[For the entire collection see Zbl 0698.00020.]
Let F be a local field, E a quadratic extension of F, and V a vector space of dimension 3 over E (hence of dimension 6 over F). Let \(\Phi\) be a skew-Hermitian form on V; \(A=Tr_{E/F}\circ \Phi\) is a symplectic form, so that \(G=U(\Phi)\) embeds in Sp(A). The corresponding metaplectic group splits over G. Let \(w_ y\) be the restriction to G of the oscillator representation associated to an additive character \(\phi\) of F; it also depends on the splitting. Any such representation is called a Weil representation.
There is a subgroup R of G that plays a central role. If \(U(\Phi)=\{g\in GL_ 3(E)|\) \(g\Phi g=\Phi\}\) for
\(\Phi=\left[\begin{matrix} 0 & 0 & 1 \\ 0 & \xi & 0 \\ 1 & 0 & 0 \end{matrix}\right],\) then \(R=R'Z\), where \(R'\) is the group of all matrices \[ \left[\begin{matrix} 1 & \xi w & ((\xi w-\bar w/2)+t) \\ 0 & \beta & \bar w \\ 0 & 0 & 1 \end{matrix}\right] \] with \(w\in E\), \(\beta \in E^ 1\), and \(t\in F\), and Z is the center of G. Since R is normal in the Borel subgroup B of G, B acts on \(R^{\wedge}\); an irreducible admissible representation \(\pi\) of G is called exceptional if \(Hom_ R(\pi,\gamma)\neq 0\) only for \(\gamma\) in a single B-orbit of \(R^{\wedge}\). The authors show that Weil representations are exceptional and that one gets a natural correspondence between the set of Weil representations and \(B/R^{\wedge}\). They also show that exceptional representations never have Whittaker models, and classify the unramified exceptional representations of G. They give similar results for \(G=GL_ 3.\)
The last three sections of the paper are concerned with computations of “local Shimura integrals”. These come from local components of global integrals giving L-functions for U(3). The interest arises from the fact that the usual methods of computing these integrals for groups other than \(GL_ n\), via Whittaker functions, does not apply.