The Adjoint L - Function for Gl

We describe two new Eulerian Rankin-Selberg integrals, using the same Eisenstein series defined on the group E8, and cuspidal representations from GL5 and GSpin11, respectively. Connections with past work of Ginzburg, Bump-Ginzburg, Jiang-Rallis and others are described. We give some details of how to relate our two integrals via formal manipulations. It is a fact of classical invariant theory that, if we consider the adjoint representation of PGLn(C) and the second fundamental representation of PSp2n(C), (often denoted ∧0) then we find that the algebras of invariants for these two representations are isomorphic. This can be seen, for example, in the table on p. 260 of [15]. More precisely, each is a free algebra having n − 1 generators of degrees 2, 3, 4, . . . , n. It was observed in [10] that when such an isomorphism of algebras of invariants exists, it is an indication that if Rankin-Selberg constructions happen to exist for the L-functions corresponding to both cases, then they are likely to make use of the same Eisenstein series. In the family at hand, there are already two examples for this: when n = 3, the adjoint L-function was constructed in [6] and the L-function for ∧0 was constructed in [10], using the same Eisenstein series, which is defined on the exceptional group G2. Similarly, for n = 4, the adjoint L-function was constructed in [1] and the Lfunction for ∧0 was constructed in [2]. These constructions use the same Eisenstein series, which is defined on the exceptional group F4. In these notes we describe some recent progress towards the construction of Rankin-Selberg integrals corresponding to the case n = 5. As far as we know, these will be the first Rankin-Selberg integrals corresponding to L group representations such that the algebra of invariants has more than three generators. This time, the Eisenstein series is defined on the exceptional group E8. Our first integral is of the form