Averages of symmetric square L-functions, and applications

We exhibit a spectral identity involving L(s,Symf) for f on SL2. Perhaps contrary to expectations, we do not treat L(s,Symf) directly as a GL3 object. Rather, we take advantage of the coincidence that the standard L-function for SL2 is the symmetric square for a cuspform on GL2 restricted to SL2. [1] As SL2 = Sp2, the integral identities obtained from Sp2n × Sp2n ⊂ Sp4n produce standard L-functions for Sp2n, giving the symmetric square for GL2 as a special case. This computation is done in an appendix. The same general argument applies to classical groups and their standard L-functions. Indeed, it is useful to note that the twist Sp∗(Φ) of Sp2n consisting of isometries of a rationally anisotropic skew-quaternion form Φ has compact arithmetic quotients, avoiding certain problems of regularization if desired.