Generalized Jacquet modules of parabolically induced representations
Let \(G\) be a connected semisimple linear Lie group with Lie algebra \(\mathfrak g\). In 1978, \textit{W. Casselman} [Proc. int. Congr. Math., Helsinki 1978, Vol. 2, 557--563 (1980; Zbl 0425.22019)] introduced the Jacquet module of a representation of \(G\). One of the tools for investigating this module is the Bruhat filtration defined by the Bruhat decomposition, see [\textit{W. Casselman, H. Hecht} and \textit{D. Miličić}, Proc. Symp. Pure Math. 68, 151--190 (2000; Zbl 0959.22010)].NEWLINENEWLINE Let \(G=KA_0N_0\) be an Iwasawa decomposition and \(P_0 = M_0A_0N_0\) a minimal parabolic subgroup together with its Langlands decomposition. Fix some parabolic subgroup \(P\) with \(P_0 \subseteq P\) and Langlands decomposition \(P=MAN\), \(A_0 \subseteq A\). Let \(\mathfrak a\) be the Lie algebra of \(A\) and \({\mathfrak a}^\ast = {\Hom}_{\mathbb C}({\mathfrak a}, {\mathbb C})\). For \(\lambda \in {\mathfrak a}^\ast\) and some irreducible representation \(\sigma\) of \(M\) set \(I(\sigma,\lambda) := {\text{ Ind}}_P^G(\sigma \otimes e^{\lambda + \rho})\), where \(\rho \in {\mathfrak a}^\ast\) is half the sum of all positive roots, as the space of \(C^\infty\)-sections of a certain vector bundle on the quotient \(G/P\). Finally, fix some character \(\eta \in {\mathfrak n}_0\) of the Lie algebra of \(N_0\) and define NEWLINE\[NEWLINE J'_\eta(I(\sigma,\lambda)) := \{ v \in I(\sigma,\lambda)'\,| \, \exists k \in {\mathbb N}\;\forall X \in {\mathfrak N}_0\;: \;(X-\eta(X))^kv=0\,\}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe author uses the Bruhat filtration on \(J'_\eta(I(\sigma,\lambda))\) for studying this very module. The investigation of the structure of the successive quotients of the Bruhat filtration are one of the main parts of the paper.NEWLINENEWLINENEWLINEConditions are stated under which these quotients become zero eventually. In the second half of the paper, inspired by the work of \textit{S. Arkhipov} [Adv. Stud. Pure Math. 40, 27--68 (2004; Zbl 1096.17003)], a generalized twisting functor is introduced and some relation between this twisting functor and the successive quotients of the Bruhat filtration is established. In the last chapter the author determines the dimension of the space of Whittaker vectors of \(I(\sigma,\lambda)'\).
scientific article; zbMATH DE number 6054397
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