## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# On the SL(2) Period Integral

U. K. Anandavardhanan and Dipendra Prasad
American Journal of Mathematics
Vol. 128, No. 6 (Dec., 2006), pp. 1429-1453
Stable URL: http://www.jstor.org/stable/40068040
Page Count: 25
Preview not available

## Abstract

Let E/F be a quadratic extension of number fields. For a cuspidal representation π of ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{E}} )$ , we study in this paper the integral of functions in π on ${\rm{SL}}_{\rm{2}} ({\rm{F}})\backslash {\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{F}} )$ . We characterize the nonvanishing of these integrals, called period integrals, in terms of π having a Whittaker model with respect to characters of ${\rm{E}}\backslash {\Bbb A}_{\rm{E}}$ which are trivial on ${\Bbb A}_{\rm{F}}$ . We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{E}} )$ whose period integral vanishes identically while each local constituent admits an ${\rm{SL}}_{\rm{2}}$ -invariant linear functional. Finally, we construct an automorphic representation π on ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{E}} )$ which is abstractly ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{F}} )$ distinguished but for which none of the elements in the global L-packet determined by it is distinguished by ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{F}} )$ .

• 1429
• 1430
• 1431
• 1432
• 1433
• 1434
• 1435
• 1436
• 1437
• 1438
• 1439
• 1440
• 1441
• 1442
• 1443
• 1444
• 1445
• 1446
• 1447
• 1448
• 1449
• 1450
• 1451
• 1452
• 1453