Let \(G(r)=\text{GL}(r,F)\) be the general linear group over a non-Archimedean local field \(F\), \(\text{char}(F)\neq 2\), and \(\widetilde G(r)\) be the metaplectic covering group of \(G(r)\). For a suitable character \(\omega\) of \(Z(\widetilde G(r))\), let \(\vartheta_{r,\omega}\) be an exceptional representation of \(\widetilde G(r)\). Let, for suitable characters \(\omega\) and \(\nu\) of \(Z(\widetilde G(r))\) and \(\pi\) a representation of \(G(r)\), \(L(\omega,\nu;\pi)\) be the space of \(G(r)\)-invariant linear functionals on \(\vartheta_{r,\omega}\otimes\vartheta_{r,\nu}\otimes\pi\).
One purpose of the article is to specify the structure of the space \(L(\omega,\nu;\pi)\) for certain classes of \(\pi\) (Section 6). A character \(\omega\) of \(Z(\widetilde G(r))\) is called suitable if it is compatible (see p. 755) with the trivial character \(\chi_0\). For an ordered partition of \(r\), \(\gamma=(\gamma_1,\dots,\gamma_r)\), and compatible characters \(\chi\) and \(\omega\) of \(Z(\widetilde G(r))\), the representation \(\vartheta_\gamma(\chi,\omega)\) is defined in p. 756 ((5.2)), and \(\vartheta_{r,\omega}=\vartheta_{(r)}(\chi_0,\omega)\) for \(\omega\) compatible with \(\chi_0\). A representation \(\pi\) of \(G(r)\) is called homogeneous if it has a unique character \(\omega\) of \(Z(G(r))\) such that there is a non-zero subquotient of \(\pi\) in which \(Z(G(r))\) acts via \(\omega\).
(1) The first main result concerns \(\pi\): homogeneous admissible representation of finite length which is general with respect to the complex numbers (Th. 6.1), and as its corollary, it is shown that for \(\pi\) a homogeneous admissible representation of finite length which is general with respect to \(1/4\), \(\dim_\mathbb{C} L(\omega,\nu;\pi)\) is at most the dimension of the space of Whittaker functionals on \(\pi\). (2) Let \(\gamma_0=(1,1,\dots,1)\) and \(G(\gamma_0)=\text{GL}(1)\times\text{GL}(1)\times\cdots\times\text{GL}(1)\), \(\chi=(\chi_1,\dots,\chi_r)\) be the character of \(g(\gamma_0)\) and \(I(\chi)\) be the principal series representation of \(G(r)\) obtained by normalized parabolic induction from \(\chi\). Defining balanced characters of \(G(\gamma_0)\) (Def. 6.2), it is shown that for \(\chi\) non-balanced, \(L(\omega,\nu;I(\chi))=\{0\}\) (Th. 6.3). (3) If \(\pi\) is an irreducible admissible representation of \(G(2)\), \(\dim_\mathbb{C}(\omega,\omega;\pi)\leq 1\), and if \(\pi\) is an irreducible admissible representation of \(G(3)\), \(\dim_\mathbb{C} L(\omega,\nu;\pi)\leq 1\) (Th. 6.5). (4) Finally, when \(\chi\) is balanced and \(\chi^2\) is regular, \(\dim_\mathbb{C} L(\omega,\nu;\pi)\leq 1\) (Th. 6.4).