##
**Shimura correspondences and quaternions.
(Correspondances de Shimura et quaternions.)**
*(French)*
Zbl 0724.11026

In this fundamental paper, the author studies the correspondences both local and global, between representations of the metaplectic group \(\mathrm{SL}_2\) and representations of the quaternion groups and between representations of \(\mathrm{SL}_2\) and representations of \(\mathrm{PGL}_2\). The study of this latter correspondence was initiated by G. Shimura [Ann. Math. (2) 97, 440–481 (1973; Zbl 0266.10022)]. The author also establishes a nonvanishing theorem for \(L\)-functions: if \(\pi\) is a cuspidal automorphic representation of \(\mathrm{PGL}_2\) whose epsilon-factor satisfies \(\varepsilon (\pi,1/2)=1\), then there exist infinitely many quadratic characters \(\chi\), with specified behavior at a finite number of places, such that \(L(\pi\otimes \chi,1/2)\neq 0\).

The first part of the article concerns the local case. Let \(\widetilde{\mathrm{SL}}_2\) denote the two-sheeted metaplectic cover of \(\mathrm{SL}_2\) over a base field \(F\) of characteristic zero \((F\neq \mathbb{C})\), and \(G'\) the group of invertible elements in the non-split quaternion algebra over \(F\), modulo the center. The author gives a complete classification of the genuine irreducible admissible unitary representations of \(\widetilde{\mathrm{SL}}_2\), and of the local correspondences. Exclude the ‘elementary’ Weil representations. He shows that the correspondence, relative to a fixed non-trivial additive character, between (genuine) irreducible principal series of \(\mathrm{SL}_2\) and those of \(\mathrm{PGL}_2\) is one-to-one. However for the special and supercuspidal representations, it is pairs of genuine representations of \(\mathrm{SL}_2\) which correspond bijectively to such representations of \(\mathrm{PGL}_2\), and to the set of irreducible finite dimensional representations of \(G'\). He shows that there is a commutative triangle relating these three sets, with the arrow between the representations of \(\mathrm{PGL}_2\) and the representations of \(G'\) being the Jacquet-Langlands correspondence.

In the second part, the author gives the global results. Fix a base field \(F\), and a continuous nontrivial additive character of the idèles of \(F\). Let \(\tilde A_{00}\) denote the orthogonal complement of the space of ‘elementary’ Weil representations in the space of irreducible cuspidal automorphic representations of \(\mathrm{SL}_2\). Two automorphic representations in \(\tilde A_{00}\) are said to be equivalent if their local factors are isomorphic at almost all places. The author shows that the equivalence classes with respect to this relation are exactly the fibres of the Shimura correspondence between \(\mathrm{SL}_2\) and \(\mathrm{PGL}_2\). Using this, he proves the multiplicity one theorem for cusp forms on \(\mathrm{SL}_2\); this result was obtained independently by S. Gelbart and I. Piatetski-Shapiro [Isr. J. Math. 44, 97–126 (1983; Zbl 0526.10026)]. He also proves similar results concerning the correspondence between automorphic representations of the groups \(\mathrm{SL}_2\) and \(G'\), as well as the nonvanishing theorem mentioned above.

In the last section of Part 2, the author turns to \(\mathrm{PGSp}_4(F)\) over a global field \(F\). This group contains a maximal parabolic subgroup whose Levi component is \(\mathrm{PGL}_2(F)\times F^{\times}\). Given \(\pi_1\) an irreducible automorphic representation of \(\mathrm{PGL}_2(F)\), the author determines necessary and sufficient conditions that there exist a Größencharacter \(\chi\) of \(F\) and an irreducible cuspidal representation \(\pi_2\) of \(\mathrm{PGSp}_4(F)\) with the following property: at all places \(v\), \(\pi_{2,v}\) is a subrepresentation of the induced representation \(\mathrm{Ind}(\pi_{1,v},\chi_v)\). In particular, such triples \((\pi_1,\pi_2,\chi)\) exist. This extends a result of I. Piatetski-Shapiro [Invent. Math. 71, 309–338 (1983; Zbl 0515.10024)], who studied the case that “all” is replaced by “almost all”. He also obtains similar results for certain twisted forms of \(\mathrm{PGL}_2\) and \(\mathrm{PGSp}_4\).

This paper, circulated in preprint form since 1982, has extensively influenced the field. Though some of the proofs given here may now be simplified using more recent results on theta series (due to Kudla, Howe, and Rallis, among others), it remains an important and elegant contribution.

The first part of the article concerns the local case. Let \(\widetilde{\mathrm{SL}}_2\) denote the two-sheeted metaplectic cover of \(\mathrm{SL}_2\) over a base field \(F\) of characteristic zero \((F\neq \mathbb{C})\), and \(G'\) the group of invertible elements in the non-split quaternion algebra over \(F\), modulo the center. The author gives a complete classification of the genuine irreducible admissible unitary representations of \(\widetilde{\mathrm{SL}}_2\), and of the local correspondences. Exclude the ‘elementary’ Weil representations. He shows that the correspondence, relative to a fixed non-trivial additive character, between (genuine) irreducible principal series of \(\mathrm{SL}_2\) and those of \(\mathrm{PGL}_2\) is one-to-one. However for the special and supercuspidal representations, it is pairs of genuine representations of \(\mathrm{SL}_2\) which correspond bijectively to such representations of \(\mathrm{PGL}_2\), and to the set of irreducible finite dimensional representations of \(G'\). He shows that there is a commutative triangle relating these three sets, with the arrow between the representations of \(\mathrm{PGL}_2\) and the representations of \(G'\) being the Jacquet-Langlands correspondence.

In the second part, the author gives the global results. Fix a base field \(F\), and a continuous nontrivial additive character of the idèles of \(F\). Let \(\tilde A_{00}\) denote the orthogonal complement of the space of ‘elementary’ Weil representations in the space of irreducible cuspidal automorphic representations of \(\mathrm{SL}_2\). Two automorphic representations in \(\tilde A_{00}\) are said to be equivalent if their local factors are isomorphic at almost all places. The author shows that the equivalence classes with respect to this relation are exactly the fibres of the Shimura correspondence between \(\mathrm{SL}_2\) and \(\mathrm{PGL}_2\). Using this, he proves the multiplicity one theorem for cusp forms on \(\mathrm{SL}_2\); this result was obtained independently by S. Gelbart and I. Piatetski-Shapiro [Isr. J. Math. 44, 97–126 (1983; Zbl 0526.10026)]. He also proves similar results concerning the correspondence between automorphic representations of the groups \(\mathrm{SL}_2\) and \(G'\), as well as the nonvanishing theorem mentioned above.

In the last section of Part 2, the author turns to \(\mathrm{PGSp}_4(F)\) over a global field \(F\). This group contains a maximal parabolic subgroup whose Levi component is \(\mathrm{PGL}_2(F)\times F^{\times}\). Given \(\pi_1\) an irreducible automorphic representation of \(\mathrm{PGL}_2(F)\), the author determines necessary and sufficient conditions that there exist a Größencharacter \(\chi\) of \(F\) and an irreducible cuspidal representation \(\pi_2\) of \(\mathrm{PGSp}_4(F)\) with the following property: at all places \(v\), \(\pi_{2,v}\) is a subrepresentation of the induced representation \(\mathrm{Ind}(\pi_{1,v},\chi_v)\). In particular, such triples \((\pi_1,\pi_2,\chi)\) exist. This extends a result of I. Piatetski-Shapiro [Invent. Math. 71, 309–338 (1983; Zbl 0515.10024)], who studied the case that “all” is replaced by “almost all”. He also obtains similar results for certain twisted forms of \(\mathrm{PGL}_2\) and \(\mathrm{PGSp}_4\).

This paper, circulated in preprint form since 1982, has extensively influenced the field. Though some of the proofs given here may now be simplified using more recent results on theta series (due to Kudla, Howe, and Rallis, among others), it remains an important and elegant contribution.

Reviewer: Solomon Friedberg (Santa Cruz)

### MSC:

11F32 | Modular correspondences, etc. |

11F27 | Theta series; Weil representation; theta correspondences |

22E50 | Representations of Lie and linear algebraic groups over local fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |