Gamma factors of intertwining periods and distinction for inner forms of \(\operatorname{GL}(n)\). (English) Zbl 1471.11171
Summary: Let \(F\) be a \(p\)-adic field, \(E\) be a quadratic extension of \(F\), \(D\) be an \(F\)-central division algebra of odd index and let \(\theta\) be the Galois involution attached to \(E / F\). Set \(H = \operatorname{GL}(m, D)\), \(G = \operatorname{GL}(m, D \otimes_F E)\), and let \(P = M U\) be a standard parabolic subgroup of \(G\). Let \(w\) be a Weyl involution stabilizing \(M\) and \(M^{\theta_w}\) be the subgroup of \(M\) fixed by the involution \(\theta_w : m \mapsto \theta(w m w)\). We denote by \(X ( M )^{w , -}\) the complex torus of \(w\)-anti-invariant unramified characters of \(M\). Following the global methods of [H. Jacquet et al., J. Am. Math. Soc. 12, No. 1, 173–240 (1999; Zbl 1012.11044)], we associate to a finite length representation \(\sigma\) of \(M\) and to a linear form \(L \in \operatorname{Hom}_{M^{\theta_w}}(\sigma, \mathbb{C})\) a family of \(H\)-invariant linear forms called intertwining periods on \(\operatorname{Ind}_P^G(\chi \sigma)\) for \(\chi \in X ( M )^{w , -} \), which is meromorphic in the variable \(\chi \). Then we give sufficient conditions for some of these intertwining periods, namely the open intertwining periods studied in [P. Blanc and P. Delorme, Ann. Inst. Fourier 58, No. 1, 213–261 (2008; Zbl 1151.22012)], to have singularities. By a local/global method, we also compute in terms of Asai gamma factors the proportionality constants involved in their functional equations with respect to certain intertwining operators. As a consequence, we classify distinguished unitary and ladder representations of \(G\), extending respectively the results of [the author, Pac. J. Math. 271, No. 2, 445–460 (2014; Zbl 1305.22024)] and
[M. Gurevich, Math. Z. 281, No. 3–4, 1111–1127 (2015; Zbl 1344.22007)] for \(D = F\), which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of the main results of [R. Beuzart-Plessis, Invent. Math. 214, No. 1, 437–521 (2018; Zbl 1409.22011)] which in the case of the group \(G\) asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for essentially square-integrable representations, but our method in fact allows us to use it only for cuspidal representations of \(G\).
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22E35 | Analysis on \(p\)-adic Lie groups |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
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