##
**Howe correspondences on a \(p\)-adic field.
(Correspondances de Howe sur un corps \(p\)-adique.)**
*(French)*
Zbl 0642.22002

Lecture Notes in Mathematics, 1291. Subseries: Mathematisches Institut der Universität und Max-Planck-Institut für Mathematik, Bonn, Vol. 11. Berlin etc.: Springer-Verlag. VII, 163 p.; DM 28.50 (1987).

Howe’s theory of dual reductive pairs was conceived around fifteen years ago, partly as an attempt to unify and extend some important (yet seemingly unrelated) developments in classical invariant theory, Weil’s theory of theta-functions on the metaplectic group, and certain “experimentally observed” relations between representations of the symplectic and orthogonal groups. In recent years, the concomitant theory of “theta-series liftings” has been sufficiently well developed so as to be viewed (at least by many experts in the field) as one of the central methods in the theory of automorphic forms (along with the trace formula and the theory of \(L\)-functions). With the appearance of this excellently prepared French monograph, and the almost simultaneous appearance of J.-L. Waldspurger’s Bourbaki talk on the same subject [Sémin. Bourbaki, Vol. 1986/87, Exp. No. 674, Astérisque 152/153, 85–99 (1987; Zbl 0634.10026)], it may now be safely asserted that Howe’s theory of the oscillator representation (also known as “Howe’s correspondence”) has finally come of age.

Roughly speaking, Howe’s local duality conjecture may be described as follows. Suppose \(W\) is a symplectic space over (a local field) \(k\), and \(U_ 1\), \(U_ 2\) are reductive subgroups of Sp(W,k) which mutually centralize each other, i.e., \(U_ 1\) and \(U_ 2\) comprise a “dual reductive pair”. If \(\omega_{\psi}\) denotes the “oscillator” or “metaplectic” representation of Sp(W), let us suppose that \(\pi_ 1\) is an irreducible representation of \(U_ 1\) which appears in \(\omega_{\psi}\), i.e., \(\operatorname{Hom}_{U_ 1}(\omega_{\psi},\pi_ 1)\neq 0\). Then Howe’s conjecture asserts the existence of a canonical irreducible representation \(\pi_ 2\) of \(U_ 2\) such that \(\operatorname{Hom}_{U_ 1\times U_ 2}(\omega_{\psi},\pi_ 1\otimes \pi_ 2)\neq 0\) (and the resulting correspondence \(\pi_ 1\to \pi_ 2=\Theta (\pi_ 1)\) is called “Howe’s correspondence”). Globally, one can consider an irreducible automorphic representation \(\pi_ 1\) of \(U_ 1\) all of whose local factors \(\pi_{1,\upsilon}\) occur in \(\omega_{\psi,\upsilon}\) and correspond (via Howe’s correspondence) to an irreducible representation \(\pi_{2,\upsilon}\) of \(U_{2,\upsilon}\); then the global version of Howe’s conjecture asserts that \(\pi_ 2\equiv \otimes \pi_{2,\upsilon}\equiv \Theta (\pi_ 1)\) is also automorphic. This global correspondence is called a “theta-series lifting” since we generally expect it to be realized by integrating a form in the space of \(\pi_ 1\) against the theta-kernel defined by \(\omega_{\psi}.\)

The purpose of the present monograph is to give a precise formulation and a detailed proof (in most cases) of Howe’s conjecture over a local p-adic field. This includes a complete treatment of the many technical problems which we brushed over in these introductory paragraphs, and which we shall refer to more carefully in the paragraphs below.

Chapter I describes the key notion of an irreducible dual reductive pair in \(\mathrm{Sp}(W,k)\), and presents the following classification of such pairs. Suppose \(D\) is a division algebra whose center is a finite separable extension of \(k\), and \(X_ 1\) and \(X_ 2\) are two \(D\)-modules; setting \(W=X\oplus X^*\), with \(X\) (resp. \(X^*\)) equal to \(X_ 1\otimes_ DX_ 2\) (resp. its dual), yields the dual pair \(U_ 1=\mathrm{GL}(X_ 1,D)\), \(U_ 2=\mathrm{GL}(X_ 2,D)\). These examples exhaust the dual pairs of type II in \(\mathrm{Sp}(W,k)\). The type I examples arise from a division algebra \(D\) with involution and two \(D\)-modules \(W_ i\) supplied with forms \(\langle\;, \rangle_ i\) which are \(D\)-sesqui-linear and \(\varepsilon_ i\)-hermitian, with \(\varepsilon_ i=\pm 1\) and \(\varepsilon_ 1\epsilon_ 2=-1\); taking \(W\approx W_ 1\otimes_ DW_ 2\), with the symplectic form \(\langle\;, \rangle=\mathrm{tr}_{D/k}(\langle\;, \rangle_ 1\langle\;, \rangle_ 2)\) gives the dual pair \(U_ 1=U(W_ 1,k)\) and \(U_ 2=U(W_ 2,k)\) (the unitary groups of \(W_ 1\) and \(W_ 2)\). The classification theorem alluded to above asserts that these two types of examples exhaust all the irreducible dual reductive pairs. Also included in Chapter I is a description of the parabolic subgroups of \(\mathrm{Sp}(W)\) in terms of the “Lagrangian” subspace of W which the parabolic stabilizes.

In Chapters II and III the authors construct and analyze the metaplectic representation \(\omega_{\psi}\) of \(\mathrm{Sp}(W,k)\) whose restriction to the dual pair product \(U_ 1\times U_ 2\) gives rise (at least conjecturally) to the Howe correspondence \(\pi_ 1\to \Theta (\pi_ 1)=\pi_ 2\). The basic topics covered are:

(a) the Stone-von Neumann theorem describing the irreducible representations of the Heisenberg group \(H(W)\), and its role in describing different realizations of the representation \(\omega_{\psi}\) (the Schrödinger, “mixed” and “lattice” models for \(\omega_{\psi})\);

(b) properties of the projective representation \(\omega_{\psi}\), including its trivialization to an ordinary representation upon restriction to (almost all) dual pair subgroups \(U_ 1\times U_ 2\);

(c) “compatibility of Howe’s correspondence with induction” (following work of S. Rallis and S. Kudla).

Although almost all these results were known prior to the appearance of this monograph, the account given by the present authors is the first which is simultaneously detailed, complete, and placed in the greatest possible generality. Concerning topic (c), whose results are elegant (but too complicated to recall here), we note only that one does not need to prove the existence of Howe’s correspondence before establishing how it must behave on “cuspidal data” (in the sense of Bernstein-Zelevinski); in fact, the authors observe that the methods of Rallis and Kudla used in studying (c) actually furnish a proof of Howe’s conjecture for cuspidal \(\pi_ 1.\)

Concerning the proof of Howe’s conjecture in general, we note first that the type II and archimedean cases were resolved by Howe several years ago. On the other hand, the last three chapters of this work center about the proof of Howe’s conjecture for non-ramified type I dual reductive pairs, i.e., pairs \(U_ 1\), \(U_ 2\) in \(\mathrm{Sp}(W,k)\) where (i) \(k\) is a \(p\)-adic local field with \(p\) odd; (ii) the additive character \(\psi\) (used to fix \(\omega_{\psi})\) is unramified; (iii) the division algebra \(D\) used to define \((U_ 1,U_ 2)\) is either \(k\) or an unramified quadratic extension of \(k\); and (iv) there exists a self-dual lattice \(L_ i\) in \(W_ i\) for \(i=1,2\).

Chapter IV presents preliminary material on conjugacy classes in certain unitary groups, culminating in a useful description of the contragredients of irreducible admissible representations of these unitary groups, and the commutativity of certain Hecke algebras.

The proof of Howe’s conjecture presented in Chapter V is indeed limited to unramified pairs as described above, and is due to Howe himself. Most recently, the reviewer has learned from Howe that a modification of these ideas yields the conjecture in general (at least in the case of odd residual characteristic); the complete proof appears in a new preprint of Waldspurger’s entitled “Démonstration d’une conjecture de dualité de Howe dans le case \(p\)-adique, \(p\neq 2\)” (see Zbl 0722.22009 and Zbl 0722.22010).

Finally, in Chapter V of this book, the authors present Howe’s (global-local) characterization of representations of the symplectic group of “small rank”. This work generalizes the earlier \(\mathrm{SL}_ 2\) theory of I. I. Piatetski-Shapiro and the reviewer [Invent. Math. 59, 145–188 (1980; Zbl 0426.10027)]; it has also subsequently been complemented by new work of S. J. Li [“Distinguishing cusp forms are theta-series”, preprint, M.I.T., 1988].

The authors are to be congratulated for making available to a wider mathematical audience a collection of difficult yet important ideas and results. Experts in the field will also be delighted to have (at last!) a convenient reference for numerous useful “facts” (such as the splitting of the metaplectic group over the unitary group) which were previously only parts of the subject’s folklore.

Roughly speaking, Howe’s local duality conjecture may be described as follows. Suppose \(W\) is a symplectic space over (a local field) \(k\), and \(U_ 1\), \(U_ 2\) are reductive subgroups of Sp(W,k) which mutually centralize each other, i.e., \(U_ 1\) and \(U_ 2\) comprise a “dual reductive pair”. If \(\omega_{\psi}\) denotes the “oscillator” or “metaplectic” representation of Sp(W), let us suppose that \(\pi_ 1\) is an irreducible representation of \(U_ 1\) which appears in \(\omega_{\psi}\), i.e., \(\operatorname{Hom}_{U_ 1}(\omega_{\psi},\pi_ 1)\neq 0\). Then Howe’s conjecture asserts the existence of a canonical irreducible representation \(\pi_ 2\) of \(U_ 2\) such that \(\operatorname{Hom}_{U_ 1\times U_ 2}(\omega_{\psi},\pi_ 1\otimes \pi_ 2)\neq 0\) (and the resulting correspondence \(\pi_ 1\to \pi_ 2=\Theta (\pi_ 1)\) is called “Howe’s correspondence”). Globally, one can consider an irreducible automorphic representation \(\pi_ 1\) of \(U_ 1\) all of whose local factors \(\pi_{1,\upsilon}\) occur in \(\omega_{\psi,\upsilon}\) and correspond (via Howe’s correspondence) to an irreducible representation \(\pi_{2,\upsilon}\) of \(U_{2,\upsilon}\); then the global version of Howe’s conjecture asserts that \(\pi_ 2\equiv \otimes \pi_{2,\upsilon}\equiv \Theta (\pi_ 1)\) is also automorphic. This global correspondence is called a “theta-series lifting” since we generally expect it to be realized by integrating a form in the space of \(\pi_ 1\) against the theta-kernel defined by \(\omega_{\psi}.\)

The purpose of the present monograph is to give a precise formulation and a detailed proof (in most cases) of Howe’s conjecture over a local p-adic field. This includes a complete treatment of the many technical problems which we brushed over in these introductory paragraphs, and which we shall refer to more carefully in the paragraphs below.

Chapter I describes the key notion of an irreducible dual reductive pair in \(\mathrm{Sp}(W,k)\), and presents the following classification of such pairs. Suppose \(D\) is a division algebra whose center is a finite separable extension of \(k\), and \(X_ 1\) and \(X_ 2\) are two \(D\)-modules; setting \(W=X\oplus X^*\), with \(X\) (resp. \(X^*\)) equal to \(X_ 1\otimes_ DX_ 2\) (resp. its dual), yields the dual pair \(U_ 1=\mathrm{GL}(X_ 1,D)\), \(U_ 2=\mathrm{GL}(X_ 2,D)\). These examples exhaust the dual pairs of type II in \(\mathrm{Sp}(W,k)\). The type I examples arise from a division algebra \(D\) with involution and two \(D\)-modules \(W_ i\) supplied with forms \(\langle\;, \rangle_ i\) which are \(D\)-sesqui-linear and \(\varepsilon_ i\)-hermitian, with \(\varepsilon_ i=\pm 1\) and \(\varepsilon_ 1\epsilon_ 2=-1\); taking \(W\approx W_ 1\otimes_ DW_ 2\), with the symplectic form \(\langle\;, \rangle=\mathrm{tr}_{D/k}(\langle\;, \rangle_ 1\langle\;, \rangle_ 2)\) gives the dual pair \(U_ 1=U(W_ 1,k)\) and \(U_ 2=U(W_ 2,k)\) (the unitary groups of \(W_ 1\) and \(W_ 2)\). The classification theorem alluded to above asserts that these two types of examples exhaust all the irreducible dual reductive pairs. Also included in Chapter I is a description of the parabolic subgroups of \(\mathrm{Sp}(W)\) in terms of the “Lagrangian” subspace of W which the parabolic stabilizes.

In Chapters II and III the authors construct and analyze the metaplectic representation \(\omega_{\psi}\) of \(\mathrm{Sp}(W,k)\) whose restriction to the dual pair product \(U_ 1\times U_ 2\) gives rise (at least conjecturally) to the Howe correspondence \(\pi_ 1\to \Theta (\pi_ 1)=\pi_ 2\). The basic topics covered are:

(a) the Stone-von Neumann theorem describing the irreducible representations of the Heisenberg group \(H(W)\), and its role in describing different realizations of the representation \(\omega_{\psi}\) (the Schrödinger, “mixed” and “lattice” models for \(\omega_{\psi})\);

(b) properties of the projective representation \(\omega_{\psi}\), including its trivialization to an ordinary representation upon restriction to (almost all) dual pair subgroups \(U_ 1\times U_ 2\);

(c) “compatibility of Howe’s correspondence with induction” (following work of S. Rallis and S. Kudla).

Although almost all these results were known prior to the appearance of this monograph, the account given by the present authors is the first which is simultaneously detailed, complete, and placed in the greatest possible generality. Concerning topic (c), whose results are elegant (but too complicated to recall here), we note only that one does not need to prove the existence of Howe’s correspondence before establishing how it must behave on “cuspidal data” (in the sense of Bernstein-Zelevinski); in fact, the authors observe that the methods of Rallis and Kudla used in studying (c) actually furnish a proof of Howe’s conjecture for cuspidal \(\pi_ 1.\)

Concerning the proof of Howe’s conjecture in general, we note first that the type II and archimedean cases were resolved by Howe several years ago. On the other hand, the last three chapters of this work center about the proof of Howe’s conjecture for non-ramified type I dual reductive pairs, i.e., pairs \(U_ 1\), \(U_ 2\) in \(\mathrm{Sp}(W,k)\) where (i) \(k\) is a \(p\)-adic local field with \(p\) odd; (ii) the additive character \(\psi\) (used to fix \(\omega_{\psi})\) is unramified; (iii) the division algebra \(D\) used to define \((U_ 1,U_ 2)\) is either \(k\) or an unramified quadratic extension of \(k\); and (iv) there exists a self-dual lattice \(L_ i\) in \(W_ i\) for \(i=1,2\).

Chapter IV presents preliminary material on conjugacy classes in certain unitary groups, culminating in a useful description of the contragredients of irreducible admissible representations of these unitary groups, and the commutativity of certain Hecke algebras.

The proof of Howe’s conjecture presented in Chapter V is indeed limited to unramified pairs as described above, and is due to Howe himself. Most recently, the reviewer has learned from Howe that a modification of these ideas yields the conjecture in general (at least in the case of odd residual characteristic); the complete proof appears in a new preprint of Waldspurger’s entitled “Démonstration d’une conjecture de dualité de Howe dans le case \(p\)-adique, \(p\neq 2\)” (see Zbl 0722.22009 and Zbl 0722.22010).

Finally, in Chapter V of this book, the authors present Howe’s (global-local) characterization of representations of the symplectic group of “small rank”. This work generalizes the earlier \(\mathrm{SL}_ 2\) theory of I. I. Piatetski-Shapiro and the reviewer [Invent. Math. 59, 145–188 (1980; Zbl 0426.10027)]; it has also subsequently been complemented by new work of S. J. Li [“Distinguishing cusp forms are theta-series”, preprint, M.I.T., 1988].

The authors are to be congratulated for making available to a wider mathematical audience a collection of difficult yet important ideas and results. Experts in the field will also be delighted to have (at last!) a convenient reference for numerous useful “facts” (such as the splitting of the metaplectic group over the unitary group) which were previously only parts of the subject’s folklore.

Reviewer: Stephen Gelbart (Rehovot)

### MSC:

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

11S40 | Zeta functions and \(L\)-functions |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

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

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11F33 | Congruences for modular and \(p\)-adic modular forms |