Theta liftings for quasi-split unitary groups
type: Item
instance of: scholarly article
title: Theta liftings for quasi-split unitary groups
zbMATH Open document ID: 0804.11037
DOI: 10.1007/BF02567700
author: Takao Watanabe
published in: Manuscripta Mathematica
publication date: 1995-01-22T00:00:00Z
full work available at URL: https://eudml.org/doc/155930
review text: 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).
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 572226
zbMATH Keywords: theta lift
zbMATH Keywords: unitary group
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: Q3312315
cites work: \(L\)-functions and Fourier-Jacobi coefficients for the unitary group \(U(3)\)
cites work: On periods of cusp forms and algebraic cycles for \(U(3)\)
cites work: On a correspondence of automorphic forms on orthogonal groups of order five
cites work: Q3762525
cites work: Hypercuspidality of automorphic cuspidal representations of the unitary group \(\mathrm{U}(2,2)\)
cites work: Theta lifting of cusp forms on the unitary group \(U(d, d)\)
OpenAlex ID: W2077470475
description: scientific article; zbMATH DE number 572226