Tohoku Mathematical Journal
2008
June
SECOND SERIES VOL. 60, NO. 2

Tohoku Math. J.
60 (2008), 227-251

Title COMMUTATION RELATIONS OF HECKE OPERATORS FOR ARAKAWA LIFTING

Author Atsushi Murase and Hiro-aki Narita

(Received January 29, 2007, revised August 31, 2007)
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 T. Arakawa, in his unpublished note, constructed and studied a theta lifting from elliptic cusp forms to automorphic forms on the quaternion unitary group of signature $(1, q)$.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.

2000 Mathematics Subject Classification. Primary 11F55.
Key words and phrases. Theta lifting, Hecke operators, Spinor $L$-functions.

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