. The authors however give a uniform proof for the archimedean and non-archimedean case except at one point where the archimedean case requires some extra work. Their proof relies on Bruhat filtration, Frobenius reciprocity and Bernstein’s localization property. All of these basic results on invariant distributions are recalled for locally compact totally disconnected spaces and for smooth manifolds.
Readers needing more background information will find that in an earlier paper of the same authors, where they showed an analogous result about general linear groups [Compos. Math. 144, No. 6, 1504–1524 (2008; Zbl 1157.22004)]. Moreover the main theorem of these two papers stands in connection with similar results for special orthogonal groups and unitary groups that have been obtained previously by B. H. Gross and D. Prasad [Can. J. Math. 44, No. 5, 974–1002 (1992; Zbl 0787.22018)] and G. van Dijk [Math. Z. 261, No. 3, 525–529 (2009; Zbl 1158.22010)], respectively.
The authors use a technique of Gelfand and Kazhdan to deduce this representation theoretic result from considerations about invariant distributions. They show that any distribution on \(\text O(W)\) which is invariant with respect to the standard two-sided action of \(\text O(V)\times\text O(V)\) is also invariant with respect to transposition. The proof of the latter result forms the main part of this paper. When \(F\) is the field of real or complex numbers or a \(p\)-adic field this has previously been obtained by G. van Dijk [Math. Z. 193, 581–593 (1986; Zbl 0613.43009)], S. Aparicio and G. van Dijk [Complex generalized Gelfand pairs (Tambov University Reports, Tambov) (2006)] and E. P. H. Bosman and G. van Dijk [Geom. Dedicata 50, No. 3, 261–282 (1994; Zbl 0829.22015)], respectively. The authors however give a uniform proof for the archimedean and non-archimedean case except at one point where the archimedean case requires some extra work. Their proof relies on Bruhat filtration, Frobenius reciprocity and Bernstein’s localization property. All of these basic results on invariant distributions are recalled for locally compact totally disconnected spaces and for smooth manifolds.
Readers needing more background information will find that in an earlier paper of the same authors, where they showed an analogous result about general linear groups [Compos. Math. 144, No. 6, 1504–1524 (2008; Zbl 1157.22004)]. Moreover the main theorem of these two papers stands in connection with similar results for special orthogonal groups and unitary groups that have been obtained previously by B. H. Gross and D. Prasad [Can. J. Math. 44, No. 5, 974–1002 (1992; Zbl 0787.22018)] and G. van Dijk [Math. Z. 261, No. 3, 525–529 (2009; Zbl 1158.22010)], respectively.
Reviewer: Roland Lötscher (Basel)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 20G25 | Linear algebraic groups over local fields and their integers |
| 11E08 | Quadratic forms over local rings and fields |
| 11E88 | Quadratic spaces; Clifford algebras |
| 15A63 | Quadratic and bilinear forms, inner products |
| 20G05 | Representation theory for linear algebraic groups |
References:
| [1] | Aparicio, S., van Dijk, G.: Complex generalized Gelfand pairs. Tambov University Reports (2006) |
| [2] | Aizenbud, A., Gourevitch, D., Sayag, E.: (GL n+1(F),GL n (F)) is a Gelfand pair for any local field. arXiv:0709.1273 (math.RT) (submitted) · Zbl 1157.22004 |
| [3] | Aizenbud, A., Gourevitch, D., Rallis, S., Schifmann, G.: Multiplicity One Theorems. arXiv:0709.4215 (math.RT) (submitted) · Zbl 1202.22012 |
| [4] | Baruch E.M. (2003). A proof of Kirillov’s conjecture. Ann. Math. 158: 207–252 · Zbl 1034.22010 · doi:10.4007/annals.2003.158.207 |
| [5] | Bernstein, J.: P-invariant Distributions on \(\mathrm{GL}(N)\) and the classification of unitary representations of \(\mathrm{GL}(N)\) (non-archimedean case). Lie group representations, II (College Park, Md., 1982/1983). Lecture Notes in Math., vol. 1041, 50–102. Springer, Berlin (1984) |
| [6] | Bosman, E.E.H., van Dijk, G.: A new class of gelfand pairs. Geom. Dedic. 50, 261–282 (1994). 261 @ 1994 Kluwer Academic Publishers. Printed in the Netherlands · Zbl 0829.22015 |
| [7] | Bernstein J. and Zelevinsky A.V. (1976). Representations of the group \(\mathrm{GL}(n, F)\) , where F is a local non-Archimedean field Uspekhi Mat. Nauk. 10(3): 5–70 |
| [8] | Gross B.H. and Prasad D. (1992). On the decomposition of a representation of SO n when restricted to SO n-1. Can. J. Math. 44(5): 974–1002 · Zbl 0787.22018 · doi:10.4153/CJM-1992-060-8 |
| [9] | Moeglin, C., Vigneras, M.-F., Waldspurger, J.-L.: Correspondances de Howe sur un corps p-adique. (French) [Howe correspondences over a p-adic field]. Lecture Notes in Mathematics, vol. 1291, pp. viii+163. Springer, Berlin (1987). ISBN: 3-540-18699-9 |
| [10] | Prasad D. (1990). Trilinear forms for representations of GL 2 and local {\(\epsilon\)} factors. Compos. Math Tome 75(1): 1–46 · Zbl 0731.22013 |
| [11] | van Dijk G. (1986). On a class of generalized Gelfand pairs. Math. Z. 193: 581–593 · Zbl 0613.43009 · doi:10.1007/BF01160476 |
| [12] | van Dijk, G.: (U(p, q), U(p, q)) is a generalized Gelfand pair (preprint) · Zbl 1158.22010 |
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