## Tohoku Mathematical Journal

### Commutation relations of Hecke operators for Arakawa lifting

#### Abstract

T. Arakawa, in his unpublished note, constructed and studied a theta lifting from elliptic cusp forms to automorphic forms on the quaternion unitary group of signature $(1, q)$. The second named author proved that such a lifting provides bounded (or cuspidal) automorphic forms generating quaternionic discrete series. In this paper, restricting ourselves to the case of $q=1$, we reformulate Arakawa's theta lifting as a theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting. As an application, we show that the theta lift of an elliptic Hecke eigenform is also a Hecke eigenform on the quaternion unitary group. We furthermore study the spinor $L$-function attached to the theta lift.

#### Article information

Source
Tohoku Math. J. (2), Volume 60, Number 2 (2008), 227-251.

Dates
First available in Project Euclid: 7 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1215442873

Digital Object Identifier
doi:10.2748/tmj/1215442873

Mathematical Reviews number (MathSciNet)
MR2428862

Zentralblatt MATH identifier
1214.11062

#### Citation

Murase, Atsushi; Narita, Hiro-aki. Commutation relations of Hecke operators for Arakawa lifting. Tohoku Math. J. (2) 60 (2008), no. 2, 227--251. doi:10.2748/tmj/1215442873. https://projecteuclid.org/euclid.tmj/1215442873

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