Galois Representations Attached to

Hilbert-Siegel Modular Forms

Claus M. Sorensen

Received: January 22, 2009 Revised: June 3, 2010

Communicated by Don Blasius

Abstract. This article is a spinoff of the book of Harris and Taylor [HT], in which they prove the local Langlands conjecture for GL(n), and its companion paper by Taylor and Yoshida [TY] on local-global compatibility. We record some consequences in the case of genus two Hilbert-Siegel modular forms. In other words, we are concerned with cusp forms π on GSp(4) over a totally real field, such that π_{∞} is regular algebraic (that is, π is cohomological). When π is globally generic (that is, has a non-vanishing Fourier coefficient), and π has a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for GSp(4) defined by Gan and Takeda in a recent preprint [GT]. Over ℚ, for π as above, this leads to a new realization of the Galois representations studied previously by Laumon, Taylor and Weissauer. We are hopeful that our approach should apply more generally, once the functorial lift to GL(4) is understood, and once the so-called book project is completed. An application of the above compatibility is the following special case of a conjecture stated in [SU]: If π has nonzero vectors fixed by a non-special maximal compact subgroup at v, the corresponding monodromy operator at v has rank at most one.

2010 Mathematics Subject Classification: 11F33, 11F41, 11F70, 11F80 Keywords and Phrases: Galois representations, Hilbert-Siegel modular forms, monodromy 1 Introduction

The interplay between modular forms and Galois representations has shown to be extremely fruitful in number theory. For example, the Hasse-Weil conjecture, saying that the L-function of an elliptic curve over ℚ has meromorphic continuation to ℂ, was proved in virtue of this reciprocity. Some of the most basic examples are Hilbert modular forms and Siegel modular forms, both very well-studied in the literature. In this paper, we study a mixture of these. First, we introduce the Siegel upper half-space (of complex dimension three):

H =^{df} {Z = X + iY ∈ M_{2}(ℂ) symmetric, with Y positive definite}.

For a moment, we will view the symplectic similitude group GSp(4) as an affine group scheme over ℤ, by choosing the non-degenerate alternating form to be

(\begin{matrix}0 & I \\ −I & 0 \end{matrix}) .

Later, beyond this introduction, we will switch to a skew-diagonal form. The similitude character is denoted by c throughout. We then consider the subgroup GSp(4, ℝ)^{+} of elements with positive similitude. It acts on H in the standard way, by linear fractional transformations. More precisely, by the formula:

gZ = (AZ + B)(CZ + D)^{−1}, g = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ GSp(4, ℝ)^{+}, Z ∈ H.

The automorphy factor CZ + D will be denoted by j(g, Z) from now on. Next, we look at the d-fold product of this setup. In more detail, we will concentrate on a discrete subgroup Γ inside GSp(4, ℝ)^{+d}, and its diagonal action on H^{d}. To exhibit examples of such Γ, we bring into play a totally real number field F, of degree d over ℚ, and label the real embeddings by σ_{i}. This ordering is used to identify GSp(4, O)^{+} with a discrete subgroup Γ. Here the plus signifies that we look at elements whose similitude is a totally positive unit in O, the ring of integers in F. Finally, in order to define modular forms for Γ, we fix weights

\underline{k}_{i} = (k_{i,1}, k_{i,2}), k_{i,1} ≥ k_{i,2} ≥ 3, i = 1, …, d.

For each i, introduce the irreducible algebraic representation ρ_{\underline{k}_{i}} of GL(2, ℂ),

Sym^{k_{i,1}−k_{i,2}}(ℂ^{2}) ⊗ det^{k_{i,2}}.

The underlying space of ρ_{\underline{k}_{i}} is just a space of polynomials in two variables, homogeneous of a given degree. Following [Bai], we then define a Hilbert-Siegel modular form for Γ, with weights \underline{k}_{i}, to be a holomorphic vector-valued function

f : H^{d} → ⨂_{i=1}^{d} Sym^{k_{i,1}−k_{i,2}}(ℂ^{2}) ⊗ det^{k_{i,2}}

satisfying the following transformation property for every \underline{Z} ∈ H^{d} and γ ∈ Γ:

f(γ\underline{Z}) = ⨂_{i=1}^{d} ρ_{\underline{k}_{i}}(c(γ_{i})^{−1}j(γ_{i}, Z_{i})) · f(\underline{Z}).

Such f are automatically holomorphic at infinity, by the Koecher principle. The form is often assumed to be cuspidal, that is, it vanishes at infinity. These modular forms have rich arithmetic properties. To exploit them, it is useful to switch to an adelic setup and instead look at automorphic representations. For instance, this immediately gives rise to a workable Hecke theory. Thus, from now on in this paper, instead of f we will focus on a cuspidal automorphic representation π of GSp(4) over the totally real field F. The analogues of Hilbert-Siegel modular forms are those π which are holomorphic discrete series at infinity. It is a basic fact that such π do not admit Whittaker models. However, it is generally believed that the L-function of π coincides with the L-function of a π′ which does admit a Whittaker model. Hence, for our purposes, there is no serious harm in assuming the existence of such a model. For a while though, let us not make this assumption, and explain in detail the expectations regarding Hilbert-Siegel modular forms and their associated Galois representations.

Let π be a cuspidal automorphic representation of GSp(4) over some totally real field F, and denote by S_{π} the set of finite places where π is ramified. For each infinite place v, we assume that π_{v} is an essentially discrete series representation, and that π_{v} has central character a ↦ a^{−w}. Here w is an integer, independent of v. Under these assumptions, and a choice of an isomorphism ι : ℚ̄_{ℓ} → ℂ, it is expected that there should be a semisimple continuous Galois representation

ρ_{π,ι} : Gal(F̄/F) → GL_{4}(ℚ̄_{ℓ})

with the following properties: ρ_{π,ι} is unramified at v ∉ S_{π} not dividing ℓ, and

L_{v}(s − \frac{3}{2}, π, spin) = det(1 − ιρ_{π,λ}(Frob_{v}) · q_{v}^{−s})^{−1}.

Here Frob_{v} is the geometric Frobenius. More prudently, the above spin Lfactor should actually lie in L[q_{v}^{−s}] for some number field L inside ℂ, and instead of ι one could focus on the finite place of L it defines. In the rational case F = ℚ, the existence of ρ_{π,ι} is now known, due to the work of many people (Chai-Faltings, Laumon, Shimura, Taylor and Weissauer). See [Lau] and [Wei] for the complete result. For arbitrary F, not much is known. Of course, when π is CAP (cuspidal associated to parabolic), or a certain functorial lift (endoscopy, base change or automorphic induction), ρ_{π,ι} is known to exist by [BRo]. However, in most of these cases ρ_{π,ι} is reducible. In the opposite case, that is, when π genuinely belongs to GSp(4), the representation ρ_{π,ι} should be irreducible. Obviously, this is the case we are interested in. Actually, we will aim higher and consider the ramified places S_{π} too. The impetus for doing so, is the recent work of Gan and Takeda [GT], in which they prove the local Langlands conjecture for GSp(4). To an irreducible admissible representation π_{v}, they associate an L-parameter

rec_{GT}(π_{v}) : W′_{F_{v}} = W_{F_{v}} × SL(2, ℂ) → GSp(4, ℂ).

The notation rec_{GT} is ours. This correspondence is natural in a number of respects. For example, it preserves the L- and ϵ-factors defined by Shahidi in the generic case [Sha]. See section 2.2.2 below for a discussion of the complete list of desiderata. Now, the representation ρ_{π,ι} should satisfy local-global compatibility. That is, for any finite place v (not dividing ℓ), the restriction ρ_{π,ι}|_{W_{Fv}} should correspond to rec_{GT}(π_{v}) through the usual dictionary [Tat]. As it stands, this is only morally true; one has to twist π_{v}. The precise folklore prediction is:

Conjecture. Let π be a cuspidal automorphic representation of GSp(4) over some totally real field F. Assume there is a cuspidal automorphic representation of GL(4) over F, which is a weak lift of π. Moreover, we assume that

π^{∘} =^{df} π ⊗ |c|^{\frac{w}{2}} is unitary, for some w ∈ ℤ.

Finally, at each infinite place v, we assume that π_{v} is an essentially discrete series representation with the same central and infinitesimal character as the finite-dimensional irreducible algebraic representation V_{μ(v)} of highest weight

t = (\begin{matrix}t_{1} \\ & t_{2} \\ & & t_{3} \\ & & & t_{4} \end{matrix}) ↦ t_{1}^{μ_{1}(v)}t_{2}^{μ_{2}(v)}c(t)^{δ(v)−w}, δ(v) =^{df} \frac{1}{2}(w − μ_{1}(v) − μ_{2}(v)).

Here μ_{1}(v) ≥ μ_{2}(v) ≥ 0 are integers such that μ_{1}(v)+μ_{2}(v) has the same parity as w. In particular, the central character ω_{π_{v}} is of the form a ↦ a^{−w} at each infinite place v. Under these assumptions, for each choice of an isomorphism ι : ℚ̄_{ℓ} → ℂ, there is a unique irreducible continuous representation

ρ_{π,ι} : Gal(F̄/F) → GSp_{4}(ℚ̄_{ℓ})

characterized by the following property: For each finite place v ∤ ℓ of F, we have

ιWD(ρ_{π,ι}|_{W_{Fv}})^{F −ss} ≃ rec_{GT}(π_{v} ⊗ |c|^{− \frac{3}{2}} ).

Moreover, π^{∘} is tempered everywhere. Consequently, ρ_{π,ι} is pure of weight

w =^{df} w + 3. The representation ρ_{π,ι} has the following additional properties:

• ρ_{π,ι}^{∨} ≃ ρ_{π,ι} ⊗ χ^{−1} where the similitude character χ = ω_{π^{∘}} · χ_{cyc}^{−w} is totally odd.

• The representation ρ_{π,ι} is potentially semistable at any finite place v|ℓ. Moreover, ρ_{π,ι} is crystalline at a finite place v|ℓ when π_{v} is unramified.

• The Hodge-Tate weights are given by the following recipe: Fix an infinite place v, and use the same notation for the place above ℓ it defines via ι.

dimℚ̄_{ℓ} gr^{j}(ρ_{π,ι} ⊗_{F_{v}} B_{dR})^{Gal(F̄_{v}/F_{v})} = 0,

unless j belongs to the set

δ(v) + {0, μ_{2}(v) + 1, μ_{1}(v) + 2, μ_{1}(v) + μ_{2}(v) + 3},

in which case the above dimension is equal to one.

The notation used will be explained carefully in the main body of the text below. Our main result is a proof of this conjecture in a substantial number of cases. The depth (and the Swan conductor) are defined below in Section 4.5, as is J_{Q}.

Main Theorem.

(a) The above conjecture holds for globally generic π such that, for some finite place v, the local component π_{v} is an unramified twist of the Steinberg representation.

(b) Let ρ_{π,ι} be the Galois representation attached to a globally generic cusp form π as in part (a). Let v ∤ ℓ be a finite place of F such that π_{v} is Iwahori-spherical and ramified. Then ρ_{π,ι}|_{I_{Fv}} acts unipotently. Moreover,

– π_{v} of Steinberg type ⟺ monodromy has rank 3.

– π_{v} has a unique J_{Q}-fixed line ⟺ monodromy has rank 2.

– π_{v} para-spherical ⟺ monodromy has rank 1.

(c) Let ρ_{π,ι} be the Galois representation attached to a globally generic cusp form π as in part (a). Let v ∤ ℓ be a finite place of F such that π_{v} is supercuspidal, and not a lift from GO(2, 2). Then ρ_{π,ι}|_{W_{Fv}} is irreducible. Furthermore, ρ_{π,ι} is trivial on some finite index subgroup of I_{F_{v}}, and

𝔣_{Swan}(ρ_{π,ι}|_{I_{Fv}}) = 4 · depth(π_{v}).

Part of our original motivation for writing this paper, was to determine the rank of the monodromy operator for ρ_{π,ι} at a place v where π_{v} is Iwahorispherical (that is, has nonzero vectors fixed by an Iwahori-subgroup). The question is completely answered by part (b). Here J_{Q} denotes the Klingen parahoric, and by π_{v} being para-spherical we mean that it has nonzero vectors fixed by a non-special maximal compact subgroup. The first two statements in (b) are part of the Conjecture on p. 11 in [GTi], while the latter statement in (b) is part of Conjecture 3.1.7 on p. 41 in [SU]. The holomorphic analogue of the latter would have applications to the Bloch-Kato conjecture for modular forms of square-free level. See [SU] and the authors thesis [Sor].

The proof of part (b) essentially follows from the local-global compatibility from part (a), and the classification of the Iwahori-spherical representations of GSp(4). The dimensions of the parahoric fixed spaces for each class of representations were tabulated in [Sch]. We only need the generic ones. For those six types of representations, the data are reproduced in Table A in section 4.5 below.

The proof of part (a) is an application of the monumental work of Harris-Taylor [HT], and its refinement by Taylor-Yoshida [TY]. Let us briefly sketch the simple strategy: First, since π is globally generic, one can lift it to an automorphic representation Π on GL(4) using theta series, by utilizing the close connection with GO(3, 3). This is a well-known, though unpublished, result of Jacquet, Piatetski-Shapiro and Shalika. Other proofs exist in the literature. For example, see [AS] for an approach using the converse theorem. We make use of Theorem 13.1 in [GT], saying that the lift π ↦ Π is strong. That is, compatible with the local Langlands correspondence everywhere. We note that, by the Steinberg assumption, Π must be cuspidal. Next, we base change Π to a CM extension E over F, and twist it by a suitable character χ to make it conjugate self-dual. We can now apply [HT] and [TY] to the representation Π_{E}(χ), in order to get a Galois representation ρ_{Π_{E}(χ),ι} over E. Since E is arbitrary, a delicate patching argument shows how to descend this collection to F, after twisting by ρ_{χ̌,ι}.

We note that the main theorem continues to hold if some π_{v} is a generalized Steinberg representation of Klingen type (see section 2 below), or a supercuspidal not coming from GO(2, 2). The point being that the local lift Π_{v} on GL(4) should remain a discrete series. However, eventually the book project of the Paris 7 GRFA seminar should make any local assumption at v superfluous. In fact, almost complete results of S.-W. Shin have been announced very recently (to pin down the Frobenius semisimplification, at the time of writing, one has to make a regularity assumption, which should be removable by a p-adic deformation argument). See Expected Theorem 2.4 in [Har]. Furthermore, the assumption that π is globally generic is used exclusively to get a strong lift to GL(4). Our understanding is that the current state of the trace formula should at least give a weak lift more generally. See [Art] in conjunction with [Whi]. In this respect, there is a very interesting preprint of Weissauer [We2], in which he proves that if π is a discrete series at infinity, it is weakly equivalent to a globally generic representation. However, apparently he needs to work over ℚ. Perhaps ideas from [Lab] will be useful in treating F of degree at least two. In any case, to get a strong lift, one would have to show that the L-packets defined in [GT] satisfy the expected character relations. This seems to be quite difficult.

In part (c) we let 𝔣_{Swan} denote the Swan conductor, closely related to the more commonly used Artin conductor, and the depth of π_{v} is defined in [MP]. The precise definitions are recalled below in section 4.5. The proof of part (c) relies on two essential ingredients. One is a formula, due to Bushnell and Frolich [BF], relating the depth to the conductor in the case of supercuspidals on GL(n). The second is a paper of Pan, showing that the local theta correspondence preserves depth [Pan]. In particular, if π_{v} has depth zero, we deduce that ρ_{π,ι} is tamely ramified at v. We finish the paper with another criterion for tame ramification, due to Genestier and Tilouine [GTi] over ℚ: Suppose π_{v}^{J_{Q},χ} is nonzero, for some non-trivial character χ of 𝔽_{v}^{∗} inflated to the units, then ρ_{π,ι} is tamely ramified:

ρ_{π,ι}|_{I_{Fv}} = 1 ⊕ 1 ⊕ χ ⊕ χ.

Here χ is the character of I_{F_{v}} obtained via local class field theory. Moreover, one can arrange for the two eigenspaces, for 1 and χ, to be totally isotropic.

From our construction of the compatible system ρ_{π,ι} in part (a), one deduces that it is motivic in the sense defined on p. 60 in [BRo]: There is a smooth projective variety X_{/F}, and an integer n, such that ρ_{π,ι} is a constituent of

H^{j}(X ×_{F} F̄, ℚ̄_{ℓ})(n),

for all ℓ, where w = j − 2n. By invoking the Weil restriction, it is enough to show the analogous result for ρ_{π,ι}|_{Γ_{E}} for some CM extension E over F. For a detailed argument, we refer to the proof of Proposition 5.2.1 on p. 86 in [BRo]. Over E, the variety is a self-product of the universal abelian variety over a simple Shimura variety. See the bottom isomorphism on p. 98 in [HT]. As in the case of Hilbert modular forms [BRo], one would like to have actual motives over E associated with π. Seemingly, one of the main obstacles in deriving this from [HT] is a multiplicity one issue for the unitary groups considered there: Is the positive integer a in part (6) on p. 12 of [TY] in fact equal to one? Conjecturally, one should even have motives over F attached to π. Even over ℚ this is not yet known. One problem is that the Hecke correspondences on a Siegel threefold do not extend to a given toroidal compactification. For a more thorough discussion of these matters, and a slightly different approach, see [H].

Many thanks are due to D. Ramakrishnan for his suggestion that I should look at the Hilbert-Siegel case by passing to a CM extension, and for sharing his insights on many occasions. I am also grateful to M. Harris for useful correspondence regarding the patching argument in section 4.3. Finally, I am thankful to C. Skinner and A. Wiles for discussions relevant to this paper, and for their encouragement and support.

2 Lifting to GL(4)

We will describe below how to transfer automorphic representations of GSp(4), of a certain type, to GL(4). Throughout, we work over a totally real base field F. Let us take π to be a globally generic cuspidal automorphic representation of GSp(4), with central character ω_{π}. We do not assume it is unitary. For v|∞,

π_{v} ≃ π_{μ(v)}^{W}, μ_{1}(v) ≥ μ_{2}(v) ≥ 0, μ_{1}(v) + μ_{2}(v) ≡ w (mod 2).

The notation is explained more carefully below. Here we fix the integer w such that ω_{π_{v}} takes nonzero a ↦ a^{−w} for all archimedean places v. In particular,

π^{∘} =^{df} π ⊗ |c|^{\frac{w}{2}} is unitary.

Using theta series, one can then associate an automorphic representation Π of GL(4) with the following properties: It has central character ω_{π}^{2}, and satisfies^{1}

• Π ⊗ ω_{π}^{−1} ≃ Π^{∨}.

• For v|∞, the L-parameter of Π_{v} has the following restriction to ℂ^{∗},

z ↦ |z|^{−w} · (\begin{matrix}(z/z̄)^{\frac{ν_{1}+ν_{2}}{2}} \\ & (z/z̄)^{\frac{ν_{1}−ν_{2}}{2}} \\ & & (z/z̄)^{− \frac{ν_{1}−ν_{2}}{2}} \\ & & & (z/z̄)^{− \frac{ν_{1}+ν_{2}}{2}} \end{matrix} ) ,

where ν_{1} = μ_{1} + 2 and ν_{2} = μ_{2} + 1 give the Harish-Chandra parameter of π_{v}. Here we suppress the dependence on v, and simply write μ_{i} = μ_{i}(v).

• L(s, Π_{v}) = L(s, π_{v}, spin), for finite v such that π_{v} is unramified.

In fact, Gan and Takeda have recently defined a local Langlands correspondence for GSp(4) such that the above lift is strong. That is, the L-parameters of π_{v} and Π_{v} coincide at all places v. For later applications, we would like Π to have a square-integrable component. Using table 2 on page 51 in [GT], we can ensure this by assuming the existence of a finite place v_{0} where π_{v_{0}} is of the form

π_{v_{0}} = \begin{cases}St_{GSp(4)}(χ) \\ St(χ, τ) \end{cases} ⟹ Π_{v_{0}} = \begin{cases}St_{GL(4)}(χ) \\ St(τ). \end{cases}

Here St_{GSp(4)}(χ) is the Steinberg representation twisted by the character χ, which need not be unramified. Also, Π_{v_{0}} is supercuspidal if π_{v_{0}} is a supercuspidal not coming from GO(2, 2). Most of the notation used here is selfexplanatory, except maybe the symbol St(χ, τ): It denotes the generalized Steinberg representation, of Klingen type, associated to a supercuspidal τ on GL(2) and a non-trivial quadratic character χ such that τ ⊗ χ = τ. We refer to page 35 in [GT] for more details. As a bonus, the existence of such a place v_{0} guarantees that Π is cuspidal: Otherwise, π is a theta lift from GO(2, 2), but by [GT] the above π_{v_{0}} do not participate here.

2.1 The archimedean case

2.1.1 Discrete series for GL(2, ℝ)

In this section, we briefly set up notation for the discrete series representations of GL(2, ℝ). Throughout we use Harish-Chandra parameters, as opposed to Blattner parameters. We will follow the notation of [Lau]. Thus, for each positive integer n, we let σ_{n} be the unique (essentially) discrete series representation of GL(2, ℝ) which has the same central character and the same infinitesimal character as the finite-dimensional irreducible representation Sym^{n−1}(ℂ^{2}).

1We normalize the isomorphism W_{ℝ}^{ab} ≃ ℝ^{∗} using the absolute value |z|_{ℂ} =^{df} |z|^{2}.

Documenta Mathematica 15 (2010) 623–670

σ_{n}(λ) =^{df} σ_{n} ⊗ | det |^{λ},

for each λ ∈ ℂ. More concretely, σ_{n} is induced from the neutral component:

σ_{n} = Ind_{GL(2,ℝ)^{+}}^{GL(2,ℝ)}(σ_{n}^{+}),

where σ_{n}^{+} is a certain representation of GL(2, ℝ)^{+} on the Hilbert space of

f : H → ℂ holomorphic, ∥f∥^{2} = ∫_{H} |f(x + iy)|^{2}y^{n−1}dxdy < ∞.

Here H denotes the upper half-plane in ℂ, and GL(2, ℝ)^{+} acts by the formula

σ_{n}^{+} (\begin{matrix}a & c \\ b & d \end{matrix}) f(z) = (ad − bc)^{n}(cz + d)^{−n−1}f(\frac{az + b}{cz + d}).

If instead of z we use z̄ in the automorphy factor, this also defines a representation σ_{n}^{−} on the anti-holomorphic functions on H. Then σ_{n} can be thought of as the direct sum σ_{n}^{+} ⊕ σ_{n}^{−}, where a non-trivial coset representative acts by reflection in the y-axis. Its Jacquet-Langlands correspondent σ_{n}^{JL} is simply Sym^{n−1}(ℂ^{2}) viewed as a representation of the Hamilton quaternions ℍ^{∗} embedded into GL(2, ℂ) in the standard fashion. Note that σ_{n} is not unitary, unless n = 1. However, after a suitable twist it becomes unitary. Moreover,

Hom_{GL(2,ℝ)^{+}}(σ_{n}^{+}(\frac{1 − n}{2}), L_{cusp}^{2}(ℝ_{+}^{∗}Γ\GL(2, ℝ)^{+}))

can be identified with the space of weight n+1 elliptic cusp forms for the discrete subgroup Γ. The weight n + 1 is the Blattner parameter of σ_{n}, describing its minimal K-types. Up to isomorphism, σ_{n} is easily seen to be invariant under twisting by the sign character of ℝ^{∗}. Consequently, σ_{n} is automorphically induced from a character on ℂ^{∗}. More generally, its twist σ_{n}(λ) is induced from

z ↦ |z|^{2λ−1}z^{n} = |z|^{n−1+2λ}(z/z̄)^{\frac{n}{2}} .

We want to write down an L-parameter for σ_{n}(λ). Thus, we let W_{ℝ} denote the Weil group of ℝ, generated by ℂ^{∗} and an element j such that j^{2} = −1 and jz = z̄j for all z ∈ ℂ^{∗}. To σ_{n}(λ) is associated a conjugacy class of homomorphisms

ϕ_{n}(λ) : W_{ℝ} → GL(2, ℂ)

with semisimple images. By the above remarks, a concrete representative is:

ϕ_{n}(λ) : z ↦ |z|^{n−1+2λ} · (\begin{matrix}(z/z̄)^{\frac{n}{2}} \\ & (z/z̄)^{− \frac{n}{2}} \end{matrix} ), ϕ_{n}(λ) : j ↦ (\begin{matrix}& 1 \\ (−1)^{n} \end{matrix}).

We note that the image of ϕ_{n}(λ) is bounded precisely when σ_{n}(λ) is unitary.

2.1.2 Discrete series for GSp(4, ℝ)

We parametrize the discrete series representations of GSp(4, ℝ) in accordance with [Lau]. Throughout, we realize symplectic groups with respect to the form

J = (\begin{matrix}& S \\ −S \end{matrix}), S = (\begin{matrix}& 1 \\ 1 \end{matrix}).

The similitude character is denoted by c. We take B to be the Borel subgroup consisting of upper triangular matrices. The maximal torus T is of the form

T = \{t = (\begin{matrix}t_{1} \\ & t_{2} \\ & & t_{3} \\ & & & t_{4} \end{matrix}) : c(t) = t_{1}t_{4} = t_{2}t_{3}\}.

We identify its group of rational characters X^{∗}(T) with the set of triples of integers μ = μ_{0} ⊕ (μ_{1}, μ_{2}), such that μ_{1} + μ_{2} ≡ μ_{0} (mod 2), using the recipe:

t^{μ} =^{df} t_{1}^{μ_{1}}t_{2}^{μ_{2}}c(t)^{\frac{μ_{0}−μ_{1}−μ_{2}}{2}} .

Its restriction to the center 𝔾_{m} takes a ↦ a^{μ_{0}}. Inside X^{∗}(T) we have the cone of B-dominant weights X^{∗}(T)^{+} consisting of all tuples μ such that μ_{1} ≥ μ_{2} ≥ 0. By a fundamental result of Chevalley, the finite-dimensional irreducible algebraic representations are classified by their highest weights. For a B-dominant weight μ as above, we let V_{μ} be the corresponding algebraic representation of GSp(4). Its central character is given by μ_{0} as described above. To describe the infinitesimal character of V_{μ}, we consider half the sum of B-positive roots:

δ =^{df} 0 ⊕ (2, 1).

The Harish-Chandra isomorphism identifies the center of the universal enveloping algebra, Z(𝔤), with the invariant symmetric algebra Sym(𝔱_{ℂ})^{W}. Under this isomorphism, the aforementioned infinitesimal character corresponds to

ν =^{df} μ + δ = μ_{0} ⊕ (ν_{1}, ν_{2}) = μ_{0} ⊕ (μ_{1} + 2, μ_{2} + 1).

Up to infinitesimal equivalence, there are precisely two essentially discrete series representations of GSp(4, ℝ) with the same central character and the same infinitesimal character as V_{μ}. Together, they form an L-packet,

{π_{μ}^{W} , π_{μ}^{H}}.

Here π_{μ}^{W} is the unique generic member, that is, it has a Whittaker model. The other member π_{μ}^{H} is holomorphic, and does not have a Whittaker model. They both have central character a ↦ a^{μ_{0}} for real nonzero a. Another way to distinguish the two representations, is to look at their (𝔤, K)-cohomology: The generic member π_{μ}^{W} has cohomology of Hodge type (2, 1) and (1, 2), whereas π_{μ}^{H} contributes cohomology of Hodge type (3, 0) and (0, 3). For example,

H^{3,0}(𝔤, K; π_{μ}^{H} ⊗ V_{μ}^{∗}) ≃ H^{0,3}(𝔤, K; π_{μ}^{H} ⊗ V_{μ}^{∗}) ≃ ℂ,

and similarly for π_{μ}^{W}. We have consistently used the Harish-Chandra parameter ν. Other authors prefer the Blattner parameter, because of its connection to the weights of Siegel modular forms. In our case, the relation is quite simple:

\underline{k} = (k_{1}, k_{2}), k_{1} = ν_{1} + 1 = μ_{1} + 3, k_{2} = ν_{2} + 2 = μ_{2} + 3.

See Theorem 12.21 in [Kn] for example. A word of caution: It is really only fair to call \underline{k} the Blattner parameter in the holomorphic case. In the generic case, it does not give the highest weight of the minimal K-type. The restriction of π_{μ}^{H} to the neutral component GSp(4, ℝ)^{+}, consisting of elements with positive similitude, decomposes as a direct sum of a holomorphic part π_{μ}^{H+} and a dual anti-holomorphic part π_{μ}^{H−}. If we assume μ_{0} = 0, so that these are unitary,

Hom_{GSp(4,ℝ)^{+}}(π_{μ}^{H+}, L_{cusp}^{2}(ℝ_{+}^{∗}Γ\GSp(4, ℝ)^{+}))

can be identified with cuspidal Siegel modular forms of weight \underline{k} for the discrete subgroup Γ. The minimal K-type of π_{μ}^{H+}, where K is isomorphic to U(2), is

Sym^{k_{1}−k_{2}}(ℂ^{2}) ⊗ det^{k_{2}}.

This is the algebraic representation of GL(2, ℂ) with highest weight t_{1}^{k_{1}}t_{2}^{k_{2}}. Next, we wish to explain how π_{μ}^{W} and π_{μ}^{H} can be described explicitly as certain theta lifts: Up to equivalence, there are precisely two 4-dimensional quadratic spaces over ℝ of discriminant one. Namely, the anisotropic space V_{4,0} and the split space V_{2,2}. The latter can be realized as M(2, ℝ) equipped with the determinant. The former can be taken to be ℍ endowed with the reduced norm. In particular,

GSO(2, 2) = (GL(2, ℝ) × GL(2, ℝ))/ℝ^{∗}, GSO(4, 0) = (ℍ^{∗} × ℍ^{∗})/ℝ^{∗}.

Here ℝ^{∗} is embedded in the centers by taking a to the element (a, a^{−1}). Thus irreducible representations of GSO(2, 2) correspond to pairs of irreducible representations of GL(2, ℝ) having the same central character. Similarly for GSO(4, 0). Now suppose σ and σ′ are irreducible representations of GL(2, ℝ) with the same central character. We say that the representation σ ⊗ σ′ is regular if σ ≠ σ′. The induced representation of the whole similitude group GO(2, 2) then remains irreducible, and we denote it by (σ ⊗ σ′)^{+} following the notation of [Rob]. On the other hand, in the invariant case where σ = σ′, there are exactly two extensions of σ ⊗ σ to a representation of GO(2, 2). By Theorem 6.8 in [Rob], precisely one of these extensions participates in the theta correspondence with GSp(4, ℝ). It is again denoted by (σ⊗σ)^{+}. The analogous results hold in the anisotropic case. Now, the following two identities can be found in several places in the literature. See [Moe] for example, or Proposition 4.3.1 in [HK]:

π_{μ}^{W} = θ((σ_{ν_{1}+ν_{2}}(\frac{1}{2}(μ_{0} − ν_{1} − ν_{2} + 1)) ⊗ σ_{ν_{1}−ν_{2}}(\frac{1}{2}(μ_{0} − ν_{1} + ν_{2} + 1)))^{+}),

and

π_{μ}^{H} = θ((σ_{ν_{1}+ν_{2}}^{JL}(\frac{1}{2}(μ_{0} − ν_{1} − ν_{2} + 1)) ⊗ σ_{ν_{1}−ν_{2}}^{JL}(\frac{1}{2}(μ_{0} − ν_{1} + ν_{2} + 1)))^{+}).

Here the equalities signify infinitesimal equivalence. Since we are dealing with theta correspondence for similitude groups, it is unnecessary to specify an additive character. As a result, we can exhibit a parameter for the μ-packet above:

ϕ_{μ} : z ↦ |z|^{μ_{0}} · (\begin{matrix}(z/z̄)^{\frac{ν_{1}+ν_{2}}{2}} \\ & (z/z̄)^{\frac{ν_{1}−ν_{2}}{2}} \\ & & (z/z̄)^{− \frac{ν_{1}−ν_{2}}{2}} \\ & & & (z/z̄)^{− \frac{ν_{1}+ν_{2}}{2}} \end{matrix} ) ,

and

ϕ_{μ} : j ↦ (\begin{matrix}& & & 1 \\ & & 1 \\ & (−1)^{μ_{0}+1} \\ (−1)^{μ_{0}+1} \end{matrix}) .

Visibly, ϕ_{μ} maps into the dual of the elliptic endoscopic group, consisting of

(\begin{matrix}a & & & b \\ & e & f \\ & g & h \\ c & & & d \end{matrix}) = (\begin{matrix}1 \\ & & 1 \\ & & & 1 \\ & 1 \end{matrix}) (\begin{matrix}a & b \\ c & d \\ & & e & f \\ & & g & h \end{matrix}) (\begin{matrix}1 \\ & & 1 \\ & & & 1 \\ & 1 \end{matrix})^{−1}

such that ad − bc equals eh − fg. Furthermore, the restriction of ϕ_{μ} ⊗ | · |^{−3} to ℂ^{∗} is a direct sum of distinct characters of the form z ↦ z^{p}z̄^{q} for two integers p and q such that p + q = μ_{0} − 3. They come in pairs: If the type (p, q) occurs, so does (q, p). We say that π_{μ}^{W} and π_{μ}^{H} are regular algebraic (up to a twist).

2.1.3 The Langlands classification for GL(4, ℝ)

The Langlands classification for GL(4, ℝ) describes all its irreducible admissible representations up to infinitesimal equivalence. The building blocks are the essentially discrete series σ_{n}(λ), and the characters sgn^{n}(λ) of the multiplicative group ℝ^{∗}. The representations of GL(4, ℝ) are then constructed by parabolic induction. For example, start out with the representations σ_{n}(λ) and σ_{n′}(λ′).

We view their tensor product as a representation of the parabolic associated with the partition (2, 2), by making it trivial on the unipotent radical. Consider

Ind_{P_{(2,2)}}^{GL(4,ℝ)}(σ_{n}(λ) ⊗ σ_{n′}(λ′)),

where we use normalized induction. Consequently, this is unitary when σ_{n}(λ) and σ_{n′}(λ′) are both unitary. By interchanging their roles, we may assume that

Re(λ) ≥ Re(λ′).

In this case, the induced representation has a unique irreducible quotient. We denote it by σ_{n}(λ) ⊞ σ_{n′}(λ′) and call it the isobaric sum. Its L-parameter is

ϕ = ϕ_{n}(λ) ⊕ ϕ_{n′}(λ′) : W_{ℝ} → GL(4, ℂ).

We want to know when it descends to a parameter for GSp(4, ℝ). First, in the case where n = n′, it maps into the Levi subgroup of the Siegel parabolic:

P = \{(\begin{matrix}A \\ & c ·^{τ} A^{−1} \end{matrix}) (\begin{matrix}1 & & x & y \\ & 1 & z & x \\ & & 1 \\ & & & 1 \end{matrix})\}.

Here τ is transposition with respect to the skew diagonal. This means the packet for GSp(4, ℝ) should be obtained by induction from the Klingen parabolic,

Q = \{(\begin{matrix}t \\ & A \\ & & t^{−1} det(A) \end{matrix}) (\begin{matrix}1 & z \\ & 1 \\ & & 1 & −z \\ & & & 1 \end{matrix}) (\begin{matrix}1 & & x & y \\ & 1 & & x \\ & & 1 \\ & & & 1 \end{matrix})\}.

To be more precise, consider the following (unitarily) induced representation:

Ind_{Q}^{GSp(4,ℝ)}(σ_{n}(λ′) ⊗ | · |^{2λ−2λ′}).

When Re(λ) is strictly greater than Re(λ′) it has a unique irreducible quotient. At the other extreme, when λ = λ′ it decomposes into a direct sum of two limits of discrete series. Secondly, in the case where n ≠ n′ we try to conjugate ϕ into the dual of the elliptic endoscopic group. The determinant condition becomes:

λ − λ′ = \frac{1}{2}(n′ − n) ∈ ℤ_{+}.  (1)

If this is satisfied, the isobaric sum descends to a packet for GSp(4, ℝ), whose members can be constructed by theta correspondence as discussed above:

\{π_{μ}^{W} , π_{μ}^{H}\}, μ_{0} = n − 1 + 2λ, ν_{1} = \frac{1}{2}(n′ + n), ν_{2} = \frac{1}{2}(n′ − n).

2.2 The non-archimedean case

2.2.1 The local Langlands correspondence for GL(n)

We will quickly review the parametrization of the irreducible admissible representations of GL(n, F), up to isomorphism, where F is a finite extension of ℚ_{p}. We will suppress F and denote this set by Π(GL(n)). This parametrization was originally conjectured by Langlands, for any connected reductive group, and recently proved for GL(n) in [HT] and [Hen] by two different methods. We let W_{F} be the Weil group of F. That is, the dense subgroup of the Galois group acting as integral powers of Frobenius on the residue field. It gets a topology by decreeing that the inertia group I_{F} is open. From local class field theory,

F^{∗} →^{∼} W_{F}^{ab}.

Here the isomorphism is normalized such that uniformizers correspond to lifts of the geometric Frobenius. It is used tacitly to identify characters of F^{∗} with characters of W_{F}. For arbitrary n, we consider the set Φ(GL(n)) consisting of conjugacy classes of continuous semisimple n-dimensional representations

ϕ : W′_{F} = W_{F} × SL(2, ℂ) → GL(n, ℂ).

The group W′_{F} is sometimes called the Weil-Deligne group. The local Langlands correspondence is then a canonical collection of bijections rec_{n}, one for each n,

rec_{n} : Π(GL(n)) →^{1:1} Φ(GL(n))

associating an L-parameter ϕ_{π} to a representation π. It satisfies a number of natural properties, which in fact determine the collection uniquely. Namely,

• The bijection rec_{1} is given by local class field theory as above.

• For any two π ∈ Π(GL(n)) and σ ∈ Π(GL(r)), we have equalities

\begin{cases}L(s, π × σ) = L(s, ϕ_{π} ⊗ ϕ_{σ}), \\ ϵ(s, π × σ, ψ) = ϵ(s, ϕ_{π} ⊗ ϕ_{σ}, ψ). \end{cases}

• The L-parameter of π ⊗ χ ∘ det equals ϕ_{π} ⊗ χ, for any character χ.

• For any π as above, det(ϕ_{π}) corresponds to its central character ω_{π}.

• For any π as above, rec_{n}(π^{∨}) is the contragredient of rec_{n}(π).

Here ψ is a non-trivial character of F, used to define the ϵ-factors. The collection rec_{n} does not depend on it. The L and ϵ-factors on the left-hand side are those from [JPS]: They are first defined for generic representations, such as supercuspidals, and then one extends the definition to all representations using the Langlands classification. For the explicit formulas, see [Kud] or [Wed].

For the right-hand side, the definition of the L and ϵ-factors can be found in [Tat]: The L-factors are given fairly explicitly, whereas the ϵ-factors are defined very implicitly. One only has an abstract characterization due to Deligne and Langlands. For a good review of these definitions, we again refer to [Wed]. It is useful to instead consider the (Frobenius semisimple) Weil-Deligne representation of W_{F} associated with a parameter ϕ as above. This is a pair (r, N) consisting of a semisimple representation r of W_{F}, and an operator N satisfying the equation

r(w) ∘ N ∘ r(w)^{−1} = |w|_{F} · N

for all w ∈ W_{F}. This N is called the monodromy operator, and it is automatically nilpotent. The correspondence relies on the Jacobson-Morozov theorem:

r(w) = ϕ(w, (\begin{matrix}|w|F^{\frac{1}{2}} \\ & |w|_{F}^{− \frac{1}{2}} \end{matrix} )), exp(N) = ϕ(1, \binom{1 1}{1}).

The local Langlands correspondence rec_{n} satisfies a number of additional natural properties, expected to hold more generally, of which we mention only a few:

• π is supercuspidal ⇔ ϕ_{π} is irreducible (and the monodromy is trivial).

• π is essentially discrete series ⇔ ϕ_{π} does not map into a proper Levi.

• π is essentially tempered ⇔ ϕ_{π}|_{W_{F}} has bounded image in GL_{n}(ℂ).

• π is generic ⇔ the adjoint L-factor L(s, Ad ∘ ϕ_{π}) has no pole at s = 1.

Here Ad denotes the adjoint representation of GL_{n}(ℂ) on its Lie algebra 𝔤𝔩_{n}(ℂ).

2.2.2 The local Langlands correspondence for GSp(4)

For GSp(4), the presence of endoscopy makes the parametrization of Π(GSp(4)) more complicated. It is partitioned into finite subsets L_{ϕ}, called L-packets, each associated with a parameter ϕ as above mapping into the subgroup GSp(4, ℂ). We use the notation Φ(GSp(4)) for the set of GSp(4, ℂ)-conjugacy classes of such ϕ. The first attempt to define these L-packets, when p is odd, is the paper [Vig]. The crucial case is when ϕ does not map into a proper Levi subgroup. In this case, Vigneras defined certain subsets L_{ϕ} by theta lifting from various forms of GO(4). However, she did not prove that these L_{ϕ} exhaust all of Π(GSp(4)). The work of Vigneras was later refined, so as to include the case p = 2, in the paper [Ro2]. More recently, Gan and Takeda [GT] were able to prove the exhaustion, for all primes p. To do that, they used work of Muic-Savin, Kudla-Rallis, and Henniart. The main theorem of [GT] gives a finite-to-one surjection

L : Π(GSp(4)) ↠ Φ(GSp(4)),

attaching an L-parameter ϕ_{π} to a representation π, and having the properties:

• π is essentially discrete series ⇔ ϕ_{π} does not map into a proper Levi.

• For any generic or non-supercuspidal π ∈ Π(GSp(4)), and σ ∈ Π(GL(r)),

\begin{cases}γ(s, π × σ, ψ) = γ(s, ϕ_{π} ⊗ ϕ_{σ}, ψ), \\ L(s, π × σ) = L(s, ϕ_{π} ⊗ ϕ_{σ}), \\ ϵ(s, π × σ, ψ) = ϵ(s, ϕ_{π} ⊗ ϕ_{σ}, ψ). \end{cases}

• The L-parameter of π ⊗ χ ∘ c equals ϕ_{π} ⊗ χ, for any character χ.

• For any π as above, c(ϕ_{π}) corresponds to its central character ω_{π}.

In the generic case, the invariants occurring on the left-hand side of the second condition are those from [Sha]. The definition can be extended to non-generic non-supercuspidals, using the Langlands classification. See page 13 in [GT]. For non-generic supercuspidals, L satisfies an additional technical identity, which we will not state here. It expresses a certain Plancherel measure as a product of four γ-factors. One has to include this last property to ensure the uniqueness of L, as long as a satisfying theory of γ-factors is absent in this setup. For completeness, let us mention a few extra properties of the map L: For a given parameter ϕ, the elements of the fiber L_{ϕ} correspond to characters of the group

A_{ϕ} = π_{0}(Z_{GSp(4,ℂ)}(imϕ)/ℂ^{∗}) = \begin{cases}ℤ/2ℤ, \\ 0. \end{cases}

Moreover, an L-packet L_{ϕ} contains a generic member exactly when L(s, Ad∘ϕ) has no pole at s = 1. If in addition ϕ|_{W_{F}} has bounded image, the members of L_{ϕ} are all essentially tempered, and the generic member is unique. It is indexed by the trivial character of A_{ϕ}. Next, we wish to at least give some idea of how the reciprocity map L is constructed in [GT]: The key is to make use of theta liftings from various orthogonal similitude groups. In analogy with the archimedean case, there are two 4-dimensional quadratic spaces over F of discriminant one. We abuse notation slightly, and continue to denote the anisotropic space by V_{4,0} and the split space by V_{2,2}. They can be realized as D equipped with the reduced norm, where D is a possibly split quaternion algebra over F. Again,

GSO(2, 2) = (GL(2, F) × GL(2, F))/F^{∗}, GSO(4, 0) = (D^{∗} × D^{∗})/F^{∗},

as previously, where D is here the division quaternion algebra. Furthermore, one looks at the 6-dimensional quadratic space D ⊕ V_{1,1}. When D is split, this is simply V_{3,3}. There is then a natural isomorphism, as given on page 9 in [GT],

GSO(3, 3) = (GL(4, F) × F^{∗})/{(a · I, a^{−2}) : a ∈ F^{∗}}.

Now, start with an irreducible representation π of GSp(4, F). By Theorem 5.3 in [GT], which relies on the work [KR] of Kudla and Rallis on the conservation conjecture, it follows that there are two possible mutually exclusive scenarios:

1. π participates in the theta correspondence with GSO(4, 0),

2. π participates in the theta correspondence with GSO(3, 3).

In the first case, one has two essentially discrete series representations σ and σ′ of GL(2, F) having the same central character, such that π is the theta lift

π = θ((σ^{JL} ⊗ σ^{′JL})^{+}) = θ((σ^{′JL} ⊗ σ^{JL})^{+}).

By the local Langlands correspondence for GL(2), we have associated parameters ϕ_{σ} and ϕ_{σ′} with equal determinants. We then conjugate their sum ϕ_{σ} ⊕ϕ_{σ′} into the dual of the elliptic endoscopic group as in the archimedean case,

ϕ_{π} = ϕ_{σ} ⊕ ϕ_{σ′} : W′_{F} → GL(2, ℂ) ×_{ℂ^{∗}} GL(2, ℂ) ⊂ GSp(4, ℂ).

In the second case, we write θ(π) as a tensor product Π ⊗ ω_{π} for an irreducible representation Π of GL(4, F). The local Langlands correspondence for GL(4) yields a parameter ϕ_{Π}. We need to know that it maps into GSp(4, ℂ) after conjugation. When π is a discrete series, this follows from a result of Muic and Savin [MS], stated as Theorem 5.4 in [GT]: Indeed, the exterior square L-factor

L(s, Π, ∧^{2} ⊗ ω_{π}^{−1})

has a pole at s = 0. When π is not a discrete series, Gan and Takeda compute θ(π) explicitly, using standard techniques developed by Kudla. For a summary, we refer to Table 2 on page 51 in their paper [GT]. It follows by inspection that ϕ_{Π} can be conjugated into a Levi subgroup of GSp(4, ℂ). Their computation works even for p = 2, and hence completes the exercise of Waldspurger [Wal]. Finally, in Proposition 11.1 of [GT], it is shown by a global argument that the above construction is consistent with that of Vigneras and Roberts.

2.3 The globally generic case

In the global situation, functoriality predicts that one should be able to transfer automorphic representations from GSp(4) to GL(4). It is widely believed that this should eventually follow by using trace formula techniques. See [Art] for a discussion on this approach. In the globally generic case, it has been known for some time that one can obtain (weak) lifts using theta series. This was first announced by Jacquet, Piatetski-Shapiro and Shalika, but to the best of our knowledge they never wrote it up. However, many of the details are to be found in [Sou]. Moreover, there is an alternative proof in [AS] relying on the converse theorem. In this section, we wish to quote a recent refinement of the above transfer, due to Gan and Takeda [GT]. First, for completeness, let us recall the notion of being globally generic: Consider the upper-triangular Borel subgroup

B = \{(\begin{matrix}s \\ & t \\ & & ct^{−1} \\ & & & cs^{−1} \end{matrix}) (\begin{matrix}1 & u \\ & 1 \\ & & 1 & −u \\ & & & 1 \end{matrix}) (\begin{matrix}1 & & x & y \\ & 1 & z & x \\ & & 1 \\ & & & 1 \end{matrix})\}.

We let N denote its unipotent radical. Now let F be a number field, and pick a non-trivial character ψ of 𝔸_{F} trivial on F. By looking at ψ(u + z) we can view it as an automorphic character of N. An automorphic representation π of GSp(4) over F is then said to be globally generic if the Whittaker functional

f ↦ ∫_{N(F)\N(𝔸_{F})} f(n)ψ^{−1}(n)dn, f ∈ π,

is not identically zero. This notion does not depend on ψ. As a consequence of the theta series approach we are about to discuss, Soudry proved the strong multiplicity one property in [Sou] for globally generic cusp forms π on GSp(4). As mentioned above, the exterior square ∧^{2} defines an isogeny between the groups GL(4) and GSO(3, 3). Thus, we need the global theta correspondence for similitude groups. For this, we refer to section 5 of [HK], noting that the normalization there differs slightly from [GT]. We will quickly review the main features of the definition: The Weil representation ω_{ψ} extends naturally to

R = {(g, h) ∈ GSp(4) × GO(3, 3) : c(g) · c(h) = 1}.

Then, a Schwartz-Bruhat function φ defines a theta kernel θ_{φ} on R by the usual formula. The theta series lifting of a form f in π is hence given by the integral

θ_{φ}(f)(h) = ∫_{Sp(4,F)\Sp(4,𝔸_{F})} θ_{φ}(gg_{h}, h)f(gg_{h})dg,

where g_{h} is any element with inverse similitude c(h). The space spanned by all such theta series θ_{φ}(f) constitute an automorphic representation of GO(3, 3), which we will denote by θ(π). It is independent of ψ. By Proposition 1.2 in [Sou], it is nonzero precisely when π is globally generic. In fact, one can express the Whittaker functional for θ(π) in terms of that for π given above. In particular, θ(π) is always generic, even though it may not be cuspidal. From now on, we will only view θ(π) as a representation of the subgroup GSO(3, 3). As such, it remains irreducible. See Lemma 3.1 in [GT] for example. In turn, via the identification ∧^{2} we view θ(π) as a representation Π ⊗ ω_{π}. We then have:

Theorem 1. The global theta lifting π ↦ θ(π) = Π ⊗ ω_{π} defines an injection from the set of globally generic cuspidal automorphic representations π of GSp(4) to the set of generic automorphic representations Π of GL(4) with central character ω_{Π} = ω_{π}^{2}. Moreover, this lifting has the following properties:

• Π ≃ Π^{∨} ⊗ ω_{π}.

• It is a strong lift, that is, ϕ_{Π_{v}} = ϕ_{π_{v}} for all places v.

• The image of the lifting consists precisely of those Π satisfying:

1. Π is cuspidal and L^{S}(s, Π, ∧^{2} ⊗ ω_{π}^{−1}) has a pole at s = 1, or

2. Π = σ ⊞ σ′ for cuspidal σ ≠ σ′ on GL(2) with central character ω_{π}.

In the latter case, π is the theta lift of the cusp form σ ⊗σ′ on GSO(2, 2).

P roof. See section 13 in [GT]. □

The refinements, due to Gan and Takeda, are primarily: The characterization of the image, and the fact that the global lift is compatible with the local Langlands correspondence at all (finite) places. The result that the lift is strong, in this sense, essentially follows from the construction of the local reciprocity map. However, locally one has to check that ϕ_{Π} is equivalent to ϕ_{σ} ⊕ϕ_{σ′}, when π is the theta lift of any σ⊗σ′ on GSO(2, 2). This is the content of Corollary 12.13 in [GT]. It is a result of their explicit determination of the theta correspondence.

3 Base change to a CM extension

In this section we will construct representations, for which the results from [HT] on Galois representations apply. For that purpose, we will fix an arbitrary CM (quadratic) extension E/F, and an arbitrary Hecke character χ of E with the property:

χ|_{𝔸_{F}^{∗}} = ω_{π}^{−1}.

Since ω_{π_{v}} is of the form a ↦ a^{−w} at each infinite place v, it follows that every such χ is automatically algebraic. For all but finitely many E, the global base change Π_{E}, to be defined below, is cuspidal. In fact, we will choose a suitable quadratic CM extension E of F such that v_{0} is totally split in E, and a suitable Hecke character χ of E. The suitability refers to the fact that the Arthur-Clozel base change Π_{E} will be cuspidal, and that its twist by χ will be conjugate self-dual,

Π_{E}(χ)^{θ} ≃ Π_{E}(χ)^{∨}.

Furthermore, Π_{E}(χ) is regular algebraic of weight zero, and we can arrange for it to have at least one square integrable component, by imposing the condition that E splits completely at v_{0}. Hence Theorem C in [HT] applies. In this section, we briefly review results of Arthur and Clozel on base change for GL(n), and discuss the compatibility with the local Langlands correspondence.

3.1 Local base change

Now that the p-adic local Langlands correspondence is available for GL(n), due to the works of Harris-Taylor and Henniart, base change makes sense for an arbitrary finite extension of local fields E/F. Indeed, a representation Π_{E} of GL(n, E) is the base change of a representation Π of GL(n, F) precisely when

ϕ_{Π_{E}} = ϕ_{Π}|_{W′_{E}}.

However, eventually we will use results from [AC]. At the time this book was written, one had to resort to a harmonic analytic definition of base change which we will review below. Fortunately, the compatibility of the two definitions has been checked by other authors. We will give precise references later. The latter definition only works for a cyclic extension E/F. For simplicity, we take it to be quadratic, and let θ be the non-trivial element in its Galois group. By lemma 1.1 in [AC], the norm map on GL(n, E), taking γ ↦ γγ^{θ}, defines an injection

N : {θ-conjugacy classes in GL(n, E)} ↪ {conjugacy classes in GL(n, F)}.

This is used to define transfer of orbital integrals: Two compactly supported smooth functions f and f_{E} are said to have matching orbital integrals when

O_{γ}(f) = \begin{cases}T O_{δθ}(f_{E}), γ = N δ, \\ 0, γ is not a norm. \end{cases}

For the definitions of the integrals involved here, we refer to page 15 in [AC]. It is the content of Proposition 3.1 in [AC] that any f_{E} has a matching function f. The fundamental lemma in this case is Theorem 4.5 in [AC]. We can now state Shintani’s definition of local base change, following Definition 6.1 in [AC]: Let Π and Π_{E} be irreducible admissible representations of GL(n, F) and GL(n, E) respectively, and assume that Π_{E}^{θ} is isomorphic to Π_{E}. Let I_{θ} be an intertwining operator between these, normalized such that I_{θ}^{2} is the identity. This determines I_{θ} up to a sign. We then say that Π_{E} is a base change of Π if and only if

tr(Π_{E}(f_{E}) ∘ I_{θ}) = c · trΠ(f)

for all matching functions f and f_{E} as above. The non-zero constant c depends only on the choice of measures, and of I_{θ}. By Theorem 6.2 in [AC] local base change makes sense for tempered representations. Using the Langlands classification, the lift then extends to all representations. For this, see the discussion on page 59 in [AC]. Since Shintani’s definition is employed in [AC], we will need:

Theorem 2. Shintani’s harmonic analytic definition of the local cyclic base change lifting is compatible with the local Langlands correspondence for GL(n).

P roof. In the non-archimedean case, this is part 5 of Lemma VII.2.6 on page 237 in [HT]. The archimedean case was settled, in general, by Clozel [Clo]. □

As an example in the archimedean case, let us base change σ_{n}(λ) ⊞ σ_{n′}(λ′) to GL(4, ℂ). For simplicity, we will stick to the case of interest in this paper where it descends to a discrete series L-packet for GSp(4, ℝ). That is, we assume (1).

−w =^{df} n − 1 + 2λ = n′ − 1 + 2λ′

Following [Kna] we let [z] denote \frac{z}{|z|}. Then the unitarily induced representation

Ind_{B}^{GL(4,ℂ)}([·]^{n} ⊗ [·]^{−n} ⊗ [·]^{n′} ⊗ [·]^{−n′}) ⊗ | det |^{−w} has a unique irreducible quotient. This is the base change we are looking for. The Langlands correspondence for GL(n, ℂ), which is much simpler than the real case, was first studied by Zelobenko and Naimark. A good reference for their results is the expository paper [Kna]. In the non-archimedean case,

St_{GL(4)}(χ)_{E} = St_{GL(4)}(χ_{E}), χ_{E} = χ ∘ N_{E/F}.

However, the generalized Steinberg representation St(τ) may not base change to a discrete series. Indeed, if ω_{E/F} denotes the associated quadratic character,

St(τ)_{E} = \begin{cases}St(τ_{E}) \\ St_{GL(2)}(ψ) ⊞ St_{GL(2)}(ψ^{θ}) \end{cases} when \begin{cases}τ ≠ τ ⊗ ω_{E/F} \\ τ = τ ⊗ ω_{E/F} . \end{cases}

Here ψ ≠ ψ^{θ} is a certain character of E^{∗}, with automorphic induction τ.

3.2 Global base change

We now let E/F denote an arbitrary CM extension of the totally real field F. Thus, the extension E/F is quadratic, and E is totally imaginary. We let θ be the non-trivial element in the Galois group. Let Π and Π_{E} be automorphic representations of GL(n, 𝔸_{F}) and GL(n, 𝔸_{E}) respectively. We will assume Π_{E} is invariant under θ. Then, we say that Π_{E} is a strong base change lift of Π if

Π_{E,w} = Π_{v,E_{w}}

for all places w|v. Here, the right-hand side is the local base change of Π_{v} to GL(n, E_{w}). When v is split in E, this lift is naturally identified with Π_{v}. By comparing trace formulas for GL(n), no stabilization required, Arthur and Clozel proved that such lifts always exist. More precisely, we have the following:

Theorem 3. There is a unique strong base change lift Π ↦ Π_{E} between isobaric automorphic representations on GL(n, 𝔸_{F}) and GL(n, 𝔸_{E}), satisfying:

• ω_{Π_{E}} = ω_{Π} ∘ N_{E/F}.

• The image of the lifting consists precisely of the θ-invariant Π_{E}.

• If Π is cuspidal, Π_{E} is cuspidal if and only if Π ≠ Π ⊗ ω_{E/F}.

P roof. This is essentially Theorem 4.2 combined with Theorem 5.1 in [AC]. Arthur and Clozel makes the assumption that Π is induced from cuspidal. This is now superfluous; the residual spectrum of GL(n) is understood by [MW]. □

Note that, if Π is cuspidal, Π_{E} is cuspidal for all but finitely many CM extensions E. Indeed, the self-twist condition is satisfied if the discriminant of E does not divide the conductor of Π. The theory of base change goes hand-in-hand with automorphic induction, which is a strong lift from isobaric automorphic representations of GL(n, 𝔸_{E}) to those of GL(2n, 𝔸_{F}) compatible with the rec_{n},

π ↦ I_{E}^{F} (π), ϕ_{I_{E}^{F} (π)_{v}} = Ind_{E_{w}}^{F_{v}}(ϕ_{π_{w}}), for all w|v. Again, this is due to Arthur and Clozel in much greater generality. See Theorem 6.2 in [AC]. For the compatibility with rec_{n} at the ramified places, we again refer to Lemma VII.2.6 in [HT]. In analogy with the above, we have the following result, which is not needed for the proof of the main theorem, but we include it for future use:

Theorem 4. There is a strong automorphic induction lift π ↦ I_{E}^{F} (π) between isobaric automorphic representations on GL(n, 𝔸_{E}) and GL(2n, 𝔸_{F}), satisfying:

• ω_{I_{E}^{F} (π)} = ω_{π}|_{𝔸_{F}^{∗}}.

• The image consists precisely of the Π such that Π = Π ⊗ ω_{E/F}.

• If π is cuspidal, I_{E}^{F} (π) is cuspidal if and only if π ≠ π^{θ}.

P roof. This is due to Arthur and Clozel. See section 6 of chapter 3 in [AC]. □

For the sake of completeness, let us mention a few links between base change and automorphic induction. For any two cuspidal π and Π as above, we have:

I_{E}^{F} (Π_{E}) = Π ⊞ (Π ⊗ ω_{E/F}), I_{E}^{F} (π)_{E} = π ⊞ π^{θ}.

3.3 Conjugate self-dual twists

Let us now take any CM extension E/F, and consider Π_{E}, where Π is the theta series lifting of our original globally generic π on GSp(4). We will assume π_{v_{0}} is of (twisted) Steinberg type at some finite place v_{0} of F. Hence Π_{E,w_{0}} is Steinberg at all places w_{0} of E dividing v_{0}, and this ensures Π_{E} is cuspidal. In [HT], one associates Galois representations to certain conjugate self-dual representations. Π_{E} itself may not satisfy this condition, when ω_{π,E} ≠ 1, but certain twists do:

Π_{E}(χ)^{θ} ≃ Π_{E}(χ)^{∨} ⇔ χ|_{𝔸_{F}^{∗}} = ω_{π}^{−1}ω_{E/F}^{n},

for n = 0 or n = 1. Such Hecke characters χ of E exist: Indeed, by Frobenius reciprocity, any Hecke character of F has infinitely many extensions to E; they are precisely the constituents of the induced representation of the compact idele class group C_{E}^{1}. By modifying this argument slightly, one can even control the ramification of the extensions if need be. Now, recall that for every place v|∞,

ω_{π_{v}}(a) = a^{−w},

for all a ∈ ℝ^{∗}. Therefore, to retain algebraicity, take n = 0 above. In this case, it follows that all the extensions χ are automatically algebraic. That is,

χ_{w}(z) = z^{a}z̄^{b}, a = a(w) ∈ ℤ, b = b(w) ∈ ℤ, a + b = w,

for each infinite place w of E, not to be confused with the weight! For such characters χ, the twist Π_{E}(χ) remains regular algebraic, and the weight is zero.

4 Galois representations

4.1 Galois representations over CM extensions

Let E be an arbitrary CM extension of the totally real field F. One of the ultimate goals of the book project [Har], is to attach an ℓ-adic Galois representation ρ_{Π,ι} to a regular algebraic conjugate self-dual cuspidal automorphic representation Π of GL(n, 𝔸_{E}), and a choice of an isomorphism ι : ℚ̄_{ℓ} → ℂ. See expected Theorem 2.4 in [Har] for a more precise formulation. In the case where Π has a square-integrable component at some finite place, pioneering work on this problem was done by Clozel [Cl] and Kottwitz [Kot], relating ρ_{Π,ι}|_{W_{Ew}} to the unramified component Π_{w} at most places w. Their work was later extended to all places w ∤ ℓ in [HT], by Harris and Taylor, in the course of proving the local Langlands conjecture. However, in [HT] the monodromy operator is ignored. This issue has been taken care of by Taylor and Yoshida in [TY], resulting in:

Theorem 5. Let E be a CM extension of a totally real field F, and let Π be a cuspidal automorphic representation of GL(n, 𝔸_{E}) satisfying the conditions:

• Π_{∞} is regular algebraic, H^{•}(𝔤, K; Π_{∞} ⊗ V^{∗}) ≠ 0.

• Π is conjugate self-dual, Π^{∨} ≃ Π^{θ}.

• Π_{w_{0}} is (essentially) square integrable for some finite place w_{0}.

Fix an isomorphism ι : ℚ̄_{ℓ} → ℂ. Then there is a continuous representation

ρ_{Π,ι} : Gal(F̄/E) → GL(n, ℚ̄_{ℓ})

such that for every finite place w of E, not dividing ℓ, we have the following:

ιWD(ρ_{Π,ι}|_{W_{Ew}})^{F −ss} ≃ rec_{n}(Π_{w} ⊗ | det |^{\frac{1−n}{2}} ).

P roof. This is Theorem 1.2 in [TY], which is a refinement of Theorem VII.1.9 in [HT]. Indeed, the former result identifies the monodromy operator: By Corollary VII.1.11 in [HT] it is known that Π_{w} is tempered for all finite places w. Therefore, by parts (3) and (4) of Lemma 1.4 in [TY], it suffices to show that ιWD(ρ_{Π,ι}|_{W_{Ew}}) is pure. That is, up to a shift, the weight filtration coincides with the monodromy filtration. This is proved in [TY] by a careful study of the Rapoport-Zink weight spectral sequence, the main new ingredient being the vanishing outside the middle-degree in Proposition 4.4 in [TY]. □

A word about the notation used in the previous Theorem: First, V denotes an irreducible algebraic representation over ℂ of the group R_{E/ℚ}GL(n), which we will denote by G. Then 𝔤 denotes the Lie algebra of G(ℝ), and K is a maximal compact subgroup times Z_{G}(ℝ). The symbol W D(ρ) stands for the Weil-Deligne representation corresponding to an ℓ-adic representation ρ of W_{E_{w}}, where w ∤ ℓ. This pair (r, N) is obtained by fixing a lift Frob_{w} of the geometric Frobenius, and a continuous surjective homomorphism t_{ℓ} : I_{E_{w}} → ℤ_{ℓ}, and then writing

ρ(Frob_{w}^{m}σ) = r(Frob_{w}^{m}σ)exp(t_{ℓ}(σ)N)

for σ ∈ I_{E_{w}} and integers m. Then r is a representation of W_{E_{w}} having an open kernel, and N is a nilpotent operator satisfying the formula mentioned above:

N ∘ r(Frob_{w}) = q_{w} · r(Frob_{w}) ∘ N.

Here q_{w} is the order of the residue field of E_{w}. The isomorphism class of (r, N) is independent of the choices made. Finally, the superscript F − ss signifies Frobenius semisimplification. That is, leave N unchanged, but semisimplify r. The representations ρ_{Π,ι} above satisfy a number of additional nice properties:

• Π is square integrable at some finite w_{0} ∤ ℓ ⟹ ρ_{Π,ι} is irreducible.

• Let w ∤ ℓ be a finite place of E, and let α be an eigenvalue of ρ_{Π,ι}(σ) for some σ ∈ W_{E_{w}}. Then α belongs to ℚ̄, and for every embedding ℚ̄ ↪ ℂ,

|α| ∈ qw^{\frac{ℤ}{2}}.

• Let w ∤ ℓ be a finite place, with Π_{w} unramified, and let α be an eigenvalue of ρ_{Π,ι}(Frob_{w}). Then α belongs to ℚ̄, and for every embedding ℚ̄ ↪ ℂ,

|α| = qw^{\frac{n−1}{2}} .

• The representation ρ_{Π,ι} is potentially semistable at any finite place w|ℓ. Moreover, ρ_{Π,ι} is crystalline at a finite place w|ℓ when Π_{w} is unramified.

The first property is clear since rec_{n}(Π_{w_{0}} ⊗ | det |^{\frac{1−n}{2}} ) is indecomposable. This was observed in Corollary 1.3 in [TY]. It is expected to continue to hold if w_{0}|ℓ (and even without the square-integrability condition, admitting the book project). The second and the third property are parts 1 and 2 of Theorem VII.1.9 in [HT]. The former is a special case of Lemma I.5.7 in [HT], which apparently follows from the Rapoport-Zink weight spectral sequence in conjunction with de Jong’s theory of alterations. The latter follows from Deligne’s work on the Weil conjectures. The last property comes down to the comparison theorems of p-adic Hodge theory. To clarify these comments, we will briefly sketch how ρ_{Π,ι} is realized geometrically in [HT]: One starts off with an n^{2}-dimensional central division algebra B over E, equipped with a positive involution ∗ such that ∗|_{E} = θ. It is assumed to satisfy a list of properties, which are irrelevant for our informal discussion. For a fixed β ∈ B such that ββ^{∗} = 1, look at the unitary similitude group G defined as follows: For a commutative ℚ-algebra R,

G(R) = {x ∈ (B^{op} ⊗_{ℚ} R)^{∗}: x^{∗}βx = c(x)β, with c(x) ∈ R^{∗}}.

The element β is chosen such that, at infinity, the derived group takes the form

G^{der}(ℝ) = U(n − 1, 1) × U(n)^{[F:ℚ]−1}.

The group G has an associated Shimura variety of PEL type. That is, for each sufficiently small compact open subgroup K inside the finite adeles G(𝔸_{f}), there is a smooth proper variety X_{K} over E classifying isogeny classes of polarized abelian schemes A of dimension [F : ℚ]n^{2}, endowed with a certain homomorphism from B into End(A)_{ℚ} and a so-called level-structure relative to K. On X_{K} one defines a ℚ̄_{ℓ}-sheaf L_{ξ} by fixing an algebraic representation ξ of G defined over ℚ̄_{ℓ}. Then, consider the following direct limit over subgroups K, endowed with natural commuting actions of G(𝔸_{f}) and of the Galois group of E:

H^{m}(X, L_{ξ}) =^{df} \underrightarrow{lim}_{K}H_{et}^{m}(X_{K} ×_{E} F̄, L_{ξ}) = ⨁_{π_{f}} π_{f} ⊗ R_{ξ}^{m}(π_{f}).

Here π_{f} runs over the irreducible admissible representations of G(𝔸_{f}), and R_{ξ}^{m}(π_{f}) is a finite-dimensional continuous Galois representation of E. To construct ρ_{Π,ι}, we first descend Π to an automorphic representation Π̃ of B^{op,∗} via the Jacquet-Langlands correspondence. Using results of Clozel and Labesse, one shows that ψ × Π̃ is a base change from G, for some algebraic Hecke character ψ of E. In this way, we end up with an automorphic representation π of G, and

R_{ξ}^{n−1}(π_{f}^{∨})^{ss} ≃ ρ_{Π,ι}^{a} ⊗ ρ_{ψ,ι}

for some positive integer a. For details see p. 228 in [HT], and p. 12 in [TY].

4.2 Hodge-Tate weights

We will now describe the Hodge-Tate weights, which are certain numerical invariants of the restriction of ρ_{Π,ι} to the Galois group of E_{w} for each place w|ℓ. We briefly recall their definition: Let K be a finite extension of ℚ_{ℓ}. Following Fontaine, we introduce the field of ℓ-adic periods B_{dR}. This is a K-algebra, it comes equipped with a discrete valuation v_{dR}, and has residue field ℂ_{ℓ}. The topology on B_{dR} is coarser than the one coming form the valuation. The Galois group of K acts continuously on B_{dR}. If t ∈ B_{dR} is a uniformizer, then we have

g · t = χ_{cyc}(g)t,

where χ_{cyc} is the ℓ-adic cyclotomic character. The valuation defines a filtration:

Fil^{j}(B_{dR}) =^{df} t^{j}B_{dR}^{+}, gr^{j}(B_{DR}) =^{df} Fil^{j}(B_{dR})/Fil^{j+1}(B_{dR}) ≃ ℂ_{ℓ}(j).

If V is a finite-dimensional continuous ℚ̄_{ℓ}-representation of Gal(ℚ̄_{ℓ}/K), we let

D_{dR}(V) =^{df} (V ⊗_{ℚ_{ℓ}} B_{dR})^{Gal(ℚ̄_{ℓ}/K)}.

This is a module over ℚ̄_{ℓ} ⊗_{ℚ_{ℓ}} K, inheriting a filtration from B_{dR}. We say that V is de Rham if this module is free of rank dimℚ̄_{ℓ}(V). In this case, for each embedding τ : K → ℚ̄_{ℓ} we introduce a multiset of integers HT_{τ} (V). It contains dimℚ̄_{ℓ}(V) elements, and j occurs with multiplicity equal to the dimension of

gr^{j}(V ⊗_{τ,K} B_{dR})^{Gal(ℚ̄_{ℓ}/K)} = gr^{j}D_{dR}(V) ⊗_{ℚ̄_{ℓ}⊗_{ℚ_{ℓ}} K,1⊗τ} ℚ̄_{ℓ}.

over ℚ̄_{ℓ}. If this is nonzero, j is called a Hodge-T ate weight for V relative to the embedding τ. Now we specialize the discussion, and take V to be the restriction of ρ_{Π,ι} as above. We will quote a result from [HT], as stated in [Har], relating the Hodge-Tate weights to the highest weights of the algebraic representation V^{∗}. Recall, this is the irreducible algebraic representation of G(ℂ) such that the tensor product Π_{∞} ⊗ V^{∗} has cohomology. By the definition of the ℚ-group G,

G(ℝ) = ∏_{σ∈Σ} GL_{n}(E ⊗_{F,σ} ℝ), G(ℂ) = ∏_{σ∈Σ} GL_{n}(E ⊗_{F,σ} ℂ),

where Σ is the set of embeddings σ : F → ℝ. For each such σ, following [Har], we let {σ̃, σ̃^{c}} denote the two complex embeddings of E extending it. We write

V^{∗} = ⊗_{σ∈Σ}V_{σ}^{∗}, V_{σ}^{∗} = V∗ ; σ̃ ⊗ V_{σ̃^{c}}^{∗}.

Here V∗ ; σ̃ is naturally identified with an irreducible algebraic representation of the group GL_{n}(ℂ), and we consider its highest weight relative to the lower triangular Borel. This is the character of the diagonal torus corresponding to

μ(σ̃) = (μ_{1}(σ̃) ≤ μ_{2}(σ̃) ≤ ⋯ ≤ μ_{n}(σ̃)).

Similarly, we get a dominant n-tuple of integers μ(σ̃^{c}) for V_{σ̃^{c}}^{∗}. It is given by:

μ_{i}(σ̃^{c}) = −μ_{n−i+1}(σ̃),

by the polarization condition. The multisets HT_{τ} for ρ_{Π,ι} are determined by:

Theorem 6. Fix an embedding s : E → ℚ̄_{ℓ}, and let w denote the associated finite place of E above ℓ. Then the Hodge-Tate weights of ρ_{Π,ι} restricted to the Galois group Gal(ℚ̄_{ℓ}/E_{w}) at w, where E_{w} = s(E)^{−}, are all of the form

j = i − μ_{n−i}(ι(s)^{c}), i = 0, …, n − 1.

In particular, the Hodge-Tate weights all occur with multiplicity one.

P roof. This is part 4 of Theorem VII.1.9 on p. 227 in [HT], but with the normalization used in [Har] in (2.6) on p. 5: The shift from ι(s) to ι(s)^{c} reflects the fact that we work with the dual of the Π in [HT]. Note that the inequalities on p. 3 in [Har] should be reversed. □

4.3 Patching

The next key step is to descend the family ρ_{Π_{E}(χ),ι} ⊗ χ^{−1} to the base field F. This is done by a patching argument, used in various guises by other authors. For example, see Proposition 4.3.1 in [BRo], or section 4.3 in [BRa]. Here we will use a variant of Proposition 1.1 in [Har], which in turn is based on the discussion on p. 230-231 in [HT]. The proof in [Har] is somewhat brief, and somewhat imprecise at the end, so we decided to include a more detailed proof below. Hopefully, this might serve as a convenient reference. In this section, we use Γ_{F} as shorthand notation for the absolute Galois group Gal(F̄/F). The setup is the following: We let I be a set of cyclic Galois extensions E, of a fixed number field F, of prime degree q_{E}. For every E ∈ I we assume we are given an n-dimensional continuous semisimple ℓ-adic Galois representation over E,

ρ_{E} : Γ_{E} → GL_{n}(ℚ̄_{ℓ}).

Here ℓ is a fixed prime. The family of representations {ρ_{E}} is assumed to satisfy:

(a) \underline{Galois invariance}: ρ_{E}^{σ} ≃ ρ_{E}, ∀σ ∈ Gal(E/F),

(b) \underline{Compatibility}: ρ_{E}|_{Γ_{EE′}} ≃ ρ_{E′}|_{Γ_{EE′}},

for all E and E′ in I. These conditions are certainly necessary for the ρ_{E} to be of the form ρ|_{Γ_{E}} for a representation ρ of Γ_{F}. What we will show, is that in fact (a) and (b) are also sufficient conditions if I is large enough. That is,

Definition 1. Following [Har], for a finite set S of places of F, we say that I is S-general if and only if the following holds: For any finite place v ∉ S, and any finite extension M of F, there is an E ∈ I linearly disjoint from M such that v splits completely in E. In this case, there will be infinitely many such E.

Recall that since E is Galois over F, it is linearly disjoint from M precisely when E ∩ M = F. Moreover, it is of prime degree, so this just means E is not contained in M. Hence, I being S-general is equivalent to: For v ∉ S, there are infinitely many E ∈ I in which v splits. A slightly stronger condition is:

Definition 2. We say that I is strongly S-general if and only if the following holds: For any finite set Σ of places of F, disjoint from S, there is an E ∈ I in which every v ∈ Σ splits completely.

To see that this is indeed stronger, we follow Remark 1.3 in [Har]: Fix a finite place v ∉ S, and a finite extension M of F. Clearly we may assume M ≠ F is Galois. Let {M_{i}} be the subfields of M, Galois over F, with a simple Galois group. For each i we then choose a place v_{i} of F, not in S, which does not split completely in M_{i}. We take Σ to be {v, v_{i}} in the above definition, and get an E ∈ I in which v and every v_{i} splits. If E was contained in M, it would be one of the M_{i}, but this contradicts the choice of v_{i}. Thus, E and M are disjoint.

Example. Let Σ = {p_{i}} be a finite set of primes. As is well-known, for odd p_{i},

p_{i} splits in ℚ(\sqrt{d}) ⟺ p_{i} ∤ d and (\frac{d}{p_{i}} ) = 1.

Here d is any square-free integer. Moreover, 2 splits when d ≡ 1 mod 8. The set of all integers d satisfying the congruences d ≡ 1 mod p_{i}, for all i, form an arithmetic progression. By Dirichlet’s Theorem, it contains infinitely many primes. Therefore, the following family of imaginary quadratic extensions

I = \{ℚ(\sqrt{−p}): almost all primes p\}

is strongly ∅-general. This gives rise to a similar family of CM extensions of any given totally real field F, by taking the set of all the composite fields F I.

The main result of this section, is a strengthening of Proposition 1.1 in [Har]:

Lemma 1. Let I be an S-general set of extensions E over F, of prime degree q_{E}, and let ρ_{E} be a family of semisimple Galois representations satisfying the conditions (a) and (b) above. Then there is a continuous semisimple

ρ : Γ_{F} → GL_{n}(ℚ̄_{ℓ}), ρ|_{Γ_{E}} ≃ ρ_{E},

for all E ∈ I. This determines the representation ρ uniquely up to isomorphism.

P roof. The proof below is strongly influenced by the proofs of Proposition 1.1 in [Har], and of Theorem VII.1.9 in [HT]. We simply include more details and clarifications. The proof is quite long and technical, so we divide it into several steps. Before we construct ρ, we start off with noting that it is necessarily unique: Indeed, for any place v ∉ S, we find an E ∈ I in which v splits. In particular, E_{w} = F_{v} for all places w of E dividing v. Thus, all the restrictions ρ|_{Γ_{Fv}} are uniquely determined. We conclude that ρ is unique, by the Cebotarev Density Theorem. For the construction of ρ, we first establish some notation used throughout the proof: We fix an arbitrary base point E_{0} ∈ I, and abbreviate

ρ_{0} =^{df} ρ_{E_{0}}, Γ_{0} =^{df} Γ_{E_{0}}, G_{0} =^{df} Gal(E_{0}/F), q_{0} =^{df} q_{E_{0}}.

We let H denote the Zariski closure of ρ_{0}(Γ_{0}) inside GL_{n}(ℚ̄_{ℓ}), and consider its identity component H^{∘}. Define M to be the finite Galois extension of E_{0} with

Γ_{M} = ρ_{0}^{−1}(H^{∘}), Gal(M/E_{0}) = π_{0}(H).

Let T be the set of isomorphism classes of irreducible constituents of ρ_{0}, ignoring multiplicities. By property (a), the group G_{0} acts on T from the right. We note that τ and τ^{σ} occur in ρ_{0} with the same multiplicity. We want to describe the G_{0}-orbits on T. First, we have the set P of fixed points τ = τ^{σ} for all σ.

The set of non-trivial orbits is denoted by C. Note that any c ∈ C has prime cardinality q_{0}. For each such c, we pick a representative τ_{c} ∈ T, and let C_{0} be the set of all these representatives {τ_{c}}. Each τ ∈ C_{0} obviously has a trivial stabilizer in G_{0}.

Step 1: The extensions of ρ_{0} to Γ_{F}.

Firstly, a standard argument shows that each τ ∈ P has an extension τ̃ to Γ_{F}. This uses the divisibility of ℚ̄_{ℓ}^{∗}, in order to find a suitable intertwining operator τ ≃ τ^{σ}. All the other extensions are then obtained from τ̃ as unique twists:

τ̃ ⊗ η, η ∈ Ĝ_{0}.

Here Ĝ_{0} is the group of characters of G_{0}. Secondly, for a τ ∈ c, we introduce

τ̃ =^{df} Ind_{Γ_{0}}^{Γ_{F}} (τ).

Since τ is not Galois-invariant, this τ̃ is irreducible. It depends only on the orbit c containing τ, and it is invariant under twisting by Ĝ_{0}. It has restriction

τ̃|_{Γ_{0}} ≃ ⨁_{σ∈G_{0}}τ^{σ}.

If we let m_{τ} denote the multiplicity with which τ ∈ T occurs in ρ_{0}, we get that

{ ⨁_{τ∈C_{0}} m_{τ} · τ̃} ⊕ {⨁_{τ∈P} ⨁_{η∈Ĝ_{0}}m_{τ̃,η} · (τ̃ ⊗ η)}

is an extension of ρ_{0} to Γ_{F} for all choices of non-negative m_{τ̃,η} ∈ ℤ such that

∑_{η∈Ĝ_{0}}m_{τ̃,η} = m_{τ}

for every fixed τ ∈ P.

Step 2: ρ_{0}(Γ_{NE_{0}}) is dense in H, when N is linearly disjoint from M over F.

To see this, let us momentarily denote the Zariski closure of ρ_{0}(Γ_{NE_{0}}) by H_{N}. N is a finite extension, so H_{N} has finite index in H. Consequently, we deduce that H_{N}^{∘} = H^{∘}. Now, NE_{0} and M are linearly disjoint over E_{0}, and therefore

Γ_{0} = Γ_{NE_{0}} · Γ_{M} ⟹ ρ_{0}(Γ_{0}) ⊂ ρ_{0}(Γ_{NE_{0}}) · H^{∘} ⊂ H_{N}.

Taking the closure, we obtain that H_{N} = H.

Step 3: If N is a finite extension of F, linearly disjoint from M over F. Then:

(1) τ|_{Γ_{NE_{0}}} is irreducible, for all τ ∈ T.

(2) τ̃|_{Γ_{N}} is irreducible, for all τ ∈ P.

(3) τ|_{Γ_{NE_{0}}} ≃ τ′|_{Γ_{NE_{0}}} ⇒ τ ≃ τ′, for all τ, τ′ ∈ T.

(4) τ̃|_{Γ_{N}} ≃ (τ̃′ ⊗ η)|_{Γ_{N}} ⇒ τ ≃ τ′ and η = 1, for all τ, τ′ ∈ P and η ∈ Ĝ_{0}.

Parts (1) and (3) follow immediately from Step 2, and obviously (1) implies (2). Also, part (3) immediately implies that τ ≃ τ′ in (4). Suppose η ∈ Ĝ_{0} satisfies:

τ̃|_{Γ_{N}} ≃ τ̃|_{Γ_{N}} ⊗ η|_{Γ_{N}}

for some τ ∈ P. In other words, η|_{Γ_{N}} occurs in End_{Γ_{NE_{0}}} (τ̃|_{Γ_{N}}), which is trivial by part (1). So, η is trivial on Γ_{N} and on Γ_{0}. Hence, η = 1 by disjointness.

Step 4: τ̃|_{Γ_{N}} is irreducible for all τ ∈ C_{0}. That is, part (2) holds for all τ ∈ T.

Since N and E_{0} are linearly disjoint over F, we see that Γ_{F} = Γ_{E} · Γ_{0}. Hence,

τ̃|_{Γ_{N}} = Ind_{Γ_{0}}^{Γ_{F}} (τ)|_{Γ_{N}} ≃ Ind_{Γ_{NE_{0}}}^{Γ_{N}} (τ|_{Γ_{NE_{0}}} ),

by Mackey theory. Now, τ|_{Γ_{NE_{0}}} is irreducible and not Galois-invariant.

Step 5: Suppose E ∈ I is linearly disjoint from M over F. Then, for a unique choice of non-negative integers m_{τ̃,η,E} with η-sum m_{τ}, we have the formula:

ρ_{E} ≃ { ⨁_{τ∈C_{0}} m_{τ} · τ̃|_{Γ_{E}}} ⊕ {⨁_{τ∈P} ⨁_{η∈Ĝ_{0}}m_{τ̃,η,E} · (τ̃ ⊗ η)|_{Γ_{E}}}.

In particular, ρ_{0} and ρ_{E} have a \underline{common} extension to Γ_{F}.

The uniqueness of the m_{τ̃,η,E} follows directly from part (4) in Step 3. Recall,

ρ_{E}|_{Γ_{EE_{0}}} ≃ ρ_{0}|_{Γ_{EE_{0}}} ≃ { ⨁_{τ∈C_{0}} m_{τ} · ⨁_{σ∈G_{0}}τ|_{Γ_{EE_{0}}}^{σ} } ⊕ {⨁_{τ∈P} m_{τ} · τ|_{Γ_{EE_{0}}} },

by the compatibility condition (b). Here all the τ|_{Γ_{EE_{0}}}^{σ} are distinct by (3). First, let us pick an arbitrary τ ∈ P. As representations of G_{0}, viewed as the Galois group of EE_{0} over E by disjointness, we have

Hom_{Γ_{EE_{0}}} (τ̃|_{Γ_{E}}, ρ_{E}) ≃ ⨁_{η∈Ĝ_{0}} dimℚ̄_{ℓ} Hom_{Γ_{E}}((τ̃ ⊗ η)|_{Γ_{E}}, ρ_{E}) · η.

The ℚ̄_{ℓ}-dimension of the left-hand side clearly equals m_{τ}, and the right-hand side defines the partition m_{τ̃,η,E} of m_{τ}. Next, let us pick an arbitrary τ ∈ C_{0}. By the same argument, using that τ̃ is invariant under twisting by Ĝ_{0}, we get:

Hom_{Γ_{EE_{0}}} (τ̃|_{Γ_{E}}, ρ_{E}) ≃ dimℚ̄_{ℓ} Hom_{Γ_{E}}(τ̃|_{Γ_{E}}, ρ_{E}) · ⨁_{η∈Ĝ_{0}}η.

Now the left-hand side obviously has dimension m_{τ} q_{0}. We deduce that τ̃|_{Γ_{E}} occurs in ρ_{E} with multiplicity m_{τ}. Counting dimensions, we obtain the desired decomposition of ρ_{E}. Note that we have not used the Galois invariance of ρ_{E}. In fact, it is a consequence of the above argument, assuming E ∩ M = F.

Step 6: Fix an E_{1} ∈ I disjoint from M over F. Introduce the representation

ρ =^{df} { ⨁_{τ∈C_{0}} m_{τ} · τ̃} ⊕ {⨁_{τ∈P} ⨁_{η∈Ĝ_{0}}m_{τ̃,η,E_{1}} · (τ̃ ⊗ η)}.

Then ρ|_{Γ_{E}} ≃ ρ_{E} for all extensions E ∈ I linearly disjoint from ME_{1} over F.

By definition, and Step 5, we have that ρ|_{Γ_{E_{1}}} ≃ ρ_{E_{1}}. Take E ∈ I to be any extension, disjoint from ME_{1} over F. We compare the decomposition of ρ|_{Γ_{E}},

ρ|_{Γ_{E}} = { ⨁_{τ∈C_{0}} m_{τ} · τ̃|_{Γ_{E}}} ⊕ {⨁_{τ∈P} ⨁_{η∈Ĝ_{0}}m_{τ̃,η,E_{1}} · (τ̃ ⊗ η)|_{Γ_{E}}},

to the decomposition of ρ_{E} in Step 5. We need to show the multiplicities match:

m_{τ̃,η,E} = m_{τ̃,η,E_{1}}, ∀τ ∈ P, ∀η ∈ Ĝ_{0}.

By property (b), for the pair {E, E_{1}}, we know that ρ|_{Γ_{E}} and ρ_{E} become isomorphic after restriction to Γ_{EE_{1}}. Once we prove EE_{1} is linearly disjoint from M over F, we are done by (2) and (4). The disjointness folllows immediately:

EE_{1} ⊗_{F} M ≃ E ⊗_{F} E_{1} ⊗_{F} M ≃ E ⊗_{F} ME_{1} ≃ EE_{1}M.

Step 7: ρ|_{Γ_{E}} ≃ ρ_{E} for all E ∈ I.

By Step 6, we may assume E ∈ I is contained in ME_{1}. Now take an auxiliary extension E ∈ I linearly disjoint from ME_{1} over F. Consequently, using (b),

ρ|_{Γ_{E}} ≃ ρ_{E} ⇒ ρ|_{Γ_{EE}} ≃ ρ_{E}|_{Γ_{EE}} ≃ ρ_{E}|_{Γ_{EE}} .

Thus, ρ|_{Γ_{E}} agrees with ρ_{E} when restricted to Γ_{EE}. It suffices to show that the union of these subgroups Γ_{EE}, as E varies, is dense in Γ_{E}. Again, we invoke the Cebotarev Density Theorem. Indeed, let w be a place of E, lying above v ∉ S. It is then enough to find an E ∈ I, as above, such that w splits completely in EE. Then Γ_{E_{w}} is contained in Γ_{EE}. We know, by the S-generality of I, that we can find an E ∈ I, not contained in ME_{1}, in which v splits completely. This E works: This follows from elementary splitting theory, as E and E are disjoint.

This finishes the proof of the patching lemma. □

Remark. From the proof above, we infer the following concrete description of the patch-up representation ρ. First fix any E_{0} ∈ I, and let P be the set of Galois-invariant constituents τ of ρ_{E_{0}}. For each such τ, we fix an extension τ̃ to F once and for all. Furthermore, let C_{0} be a set of representatives for the non-trivial Galois orbits of constituents of ρ_{E_{0}}. Then ρ is of the following form

ρ ≃ { ⨁_{τ∈C_{0}} m_{τ} · Ind_{Γ_{0}}^{Γ_{F}} (τ)} ⊕ {⨁_{τ∈P} ⨁_{η∈Gal(E_{0}/F)^{∧}}m_{τ̃,η} · (τ̃ ⊗ η)}.

Here the m_{τ̃,η} are some non-negative integers with η-sum m_{τ}, the multiplicity of τ in ρ_{E_{0}}. This fairly explicit description may be useful in deriving properties of ρ from those of ρ_{E_{0}}.

For future reference, we finish this section with a few remarks on the generalization of the patching lemma to solvable extensions. Thus, I now denotes a collection of solvable Galois extensions E over F, and we assume we are given Galois representations ρ_{E}, as above, satisfying (a) and (b). For any L over F,

I_{L} =^{df} {E ∈ I : L ⊂ E}.

Loosely speaking, we say that I is S-general if it is S-general in prime layers:

Definition 3. Following [Har], for a finite set S of places of F, we say that I is S-general if and only if the following holds: For every L such that I_{L} ≠ ∅,

{prime degree extensions K/L, with I_{K} ≠ ∅}

is S(L)-general in the previous sense. S(L) denotes the places of L above S.

From now on, we will make the additional hypothesis that all the extensions E ∈ I have uniformly bounded heights. That is, there is an integer H_{I} such that every index [E : F] has at most H_{I} prime divisors (not necessarily distinct).

Lemma 2. Assume the collection I has uniformly bounded heights. Then I is S-general if and only if the following condition holds for every L with I_{L} ≠ ∅: Given a finite place w ∉ S(L) and a finite extension M over L, there is an extension E ∈ I_{L} linearly disjoint from M over L, in which w splits completely.

P roof. The if part follows immediately by unraveling the definitions. The only if part is proved by induction on the maximal height of the collection I_{L} over L, the height one case being the definition. Suppose I_{L} has maximal height H, and assume the lemma holds for smaller heights. Let w and M be as above. By S-generality, there is a prime degree extension K over L with I_{K} ≠ ∅, disjoint from M over L, in which w splits. Fix a place w̃ of K above w. Now, I_{K} clearly has maximal height less than H. By the induction hypothesis there is an E ∈ I_{K}, disjoint from MK over K, in which w̃ splits. This E works. □

Under the above assumptions on I, a given place w ∉ S(L) splits completely in infinitely many E ∈ I_{L}, unless L belongs to I. One has a stronger notion:

Definition 4. We say that I is strongly S-general if and only if the following holds: For any L such that I_{L} ≠ ∅, and any finite set Σ of places of L disjoint from S(L), there is an E ∈ I_{L} in which every v ∈ Σ splits completely.

As in the prime degree case, treated above, one shows that this is indeed a stronger condition. Our next goal is to prove the following generalization of the patching lemma to certain collections of solvable extensions:

Theorem 7. Let I be an S-general collection of solvable Galois extensions E over F, with uniformly bounded heights, and let ρ_{E} be a family of n-dimensional continuous semisimple ℓ-adic Galois representations satisfying the conditions (a) and (b) above. Then there is a continuous semisimple representation

ρ : Γ_{F} → GL_{n}(ℚ̄_{ℓ}), ρ|_{Γ_{E}} ≃ ρ_{E},

for all E ∈ I. This determines the representation ρ uniquely up to isomorphism.

P roof. Uniqueness is proved by paraphrasing the argument in the prime degree situation. The existence of ρ is proved by induction on the maximal height of I over F, the height one case being the previous patching lemma. Suppose I has maximal height H, and assume the Theorem holds for smaller heights. Take an arbitrary prime degree extension K over F, with I_{K} ≠ ∅. Clearly I_{K} is an S(K)-general set of solvable Galois extensions of K, of maximal height strictly smaller than H. Moreover, the subfamily {ρ_{E}}_{E∈I_{K}} obviously satisfies (a) and (b). By induction, we find a continuous semisimple ℓ-adic representation

ρ_{K} : Γ_{K} → GL_{n}(ℚ̄_{ℓ}), ρ_{K}|_{Γ_{E}} ≃ ρ_{E},

for all E ∈ I_{K}. We then wish to apply the prime degree patching lemma to the family {ρ_{K}}, as K varies over extensions as above. By definition, such K do form an S-general collection over F. It remains to show that {ρ_{K}} satisfies (a) and (b). To check property (a), take any σ ∈ Γ_{F}, and note that ρ_{K}^{σ} agrees with ρ_{K} after restriction to Γ_{E} for an arbitrary extension E ∈ I_{K}. The union of these Γ_{E} is dense in Γ_{K} by the Cebotarev Density Theorem: Every place w of K, outside S(K), splits in some E ∈ I_{K}, so the union contains Γ_{K_{w}}. To check property (b), fix prime degree extensions K and K′ as above. Note that

(ρ_{K}|_{Γ_{KK′}})|_{Γ_{EE′}} ≃ (ρ_{K′}|_{Γ_{KK′}})|_{Γ_{EE′}}, ∀E ∈ I_{K}, ∀E′ ∈ I_{K′}.

We finish the proof by showing that the union of these Γ_{EE′} is dense in Γ_{KK′}. Let w be an arbitrary place of KK′ such that w|_{F} does not lie in S. Choose an extension E ∈ I_{K} linearly disjoint from KK′ over K, in which w|_{K} splits. Then pick an extension E′ ∈ I_{K′} linearly disjoint from EK′ over K′, in which w|_{K′} splits. By elementary splitting theory, w splits in EK′, and any place of EK′ above w splits in EE′. Consequently, w splits in EE′, see the diagram:

[figure: EE′; EK′; E′; E; KK′; K; K′]

The union then contains the Galois group of (KK′)_{w}. Done by Cebotarev. □

The previous result should be compared to Corollary 1.2 in [Har].

4.4 Galois representations associated to π

Let π be the globally generic cusp form on GSp(4) introduced earlier, Π its lift to GL(4), and let Π_{E} be the base change of Π to GL(4) over a CM extension E of F. Recall that, for certain algebraic Hecke characters χ of E, the twisted representation Π_{E}(χ) is conjugate self-dual. We consider the representations

ρ_{π,ι,E} =^{df} ρ_{Π_{E}(χ),ι} ⊗ ρ_{χ̌,ι}.

Up to isomorphism, this is independent of χ. Indeed, for each place w ∤ ℓ of E,

ιWD(ρ_{π,ι,E}|_{W_{Ew}})^{F −ss} ≃ rec_{4}(Π_{E,w} ⊗ | det |^{− \frac{3}{2}} ).

We only consider CM extensions E, in which v_{0} splits, such that Π_{E} is cuspidal. Here v_{0} is the place of F where π_{v_{0}} is of Steinberg type. This collection I is certainly strongly ∅-general, according to the example in the previous section. Moreover, the family of 4-dimensional Galois representations ρ_{π,ι,E} satisfies the patching conditions (a) and (b). For example, to check the Galois invariance,

ρ_{π,ι,E}^{θ} ≃ ρ_{Π_{E}(χ)^{θ},ι} ⊗ ρ_{χ̌^{θ},ι} ≃ ρ_{Π_{E}(χ)^{∨},ι} ⊗ ρ_{χ̌^{θ},ι} ≃ ρ_{Π_{E}(χ^{θ}),ι} ⊗ ρ_{χ̌^{θ},ι},

by our choice of χ. Taking χ^{θ} instead of χ, then shows that ρ_{π,ι,E}^{θ} is isomorphic to ρ_{π,ι,E} by the aforementioned independence. Alternatively, one can use the local description of ρ_{π,ι,E} above at the unramified places, and the fact that Π_{E} is a base change from F. To check the compatibility, note that for w ∤ ℓ,

ιWD(ρ_{π,ι,E}|_{W_{(EE′)w}})^{F −ss} ≃ rec_{4}(Π_{v,(EE′)_{w}} ⊗ | det |^{− \frac{3}{2}} ),

and similarly for ρ_{π,ι,E′}. See Lemma VII.2.6 in [HT]. Now (b) follows from Cebotarev. By the patching lemma, we finally get a continuous representation

ρ_{π,ι} : Γ_{F} → GL_{4}(ℚ̄_{ℓ}), ρ_{π,ι}|_{Γ_{E}} ⊗ ρ_{χ,ι} ≃ ρ_{Π_{E}(χ),ι}.

It is irreducible, since ρ_{Π_{E}(χ),ι} is known to be irreducible [TY], and satisfies:

ιWD(ρ_{π,ι}|_{W_{Fv}})^{F −ss} ≃ rec_{GT}(π_{v} ⊗ |c|^{− \frac{3}{2}} ),

at each finite place v ∤ ℓ of F. Here rec_{GT} is the local Langlands correspondence for GSp(4), as defined by Gan and Takeda in [GT]. To see this, pick any E in which v splits, and use the local description of ρ_{π,ι,E} together with the fact that Π is a strong lift of π. From the list of properties of ρ_{Π_{E}(χ),ι}, we then read off:

• Let v ∤ ℓ be a finite place of F, and let α be an eigenvalue of ρ_{π,ι}(σ) for some σ ∈ W_{F_{v}}. Then α belongs to ℚ̄, and for every embedding ℚ̄ ↪ ℂ,

|α| ∈ qv^{\frac{ℤ}{2}} .

• Let v ∤ ℓ be a finite place, with π_{v} unramified, and let α be an eigenvalue of ρ_{π,ι}(Frob_{v}). Then α belongs to ℚ̄, and for every embedding ℚ̄ ↪ ℂ,

|α| = qv^{\frac{w+3}{2}} .

• The representation ρ_{π,ι} is potentially semistable at any finite place v|ℓ. Moreover, ρ_{π,ι} is crystalline at a finite place v|ℓ when π_{v} is unramified.

For the second part, we recall that χ is an algebraic Hecke character with infinity types z^{a}z̄^{b}, where a + b = w. In particular, for the unitary twist π^{∘} we have:

L_{v}(s − \frac{1}{2}(w + 3), π^{∘}, spin) = det(1 − ιρ_{π,ι}(Frob_{v}) · q_{v}^{−s})^{−1}

at all places v ∤ ℓ where π^{∘} is unramified. Note that, by twisting ρ_{π,ι} with integral powers of the cyclotomic character χ_{cyc}, we may alter the motivic weight w+3 by any even integer. We compare with the motivic weight k_{1}+k_{2}−3 in [Wei].

Temperedness of π^{∘}: From the above, it follows immediately that π^{∘} has unitary Satake parameters at all places v ∤ ℓ where π^{∘} is unramified. In fact, π^{∘} is tempered at every place v: Indeed, by Corollary VII.1.11 in [HT], we know that Π_{E} is essentially tempered everywhere. That is, ϕ_{π_{v}}|_{W_{Ew}} has bounded image in GL_{4}(ℂ) for every finite place w of E. Consequently, the same holds for ϕ_{π_{v}}|_{W_{Fv}}.

The image of ρ_{π,ι}: Since the eigenvalues of ρ_{π,ι}(Frob_{v}) coincide with the integral Satake parameters of π_{v}, for finite v ∤ ℓ where π_{v} is unramified, Cebotarev yields:

ρ_{π,ι}^{∨} ≃ ρ_{π,ι} ⊗ χ^{−1}, χ =^{df} ω_{π^{∘}} · χ_{cyc}^{−w−3},

where we confuse ω_{π^{∘}} with its finite order ℓ-adic avatar. In other words, the space of ρ_{π,ι} has a non-degenerate bilinear form preserved by Γ_{F} with similitude χ. We have already observed that ρ_{π,ι} is irreducible, so by Schur’s lemma this bilinear form must be symmetric or symplectic. Thus, the image of ρ_{π,ι} can always be conjugated into GO(4) or GSp(4). Under our running assumptions on π, in fact into the latter: Otherwise, by local-global compatibility at the place v_{0} ∤ ℓ, the L-parameter of Π_{v_{0}} is of orthogonal type. That is, it maps

rec(Π_{v_{0}}) : W′_{F_{v_{0}}} = W_{F_{v_{0}}} × SL(2, ℂ) → GO(4, ℂ).

However, Π_{v_{0}} is the transfer of π_{v_{0}}, so rec(Π_{v_{0}}) also preserves a symplectic form on ℂ^{4}. Now, Π_{v_{0}} is a generalized Steinberg representation, and one verifies that

Z_{GL(4,ℂ)}(im(rec(Π_{v_{0}}))) = ℂ^{∗}

by an easy computation. Indeed, rec(Π_{v_{0}}) is of the form ϕ ⊠ S_{d}, where ϕ is an irreducible representation of W_{F_{v_{0}}} , and S_{d} is the d-dimensional irreducible representation of SL(2, ℂ). Ergo, the above symplectic form must agree with the orthogonal form up to a scalar. This is a contradiction. The symplecticity of ρ_{π,ι}, just shown, is a special case of Theorem F on p. 6 in [CCl] when ω_{π} is trivial. This result from [CCl] has recently been generalized to the CM case in [BCh]. When F = ℚ, the symplecticity of ρ_{π,ι} is shown in [Wei], for globally generic π, using Poincare duality. Indeed, by [Sou], π occurs with multiplicity one, so ρ_{π,ι} can be realized as the π_{f}-isotypic component of H^{3} of a Siegel threefold. The cup product pairing then provides the desired symplectic form.

Baire category theory: To check that the image of ρ_{π,ι} is in fact contained in GSp_{4}(L), for some finite extension L over ℚ_{ℓ}, we invoke the Baire Category Theorem: Every locally compact Hausdorff space is a Baire space (that is, the union of any countable collection of closed sets with empty interior has empty interior). We will apply it to the compact subgroup ρ_{π,ι}(Γ_{F}) inside GSp_{4}(ℚ̄_{ℓ}).

ρ_{π,ι}(Γ_{F}) = ⋃_{L/ℚ_{ℓ} finite}ρ_{π,ι}(Γ_{F}) ∩ GSp_{4}(L)

is a countable union of closed subgroups, since each L is complete. Therefore,

ρ_{π,ι}(Γ_{F}) ∩ GSp_{4}(L) has non-empty interior,

for some L, and hence this is an open subgroup. That is, the image of Γ_{M} for some finite extension M over F. In particular, it has finite index in ρ_{π,ι}(Γ_{F}). By enlarging L, to accomodate the finitely many coset representatives, we can arrange for the image of ρ_{π,ι} to be contained in the L-rational points GSp_{4}(L).

Total oddness: χ(c) = −1 for every complex conjugation c ∈ Γ_{F} from ℚ̄ ↪ ℂ.

Hodge-Tate weights: Let us fix an embedding s : F → ℚ̄_{ℓ}, and let v be the associated place of F above ℓ. We wish to compute the Hodge-Tate weights of ρ_{π,ι} restricted to Γ_{F_{v}}, where F_{v} = s(F)^{−}. That is, for each integer j, evaluate

dimℚ̄_{ℓ} gr^{j}(ρ_{π,ι} ⊗_{F_{v}} B_{dR})^{Γ_{Fv}} .

We will reduce this to the analogous result for ρ_{Π_{E}(χ),ι} already mentioned. Thus, we fix a CM extension E, in which v splits. Once and for all, we fix a divisor w of v, and look at a corresponding embedding s̃: E → ℚ̄_{ℓ} over s. This canonically identifies E_{w} = s̃(E)^{−} with F_{v}. Now note that, for characters χ as above,

ρ_{π,ι}|_{Γ_{Fv}} ⊗ ρ_{χ,ι}|_{Γ_{Ew}} ≃ ρ_{Π_{E}(χ),ι}|_{Γ_{Ew}} .

Therefore, we first record the Hodge-Tate weight of ρ_{χ,ι}|_{Γ_{Ew}}. The associated complex embedding ι(s̃) defines an infinite place of E, where χ has the form z^{a}z̄^{b}. As is well documented elsewhere in the literature, for example in [Bla], the Hodge-Tate weight is then −b, with our choice of normalization. Therefore,

dimℚ̄_{ℓ} gr^{j}(ρ_{π,ι} ⊗_{F_{v}} B_{dR})^{Γ_{Fv}} = dimℚ̄_{ℓ} gr^{j−b}(ρ_{Π_{E}(χ),ι} ⊗_{E_{w}} B_{dR})^{Γ_{Ew}} ,

since D_{dR} is a ⊗-functor. It remains to find the highest weight of the V with

H^{•}(𝔤, K; Π_{E}(χ)_{∞} ⊗ V^{∗}) ≠ 0.

More precisely, we let σ = ι(s) ∈ Σ, and consider the two complex embeddings {σ̃, σ̃^{c}} of E extending σ. Here σ̃ = ι(s̃). In our earlier notation, we need to compute the quadruple μ(σ̃^{c}). For this, we follow the proof of Lemma 3.14 on p. 114 in [Cl2]: We consider the local component of Π_{E}(χ) at the infinite place of E above σ. We know its L-parameter, so according to p. 113 in [Cl2]:

\begin{cases}μ_{1}(σ̃^{c}) = b + 3 − \frac{1}{2}(w + n′), \\ μ_{2}(σ̃^{c}) = b + 2 − \frac{1}{2}(w + n), \\ μ_{3}(σ̃^{c}) = b + 1 − \frac{1}{2}(w − n), \\ μ_{4}(σ̃^{c}) = b + 0 − \frac{1}{2}(w − n′). \end{cases}

Here we have introduced n = ν_{1} − ν_{2} and n′ = ν_{1} + ν_{2}. Moreover, the motivic weight w + 3 is denoted by w. From the above, and the result from section 4.2, we deduce that the Hodge-Tate weights of ρ_{π,ι}|_{Γ_{Fv}} are given by the sequence:

\frac{1}{2}(w − n′) < \frac{1}{2}(w − n) < \frac{1}{2}(w + n) < \frac{1}{2}(w + n′).

In particular, they are distinct. We will rewrite this slightly. For each σ ∈ Σ,

δ = δ(σ) =^{df} \frac{1}{2}(w − n′) = \frac{1}{2}(w − μ_{1} − μ_{2}) ∈ ℤ.

With this notation, the set of Hodge-Tate weights takes the following form:

HT(ρ_{π,ι}|_{Γ_{Fv}}) = {δ, ν_{2} + δ, ν_{1} + δ, ν_{1} + ν_{2} + δ}.

In the case F = ℚ it is customary to take w = k_{1} + k_{2} − 3, that is, δ = 0. In this case, we recover the Hodge types given in Theorem III on p. 2 in [Wei].

4.5 Consequences of local-global compatibility

Parahoric subgroups: We fix a finite place v of F, and define certain compact open subgroups of GSp_{4}(F_{v}), known as the parahoric subgroups. They arise as stabilizers of points in the Bruhat-Tits builidng. We refer to [Tit] for a general discussion. First, we have the hyperspecial maximal compact subgroup

K = K_{v} =^{df} GSp_{4}(O_{v}).

Inside of it, we have the pullbacks of the two parabolics via the reduction map:

\begin{cases}J_{P} = J_{P,v} =^{df} \{k ∈ K: k (mod v) ∈ P (𝔽_{v})\}, \\ J_{Q} = J_{Q,v} =^{df} \{k ∈ K: k (mod v) ∈ Q(𝔽_{v})\}. \end{cases}

They are usually called the Siegel and Klingen parahoric, respectively. Moreover,

I = I_{v} =^{df} J_{P} ∩ J_{Q} = {k ∈ K: k (mod v) ∈ B(𝔽_{v})}

is called the (upper-triangular) Iwahori subgroup. Furthermore, let us introduce

η = η_{v} =^{df} (\begin{matrix}& & 1 \\ & & & 1 \\ ϖ_{v} \\ & ϖ_{v} \end{matrix}), c(η) = −ϖ_{v}, η^{2} = ϖ_{v} · I_{4}.

This η is occasionally referred to as the Atkin-Lehner element. It depends on the choice of a uniformizer ϖ_{v}, but it is well-defined modulo the maximal compact subgroup of the torus T_{c}. An explicit calculation shows that the η-conjugate ηKη^{−1} is another hyperspecial maximal compact subgroup containing the Siegel parahoric. There is one more maximal compact subgroup containing I. Namely,

K̃ = K̃_{v} =^{df} subgroup generated by J_{Q} and its η-conjugate.

This is the paramodular, or non-special, subgroup. We have thus described all parahoric subgroups of GSp_{4}(F_{v}) up to conjugacy. Note that η normalizes K̃ and J_{P}, hence I. Therefore, if J is any one of these subgroups, and π is an irreducible admissible representation, π(η) is a well-defined operator on the Jinvariants π^{J}. Its square is multplication by ω_{π}(ϖ_{v}). In particular, if ω_{π} = 1, the eigenvalues of π(η) are ±1. These are tabulated in table 3 on p. 16 in [Sch].

Nilpotent orbits: Suppose π is an irreducible representation of GSp_{4}(F_{v}), assumed to be Iwahori-spherical. That is, π^{I} ≠ 0. Then its L-parameter is given by a semisimple element in GSp_{4}(ℂ) plus a compatible nilpotent element N in the Lie algebra 𝔤𝔰𝔭_{4}(ℂ), both viewed up to conjugacy. Here we list the finitely many possibilities for N. First, by the theory of the Jordan normal form,

(\begin{matrix}0 & 1 \\ & 0 \\ & & 0 \\ & & & 0 \end{matrix}), (\begin{matrix}0 & 1 \\ & 0 \\ & & 0 & 1 \\ & & & 0 \end{matrix}), (\begin{matrix}0 & 1 \\ & 0 & 1 \\ & & 0 \\ & & & 0 \end{matrix}), (\begin{matrix}0 & 1 \\ & 0 & 1 \\ & & 0 & 1 \\ & & & 0 \end{matrix})

represent the non-trivial nilpotent classes in 𝔤𝔩_{4}(ℂ). An explicit computation verifies that the third representative cannot be conjugated into 𝔤𝔰𝔭_{4}(ℂ). Recall that 𝔰𝔭_{4}(ℂ) consists of X such that JX is symmetric. Here J is the antidiagonal symplectic form fixed throughout the paper. However, the other three representatives are in fact symplectic in this sense: They are conjugate to

N_{1} =^{df} (\begin{matrix}0 \\ & 0 & 1 \\ & & 0 \\ & & & 0 \end{matrix}), N_{2} =^{df} (\begin{matrix}0 & 1 \\ & 0 \\ & & 0 & −1 \\ & & & 0 \end{matrix}), N_{3} =^{df} (\begin{matrix}0 & 1 \\ & 0 & 1 \\ & & 0 & −1 \\ & & & 0 \end{matrix})

respectively. Again, this follows immediately from the theory of normal forms. Each N_{i} is contained in 𝔰𝔭_{4}(ℂ). Note that N_{i} has rank i. It may be useful to observe that N_{1} is a root vector for the long simple root β(t) = 2t_{2}, whereas N_{2} is a root vector for the short simple root α(t) = t_{1} − t_{2}. Their sum is N_{3}. By the Jacobson-Morozov theorem, [Jac, Theorem 3], the N_{i} in fact represent the GSp_{4}(ℂ)-conjugacy classes of nilpotent elements in 𝔤𝔰𝔭_{4}(ℂ). To aid comparison with [Sch], let us make the following remark: For the L-parameter of π to have a more transparent semisimple part, one often takes a different set of representatives, see p. 6 in [Sch]. For example, the two nilpotent elements

N′_{1} =^{df} (\begin{matrix}0 & & & 1 \\ & 0 \\ & & 0 \\ & & & 0 \end{matrix}), N′_{2} =^{df} (\begin{matrix}0 & & & 1 \\ & 0 & 1 \\ & & 0 \\ & & & 0 \end{matrix}),

are GSp_{4}(ℂ)-conjugate to N_{1} and N_{2} respectively. This is easy to check.

Iwahori-spherical generic representations: By a well-known result of Casselman, see [Car] for a nice review, the Iwahori-spherical π are precisely the constituents of unramified principal series. The way they decompose, in the case of GSp_{4}, was written out explicitly in [ST]. One gets 17 families of representations,

I, IIa, IIb, IIIa, IIIb, IV a − IV d, V a − V d, V Ia − V Id,

according to how the unramified principal series breaks up. We refer to pages 6-8 in [Sch] for a precise definition of each class of representations. Fortunately, here we are only interested in the generic representations. That is, the 6 classes

I, IIa, IIIa, IV a, V a, V Ia.

We will briefly recall the definition of each of these classes. To do that, we will first introduce the notation used in [ST]: For three quasi-characters χ_{i}, let

χ_{1} × χ_{2} ⋊ χ_{3} =^{df} Ind_{B}(χ_{1} ⊗ χ_{2} ⊗ χ_{3})

be the principal series for GSp_{4} obtained by normalized induction from

χ_{1} ⊗ χ_{2} ⊗ χ_{3} : t ↦ χ_{1}(t_{1})χ_{2}(t_{2})χ_{3}(c(t)).

Similarly, if τ is an irreducible representation of GL_{2}, and χ is a character, let

τ ⋊ χ =^{df} Ind_{P} (τ ⊗ χ), χ ⋊ τ =^{df} Ind_{Q}(χ ⊗ τ).

Again, we use normalized induction, and we identify the Levi subgroups of P and Q with GL_{2} × GL_{1} in the natural way. Having this notation at hand, we describe the generic classes of Iwahori-spherical representations discussed above:

(I) π = χ_{1} × χ_{2} ⋊ χ_{3},

(IIa) π = St_{GL(2)}(χ_{1}) ⋊ χ_{2},

(IIIa) π = χ_{1} ⋊ St_{GL(2)}(χ_{2}),

(IV a) π = St_{GSp(4)}(χ),

(V a) π ⊂ St_{GL(2)}(ν^{1/2}ξ_{0}) ⋊ ν^{−1/2}χ

(V Ia) π ⊂ 1 ⋊ St_{GL(2)}(χ)

Here all the characters χ and χ_{i} are unramified. Moreover, ν denotes the normalized absolute value, and ξ_{0} is the non-trivial unramified quadratic character. According to Table 1 on p. 9 in [Sch], only IV a and V a are discrete series. Type V Ia representations are the analogues of limits of discrete series. Table 3 on p. 16 in [Sch] lists the dimensions of the parahoric fixed spaces for all 17 families above, plus additional data such as the Atkin-Lehner eigenvalues when ω_{π} = 1. Below, we concatenate parts of Table 2 and parts of Table 3 in [Sch]. That is,

\begin{matrix}type & N & K & K̃ & J_{P} & J_{Q} & I \\ I & 0 & 1 & 2 & 4 & 4 & 8 \\ IIa & N_{1} & 0 & 1 & 1 & 2 & 4 \\ IIIa & N_{2} & 0 & 0 & 2 & 1 & 4 \\ IV a & N_{3} & 0 & 0 & 0 & 0 & 1 \\ V a & N_{2} & 0 & 0 & 0 & 1 & 2 \\ V Ia & N_{2} & 0 & 0 & 1 & 1 & 3 \end{matrix}

Table A: Parahoric fixed spaces and monodromy

We note that the assignment of monodromy operators in [Sch] is compatible with the local Langlands correspondence, as defined by Gan and Takeda via theta correspondence. This follows from the explicit calculations in section 12 of [GT], see their remarks on p. 33. By local-global compatibility, we deduce:

Corollary 1. Let ρ_{π,ι} be the Galois representation attached to a globally generic cusp form π as above. Let v ∤ ℓ be a finite place of F such that π_{v} is Iwahori-spherical and ramified. Then ρ_{π,ι}|_{I_{Fv}} acts unipotently. In fact, the image ρ_{π,ι}(I_{F_{v}}) is topologically generated by a GSp_{4}(ℚ̄_{ℓ})-conjugate of exp(N_{i}), where

i = \begin{cases}1, π_{v} of type IIa, \\ 2, π_{v} of type IIIa, Va, or VIa, \\ 3, π_{v} of type IVa. \end{cases}

In particular, we have the following consequences conjectured in [GTi] and [SU]:

• π_{v} of Steinberg type ⟺ monodromy has rank 3.

• π_{v} has a unique J_{Q}-fixed line ⟺ monodromy has rank 2.

• π_{v} para-spherical ⟺ monodromy has rank 1.

P roof. This follows immediately from Table A above. □

The first two consequences are part of the Conjecture on p. 11 in [GTi]. Note that part 4 of that Conjecture is false: If π_{v} has a unique J_{P} -fixed line, one cannot deduce that monodromy has rank one. The last consequence is Conjecture 3.1.7 on p. 41 in [SU], for globally generic π as above. Skinner and Urban used the holomorphic analogue as a substitute for deep results of Kato, in order to study Selmer groups for certain modular forms of square-free level. Another application to the Bloch-Kato conjecture in this context, contingent on the holomorphic analogue of the third consequence above, was given in [Sor].

Supercuspidal generic representations: According to Table 2 on p. 51 in [GT], there are two types of supercuspidal generic representations π of GSp_{4}. Firstly,

π = θ((σ ⊗ σ′)^{+}),

for distinct supercuspidal representations σ ≠ σ′ on GL_{2}. In this case, the lift to GL_{4} is the isobaric sum σ ⊞ σ′. Secondly, if π is not a lift from GO_{2,2}, when lifted to GL_{4} it remains supercuspidal. Again, by local-global compatibility:

Corollary 2. Let ρ_{π,ι} be the Galois representation attached to a globally generic cusp form π as above. Let v ∤ ℓ be a finite place of F such that π_{v} is supercuspidal. Then ρ_{π,ι} is trivial on some finite index subgroup of I_{F_{v}}. Moreover,

π_{v} is \underline{not} a lift from GO_{2,2} ⟺ ρ_{π,ι}|_{W_{Fv}} is irreducible.

On the contrary, when π_{v} \underline{is} a lift from GO_{2,2}, the restriction ρ_{π,ι}|_{W_{Fv}}^{ss} breaks up as a sum of two non-isomorphic irreducible two-dimensional representations.

P roof. This follows from the foregoing discussion. □

For a moment, let us continue with the setup of the previous Corollary. The exponent of the Artin conductor of ρ_{π,ι}|_{I_{Fv}} is defined by the standard formula:

𝔣(ρ_{π,ι}|_{I_{Fv}}) =^{df} ∑_{i=0}^{∞} \frac{1}{[Ĩ_{F_{v}} : Ĩ_{F_{v},i}]} · codimℚ̄_{ℓ}(ρĨ_{Fv ,i} ; π,ι) ∈ ℤ,

where Ĩ_{F_{v},i} is the ith ramification group in Ĩ_{F_{v}}, in turn some finite quotient of I_{F_{v}} through which ρ_{π,ι} factors. This sum is finite. The exponent of the Swan conductor, 𝔣_{Swan}, is given by the same formula except the summation starts at i = 1. By irreducibility, it is easy to see that ρ_{π,ι} has no nonzero I_{F_{v}}-invariants:

𝔣_{Swan}(ρ_{π,ι}|_{I_{Fv}}) = 𝔣(ρ_{π,ι}|_{I_{Fv}}) − 4.

We wish to relate this to the depth of π_{v}, a non-negative rational number measuring its wild ramification. We very briefly recall the definition: Let G be (the rational points of) a connected reductive group over F_{v}, and let x be a point on its extended Bruhat-Tits building [Tit]. Its stabilizer G_{x} is the corresponding parahoric subgroup. In [MP], Moy and G. Prasad defined an exhaustive descending filtration of G_{x}, consisting of open subgroups G_{x,r} parametrized by non-negative real numbers r. They then defined a pro-p group G_{x,r^{+}} to be the union of the G_{x,s} for s > r. The breaks r, where G_{x,r^{+}} is a proper subgroup of G_{x,r}, is known to form an unbounded discrete subset of ℝ. The depth of π is

depth(π) =^{df} inf{r: π^{G_{x,r^{+}}} ≠ 0, some x} ∈ ℚ.

Here π is any irreducible admissible representation of G. Our goal is to show:

Proposition 1. Let ρ_{π,ι} be the Galois representation attached to a globally generic cusp form π as above. Let v ∤ ℓ be a finite place of F such that π_{v} is supercuspidal, and \underline{not} a lift from GO_{2,2}. Then we have the following identity:

𝔣_{Swan}(ρ_{π,ι}|_{I_{Fv}}) = 4 · depth(π_{v}).

P roof. As is well-known, see [Tat], the Artin conductor fits into the ϵ-factor:

ϵ(s, ιWD(ρ_{π,ι}|_{W_{Fv}}), ψ) = ϵ(0, ιWD(ρ_{π,ι}|_{W_{Fv}}), ψ) · q_{v}^{−s(𝔣(ρ_{π,ι}|_{I_{Fv}})+4n(ψ))}.

Here ψ is some fixed non-trivial character of F_{v}, and n(ψ) is the largest n such that ψ is trivial on 𝔭_{v}^{−n}. Similarly, if Π_{v} is the supercuspidal lift of π_{v} to GL_{4},

ϵ(s, Π_{v}, ψ) = ϵ(0, Π_{v}, ψ) · q_{v}^{−s(𝔣(Π_{v})+4n(ψ))}.

Here 𝔣(Π_{v}) is the standard conductor of Π_{v}, that is, the smallest f such that Π_{v} has nonzero vectors fixed by the subgroup consisting of elements in GL_{4}(O_{v}) whose last row is congruent to (0, …, 0, 1) modulo 𝔭_{v}^{f}. Hence, we deduce that

𝔣(ρ_{π,ι}|_{I_{Fv}}) = 𝔣(Π_{v}),

since the local Langlands correspondence for GL_{4} preserves ϵ-factors. The determinant twist can be ignored. Now, the key ingredient is the following formula due to Bushnell and Frölich [BF], which holds for supercuspidals on any GL_{n},

𝔣(Π_{v}) = n · depth(Π_{v}) + n, n = 4.

We note, in passing, that this formula was generalized to the square integrable case in [LR]. It remains to explain why π_{v} and Π_{v} have the same depth. Keep in mind that Π_{v} ⊗ ω_{π_{v}} is the theta lift of π_{v} to GSO_{3,3}. Now invoke the main result from [Pan], suitably extended to incorporate similitudes. For this last step, Lemma 2.2 on p. 7 in [GT] is very useful. We omit the details. □

Tame ramification: In the previous Proposition, let us take π_{v} to have depth zero. In other words, π_{v} has nonzero vectors fixed by the pro-unipotent radical of some parahoric subgroup. In this special case, ρ_{π,ι}|_{I_{Fv}} factors through the tame quotient I_{F_{v}}^{t}, a pro-cyclic group of pro-order prime-to-p. More concretely, ρ_{π,ι}|_{I_{Fv}} is the direct sum of four ℓ-adic characters of I_{F_{v}}, of finite order primeto-p. Below, we will state a related result, due to Genestier and Tilouine in the rational case. Let χ be any complex character of 𝔽_{v}^{∗} and view it as a character

χ : J_{Q,v} → GL(2, 𝔽_{v}) × 𝔽_{v}^{∗} → 𝔽_{v}^{∗} → ℂ^{∗},

where the second map is projection. If π is an irreducible admissible representation of GSp_{4}, we will be looking at the space π^{J_{Q},χ} of vectors on which J_{Q} acts via the character χ. When χ ≠ 1, this is non-trivial only for principal series:

Lemma 3. Let π be an irreducible generic representation of GSp_{4}(F_{v}), and let χ be a non-trivial character of 𝔽_{v}^{∗}, viewed as a character of J_{Q} as above. Then π^{J_{Q},χ} is nonzero if and only if π is a tamely ramified principal series of the form

π = χ̃ × (unram.) ⋊ (unram.),

for some extension χ̃ of χ inflated to a tamely ramified character of O_{v}^{∗}.

P roof. First, let us assume π^{J_{Q},χ} contains nonzero vectors. On such vectors, the Iwahori subgroup I acts via the character χ⊗1⊗1. By Roche’s construction of types for principal series [Roc], see the formulation on p. 10 in [So2], we deduce that π must be a subquotient of a principal series representation of the form

χ̃ × χ_{1} ⋊ χ_{2}

as in the Lemma. That is, both χ_{i} are unramified, and χ̃ extends χ. Our goal is to show that this principal series is necessarily irreducible. For that, we use the criterion from Theorem 7.9 in [Tad]. Since χ̃ is ramified, the only way it could be reducible is if χ_{1} = ν^{±1}. Recall that ν denotes the normalized absolute value on F_{v}. For simplicity, let us assume that χ_{1} = ν. The other case is taken care of by taking the dual. Then, by Lemmas 3.4 and 3.9 in [ST], there is a sequence

0 → χ̃ ⋊ St_{GL(2)}(σ) → χ̃ × ν ⋊ ν^{−1/2}σ → χ̃ ⋊ 1_{GL(2)}(σ) → 0,

where we write χ_{2} as ν^{−1/2}σ. Both constituents are irreducible. However, the quotient is non-generic, so π must be the subrepresentation. It remains to check

(χ̃ ⋊ St_{GL(2)}(σ))^{J_{Q},χ} = 0.

This is done by explicit calculation: We fix a complete set of representatives,

Q\G/J_{Q} = \{1, s_{1}, s_{1}s_{2}s_{1}\}, s_{1} =^{df} (\begin{matrix}& 1 \\ 1 \\ & & & 1 \\ & & 1 \end{matrix}), s_{2} =^{df} (\begin{matrix}1 \\ & & 1 \\ & −1 \\ & & & 1 \end{matrix}).

We are inducing from the Klingen parabolic, and an easy argument shows that the χ-vectors in the induced representation are given by the χ-vectors in the representation of the Levi subgroup. In our case, we get the three contributions:

• (χ̃ ⊗ St_{GL(2)}(σ))^{Q∩J_{Q},χ} = χ̃^{O^{∗},χ} ⊗ St_{GL(2)}(σ)^{GL_{2}(O)} = 0,

• (χ̃ ⊗ St_{GL(2)}(σ))^{Q∩s_{1}J_{Q}s_{1},χ} = χ̃^{O^{∗}} ⊗ St_{GL(2)}(σ)^{I,χ} = 0,

• (χ̃ ⊗ St_{GL(2)}(σ))^{Q∩s_{1}s_{2}s_{1}J_{Q}s_{1}s_{2}^{−1}s_{1},χ} = χ̃^{O^{∗},χ′} ⊗ St_{GL(2)}(σ)^{GL_{2}(O)} = 0.

Here χ′ is some irrelevant character. This proves the only if part of the Lemma. The converse is easier. Indeed, by the same observation, it suffices to check that

(χ̃ ⊗ χ_{1} ⊗ χ_{2})^{B∩J_{Q},χ} = χ̃^{O^{∗},χ} ⊗ χ_{1}^{O^{∗}} ⊗ χ_{2}^{O^{∗}} ≠ 0,

as follows from our assumptions on these characters. This finishes the proof. □

As a last application of local-global compatibility, in conjunction with the previous Lemma, we obtain the following result due to Genestier and Tilouine in the rational case F = ℚ; compare to the second part of Theorem 2.2.5 in [GTi].

Corollary 3. Let ρ_{π,ι} be the Galois representation attached to a globally generic cusp form π as above. Let v ∤ ℓ be a finite place of F such that π_{v}^{J_{Q},χ} is \underline{nonzero} for some \underline{non-trivial} tamely ramified character χ of O_{v}^{∗}. It follows that

ρ_{π,ι}|_{I_{Fv}} = 1 ⊕ 1 ⊕ χ ⊕ χ.

Here χ is the character of I_{F_{v}} obtained via local class field theory. Moreover, one can arrange for the two eigenspaces, for 1 and χ, to be totally isotropic.

P roof. The previous Lemma. See also (vi) of Proposition 12.15 in [GT]. □

In [GTi], this result was proved by a careful study of the bad reduction of Siegel threefolds with Klingen level structure at v. The eigenspace polarization comes from the cohomology of each irreducible component of the special fiber.

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Claus M. Sorensen

Department of Mathematics Princeton University USA claus@princeton.edu