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.

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 |