On Jacquet modules of induced representations of \(p\)-adic symplectic groups.

*(English)*Zbl 0760.22008
Harmonic analysis on reductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 305-314 (1991).

[For the entire collection see Zbl 0742.00061.]

Jacquet modules are a useful tool in the representation theory of reductive \(p\)-adic groups. The technique of Jacquet modules is very convenient for the analysis of parabolically induced representations. However, in this case it can be very difficult to understand the complete structure of the Jacquet modules. One can study Jacquet modules for different parabolic subgroups, and then compare them. In this way it is possible to get fairly explicit information about the induced representations. So it is very important to give a more explicit description of Jacquet modules of induced representations.

In this paper the author explains one possible approach to this problem for the case of symplectic groups. An application of this approach to the construction of square integrable representations is given. The paper contains also an application of it to reducibility problems. Results are given without proofs. The author states that complete proofs are to appear elsewhere.

Jacquet modules are a useful tool in the representation theory of reductive \(p\)-adic groups. The technique of Jacquet modules is very convenient for the analysis of parabolically induced representations. However, in this case it can be very difficult to understand the complete structure of the Jacquet modules. One can study Jacquet modules for different parabolic subgroups, and then compare them. In this way it is possible to get fairly explicit information about the induced representations. So it is very important to give a more explicit description of Jacquet modules of induced representations.

In this paper the author explains one possible approach to this problem for the case of symplectic groups. An application of this approach to the construction of square integrable representations is given. The paper contains also an application of it to reducibility problems. Results are given without proofs. The author states that complete proofs are to appear elsewhere.

Reviewer: A.Klimyk (Kiev)

##### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

20G05 | Representation theory for linear algebraic groups |

22E50 | Representations of Lie and linear algebraic groups over local fields |