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Theta Lifts of Tempered Representations for Dual Pairs (Sp2n,O(V))

Published online by Cambridge University Press:  20 November 2018

Goran Muić
Goran Muić*
Affiliation:
Department of Mathematics, University of Zagreb, 10000 Zagreb, Croatia e-mail:[email protected]
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Abstract

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This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs $\left( {{\text{S}}_{{{\text{p}}_{2n}},}}\,O\left( V \right) \right)$ over a non-archimedean field $F$ of characteristic different than 2, where $n$ is the split rank of ${{\text{S}}_{{{\text{p}}_{2n}}}}$ and the dimension of the space $V$ (over $F$) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.

Keywords

Primary 22E3secondary: 22E5011F70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 6 , 01 December 2008 , pp. 1306 - 1335
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Bernstein, J., Second adjointness for representations of p-adic reductive groups. Preprint (Harvard 1987).Google Scholar
[2] Bernstein, I. N. and Zelevinsky, A. V., Induced representations of reductive p-adic groups. I. Ann. Sci.É cole Norm Sup. 10(1977), no. 4, 441–472.Google Scholar
[3] Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups. Preprint, www.math.ubc.ca/˜ cass/research/p-adic-book-dvi.Google Scholar
[4] Goldberg, D., Reducibility of induced representations for Sp(2n) and SO(n). Amer. J. Math. 116(1994), no. 5, 1101—1151.Google Scholar
[5] Kudla, S. S., On the local theta-correspondence. Invent. Math. 83(1986), no. 2, 229–255.Google Scholar
[6] Kudla, S. S., On the Theta Correspondence. (lectures at European School of Group Theory, Beilngries, 1996), www.math.toronto.ca/˜ skudla/castle/pdf.Google Scholar
[7] Lapid, E., Muić, G., and Tadić, M., On the generic unitary dual of quasi-split classical groups. Int. Math. Res. Not. 2004, no. 26, 1335–1354.Google Scholar
[8] Moeglin, C., Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc. (JEMS) 4(2002), no. 2, 143–200.Google Scholar
[9] Moeglin, C. and Tadić, M., Construction of discrete series for classical p-adic groups. J. Amer. Math. Soc 15(2005), no. 3, 715–786.Google Scholar
[10] Moeglin, C., Vignéras, M.-F., and Waldspurger, J.L., Correspondence de Howe sur un corps p-adique. Lecture Notes inMathematics 1291, Springer-Verlag, Berlin, 1987.Google Scholar
[11] Muić, G., Howe correspondence for discrete series representations; the case of (Sp(n),O(V)). J. Reine Angew. Math. 567(2004), 99–150.Google Scholar
[12] Muić, G., Reducibility of standard representations. Pacific J. Math. 222(2005), no. 1, 133–168.Google Scholar
[13] Muić, G., On the structure of the full lift for the Howe correspondence of (Sp(n),O(V)) for rank-one reducibilities. Canad. Math. Bull. 49(2006), no. 4, 578–591.Google Scholar
[14] Muić, G., On the structure of theta lifts of discrete series for dual pairs (Sp(n),O(V)). Israel J. Math. 164(2008), 87–124.Google Scholar
[15] Muić, G. and Savin, G., Symplectic-orthogonal theta lifts of generic discrete series. Duke Math. J. 101(2000), no. 2, 317–333.Google Scholar
[16] Rallis, S., On the Howe duality conjecture. CompositioMath. 51(1984), no. 3, 333–399.Google Scholar
[17] Tadić, M., On regular square integrable representations of p-adic groups. Amer. J. Math 120(1998), no. 1, 159–210.Google Scholar
[18] Waldspurger, J.-L., Démonstration d’une conjecture de duality de Howe dans le case p-adiques, p 6= 2. Festschrift in Honor of I.I. Piatetski-Shapiro, Part II, Israel Math. Conf. Proc. 2,Weizmann, Jerusalem, 1990, pp. 267–324.Google Scholar
[19] Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra. J. Inst. Math. Jussieu 2(2003), no. 2, 235–333.Google Scholar
[20] Zelevinsky, A. V., Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. Ecole Norm. Sup. 13(1980), no. 2, 165–210.Google Scholar