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On square-integrable representations of classical \(p\)-adic groups. (English) Zbl 0995.22003

The author studies the problem of classification of non-cuspidal irreducible square integrable representations of the classical \(p\)-adic groups \(Sp_{2n}(F)\) and \(SO_{2n+1}(F)\). In the paper under review, the author constrains the representations induced by irreducible cuspidal representations where one needs to look for square integrable representations which show up as subquotients. The author first reduces the problem to (square integrable) representations supported in ”cuspidal lines”. Then he picks a subquotient \(\chi_0(\pi)\) in a minimal non-trivial Jacquet module of an irreducible representation \(\pi\), in a way that \(\chi_0(\pi)\) is minimal for an appropriate ordering (related to the Casselman’s square integrability criterion). He uses this subquotient to produce an irreducible essentially square integrable representation \(\delta_0(\pi)=\delta_1\otimes\dots\otimes\delta_l\otimes\sigma\) of a Levi subgroup, where \(\sigma\) is an irreducible cuspidal representation of a classical group (\(\delta_i\) are irreducible essentially square integrable representations of general linear groups). For this representation, \(\pi\hookrightarrow \text{Ind}(\delta_0(\pi))\). Write \(\delta_i=|\text{det}|_F^{e_i} \delta_i^u\), where \(e_i\in \mathbb R\) and \(\delta_i^u\) has unitary central character (which implies that \(\delta^u_i\) is unitarizable). The author proves the following criterion for square integrability: \(\pi\) is square integrable if and only if all \(e_i>0\). The problem which now remains is to study which \(\delta_0\) can show up as \(\delta_0(\pi)\) of an irreducible square integrable representation of a classical group. Recently, by different methods, in the paper [J. Eur. Math. Soc. (JEMS) 4, 143-200 (2000; Zbl 1002.22009)]C. Mœglin has solved the above question (i.e. where square integrable subquotients can show up, and what they are). She uses in her approach a natural technical assumption, which is proved in some cases, and is expected to hold in general.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields

Citations:

Zbl 1002.22009
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