##
**Symplectic local root numbers, central critical \(L\)-values, and restriction problems in the representation theory of classical groups.**
*(English)*
Zbl 1280.22019

Gan, Wee Teck et al., Sur les conjectures de Gross et Prasad. I. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-348-5/pbk). Astérisque 346, 1-109 (2012).

The Gan-Gross-Prasad conjectures about branching laws, initially formulated for orthogonal groups, had a significant impact on the research in the last twenty years. In this paper, the authors generalize the problem of restricting irreducible representations of \(\mathrm{SO}_n\) to \(\mathrm{SO}_{n-1}\) – the problem for which Gross and Prasad proposed a “rather speculative approach... some twenty years ago” – to the case of the classical groups and their pure inner forms and restrictions to the subgroups of the same types which are relevant. This is a setting which uses Vogan’s description of \(L\)-packets, and roughly, the authors describe the members of the packet (a representation of a pure inner form of the group in question) for which the restriction to the subgroup \(H\) contains its certain fixed representation. The authors conjecture that there is such a unique representation in each generic Vogan \(L\)-packet. Also, the authors prove many local statements when treating the local case (on local root numbers, on \(L\)-parameters of classical groups, how the uniqueness of the general Bessel and Fourier-Jacobi models follow from the basic cases, i.e. when the aforementioned group \(H\) is essentially just the classical group of the same type, but attached to the space of the codimension one). Besides these local conjectures, the authors also discuss the global ones. The definitions are quite analogous, and for a specific fixed representation of the group \(H(\mathbf{A})\) (here \(\mathbf{A}\) is the ring of adeles) and an irreducible tempered global representation \(\pi\) (appearing with multiplicity one in the space of cups forms) of \(G(\mathbf{A})\), they conjecture that the restriction of a certain linear form on the representation \(\pi\) (playing an analogous role to the Hom-spaces in the local setting) is non-zero, can be expressed in terms of the central critical value of a certain \(L\)-function (attached to a standard representation of \(\hat{G}\)). The classical groups in question are orthogonal, symplectic and unitary, so there is naturally a field \(k_0\) and its quadratic extension \(k\) with the involution on \(k\) fixing \(k_0\) (the case of \(k=k_0\) and trivial involution is, of course, allowed). In the specific situation, where \(k\) is not a field, but rather an algebra \(k_0\times k_0\), many conjectures in this paper are actually theorems.

Now we describe the content of the paper more thoroughly. In the introduction, the authors review local and global settings and roughly state the problems. In the 2. Section, they describe the classical groups \(G(V)\) they are studying, and describe the pure inner forms of these groups (orthogonal, symplectic, unitary) in terms of the first cohomology group of these groups. Here, \(V\) is a vector space over \(k\), and \(G(V)\) stabilizes a sesquilinear form on \(V.\) Let \(W\) be a non-degenerate subspace of \(V\) (with some additional conditions). Then, the subgroups \(H\) of the groups \(G=G(V)\times G(W)\) are introduced; these are the subgroups to be studied in the paper. In the 3. Section the self-dual and conjugate dual representations of the Weil-Deligne group \(\mathrm{WD}(k)\) are studied. These representations give Langlands parameters for the irreducible representations of \(G(V)\) that are studied here. In the 4. Section, the authors give the centralizers of the representations of \(\mathrm{WD}(k)\) on the vector space \(M\), introduced in the previous section, with special emphasis on the situation where the representation of \(\mathrm{WD}(k)\) is self-dual or conjugate dual (meaning there is a bilinear form on \(M\), preserved by the representation or the conjugate of it) and the objects of study are the centralizers inside the autmorphism group of \(M\) which preserve the bilinear form on \(M.\) After explicitly describing such centralizers, the description of their component group follows easily. The idea is to define signed invariants for the representations of \(\mathrm{WD}(k)\), which will, in turn, give the characters of the component group (of the centralizer) (e.g, the dimension (mod 2) of the \(-1\) eigenspace in \(M\) for a semisimple element of the component group). Some more sophisticated signed invariants are discussed in the 5. Section, which recalls the theory of local root numbers (for the representations of the local Weil-Deligne group \(\mathrm{WD}(k)\)), following Deligne. More information is given for orthogonal and conjugate-orthogonal root numbers (which are, in general, better understood). In the next, the 6. Section, the theory of the previous section (the root numbers) is applied to construct characters of the component groups. They study the behaviour of the root numbers of the tensor product of two \(\mathrm{WD}(k)\) representations, say \(M\) and \(N\), and how it defines a character of the component group \(A_M\times A_N\). Here, \(M\) and \(N\) are both selfdual or conjugate dual. In the more complicated case of selfdual representations, in order to get a quadratic character and indeed a character of the component group, one considers only the case when \(M\) and \(N\) are both of even dimension, and then the root character defines a character on the subgroup (of index at most two) of the component group. Now, in the 7. Section, the authors relate these representations of \(\mathrm{WD}(k)\) to Langlands parameters for classical groups. They place special emphasis on the treatment of the unitary group, i.e. they explain how in each case the dual and the \(L\)-group can be equipped with a standard representation. Also, they describe \(L\)-groups for the classical groups with the aid of invariant theory, which is straightforward in the symplectic and orthogonal case, but a bit more involved (including Asai representations) for unitary groups. The next section enables one to relate Langlands parameters (defined as homomorphisms from \(\mathrm{WD}(k_0)\) to the \(L\)-group, satisfying some additional requirements) defined there with the selfdual or conjugate dual representations of \(\mathrm{WD}(k)\). This is given in Theorem 8.1 (as before, there are some subtleties with the unitary group since then \(\mathrm{WD}(k_0)\neq \mathrm{WD}(k)\)). The authors then, in the 9. Section, recall the Langlands conjecture in a form proposed by Vogan (for the general case of quasi-split connected groups over a local field \(k_0\)), where the representations of all pure inner forms of the group appear simultaneously and then they make the desiderata more explicit in the 10. Section, dealing with the classical groups. In the 11. Section, assuming the Langlands-Vogan parametrization for the odd orthogonal groups, the authors through this parametrization and the Howe correspondence obtain a Langlands-Vogan parametrization for the metaplectic group (of course, except in the complex case). They recall the results of Adams-Barbasch, Kudla-Rallis, and Gan-Savin in which the parametrization of the irreducible genuine representations of the metaplectic group (of semisimple rank \(n\)) is given by the irreducible representations of \(\mathrm{SO}(V),\) where the dimension of \(V\) is \(2n+1\) and \(\mathrm{SO}(V)\) ranges over the groups for which the discriminant of \(V\) equals 1. They elaborate the non-Archimedean case. This parametrization depends on the non-trivial additive character of \(k\) (which enters the theta correspondence). Also, the assumption is not only that the characteristic of the field \(k\) is different from 2 (like in the rest of the paper), but for Howe duality to hold, one needs that the residual characteristic of \(k\) is different from 2. The issue of the dependence of the Vogan parameters on the additive character and of the parameter on the conjugation of a representation of the metaplectic group by an element of the similitude group is also discussed (and some conjectures are stated). A basic conjecture is that the dimension of the space \(\mathrm{dim}_{\mathbf{C}}(\pi)\otimes \overline{\nu},\mathbf{C})\) is at most one. The basic cases (\(\mathrm{dim} W^{\perp}=0\) or \(1\)) are solved in many cases (Archimedean instances seem to pose bigger problems). In Section 15 it is shown how the multiplicity one theorem for general Bessel models follows from the basic case (\(\mathrm{dim} W^{\perp}=0\) or \(1\)) for \(k\) non-Archimedean (which is known to be true). Analogously, in Section 16 it is shown how the multiplicity one theorem for general Fourier-Jacobi models follows from the basic cases (which are known to hold) for non-Archimedean \(k\). In Section 17 local conjectures are given, i.e. the criterion for the Hom space (discussed above) to be non-zero. The criterion (conjectural) says that in each generic Vogan \(L\)-packet there is a unique representation \(\pi=\pi(\phi,\chi)\) (possibly of a pure inner form of \(G\)) such that Hom space is non-zero. Here authors fully describe the character \(\chi\) of the component group in each case (for every classical group). In Section 18 the authors check the internal consistency of the conjecture, because the parametrization in different cases depends on some choices (of additive characters, scaling of the bilinear forms defining the classical groups, additive characters defining the Howe correspondence etc.). In Section 19 it is shown how the conjecture in Section 17 (the explicit description of the representation having non-zero models inside the Vogan \(L\)-packet) reduces to the basic cases (\(\mathrm{dim} W^{\perp}=0\) or \(1\)) in the non-Archimedean case. In Section 20 another variant of the conjecture for Section 17 is given. This variant is not so precise about the parametrization of the elements of a Vogan \(L\)-packet by the characters of the component group. To approach global conjectures, it is essential to have a good understanding of the unramified situation. So, in Section 21, the authors explicitly describe unramified representations of \(\mathrm{WD}(k)\), and determine when these representations are selfdual or conjugate-dual. In this case distinguished characters needed for the conjecture can be given very explicitly, and they refine the restriction conjectures in the unramified case. In Section 22, the authors start treating the global case, and in this section they recall the construction of adelic algebraic groups and adelic metaplectic groups. In Section 23 they formulate the global restriction problem. In the orthogonal and Hermitian case, a (global) Bessel coefficient of the tempered cuspidal representation of the global group \(G=G(V)\times G(W)\) is defined; analogously, for the symplectic and skew-Hermitian cases a Fourier-Jacobi coefficient is defined. So, the global problem is to see when these coefficients are non-zero when restricted to a tempered cuspidal representation \(\pi\) of \(G(A)\). In Section 24 the global conjectures related to this global restriction problem are formulated. The first form of the conjecture includes a global \(L\)-function related to the distinguished symplectic representation \(R\) of the \(L\)-group. Roughly, the conjecture states that the global coefficient (Bessel or Fourier-Jacobi) does not vanish for a tempered automorphic cuspidal representation \(\pi\) which occurs with multiplicity one in the space of cuspidal automorphic forms if \(L(\pi,R,s)\) does not vanish at \(s=1/2\) and if all the appropriate local Hom-spaces are non-zero. Next, the authors present their conjectures in the light of the Langlands and Arthur conjectures about the multiplicities of automorphic representations in the discrete spectrum. So, in Section 25 the global Arthur discrete parameters are defined, and the attached data is analyzed (a global component group, local \(L\)-parameter, a global Vogan packet and so on) and Arthur’s multiplicity conjectures are given, specifically for the classical groups (many of these claims have been proven by now). Specifically, the authors discuss the coherence of the set of all local data to define a global representation of a pure inner form of a group defined over a global field. Also, they discuss the same problem for the global metaplectic group, giving conjectures about a global multiplicity formula for a discrete global Arthur parameter for the metaplectic group. Next, in Section 26, the authors start from a global discrete parameter for the group \(G_0=G(V_0)\times G(W_0)\); there is a corresponding submodule in the automorphic discrete spectrum and the question is: is the restriction of the appropriate functional (Bessel or Fourier-Jacobi) non-zero on this subspace? Now, passing from global to local parameters (and taking everywhere the distinguished local representation discussed above), there are several questions: first, is the local data coherent to correspond to a representation of a group over the adeles? If this is so, is the corresponding representation cuspidal? And if it is, is our linear form non-zero on this representation?

The answers are then expressed in terms of a character corresponding to this global representation and epsilon factors for the first question; for the second, the global epsilon factor has to be equal to 1 at \(s=1/2\) and for the third, again we have the non-vanishing of the global \(L\)-function mentioned above at \(s=1/2\).

The last section deals with the same situation as in the previous section, but with the difference that the local data are not coherent. But, with some additional assumptions, the authors can make modifications on certain local places to actually get some coherent global groups. Then, they state a conjecture for the finite part of the global representation to appear in the Chow group cycles (instead of the spaces of automorphic forms) and have a non-zero linear form (analogous to the defining models above) on them: the condition is that the first derivative of the associated \(L\)-function is non-zero at \(s=1/2\).

For the entire collection see [Zbl 1257.22001].

Now we describe the content of the paper more thoroughly. In the introduction, the authors review local and global settings and roughly state the problems. In the 2. Section, they describe the classical groups \(G(V)\) they are studying, and describe the pure inner forms of these groups (orthogonal, symplectic, unitary) in terms of the first cohomology group of these groups. Here, \(V\) is a vector space over \(k\), and \(G(V)\) stabilizes a sesquilinear form on \(V.\) Let \(W\) be a non-degenerate subspace of \(V\) (with some additional conditions). Then, the subgroups \(H\) of the groups \(G=G(V)\times G(W)\) are introduced; these are the subgroups to be studied in the paper. In the 3. Section the self-dual and conjugate dual representations of the Weil-Deligne group \(\mathrm{WD}(k)\) are studied. These representations give Langlands parameters for the irreducible representations of \(G(V)\) that are studied here. In the 4. Section, the authors give the centralizers of the representations of \(\mathrm{WD}(k)\) on the vector space \(M\), introduced in the previous section, with special emphasis on the situation where the representation of \(\mathrm{WD}(k)\) is self-dual or conjugate dual (meaning there is a bilinear form on \(M\), preserved by the representation or the conjugate of it) and the objects of study are the centralizers inside the autmorphism group of \(M\) which preserve the bilinear form on \(M.\) After explicitly describing such centralizers, the description of their component group follows easily. The idea is to define signed invariants for the representations of \(\mathrm{WD}(k)\), which will, in turn, give the characters of the component group (of the centralizer) (e.g, the dimension (mod 2) of the \(-1\) eigenspace in \(M\) for a semisimple element of the component group). Some more sophisticated signed invariants are discussed in the 5. Section, which recalls the theory of local root numbers (for the representations of the local Weil-Deligne group \(\mathrm{WD}(k)\)), following Deligne. More information is given for orthogonal and conjugate-orthogonal root numbers (which are, in general, better understood). In the next, the 6. Section, the theory of the previous section (the root numbers) is applied to construct characters of the component groups. They study the behaviour of the root numbers of the tensor product of two \(\mathrm{WD}(k)\) representations, say \(M\) and \(N\), and how it defines a character of the component group \(A_M\times A_N\). Here, \(M\) and \(N\) are both selfdual or conjugate dual. In the more complicated case of selfdual representations, in order to get a quadratic character and indeed a character of the component group, one considers only the case when \(M\) and \(N\) are both of even dimension, and then the root character defines a character on the subgroup (of index at most two) of the component group. Now, in the 7. Section, the authors relate these representations of \(\mathrm{WD}(k)\) to Langlands parameters for classical groups. They place special emphasis on the treatment of the unitary group, i.e. they explain how in each case the dual and the \(L\)-group can be equipped with a standard representation. Also, they describe \(L\)-groups for the classical groups with the aid of invariant theory, which is straightforward in the symplectic and orthogonal case, but a bit more involved (including Asai representations) for unitary groups. The next section enables one to relate Langlands parameters (defined as homomorphisms from \(\mathrm{WD}(k_0)\) to the \(L\)-group, satisfying some additional requirements) defined there with the selfdual or conjugate dual representations of \(\mathrm{WD}(k)\). This is given in Theorem 8.1 (as before, there are some subtleties with the unitary group since then \(\mathrm{WD}(k_0)\neq \mathrm{WD}(k)\)). The authors then, in the 9. Section, recall the Langlands conjecture in a form proposed by Vogan (for the general case of quasi-split connected groups over a local field \(k_0\)), where the representations of all pure inner forms of the group appear simultaneously and then they make the desiderata more explicit in the 10. Section, dealing with the classical groups. In the 11. Section, assuming the Langlands-Vogan parametrization for the odd orthogonal groups, the authors through this parametrization and the Howe correspondence obtain a Langlands-Vogan parametrization for the metaplectic group (of course, except in the complex case). They recall the results of Adams-Barbasch, Kudla-Rallis, and Gan-Savin in which the parametrization of the irreducible genuine representations of the metaplectic group (of semisimple rank \(n\)) is given by the irreducible representations of \(\mathrm{SO}(V),\) where the dimension of \(V\) is \(2n+1\) and \(\mathrm{SO}(V)\) ranges over the groups for which the discriminant of \(V\) equals 1. They elaborate the non-Archimedean case. This parametrization depends on the non-trivial additive character of \(k\) (which enters the theta correspondence). Also, the assumption is not only that the characteristic of the field \(k\) is different from 2 (like in the rest of the paper), but for Howe duality to hold, one needs that the residual characteristic of \(k\) is different from 2. The issue of the dependence of the Vogan parameters on the additive character and of the parameter on the conjugation of a representation of the metaplectic group by an element of the similitude group is also discussed (and some conjectures are stated). A basic conjecture is that the dimension of the space \(\mathrm{dim}_{\mathbf{C}}(\pi)\otimes \overline{\nu},\mathbf{C})\) is at most one. The basic cases (\(\mathrm{dim} W^{\perp}=0\) or \(1\)) are solved in many cases (Archimedean instances seem to pose bigger problems). In Section 15 it is shown how the multiplicity one theorem for general Bessel models follows from the basic case (\(\mathrm{dim} W^{\perp}=0\) or \(1\)) for \(k\) non-Archimedean (which is known to be true). Analogously, in Section 16 it is shown how the multiplicity one theorem for general Fourier-Jacobi models follows from the basic cases (which are known to hold) for non-Archimedean \(k\). In Section 17 local conjectures are given, i.e. the criterion for the Hom space (discussed above) to be non-zero. The criterion (conjectural) says that in each generic Vogan \(L\)-packet there is a unique representation \(\pi=\pi(\phi,\chi)\) (possibly of a pure inner form of \(G\)) such that Hom space is non-zero. Here authors fully describe the character \(\chi\) of the component group in each case (for every classical group). In Section 18 the authors check the internal consistency of the conjecture, because the parametrization in different cases depends on some choices (of additive characters, scaling of the bilinear forms defining the classical groups, additive characters defining the Howe correspondence etc.). In Section 19 it is shown how the conjecture in Section 17 (the explicit description of the representation having non-zero models inside the Vogan \(L\)-packet) reduces to the basic cases (\(\mathrm{dim} W^{\perp}=0\) or \(1\)) in the non-Archimedean case. In Section 20 another variant of the conjecture for Section 17 is given. This variant is not so precise about the parametrization of the elements of a Vogan \(L\)-packet by the characters of the component group. To approach global conjectures, it is essential to have a good understanding of the unramified situation. So, in Section 21, the authors explicitly describe unramified representations of \(\mathrm{WD}(k)\), and determine when these representations are selfdual or conjugate-dual. In this case distinguished characters needed for the conjecture can be given very explicitly, and they refine the restriction conjectures in the unramified case. In Section 22, the authors start treating the global case, and in this section they recall the construction of adelic algebraic groups and adelic metaplectic groups. In Section 23 they formulate the global restriction problem. In the orthogonal and Hermitian case, a (global) Bessel coefficient of the tempered cuspidal representation of the global group \(G=G(V)\times G(W)\) is defined; analogously, for the symplectic and skew-Hermitian cases a Fourier-Jacobi coefficient is defined. So, the global problem is to see when these coefficients are non-zero when restricted to a tempered cuspidal representation \(\pi\) of \(G(A)\). In Section 24 the global conjectures related to this global restriction problem are formulated. The first form of the conjecture includes a global \(L\)-function related to the distinguished symplectic representation \(R\) of the \(L\)-group. Roughly, the conjecture states that the global coefficient (Bessel or Fourier-Jacobi) does not vanish for a tempered automorphic cuspidal representation \(\pi\) which occurs with multiplicity one in the space of cuspidal automorphic forms if \(L(\pi,R,s)\) does not vanish at \(s=1/2\) and if all the appropriate local Hom-spaces are non-zero. Next, the authors present their conjectures in the light of the Langlands and Arthur conjectures about the multiplicities of automorphic representations in the discrete spectrum. So, in Section 25 the global Arthur discrete parameters are defined, and the attached data is analyzed (a global component group, local \(L\)-parameter, a global Vogan packet and so on) and Arthur’s multiplicity conjectures are given, specifically for the classical groups (many of these claims have been proven by now). Specifically, the authors discuss the coherence of the set of all local data to define a global representation of a pure inner form of a group defined over a global field. Also, they discuss the same problem for the global metaplectic group, giving conjectures about a global multiplicity formula for a discrete global Arthur parameter for the metaplectic group. Next, in Section 26, the authors start from a global discrete parameter for the group \(G_0=G(V_0)\times G(W_0)\); there is a corresponding submodule in the automorphic discrete spectrum and the question is: is the restriction of the appropriate functional (Bessel or Fourier-Jacobi) non-zero on this subspace? Now, passing from global to local parameters (and taking everywhere the distinguished local representation discussed above), there are several questions: first, is the local data coherent to correspond to a representation of a group over the adeles? If this is so, is the corresponding representation cuspidal? And if it is, is our linear form non-zero on this representation?

The answers are then expressed in terms of a character corresponding to this global representation and epsilon factors for the first question; for the second, the global epsilon factor has to be equal to 1 at \(s=1/2\) and for the third, again we have the non-vanishing of the global \(L\)-function mentioned above at \(s=1/2\).

The last section deals with the same situation as in the previous section, but with the difference that the local data are not coherent. But, with some additional assumptions, the authors can make modifications on certain local places to actually get some coherent global groups. Then, they state a conjecture for the finite part of the global representation to appear in the Chow group cycles (instead of the spaces of automorphic forms) and have a non-zero linear form (analogous to the defining models above) on them: the condition is that the first derivative of the associated \(L\)-function is non-zero at \(s=1/2\).

For the entire collection see [Zbl 1257.22001].

Reviewer: Marcela Hanzer (Zagreb)

### MSC:

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

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

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

11R39 | Langlands-Weil conjectures, nonabelian class field theory |