Structure arising from induction and Jacquet modules of representations of classical \(p\)-adic groups
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.
scientific article; zbMATH DE number 848960