Automorphic induction for elliptic representations (Q6620010)

In the paper under review, the author studies the automorphic induction for elliptic representations. The author starts by studying the intertwining operators between the automorphic and parabolic induced representations using the Whitaker functionals.\N\NConsider an cyclic extension \(E\) of a field \(F\) of degree \(d\) defined by a character \(\kappa: F^\times \to \mathbb C^\times\) and the norm \(N_{E/F} : E^\times \to F^\times\); \(\ker(\kappa) = N_{E/F}(E^\times)\). A local automorphic induction associates to a stable irreducible smooth representation \(\tau\) of the general linear group \(H=\mathrm{GL}_m(E)\) over \(E\) a \(\kappa\)-stable irreducible smooth representation \(\pi\cong (\tau \circ \det) \otimes \pi\) of \(G=\mathrm{GL}_{md}(F)\) over \(F\) via the local Langlands correspondence. Following the Bernstein-Zelevinsky classification, every elliptic, i.e. square-integrable, irreducible smooth representation \(\delta\) of \(G\) is defined by a unique pair \((\rho,k)\) consisting of a divisor \(k\) of \(n=md\) and a cuspidal unitary irreducible smooth representation \(\rho\) of \(\mathrm{GL}_{n/k}(F)\) as an irreducible sub-representation \(\delta(\rho,k)\) of \(\nu^{(k-1)/2}\rho \times \nu^{(k-1)/2-1}\rho\times \dots \times \nu^{-(k-1)/2}\rho\) associated with the Levi component of the standard parabolic subgroup \(\underbrace{\mathrm{GL}_{n/k}(F) \times \dots \times \mathrm{GL}_{n/k}(F)}_{k}\), where \(\nu = |.|_F \circ \det(.)\). Following the local Langlands correspondence, every elliptic representation of \(H\) is a quotient of a parabolic induced representation \(\mathrm{ind}_{L_{E,I}}^H(\tau_{E,I})\) from a square-integrable inducing representation \(\tau_{E,I}=\mathrm{ind}_{L_E}^{L_{E,I}}(\nu_E^{k-1} \rho_E\otimes \nu_E^{k-2} \rho_E\otimes\dots\otimes \rho_E )\) of the Levi subgroup \(L_{E,I}\) from the Levi subgroup \(L_E\). Here, \(\rho_E\) admits a \(\kappa\)-recovery \(\rho_1 \times\kappa \rho_1 \times\dots \times \kappa^{d_1-1}\rho_1\) to \(G\) of an irreducible cuspidal representation \(\rho_1\) of \(\mathrm{GL}_{n_1}(F), n_1 = m/k\). Similarly, the representation \(\tau_E = \tau_1 \times \kappa \tau_1 \times \dots\times \kappa^{d_1 -1}\tau_1\), where \(\tau_1\) is a square-integrable representation of \(\mathrm{GL}_{n_1}(F)\), \(n_1 = m/k\). The pair \((\tau_1, I)\) defines an elliptic representation \(\pi_I\) of \(G_1=\mathrm{GL}_{n_1k}(F)\) (Section 4) such that: the subset \(I\subset \{1, \dots, k\}\) defines a Levi subgroup \(L_1\) of \(G_1\), containing \(L_1= \mathrm{GL}_{n_1}(F) \times \dots \times \mathrm{GL}_{n_1}(F) \); the representation \(\tau_{1,I}\) is the unique irreducible sub-representation of \(\tau_{E,I}=\mathrm{ind}_{L_E}^{L_{E,I}}(\nu_E^{k-1} \rho_E\otimes \nu_E^{k-2} \rho_E\otimes\dots\otimes \rho_E )\) and \(\pi_{1,I}\) is the Langlands quotient of the parabolic induced representation \(X_{1,I} = \mathrm{ind}_{L_{1,I}}^{G_1}(\tau_{1,I})\).\N\N\NThe main result of the paper is Theorem 7.1 stating that \(\pi_{I}=\pi_{1,I}\times \kappa\pi_{1,I}\times\dots\times\kappa^{d_1-1}\pi_{1,I} \) is just such a \(\kappa\)-recovery of \(\pi_{E,I}\) from \(G_1\) to \(G\).