Twisted symmetric square L-functions for $$\mathrm {GL}_n$$ GL n and invariant trilinear forms

Abstract

Following the works of Bump and Ginzburg and of Takeda, we develop a theory of twisted symmetric square L-functions for \(\mathrm {GL}_n\). We characterize their pole in terms of certain trilinear period integrals, determine all irreducible summands of the discrete spectrum of \(\mathrm {GL}_n\) having nonvanishing trilinear periods, and construct nonzero local invariant trilinear forms on a certain family of induced representations.