Theta liftings for quasi-split unitary groups
This paper is concerned with the theta lift from one unitary group to another defined with respect to the same quadratic extension. More precisely consider a vector space \(V\) over a field \(F\); let \(E\) be a quadratic extension of \(F\) and let \(V_ E = V \otimes E\). Let \(V^*\) be the dual space and let \(\langle , \rangle\) denote the canonical pairing. Let \(\sigma\) denote the Galois involution of \(E\) over \(F\). Then we obtain a Hermitian structure on \(V_ E \oplus V^*_ E \oplus E\) by \(q_ 1 ((x,y,z)) = \text{Tr}_{E/F} (\langle\sigma (x),y\rangle) + N_{E/F}(z)\). Also we obtain analogously such a structure on \(V_ E \oplus V^*_ E\) by \(q((x,y)) = \text{Tr}_{E/F} (\langle\sigma (x),y\rangle)\). The author describes in detail lifts of automorphic forms on unitary groups of the form \(U(q_ 1)\) to those on groups of the form \(U(q)\). This generalizes on the one hand work of Rallis on the pairs \((Sp(2n), O(m))\) and of Gelbart and Piatetski-Shapiro (for the case as above when \(V\) is one-dimensional).
scientific article; zbMATH DE number 572226