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Exterior square $$L$$-functions. (English) Zbl 0695.10025
Automorphic forms, Shimura varieties, and L-functions. Vol. II, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 11, 143-226 (1990).
The authors discuss an integral representation for the (partial) Langlands $$L$$-function $$L(s,\pi,\tau)$$, were $$\pi$$ is an automorphic cuspidal (unitary) representation of $$\text{GL}_n({\mathbb A}_F)$$, and $$\tau$$ is the exterior square representation of $$\text{GL}_n({\mathbb C})$$ in the space of antisymmetric tensors. In particular, if $$n$$ is even, this $$L$$-function is shown to have a pole at $$s=1$$ if and only if certain period integrals – which are residues of the integral representation – are non-zero. This is the main result of the paper, and it is motivated by the following ideas.
Suppose $$\pi$$ is indexed by an irreducible $$n$$-dimensional representation $$\tau$$ of the Galois group of $$F$$; then the Artin $$L$$-function $$L(s,r)$$ agrees with $$L(s,\pi)$$, and $$L(s,\pi,\tau)=L(s,\tau \circ r)$$. Now suppose also that $$L(s,\pi,\tau)$$ (and hence $$L(s,\tau\circ r))$$ has a pole at $$s=1$$. This means that $$\tau\circ r$$ must contain the trivial representation, i.e., the image of $$r$$ is contained in a conjugate of the symplectic group. Thus it is suggested (via Langlands’ conjecture) that $$\pi$$ is the functorial image of some automorphic representation $$\pi '$$ of the group $$G'$$ whose $$L$$-group is the symplectic group. When $$n=4$$, one can in fact use the Weil representation (and theta-series liftings) to carry out this idea; the group $$G'$$ is then $$\text{GSp}(4)$$, the groups $$\text{GL}(4)$$ and $$G'$$ form a dual reductive pair, and the period integrals alluded to above are precisely those needed to show that $$\pi$$ is the theta-series lifting of some $$\pi '$$ in $$G'$$. (This represents work in progress of the authors and Piatetski-Shapiro.) In general, one hopes to prove such a functorial lifting result using the trace formula.
It should be mentioned that an alternate integral representation for $$L(s,\pi,\tau)$$ has been recently presented by D. Bump and S. Friedberg [Festschrift in Honor of I. Piatetski-Shapiro, Isr. Math. Conf. Proc. 3, 47–65 (1990; Zbl 0712.11030)].
For the entire collection see Zbl 0684.00004.
Reviewer: S. Gelbart

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings