The conservation relation for cuspidal representations
Let π ∈ Cusp (U(Vn)) be a smooth cuspidal irreducible representation of a unitary group U(Vn) of dimension n over a non-Archimedean locally compact field. Let W± m the two isomorphism classes of Hermitian spaces of dimension m, and the denote by τ + ∈ Cusp U(W+m+) and τ− ∈ Cusp U(W−m−) the first non...
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| Format: | article |
| Status: | Versión enviada para evaluación y publicación |
| Publication Date: | 2012 |
| Country: | España |
| Institution: | Universidad de Sevilla (US) |
| Repository: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/47611 |
| Online Access: | http://hdl.handle.net/11441/47611 https://doi.org/10.1007/s00208-011-0636-5 |
| Access Level: | Open access |
| Keyword: | Theta correspondence Theta dichotomy Representations of p-adic groups |
| Summary: | Let π ∈ Cusp (U(Vn)) be a smooth cuspidal irreducible representation of a unitary group U(Vn) of dimension n over a non-Archimedean locally compact field. Let W± m the two isomorphism classes of Hermitian spaces of dimension m, and the denote by τ + ∈ Cusp U(W+m+) and τ− ∈ Cusp U(W−m−) the first non-zero theta lifts of π. In this article we prove that m+ + m− = 2n + 2, which was conjectured in [M. Harris, S. S. Kudla, J. Sweet, Theta dichotomy for unitary groups, Journal of the AMS, vol. 9, pp. 941-1004 (1996), Speculations 7.5 and 7.6]. We prove similar equalities for the other dual pairs of type I : the symplecticorthogonal dual pairs and the quaternionic dual pairs. |
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