Descent Construction for Gspin Groups: Main Results and Applications

The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram on cuspidality of the exterior square lift for representations of GL4. Complete proofs of the results announced in the present note will appear in our forthcoming articles.


Preliminaries
1.1. GSpin groups and their quasisplit forms. Let F be a number field. By the classification results in Chapter 16 of [Spr98](see especially 16.3.2, 16.3.3, 16.4.2), and the definition of the L group, there is a unique quasisplit F group G such that the connected component of the identity in L G is GSp 2n (C). This is GSpin 2n+1 .
Similarly, there is a 1-1 correspondence between quasisplit F groups G such that L G 0 = GSO 2n (C) and homomorphisms from Gal(F /F ) to the group with two elements, and hence, by class field theory, with quadratic characters of χ : A × F /F × → {±1} (the case n = 4 is no different, see Section 5). The unique split group G such that L G 0 = GSO 2n (C) corresponds to the trivial character. We denote this group GSpin 2n . The finite Galois form of its L group is GSO 2n (C). The form corresponding to the nontrivial character χ we denote by GSpin χ 2n . The finite Galois form of its L group is GSO 2n (C) ⋊ Gal(E/F ) where E is the quadratic extension of F corresponding to χ.
Similarly, one expects a lifting of automorphic representations of GSpin 2n (A) to automorphic representations of GL 2n (A) corresponding to the inclusion For (globally) generic representations, the existence of these liftings is proved in [AS06]. Now consider GSpin χ 2n for χ = 1. Regardless of χ, the L group GSO 2n (C) ⋊ Gal(E/F ), with E as above, is isomorphic to GO 2n (C), and a specific isomorphism can be fixed by mapping the nontrivial element of Gal(E/F ) to One then expects a lifting of automorphic representations of GSpin χ 2n (A) to automorphic representations of GL 2n (A) corresponding to the inclusion GO 2n (C) ֒→ GL 2n (C).
We understand that this is the subject of [AS08]. We denote all of these liftings by AS.

Main results
2.1. The odd case.
Theorem A. Let ω be a Hecke character. Suppose n 1 , . . . , n m ∈ N, and that, for each 1 ≤ i ≤ m, τ i is an irreducible cuspidal automorphic representation of GL 2n i (A) such that L S (s, τ i , ∧ 2 × ω −1 ) has a pole at s = 1. Suppose furthermore that the representations τ i are all distinct. Let n = n 1 + · · · + n m . Then there exists a globally generic irreducible cuspidal automorphic representation Pn(A) (τ 1 ⊗ · · · ⊗ τ m ) (normalized induction), where P n is the standard parabolic of GL 2n corresponding to the ordered partition 2n = 2n 1 + · · · + 2n m of 2n. Furthermore, the central character of σ is ω.
has a pole at s = 1, one may not deduce that ℓ is even. However, one may deduce that τ ∼ =τ ⊗ ω, whence ω ℓ = ω 2 τ (where ω τ is the central character of τ ). If ℓ is odd, it then follows that ω is the square of another global character η, and that τ ′ = τ ⊗ η −1 is self dual, with L S (s, τ ′ , sym 2 ) having a pole at s = 1. Thus, the case when ω is a square reduces to the self-dual case, and in the case when ω is not a square we can deduce that ℓ is even and that ω τ /ω ℓ 2 is quadratic.
2.2. The even case. For the statement of the next main result, it will be convenient to define GSpin χ 2n := GSpin 2n when χ is the trivial character. Theorem B. Let ω be a Hecke character which is not the square of another Hecke character. Suppose n 1 , . . . , n m ∈ N, and that, for each i, τ i is an irreducible cuspidal automorphic representation of GL 2n i (A) such that L S (s, τ i , sym 2 ×ω −1 ) has a pole at s = 1. Suppose furthermore that the representations τ i are all distinct. Let n = n 1 +· · ·+n m , and, for each i, let Then there exists a globally generic irreducible cuspidal automorphic representa- where P n is the standard parabolic of GL 2n corresponding to the ordered partition 2n = 2n 1 + · · · + 2n m of 2n. Furthermore, the central character of σ is ω.

Applications
3.1. The image of the weak lift AS. We now concentrate on the case of split general Spin groups. In [AS06] , the authors show the existence of functorial lifts from automorphic representations of GSpin 2n (A) or GSpin 2n+1 (A) to GL 2n (A). They show that the images consist of automorphic representations which satisfy the essential self-duality condition at almost all places.
Based on the self-dual case, (cf. Theorem A of [GRS01]) one expects that the image of each Asgari-Shahidi lifting consists of isobaric sums of distinct essentially self dual cuspidal representations satisfying the appropriate L-function condition. For example, any representation in the image of the lift from GSpin 2n+1 should be an isobaric sum of distinct ω-symplectic cuspidals, for some Hecke character ω.
Our results support this expectation. We provide a "lower bound" for the image of the Asgari-Shahidi lifting, by showing that any isobaric sum of distinct essentially self dual cuspidal representations satisfying the appropriate L-function condition is in the image of the appropriate lift.
3.2. The image of the exterior square lift: GL 4 to GL 6 . The existence of an exterior square lift of a cuspidal automorphic representation of GL 4 (A) as an automorphic representation of GL 6 (A) was established by Kim in [Kim03]. Recently, Asgari-Raghuram provided an explicit description of those cuspidal automorphic representations of GL 4 (A) whose exterior square lift to GL 6 (A) is not cuspidal. Among other things their argument requires the following special case of Theorem B.
Corollary 3.2.1. Let Π be a cuspidal representation of GL 4 (A) and let ω be any character of GL 1 (F )\GL 1 (A). Assume that the partial L-function L S (s, Π, sym 2 ⊗ ω −1 ) has a pole at s = 1 for a sufficiently large finite set S of places of F . Let χ = ω Π ω −2 . Then there exists a globally generic cuspidal representation π of GSpin χ 4 (A) such that π transfers to Π. Roughly speaking, Asgari and Raghuram prove that the exterior square lift of a cuspidal representation Π of GL 4 is cuspidal unless Π is isomorphic to a twist of either itself or its contragredient, and that this occurs only if Π is itself in the image of one of four functorial lifts. For the precise statement, see [A-R], Theorem 1.1, p.2. For the precise relationship with our results, see p. 12.

Scheme of Proof
The proofs of Theorem A and Theorem B are obtained by adapting (the special orthogonal group case of) the descent method of Ginzburg, Rallis, and Soudry [GRS99b, GRS99a, GRS99c, GRS01, GRS02]. The adaptation is reasonably straightforward owing to two observations: (1) There is a surjective homomorphism, defined over F, from GSpin m to SO m , which restricts to an isomorphism between the unipotent subvarieties.
(2) The kernel of this projection is contained in the center of GSpin m . Thus, the action GSpin m on itself by conjugation factors through the projection. In what follows we detail the steps needed to prove Theorem B. The proof of Theorem A is similar and technically simpler.
We can conveniently describe the method in the following steps: (1) Construction of a family of descent representations of GSpin χ 4n+1−2ℓ (A) for ℓ ≥ n.
(4) Matching of spectral parameters at unramified places. The construction of the descent representations relies on the notion of Fourier coefficient, as defined in [GRS03], [G] (cf. also the "Gelfand-Graev" coefficients of [So]). For purposes of presenting certain of the global arguments, it seems convenient to embed these Fourier coefficients into a slightly larger family of functionals, which we shall refer to as "unipotent periods." Suppose that U is a unipotent subgroup of GSpin 4n+1 and ψ is a character of U (F )\U (A). We define the corresponding unipotent period to be the map from smooth, left U (F )-invariant functions on GSpin 4n+1 (A) to smooth, left (U (A), ψ)-equivariant functions, given by Each unipotent period has a local analogue at each finite place, which is a twisted Jacquet functor.
Suppose now that U is the unipotent radical of a standard parabolic subgroup, and let M denote the Levi. The characters of U (F )\U (A) may be identified with the points of an F -vector space, so that the stabilizer Stab M (ψ) makes sense as an algebraic group defined over F. We assume that ψ corresponds to a point in general position. Then the map is indeed a "Fourier coefficient," as defined (and associated to a nilpotent orbit) in [GRS03,G].It maps smooth functions of moderate growth on GSpin 4n+1 (F )\GSpin 4n+1 (A) to smooth functions of moderate growth on Stab M (ψ)(F )\Stab M (ψ)(A).
Let S be a set of unipotent periods. We will say that another unipotent period (U, ψ) is spanned by S if We are now ready to describe each of the four steps listed above in more detail.
Step one: Construction of the descent representations Using τ 1 , . . . , τ m , a space of Eisenstein series E τ ,ω (g, s) on GSpin 4n+1 (A) is constructed-corresponding to a representation induced from the standard parabolic subgroup P = M U of GSpin 4n+1 for which M ∼ = GL 2n 1 × · · · × GL 2nm × GL 1 . The partial L functions appear in the constant terms of elements of this space. As a consequence, some of them have non-vanishing multi-residues at a certain point s 0 , precisely because of the L-function hypothesis on τ . In this fashion we obtain a residual representation-which lies in the discrete spectrum of L 2 (GSpin 4n+1 (F )\GSpin 4n+1 (A))-the nontriviality of which depends intrinsically on this Lfunction condition. We denote this representation by E −1 (τ , ω). Now, GSpin 4n+1 contains a family of parabolic subgroups Q ℓ = L ℓ N ℓ , ℓ = 1 to 2n, with L ℓ isomorphic to GL ℓ 1 × G 4n−2ℓ+1 , having the crucial property that for each character ψ of N ℓ in general position, the identity component of the group Stab L ℓ (ψ) is isomorphic to one of the groups GSpin χ 4n−2ℓ . Fixing specific isomorphisms, we may pull back each Fourier coefficient F C ψ (E −1 (τ , ω)) as described above, to a space of functions defined on GSpin χ 4n−2ℓ (A). There are many characters ψ for a given value of ℓ and χ, but they comprise a single orbit for the action of L ℓ (F ) by conjugation, and the various spaces F C ψ (E −1 (τ , ω)) all pull back to the same space of functions on GSpin χ 4n−2ℓ (A), regardless of the choice of ψ in this orbit and regardless of the choice of isomorphism GSpin χ 4n−2ℓ → Stab L ℓ (ψ) 0 . (For this, we require the extension of meromorphic continuation of Eisenstein series to non-K-finite sections, provided in [La08].) In this manner we obtain a space of functions on GSpin χ 4n−2ℓ (A) for each value of χ. The family of representations thus obtained comprises the descent representations.
Step two: Vanishing of other descents For ℓ > n, one shows that the above Fourier coefficients vanish identically on our residue representation E −1 (τ , ω). The reason is local: the corresponding twisted Jacquet module of the unramified constituent of the corresponding local induced representation vanishes. The same is true if ℓ = n, at any unramified place v such that the identity component of Stab Ln (ψ) is not isomorphic to GSpin χ 2n over F v . The remaining descent representation, corresponding to ℓ = n and χ = χ τ , may now be referred to as "the" descent without ambiguity.
Step Three: Cuspidality and genericity of the descent Next we appeal to global arguments which may be presented in terms of "identities of unipotent periods." Consider the unipotent period on C ∞ (GSpin 4n+1 (F )\GSpin 4n+1 (A)) which consists of taking the constant term with respect to the maximal parabolic with Levi isomorphic to GL 2n × GL 1 , and then taking a Whittaker integral on the GL 2n Levi. It can be shown that this unipotent period does not vanish on our residue representation E −1 (τ , ω). One shows that this unipotent period is, in fact, spanned by the periods corresponding to Whittaker integrals on the descent representations (as ℓ ≥ n and χ vary).
Having proved by local arguments that these periods vanishes identically on the residue representation E −1 (τ , ω), whenever ℓ > n or χ = χ τ , we deduce that they do not vanish identically when ℓ = n and χ = χ τ . This shows that the space F C ψ (E −1 (τ , ω)) is not only nontrivial, but supports a nontrivial global Whittaker integral.
Next, consider the unipotent periods on C ∞ (GSpin 4n+1 (F )\GSpin 4n+1 (A)) which consist of taking the constant term with respect to the maximal parabolic with Levi isomorphic to GL k × GSpin 4n−2k+1 for some k, and then, if k is even, performing the integral one would use to define a descent representation of GSpin 4n−2k+1 , with some value of ℓ larger than n − k 2 . Combining the vanishing results of Step two with well-known facts from the theory of Eisenstein series, we deduce that all of these periods vanish identically on the residue representation E −1 (τ , ω). We then show that the unipotent period which corresponds to taking the constant term of one of the functions in F C ψ (E −1 (τ , ω)) is in their span.
It follows that all the functions in the descent are cuspidal. At this point, we may deduce that the descent representation decomposes discretely as a direct sum of irreducible cuspidal automorphic representations, at least one of which is generic. We select one such component for the representation σ of Theorem B. What remains is to show that σ lifts weakly to Ind GL 2n (A) Pn(A) (τ 1 ⊗ · · · ⊗ τ m ).
Step Four: Matching of spectral parameters at unramified places For ℓ = n, at an unramified place, where the identity component of Stab Ln (ψ) is isomorphic to GSpin χτ 2n , the twisted Jacquet module of the unramified constituent of the local induced representation is isomorphic, as a Stab Ln (ψ)(F v )-module to a certain induced representation of Stab Ln (ψ)(F v ). When restricted to the identity component, this representation may not be irreducible. Nevertheless, we are able to deduce that any nonzero irreducible component of the Fourier coefficient must lift weakly to Ind GL 2n (A) Pn(A) (τ 1 ⊗ · · · ⊗ τ m ).

Final Remarks
(1) When considering the identification of GO 2n (C) with GSO 2n (C) ⋊ Gal(E/F ), one could also map the nontrivial element to This produces a slightly different functorial lift corresponding to the twist of the one we have chosen above by the quadratic character χ. Theorem B is, of course, true for this "alternate" lifting, as well, since one may "untwist." (2) These are essentially the only distinct extensions of the inclusion GSO 2n (C) ֒→ GL 2n (C) to GSO 2n (C) ⋊ Gal(E/F ) in the following sense. Suppose V 1 and V 2 are two 2n dimensional representations of GSO 2n (C) ⋊ Gal(E/F ) such that the restriction of either to GSO 2n (C) is the standard representation. Then one may show that V 2 is isomorphic to either V 1 or the twist of V 1 by the unique nontrivial character of Gal(E/F ). (3) A natural question arises in the case n = 4: does the 3-fold symmetry of the Dynkin diagram of GSpin 8 lead to additional quasi-split forms? The answer is no, because the 3-fold symmetry of the D 4 root system does not extend to a symmetry of the root data of GSO 8 and GSpin 8 .