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Structure arising from induction and Jacquet modules of representations of classical $$p$$-adic groups. (English) Zbl 0874.22014
Jacquet modules are used for the study of parabolically induced representations of reductive groups over a $$p$$-adic field $$F$$. There exists a description of factors of certain filtrations on these induced representations. In the case of general linear groups, the functor of parabolic induction and the Jacquet functor induce the structure of a $${\mathbb{Z}}_+$$-graded Hopf algebra on the sum $$R$$ of Grothendieck groups of categories of smooth representations of $$GL(n,F)$$ of finite length. The multiplication $$m: R\times R\to R$$ is defined using parabolic induction and the comultiplication $$m^*: R\to R\otimes R$$ is defined in terms of Jacquet modules. The most interesting part of the structure is the property that $$m^*: R\to R\otimes R$$ is a ring homomorphism. This enables one to compute composition series of parabolically induced representations in a very simple way. The author defines the direct sum of Grothendieck groups $$R(S)$$, which corresponds either to the series $$Sp(n,F)$$ or $$SO(2n+1,F)$$ in a way similar to that in which $$R$$ was defined for general linear groups. The action $$\triangleright$$ of $$R$$ on $$R(S)$$ is defined using parabolic induction. In this way $$R(S)$$ becomes a $${\mathbb{Z}}_+$$-graded comodule over $$R$$. The comodule structure map $$\mu ^*: R(S)\to R\otimes R(S)$$ is again defined using the Jacquet modules, in a way similar to that in the case of $$GL(n,F)$$. The author determines the structure of $$R(S)$$ over $$R$$ (note that $$R(S)$$ is not a Hopf module over $$R$$). The contragredient functor defines an automorphism $$\sim : R\to R$$ in a natural way. Let $$M^*=(m\otimes 1)\circ (\sim \otimes m^*) \circ s\circ m^* : R\to R\otimes R$$, where $$s$$ is the flip. It is shown that $$R(S)$$ is an $$M^*$$-Hopf module over $$R$$. Let $$\pi$$ be an irreducible smooth representation of $$GL(n,F)$$ and let $$\sigma$$ be a similar representation of $$Sp(n,F)$$ or $$SO(2n+1,F)$$. Then $$\mu ^* (\pi \triangleright \sigma )=M^*(\pi )\triangleright \mu ^*(\sigma )$$. This formula connects the module and comodule structures on $$R(S)$$. This is a combinatorial formula which enables one to obtain in a similar manner factors of filtrations of Jacquet modules of parabolically induced representations.
Reviewer: A.Klimyk (Kiev)

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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