Factorization of Period Integrals

✍ Scribed by Hervé Jacquet

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 231 KB
• Volume: 87
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2584

✦ Synopsis

to the memory of yasuko jacquet

We show that for certain quadratic extensions EÂF of number fields the period integral of a cusp form of GL(3, E) over the unitary group H 0 in three variables is a product of local linear forms.

2001 Academic Press

Contents.

1. Global results

2. L 2 -norm of a pure tensor. 3. Matching of orbital integrals. 4. Proof of the main theorem. 5. Local results: supercuspidal case. 6. Supercuspidal representations for GL(2, E). 7. Representations induced from a cuspidal representation. 8. Concluding remarks.

1. GLOBAL RESULTS

Let EÂF be a quadratic extension of number fields. We will denote by _ the non-trivial element of the Galois group of EÂF and will often write _(z)=zÄ . We will denote by U 1 the unitary group in 1 variable. We assume that every Archimedean place of F splits in E. We let H 0 be the unitary group for the 3_3 identity matrix. Recall that a cuspidal automorphic representation 6 of GL(3, E A ) is said to be distinguished by H 0 if the linear form: P(,) := | H 0 (F )"H 0 (F A ) ,(h) dh,