Relative non-cuspidality of representations induced from split parabolic subgroups (Q1984013)

Let \(G/H\) be a symmetric space over a \(p\)-adic field \(F\). A smooth representation of \(G\) is called \(H\)-distinguished if it carries a nonzero \(H\)-invariant linear form. If \(\pi\) is an \(H\)-distinguished representation, \(\Lambda\) is an \(H\)-invariant linear form on the space \(E\) of \(\pi\), and \(v\) is a vector in \(E\), one can form the matrix coefficient \(g \mapsto \Lambda(\pi(g)v)\). The representation \(\pi\) is called \textit{\(H\)-relatively cuspidal} if all such matrix coefficients have compact support modulo \(Z_GH\), where \(Z_G\) is the center of \(G\). The main result of this paper is that if \(\pi\) is an \(H\)-distinguished representation that arises from a ``generic'' distinguished representation of a proper parabolic subgroup of \(G\), then \(\pi\) cannot be \(H\)-relatively cuspidal in the sense above. More precisely, let \(\sigma\) be an \(F\)-involution of \(G\) with fixed-point-set \(H\), let \(P\) be a parabolic subgroup such that \(P\) and \(\sigma(P)\) are opposite, let \(M\) be the Levi factor \(P \cap \sigma(P)\), and let \(\varrho\) be an irreducible \(M \cap H\) distinguished representation of \(M\). Let \(\mathcal{X}_{M, \sigma}\) be the identity component of the group of unramified characters \(\chi\) of \(M\) satisfying \(\chi \circ \sigma = \chi^{-1}\); this is an algebraic variety, and \(\varrho \otimes \chi\) is \(M \cap H\)-distinguished for every \(\chi \in \mathcal{X}_{M, \sigma}\). A more formal statement of the main theorem is then: the set of \(\chi \in \mathcal{X}_{M, \sigma}\) such that \(\mathrm{Ind}_{P}^G(\varrho \otimes \chi)\) is \(H\)-relatively cuspidal must be a finite union of proper closed subvarieties of \(\mathcal{X}_{M, \sigma}\). The theorem is deduced from work of Blanc-Delorme decomposing the space of \(H\)-invariant linear functionals for \(\mathrm{Ind}_{P}^G(\varrho \otimes \chi)\) and generic \(\chi\) [\textit{P. Blanc} and \textit{P. Delorme}, Ann. Inst. Fourier 58, No. 1, 213--261 (2008; Zbl 1151.22012)], and previous work of the authors [\textit{S. I.~Kato} and \textit{K.~Takano}, Int. Math. Res. Not. 2008, rnn028 (2008; Zbl 1197.22007)]. The last two pages focus on the example of \(G/H=\mathrm{GL}(4,F)/\mathrm{GL}(2) \times \mathrm{GL}{2}\), and show how the key lemma can be used to prove that a large family of induced representations are non-\(H\)-relatively cuspidal.