PULLBACKS OF EISENSTEIN SERIES FROM GU(3,3) AND CRITICAL L-VALUES FOR GSp(4) x GL(2)

Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight l. We prove a pullback formula for certain Eisenstein series-thus generalizing a construction of Shimura-and use this to derive an explicit integral representation for the degree eight L-function L(s, F x g). This integral representation involves the pullback of a simple Siegel-type Eisenstein series on the unitary group GU(3, 3). As an application, we prove a reciprocity law-predicted by Deligne's conjecture-for the critical special values L(m, F x g) where m is an element of Z with 2 <= m <= l/2 = 1.