SPECTRAL MOMENT FORMULAE FOR G L ( 3 ) × G L ( 2 ) L ${GL}(3)\times {GL}(2)\ {L}$ upper G upper L left parenthesis 3 right parenthesis times upper G upper L left parenthesis 2 right parenthesis upper L-FUNCTIONS II: THE EISENSTEIN CASE

Abstract This work is the second in a series, following Part I [Kwa24] ( Algebra Number Theory 18 .10 (2024)) and preceding Part III [Kwa25] ( Math. Ann. 391 .1 (2025)). We continue our investigation of spectral moments of GL ( 3 ) × GL ( 2 ) L $\text {GL}(3)\times \text {GL}(2)\ L$ GL left parenthesis 3 right parenthesis times GL left parenthesis 2 right parenthesis upper L -functions from the perspective of period integrals. Using an identity between two distinct periods for the GL ( 3 ) $\text {GL}(3)$ GL left parenthesis 3 right parenthesis Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of GL ( 2 ) L $\text {GL}(2)\ L$ GL left parenthesis 2 right parenthesis upper L -functions to the shifted fourth moment of GL ( 1 ) L $\text {GL}(1) L$ GL left parenthesis 1 right parenthesis upper L -functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures ( Proc. Lond. Math. Soc. (3) 91 (2005)).