Galois Representations for General Symplectic Groups 3

Home , Sug Woo Shin

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5 GSpin-valued Galois Representations 30

6 The trace formula with ﬁxed central character 32

7 Cohomology of certain Shimura varieties of abelian type 40

8 Galois representations in the cohomology 45

9 Construction of the GSpin-valued representation 53

10 Compatibility at unramiﬁed places 59

11 Galois representations for the exceptional group G2 62

12 Automorphic multiplicity 65

13 Meromorphic continuation of the Spin L-function 68

A Lefschetz functions 70

B Conjugacy in the standard representation 81

Introduction

Let G be a connected reductive group over a number ﬁeld F . The conjectural global Langlands correspondence for G predicts a correspondence between certain automorphic representations of G(AF ) and certain ℓ-adic Galois representations valued in the L-group of G. Let us recall from [BG14, 3.2] a rather precise conjecture § on the existence of Galois representations for a connected reductive group G over a number ﬁeld F . Let π be a cuspidal L-algebraic automorphic representation of

G(AF ). (We omit their conjecture in the C-normalization, cf. [BG14, Conj. 5.40], but see Theorem 9.1 below.) Denote by G(Qℓ) the Langlands dual group of G over L Qℓ, and by G(Qℓ) the L-group of G formed by the semi-direct product of G(Qℓ) with Gal(F /F ). According to their conjecture,b for each prime ℓ and each field isomorphism ι : C Q , there should exist a continuous representation b ≃ ℓ ρ : Gal(F/F ) LG(Q ), π,ι → ℓ which is a section of the projection LG(Q ) Gal(F/F ), such that the following ℓ → holds: at each place v of F where πv is unramified, the restriction ρπ,ι,v : Gal(F v/Fv) L → G(Qℓ) corresponds to πv via the unramified Langlands correspondence. Moreover

ρπ,ι should satisfy other desiderata, cf. Conjecture 3.2.2 of loc. cit. For instance at Galois representations for general symplectic groups 3

places v of F above ℓ, the localizations ρπ,ι,v are potentially semistable and have Hodge-Tate cocharacters determined by the infinite components of π. Note that if G is a split group over F , we may as well take ρπ,ι to have values in G(Qℓ). To simplify notation, we often fix ι and write ρπ and ρπ,v for ρπ,ι and ρπ,ι,v, understanding that these representations do depend on the choice of ι in general. b Our main result confirms the conjecture for general symplectic groups over totally real fields in a number of cases (up to Frobenius semisimplification, meaning that we take the semisimple part in (ii) of Theorem A below). We find these groups interesting for two reasons. Firstly they naturally occur in the moduli spaces of polarized abelian varieties and their automorphic/Galois representations have been useful for arithmetic applications (such as the study of L-functions, modularity and the Sato-Tate conjecture). Secondly new phenomena (as the semisimple rank grows) make the above conjecture sufficiently nontrivial, stemming from the nature of the dual group of a general symplectic group: e.g. faithful representations have large dimensions and locally conjugate representations may not be globally conjugate. Let F be a totally real number field. Let n 2. Let GSp denote the split general ≥ 2n symplectic group over F , equipped with similitude character sim : GSp G over 2n → m F . The dual group GSp2n is the general spin group GSpin2n+1, which we view over

Qℓ (or over C via ι), admitting the spin representation d spin: GSpin GL n . 2n+1 → 2 Consider the following hypotheses on π.

(St) There is a ﬁnite F -place vSt such that πvSt is the Steinberg represen-

tation of GSp2n(FvSt ) twisted by a character. (L-coh) π sim n(n+1)/4 is ξ-cohomological for an irreducible algebraic repre- ∞| | sentation ξ = ξ of the group (Res GSp ) C GSp y:F֒→C y F/Q 2n ⊗Q ≃ y:F֒→C 2n,C⊗ (Definition 1.12 below). Q A trace formula argument shows that there are plenty of (in particular infinitely many) π satisfying (St) and (L-coh), cf. [Clo86]. Let Sbad denote the finite set of rational primes p such that either p = 2, p ramifies in F , or πv ramifies at a place v of F above p. Theorem A (Theorem 10.3). Suppose that π satisfies hypotheses (St), (L-coh). Let ℓ be a prime number and ι: C ∼ Q a field isomorphism. Then there exists a repre- → ℓ sentation ρ = ρ : Gal(F /F ) GSpin (Q ), π π,ι → 2n+1 ℓ unique up to GSpin2n+1(Qℓ)-conjugation, attached to π and ι such that the following hold. 4 Arno Kret, Sug Woo Shin

(i) The composition

ρ Gal(F /F ) π GSpin (Q ) SO (Q ) GL (Q ) → 2n+1 ℓ → 2n+1 ℓ ⊂ 2n+1 ℓ

is the Galois representation attached to a cuspidal automorphic Sp2n(AF )-subrepresentation π♭ contained in π. Further, the composition

ρ Gal(F /F ) π GSpin (Q )/Spin (Q ) GL (Q ) → 2n+1 ℓ 2n+1 ℓ ≃ 1 ℓ corresponds to the central character of π via class ﬁeld theory and ι.

(ii) For every ﬁnite place v which is not above S ℓ , the semisimple part bad ∪{ } of ρπ(Frobv) is conjugate to ιφπv (Frobv) in GSpin2n+1(Qℓ), where φπv is the unramiﬁed Langlands parameter of πv.

(iii) For every v ℓ, the representation ρ is de Rham (in the sense that r ρ is | π,v ◦ π,v de Rham for all representations r of GSpin ). Moreover 2n+1,Qℓ

(a) The Hodge-Tate cocharacter of ρπ,v is explicitly determined by ξ. More precisely, for all y : F C such that ιy induces v, we have → n(n+1) µHT(ρπ,v, ιy)= ιµHodge(ξy)+ 4 sim

(for µHT and µHodge see Deﬁnitions 1.10 and 1.14 below).

(b) If πv has nonzero invariants under a hyperspecial (resp. Iwahori) subgroup

of GSp2n(Fv) then either ρπ,v or a quadratic character twist is crystalline (resp. semistable). (c) If ℓ / S then ρ is crystalline. ∈ bad π,v (iv) For every v , ρ is odd (see Deﬁnition 1.8 and Remark 10.4 below). |∞ π,v

(v) The Zariski closure of the image of ρπ maps onto one of the following three

subgroups of SO2n+1: (a) SO2n+1, (b) the image of a principal SL2 in SO2n+1,

or (c) (only possible when n =3) G2 embedded in SO7.

′ (vi) If ρ : Gal(F /F ) GSpin2n+1(Qℓ) is any other continuous morphism such → ′ that, for almost all F -places v where ρ and ρπ are unramiﬁed, the semisimple ′ ′ parts ρ (Frobv)ss and ρπ(Frobv)ss are conjugate, then ρ and ρ are conjugate.

Our theorem is new when n 3. When n = 2, a better and fairly complete result ≥ without conditions (St) has been known by [Tay93,Lau05,Wei05,Urb05,Sor10,Jor12, GT19]. (Often π is assumed globally generic in the references but this is no longer necessary thanks to [GT19].) Galois representations for general symplectic groups 5

The above theorem in particular associates a weakly compatible system of λ-adic representations with π. See also Proposition 13.1 below for precise statements on the weakly compatible system consisting of spin ρ . It is worth noting that the unique- ◦ π ness in (vi) would be false for general GSpin2n+1(Qℓ)-valued Galois representations in view of Larsen’s example in 5 below.3 Our proof of (vi) relies heavily on the fact § that ρπ contains a regular unipotent element in the image, coming from condition (St). When n = 3 and F = Q, we employ the strategy of Gross and Savin [GS98] to construct a rank 7 motive over Q with Galois group of exceptional type G2 in the cohomology of Siegel modular varieties of genus 3. The point is that ρπ as in the above theorem factors through G (Q ) ֒ GSpin (Q ) if π comes from an 2 ℓ → 7 ℓ automorphic representation on (an inner form of) G2(A) via theta correspondence. In particular we get yet another proof aﬃrmatively answering a question of Serre in the case of G2, cf. [KLS10, DR10, Yun14, Pat16] for other approaches to Serre’s question (none of which uses Siegel modular varieties). Along the way, we also obtain some new instances of the Buzzard–Gee conjecture for a group of type G2. Our result also provides a solid foundation for investigating the suggestion of Gross–Savin that a certain Hilbert modular subvariety of the Siegel modular variety should give rise to the cohomology class for the complement of the rank 7 motive of type G2 in the rank 8 motive cut out by π, as predicted by the Tate conjecture. See Theorem 11.1, Corollary 11.3, and Remark 11.5 below for details. We mention that Magaard and Savin obtained similar results (using a diﬀerent method) in a new version of [MS]. As another consequence of our theorems, we deduce multiplicity one theorems for automorphic representations for inner forms of GSp2n,F under similar hypotheses from the multiplicity formula by Bin Xu [Xu18]. As his formula suggests, multiplicity one is not always expected when all hypotheses are dropped.

Theorem B (Theorem 12.1). Let π be a cuspidal automorphic representation of

GSp2n(AF ) satisfying (St) and (L-coh). The automorphic multiplicity of π is equal to 1.

By part (vi) of Theorem A asserting uniqueness, we have (a version of) strong multiplicity one for the L-packet of π. In Proposition 6.3 we prove a weak Jacquet- Langlands transfer for π in Theorem B. This allows us to propagate multiplicity one from π as above to the corresponding automorphic representations on a certain inner form. See Theorem 12.2 below for details. Note that (weak and strong) multiplicity

3 In contrast, for cuspidal automorphic representations of the group Sp2n(AF ), the corresponding analogue of statement (vi) does hold (cf. Proposition B.1). So the possible failure of (vi) is a new phenomenon for the cuspidal spectrum of the similitude group GSp2n(AF ). 6 Arno Kret, Sug Woo Shin

one theorem for globally generic cuspidal automorphic representations of GSp4(AF ) has been known by Jiang and Soudry [Sou87, JS07]. Our results yield a potential version of the spin functoriality, thus also a meromorphic continuation of the spin L-function, for cuspidal automorphic representations of GSp2n(AF ) satisfying (St), (L-coh), and the following strengthening of (spin-reg):

(spin-REG) The representation πv is spin-regular at every inﬁnite place v of F .

The last condition means that the Langlands parameter of πv maps to a regular parameter for GL2n by the spin representation, cf. Definitions 1.6 and 1.7 below. Thanks to the potential automorphy theorem of Barnet-Lamb, Gee, Geraghty, and Taylor [BLGGT14, Thm. A] it suffices to check the conditions for their theorem to apply to spin ρ (for varying ℓ and ι). This is little more than Theorem A; the ◦ π details are explained in 13 below. § Theorem C (Corollary 13.4). Under hypotheses (St), (L-coh), and (spin-REG) on π, there exists a finite totally real extension F ′/F (which can be chosen to be dis- joint from any prescribed finite extension of F in F ) such that spin ρ ′ is ◦ π|Gal(F /F ) automorphic. More precisely, there exists a cuspidal automorphic representation Π of GL2n (AF ′ ) such that

• ′ −1 for each finite place w of F not above Sbad, the representation ι spin ρπ W ′ ◦ | Fw is unramified and its Frobenius semisimplification is the Langlands parameter

for Πw,

• at each inﬁnite place w of F ′ above a place v of F , we have φ spin Πw |WC ≃ ◦ φ . πv |WC In particular the partial spin L-function LS(s, π, spin) admits a meromorphic continuation and is holomorphic and nonzero in a right half plane.

We can be precise about the right half plane in terms of π: For instance it is given by Re(s) 1 if π has unitary central character. Due to the limitation of our ≥ method, we cannot control the poles. Before our work, the analytic properties of the spin L-functions have been studied mainly via Rankin-Selberg integrals; some partial results have been obtained for GSp for 2 n 5 by Andrianov, Novodvorsky, 2n ≤ ≤ Piatetski-Shapiro, Vo, Bump–Ginzburg, and more recently by Pollack–Shah and Pollack, cf. [PS97, Vo97, BG00, PS18, Pol17]. See [Pol17, 1.3] for remarks on spin L-functions with further references. Finally we comment on the hypotheses of our theorems. The statements (ii) and (iii.c) are not optimal in that we exclude a little more than the ﬁnite places v where

πv is unramified. This is due to the fact that the Langlands–Rapoport conjecture Galois representations for general symplectic groups 7 has been proved by [Kis17] at a p-adic place only when p > 2 and the defining data (including the level) are unramified at all p-adic places4. Condition (L-coh) is essential to our method but it is perhaps possible to prove Theorem A under a slightly weaker condition that π appears in the coherent cohomology of our Shimura varieties. The rather strong condition (spin-REG) in Theorem C is necessary due to the current limitation of potential automorphy theorems. Condition (St) in Theorem A is believed to be superfluous. Significant work and ideas would be needed to get around it. On the other hand, (St) is harmless to assume for local applications, which we intend to pursue in future work. Idea of proof. Our proof of Theorem A relies crucially on Arthur’s book [Art13]. His results used to be conditional on the stabilization of the twisted trace formula, whose proof has been completed by Moeglin and Waldspurger; see [MW16a, MW16b] for the last one. Thus Arthur’s results are essentially unconditional.5 Since Theorems B and C depend in turn on Theorem A, all our results count on Arthur’s book. Let us sketch our proof with a little more detail. We prove Theorem A by combin- ing two approaches: (M1) Shimura varieties for inner forms of GSp2n, and (M2) Lift- ing to GSpin2n+1(Qℓ) the SO2n+1(Qℓ)-valued Galois representations as constructed by the works of Arthur and Harris-Taylor. ∗ First Method. Consider the inner form G over F of G := GSp2n,F , such that the local group Gv is

non-split if v = vSt and [F = Q] is even, anisotropic modulo center if v , v = v ,  |∞ 6 0 split otherwise.  We consider Shimura varieties arising from the group ResF/QG (and the choice of X as in 7 below). Note that (the Q-points of) Res G has factor of similitudes in § F/Q F ×, as opposed to Q×, and that our Shimura varieties are not of PEL type (when F = Q) but of abelian type. This should already be familiar for n = 1 (though we 6 assume n 2 for our main theorem to be interesting), where we obtain the usual ≥ Shimura curves, cf. [Car86a]. In case F = Q our Shimura varieties are the classical Siegel modular varieties. The idea is to consider (the semisimpliﬁcation of) the compactly supported ´etale

4When F = Q, our argument uses Siegel modular varieties, so we may allow p = 2 by appealing to Kottwitz’s work on PEL-type Shimura varieties instead of [Kis17]. 5Strictly speaking, Arthur postponed some technical details in harmonic analysis to future articles ([A25, A26, A27] in the bibliography of [Art13]), which have not appeared yet. The weighted fundamental lemma has been proved by Chaudouard–Laumon but the proof has appeared for split groups thus far. 8 Arno Kret, Sug Woo Shin cohomology i i Hc(ShK, ιξ) = lim Hc(ShK, ιξ), i 0, L ∞ L ≥ K⊂−→G(AF ) where is the ℓ-adic local system attached to some irreducible complex represen- Lιξ tation ξ of G via ι. Then Hi (Sh , ) has an action of G(A∞) Gal(F /F ); and one c K Lιξ F × hopes to prove that through this action the module Hi (Sh , ) realizes the Lang- c K Lιξ lands correspondence. In particular, one tries to attach to a cuspidal automorphic representation π of GSp2n(AF ) the following virtual Galois representation (see also Equation (7.3))

′ def i ′,∞ i π π Hπ = ( 1) [HomG(A∞)(π , Hc(ShK, ιξ)ss)], 7→ − F Qℓ L i≥0 X where the first mapping is a weak transfer of π from G∗ to G. (We need to twist π by n(n+1)/4 to have H = 0 but this twist will be ignored in the introduction.) |·| π 6 The subscript ( ) denotes the semisimplification as a G(A∞)-module. · ss F The construction of Hπ has two issues. Firstly there are the usual issues coming from endoscopy. Our assumption (St) circumvents this difficulty (and helps us at other places). Secondly, even without endoscopy, one does not expect Hπ to realize the representation ρπ itself; rather it realizes (up to sign, twist, and multiplicity) the composition ρπ spin Gal(F /F ) GSpin (Q ) GL n (Q ). → 2n+1 ℓ → 2 ℓ In particular if one wants to use H to construct ρ , one has to show that r : Gal(F /F ) π π → GL (Hπ) GL2n (Q ) has (up to conjugation) image in the group GSpin (Q ) Qℓ ≃ ℓ 2n+1 ℓ ⊂ GL2n (Qℓ). With point counting methods it can be shown that this is true for the

Frobenius elements, but since these elements are only deﬁned up to GL2n (Qℓ)- conjugation, we are unable to deduce directly that the entire representation r has image in GSpin2n+1(Qℓ). More information seems to be required. Second Method. Consider a continuous representation ρ: Gal(F/F ) SO (Q ), → 2n+1 ℓ and the exact sequence

× 1 Q GSpin (Q ) SO (Q ) 1. → ℓ → 2n+1 ℓ → 2n+1 ℓ → Using the theorem of Tate that H2(F, Q/Z) vanishes, it is not hard to show that ρ has a continuous lift ρ: Gal(F/F ) GSpin2n+1(Qℓ). In particular one could try to → ♭ attach to π a (non-unique) automorphic Sp2n(AF )-subrepresentation π π, and ♭ ⊂ then to π the Galoise representation ρ ♭ : Gal(F /F ) SO (Q ), constructed by π → 2n+1 ℓ the combination of results of Arthur and others (Theorem 2.4). One now hopes to construct ρπ : Gal(F /F ) GSpin2n+1(Qℓ) as a twist ρπ♭ χ for some continuous → × ⊗ character χ: Gal(F /F ) Q . However it is a priori unclear where χ should come → ℓ from. (The central character of π only determines the squaref of χ.) Galois representations for general symplectic groups 9

Construction. We ﬁnd the character χ by comparing ρπ♭ with the representation

Hπ. Consider the diagram f ρ2 ρ ( 1 + ( / / Gal(F /F ) / GSpin (Q ) / GL n (Q ) g 2n+1 ℓ spin 2 ℓ ρπ♭

ρ . / π♭ . SO2n+1(Qℓ) / PGL2n (Qℓ) spin where ρ = spin ρ ♭ for some choice of lift ρ ♭ of ρ ♭ , and we construct ρ from 1 ◦ π π π 2 the representation Hπ. We show in three steps that ρ1 and ρ2 are conjugate up to twisting by the soughtf after character χ. f Step 1: Show connectedness of image. More precisely we show that the image of

ρ ♭ (thus also the image of spin ρ ♭ ) has connected Zariski closure. Because π is a π ◦ π twist of the Steinberg representation at a ﬁnite place, we can assure that ρ ♭ π |Gal(F /E) has a regular unipotent element N in its image. It then follows from results of Saxl–

Seitz that the reductive subgroups of SO2n+1(Qℓ) containing N are connected (see Proposition 3.5).

Step 2: Construct ρ2. We compare the point counting formula for (ResF/QG, H) with the Arthur-Selberg trace formula for G/F . Since the datum (ResF/QG, H) is not of PEL type (unless F = Q), the classical work of Kottwitz [Kot92b,Kot90] does not apply. Instead we use the counting point formula as derived in [KSZ] from Kisin’s recent proof of the Langlands–Rapoport conjecture for Shimura data of abelian type, so in particular for (Res G, H). Consequently, we have H = a spin ιφ F/Q π|Gal(F v/Fv) · ◦ πv at the unramiﬁed places v, where φπv is the unramiﬁed L-parameter of πv and a Z is essentially the automorphic multiplicity of π. (In fact, we show that ∈ >0 a = 1 together with Theorem B but only after the construction of ρπ is done. See 12 below.) § Step 3: Produce χ. To do this we prove:

Lemma. Let r ,r : Gal(F /F ) GL (Q ) two continuous representations, which 1 2 → m ℓ are unramiﬁed almost everywhere, r1 is has Zariski connected image modulo center, and for almost all F -places

r1(Frobv)ss is conjugate to r2(Frobv)ss in PGLm(Qℓ).

× Then r r χ for a continuous character χ: Gal(F /F ) Q . 1 ≃ 2 ⊗ → ℓ

By counting points on Shimura varieties we control ρ2 at the unramiﬁed places.

By a different argument ρ1 is, up to scalars, also controlled at the unramified places. Hence the lemma applies, and allows us to find a character χ such that ρ ρ χ. 2 ≃ 1 ⊗ 10 Arno Kret, Sug Woo Shin

To prove Theorem A we deﬁne

def ρ = ρ ♭ χ: Gal(F /F ) GSpin (Q ), π π ⊗ → 2n+1 ℓ and check that ρπ satisfiesf the desired properties stated in the theorem. Acknowledgments: The authors wish to thank Fabrizio Andreatta, Alexis Bouthier, Laurent Clozel, Wushi Goldring, Christian Johansson, Kai-Wen Lan, Brandon Levin, David Loeffler, Judith Ludwig, Giovanni Di Matteo, Sophie Morel, Stefan Patrikis, Pe- ter Scholze, Benoit Stroh, Richard Taylor, Olivier Ta¨ıbi, Jack Thorne, Jacques Tilouine, David van Overeem and David Vogan for helpful discussions and answering our questions. A.K. thanks S.W.S. and the Massachusetts Institute of Technology for the invitation to speak in the number theory seminar, which led to this work. Furthermore he thanks the Institute for Advanced Study, the Mathematical Sciences Research Institute, the Max Planck Institute for Mathematics, Korea Institute for Advanced Study, and the University of Amsterdam, where this work was carried out. During his time at the IAS (resp. MSRI) A.K. was supported by the NSF under Grant No. DMS 1128155 (resp. No. 0932078000). S.W.S. was partially supported by NSF grant DMS-1449558/1501882 and a Sloan Fellow- ship. Both authors are grateful to an anonymous referee for an extensive list of comments and simplifications. In particular he/she explained Proposition 4.6 to us and suggested a significant simplification based on the fact that the SO2n+1-valued Galois representation ρπ♭ has connected image (without extra regularity hypotheses on the component of π at infinity). Following the suggestion we dispensed with the eigenvariety and patching arguments in an earlier version (arXiv:1609.04223v1) of our paper. We thank another anonymous reviewer for sending us corrections and suggesting improvements on the next version (arXiv:1609.04223v2), especially in 5. § Notation

We ﬁx the following notation and convention.

• “Almost all” always means “all but ﬁnitely many”.

• n 2 is an integer. ≥ • F is a totally real number ﬁeld, embedded into C.

• is the ring of integers of F . OF • [1/S] is the localization of with finite primes in S inverted, where S is OF OF a finite set of places of F . (Finite places are used interchangeably with finite primes.) Galois representations for general symplectic groups 11

• A is the ring of adèles of F , i.e. A := (F R) (F Z). F F ⊗Q × ⊗Z • If Σ is a finite set of F -places, then AΣ A is the ring of adèlesb with trivial F ⊂ F components at the places in Σ, and F := F ; F := F R. Σ v∈Σ v ∞ ⊗Q Q • If p is a prime number, then F := F Q . p ⊗Q p • ℓ is a fixed prime number (different from p).

• Q is a ﬁxed algebraic closure of Q , and ι: C ∼ Q is an isomorphism. ℓ ℓ → ℓ

• For each prime number p we ﬁx the positive root √p R>0 C. From ι ∈ ⊂ x we then obtain a choice for √p Qℓ. Thus, if q is a prime power, then q is ∈ 1 n(n+1)/4 well-deﬁned in C as well as in Q for all half integers x Z (e.g. qv in ℓ ∈ 2 Corollary 8.7)

• If π is a representation on a complex vector space then we set ιπ := π Q . ⊗C,ι ℓ Similarly if φ is a local L-parameter of a reductive group G so that φ maps L L into G(C) then ιφ is the parameter with values in G(Qℓ) obtained from φ via ι.

• Γ := Gal(F /F ) is the absolute Galois group of F .

• Γv := Gal(F v/Fv) is (one of) the local Galois group(s) of F at the place v.

• := Hom (F, R) is the set of infinite places of F . V∞ Q • c Γ is the complex conjugation (well-defined as a conjugacy class) induced v ∈ . by any embedding F ֒ C extending v → ∈V∞ • If G is a locally profinite group equipped with a Haar measure, then we write (G) for the Hecke algebra of locally constant, complex valued functions with H compact support. We write (G) for the same algebra, but now consisting HQℓ of Qℓ-valued functions.

• We normalize parabolic induction by the half power of the modulus character as in [BZ77, 1.8], so as to preserve unitarity. The Satake transform and parameters are normalized similarly, e.g. as in [BG14, 2.2].

• We normalize class ﬁeld theory so that geometric Frobenius elements correspond to local uniformizers. Our normalization of the local Langlands corre-

spondence for GLn is the same as in [HT01]. 12 Arno Kret, Sug Woo Shin

• If H/Q is a reductive group, a group-valued representation Γ H(Q ) means, ℓ → ℓ by deﬁnition, a continuous group morphism for the Krull topology on Γ and

the ℓ-adic topology on H(Qℓ). Similarly every character is assumed to be continuous throughout the paper.

The (general) symplectic group

Write An for the n n-matrix with zeros everywhere, except on its antidiagonal, × An where we put ones. Write J := ( −An ) GL (Z). We deﬁne GSp as the alge- n ∈ 2n 2n braic group over Z, such that for all rings R,

GSp (R)= g GL (R) tg J g = x J for some x R× . 2n { ∈ 2n | · n · · n ∈ } The factor of similitude x R× induces a morphism sim: GSp G . Write ∈ 2n → m T GSp for the diagonal maximal torus. Then X∗(T )= n Ze where GSp ⊂ 2n GSp i=0 i e : diag(a ,...,a , ca−1,...,ca−1) a (i> 0)L, i 1 n n 1 7→ i e : diag(a ,...,a , ca−1,...,ca−1) c. 0 1 n n 1 7→ We let B GSp be the upper triangular Borel subgroup. We have the following GSp ⊂ 2n corresponding simple roots and coroots,

α = e e , ..., α = e e , α =2e e X∗(T ), 1 1 − 2 n−1 n−1 − n n n − 0 ∈ GSp α∨ = e∗ e∗, ..., α∨ = e∗ e∗ , α∨ = e∗ X (T ). 1 1 − 2 n−1 n−1 − n n n ∈ ∗ GSp We deﬁne Sp = ker(sim). Write B = B Sp and T = T Sp . 2n Sp GSp ∩ 2n Sp GSp ∩ 2n The (general) orthogonal group

Let m Z and let GO be the algebraic group over Q such that for all Q-algebras ∈ ≥1 m R, GO (R)= g GL (R) tg A g = x A for some x R× , m { ∈ m | · m · · m ∈ } where A is the m m-antidiagonal unit matrix. We have the factor of similitude m × sim: GO G and put O = ker(sim) and SO =O SL . Let T GO , 2n+1 → m m m m∩ m GO ⊂ m T SO be the diagonal tori. We write std: GO GL for the standard SO ⊂ m m → m representation. If m = 2n +1 is odd, the root datum of SO2n+1 is dual to Sp2n. In ∗ particular, we identify X∗(TSO)= X (TSp).

The (general) spin group

Consider the symmetric form

x, y = x y + x y + + x y = tx A y h i 1 2n+1 2 2n ··· 2n+1 1 · 2n+1 · Galois representations for general symplectic groups 13 on Q2n+1. The associated quadratic form is Q(x)= x x + x x + + x x . 1 2n+1 2 2n ··· 2n+1 1 Let C be the Clifford algebra associated to (Q2n+1, Q). It is equipped with an embedding Q2n+1 C which is universal for maps f : Q2n+1 A into associative rings ⊂ → A satisfying f(x)2 = Q(x) for all x Q2n+1. Let b ,...,b be the standard basis ∈ 1 2n+1 of Q2n+1. The products B = b for I 1, 2,..., 2n +1 form a basis of C. I i∈I i ⊂ { } The algebra C has a Z/2Z-grading, C = C+ C−, induced from the grading on the Q ⊕ tensor algebra. On the Clifford algebra C we have an unique anti-involution that ∗ is determined by (v v )∗ =( 1)rv v for all v ,...,v V . We define for all 1 ··· r − r ··· 1 1 r ∈ Q-algebras R,

GSpin (R)= g (C+ R)× g Rn g∗ = Rn . 2n+1 { ∈ ⊗ | · · } The spinor norm on C induces a character : GSpin G . The action of the N 2n+1 → m group GSpin stabilizes Q2n+1 C and we obtain a surjection q′ : GSpin 2n+1 ⊂ 2n+1 → ։ ′ GO2n+1. We write q for the surjection GSpin2n+1 SO2n+1 obtained from q . We write T GSpin for the torus (q′)−1(T ). We then have GSpin ⊂ 2n+1 GSpin X∗(T )= Ze∗ Ze∗ Ze∗ (= X (T )), GSpin 0 ⊕ 1 ⊕···⊕ n ∗ GSp X (T )= Ze Ze Ze (= X∗(T )). ∗ GSpin 0 ⊕ 1 ⊕···⊕ n GSp The group GSpin (C) is dual to GSp (C). The character sim : GSp G 2n+1 2n 2n → m induces a central embedding G GSpin , still denoted by sim. m → 2n+1 The spin representation

Let k be an algebraically closed ﬁeld of characteristic zero. Write W := n k b ⊕i=1 · i and • W for the exterior algebra of W . Then W is an n-dimensional isotropic subspace of (Q2n+1, Q). We have End( • W ) C+, and hence • W is a 2n- V ≃ dimensional representation of GSpin , called the spin representation (cf. [FH91, 2n+1,kV V (20.18)]). The composition of spin with GL n PGL n induces a morphism 2 ,k → 2 ,k spin: SO PGL n . 2n+1,k → 2 ,k Lemma 0.1. When n mod 4 is 0 or 3 (resp. 1 or 2), there exists a symmetric (resp. symplectic) form on the 2n-dimensional vector space underlying the spin representation such that the form is preserved under GSpin2n+1(k) up to scalars. The resulting map GSpin GO n (resp. GSpin GSp n ) over k followed by the simili- 2n+1 → 2 2n+1 → 2 tude character of GO n (resp. GSp n ) coincides with the spinor norm . 2 2 N Proof. We may identify the 2n-dimensional space with • W . Write for the main ∗ involution on C+ as well as on • W . Given s, t • W , write β(s, t) k for the k ∈ V ∈ projection of s∗ t • W onto n W = k. It is elementary to check that β( , ) is ∧ ∈ V V · · V V 14 Arno Kret, Sug Woo Shin symmetric if n mod 4 is 0 or 3 and symplectic otherwise, cf. [FH91, Exercise 20.38]. Now let x GSpin (k), also viewed as an element of C+. Note that x∗x k× ∈ 2n+1 k ∈ is the spinor norm of x. Then β(xs, xt)=(xs)∗ (xt)=(x∗x)s t = x∗xβ(s, t), ∧ ∧ completing the proof.

1 Conventions and recollections

Let G/Q be a reductive group, T G a maximal torus and W its Weyl group in ℓ ⊂ G. Recall that by highest weight theory the trace characters of irreducible representations form a basis of the algebra (T/W ). We call a set S of representations of O G/Qℓ a fundamental set if the trace characters of all exterior powers of the representations in S generate the algebra (T/W ) of global sections of the variety T/W . O Here are some examples of such S, where we deﬁne the standard representation std of GSpin (m even or odd) to be the composition GSpin ։ GSO ֒ GL . For m m m → m the group G2, write std and ad for the irreducible 7-dimensional and 14-dimensional (adjoint) representations, respectively.

G GLn GSp2n GSO2n+1 GO2n GSpin2n+1 GSpin2n G2 S std std, sim std, sim std, sim spin, std, spin+, spin−, std, std, ad N N Justiﬁcation of Table. The justiﬁcation for the groups G = GO2n and GSO2n+1 follows from appendix B. Notice that the remaining groups are all connected. The fundamental representations of a simply connected semisimple group G are those irreducible representations whose highest weights (for some Borel) are dual to the basis of coroots of G; the trace characters of fundamental representations generate (T/W ) as an algebra. Exercises V.28–30 and 33 of [Kna02, p. 344] show that O for the Lie algebras sl , so , so and g the representations istd (1 i n 2n+1 2n 2 {∧ ≤ ≤ n) , istd (1 i n 1), spin , istd (1 i n 2), spin+, spin− and } {∧ ≤ ≤ − } {∧ ≤ ≤ − } std, ad (respectively) are the fundamental representations of these Lie algebras. { } The statement for the corresponding semisimple simply connected groups follow from this. It is then routine to derive the sets S listed above for the groups GLn,

GSpin2n+1, GSpin2n and G2. This leaves us with the group GSp2n. In this case the representations istd of sp are reducible, but it follows from the maps ϕ and ∧ 2n k theorem 17.5 in [FH91, p. 260] that they generate the fundamental representations of sp2n. Lemma 1.1. Let g ,g G(Q ) be two semisimple elements. Then g g G(Q ) 1 2 ∈ ℓ 1 ∼ 2 ∈ ℓ if and only if ρ(g ) ρ(g ) GL(V ) for all (ρ, V ) S. 1 ∼ 2 ∈ ∈ Proof. “ ” is obvious. We prove “ ”. Since T/W is an algebraic variety, two Q - ⇒ ⇐ ℓ points x1, x2 of it are the same if and only if f(x1)= f(x2) for the algebraic functions Galois representations for general symplectic groups 15 f (T/W ). Since ρ(g ) ρ(g ) for all ρ S, we also have Tr n ρ(g ) = ∈ O 1 ∼ 2 ∈ 1 Tr n ρ(g ) for all n 1. Since the functions f = Tr n ρ in (T/W ) generate this 2 ≥ O V algebra the images of g and g in T/W are the same. V 1 2 V Remark 1.2. Gauger [Gau77] and Steinberg [Ste78, Thm. 3] proved Lemma 1.1 also for non-semisimple elements g ,g G(Q ) and (under various assumptions) 1 2 ∈ ℓ algebraically closed ﬁelds of characteristic p 0. ≥ Let r ,r : Γ G(Q ) be two semisimple representations that are unramiﬁed 1 2 → ℓ at almost all places. (As remarked before, all representations are continuous by convention.)

Lemma 1.3. The following statements are equivalent:

1. for a Dirichlet density one set of ﬁnite F -places v where r1,r2 are unramiﬁed we have r (Frob ) r (Frob ) G(Q ); 1 v ss ∼ 2 v ss ∈ ℓ 2. there exists a dense subset Σ Γ such that for all σ Σ we have r (σ) ⊂ ∈ 1 ss ∼ r (σ) G(Q ); 2 ss ∈ ℓ 3. for all σ Γ we have r (σ) r (σ) G(Q ); ∈ 1 ss ∼ 2 ss ∈ ℓ 4. for all linear representations ρ: G GL the representations ρ r and ρ r → N ◦ 1 ◦ 2 are isomorphic.

5. for a fundamental set of irreducible linear representations ρ: G GL the → N representations ρ r and ρ r are isomorphic. ◦ 1 ◦ 2 Proof. (3) (1) is tautological, (1) (2) follows from the Chebotarev density ⇒ ⇒ theorem, and (2) (3) follows from the continuity of the map G(Q ) (T/W )(Q ) ⇒ ℓ → ℓ taking the semisimple part. The implication (3) (4) follows from the Brauer- ⇒ Nesbitt theorem, (4) (5) is obvious, and (5) (3) is Lemma 1.1. ⇒ ⇒

Definition 1.4. If one of the conditions in Lemma 1.3 holds, then r1 and r2 are said to be locally conjugate, and we write r r . 1 ≈ 2 Definition 1.5. Let T be a maximal torus in a reductive group G over an algebraically closed field. A weight ν X∗(T ) is regular if α∨, ν = 0 for all coroots α∨ of T in ∈ h i 6 G.

Deﬁnition 1.6. Let φ: WR GSpin2n+1(C) be a Langlands parameter. Denote by T → × the diagonal maximal torus in GL n and by T its dual torus. We have W = C 2 C ⊂ WR. The composition b × spin C W GSpin (C) GL n (C) ⊂ R → 2n+1 → 2 16 Arno Kret, Sug Woo Shin is conjugate to the cocharacter z µ (z)µ (z) given by some µ ,µ X (T ) C = 7→ 1 2 1 2 ∈ ∗ ⊗Z X∗(T ) C such that µ µ X∗(T ). Then φ is spin-regular if µ is regular ⊗Z 1 − 2 ∈ 1 (equivalently if µ is regular; note that µ and µ are swapped if spin φ is conjugatedb 2 1 2 ◦ by the image of an element j W such that j2 = 1 and jwj−1 = w for w W ). ∈ R − ∈ C

Deﬁnition 1.7. An automorphic representation π of GSp2n(AF ) is spin-regular at v∞ if the Langlands parameter of the component πv∞ is spin-regular.

Let H be a connected reductive group over Qℓ for the following two deﬁnitions

(which could be extended to disconnected reductive groups). Let hder denote the Lie algebra of its derived subgroup. Write c for the nontrivial element of Gal(C/R).

Deﬁnition 1.8 (cf. [Gro]). A representation ρ : Gal(C/R) H(Q ) is odd if the → ℓ trace of c on h through the adjoint action of ρ(c) is equal to rank(h ). der − der We remark that the mapping GL SO GO is an isomorphism, in 1 × 2n+1 → 2n+1 particular the latter is connected.

Lemma 1.9. Let ρ♭ : Gal(C/R) GO (Q ) be a representation. Write ρ♯ : Γ → 2n+1 ℓ → GL (Q ) for the composition of ρ♭ with the standard embedding. If Tr ρ♯(c) 1 2n+1 ℓ ∈ {± } then ρ♭ is odd.

Proof. We may choose a model for the Lie algebra of SO2n+1 to consist of X t −1 ∈ GL2n+1 such that X + A2n+1XA2n+1 = 0. Such an X =(xi,j) is characterized by the condition x + x = 0 for every 1 i, j 2n + 1. Write t := ρ♭(c). By i,j n+1−j,n+1−i ≤ ≤ conjugation and multiplying with 1 GL if necessary, we can assume that t − ∈ 2n+1 is in the diagonal maximal torus in SO2n+1 (not only in GO2n+1, using the fact that the latter is the product of SO with center) of the form diag(1 , 1 , 1, 1 , 1 ), 2n+1 a − b − b a where 0 a, b n and a + b = n. Since Tr ρ♯(c ) 1 we have a = b if n is even ≤ ≤ v ∈ {± } and b a = 1 if n is odd. Now an explicit computation shows that the trace of the − adjoint action of t on Lie (SO ) has trace 2(a b)2 +2(a b) n, which is equal 2n+1 − − − to n = rank(SO ) in all cases. − − 2n+1

Let K be a finite extension of Qℓ. Fix its algebraic closure K and write K for its completion. b Definition 1.10 (cf. [BG14, 2.4]). Let ρ : Gal(K/K) H(Q ) be a representation. → ℓ We say that ρ is crystalline/semistable/de Rham/Hodge-Tate if for some (thus every) faithful algebraic representation ξ : H GL over Q , the composition ξ ρ is → N ℓ ◦ crystalline/semistable/de Rham/Hodge-Tate. Now suppose that ρ is Hodge-Tate. For each field embedding i: Q K, a cocharacter µ (ρ, i) b : G H over ℓ → HT K m → K is called a Hodge-Tate cocharacter for ρ and i if for some (thus every) faithful b b Galois representations for general symplectic groups 17 algebraic representation ξ : H GL(V ) on a finite dimensional Q -vector space V , → ℓ the cocharacter ξ ρ induces the Hodge-Tate decomposition of semilinear Gal(K/K)- ◦ modules on K-vector spaces:

b V K = Vk. ⊗Qℓ,i Mk∈Z b Namely V is the weight k space for the G -action through ξ µ (ρ, i), while k m ◦ HT Vk is also the K-linear span of the K-subspace in V K on which Gal(K/K) ⊗Qℓ,i acts through the ( k)-th power of the cyclotomic character. (So our convention is b − b that the Hodge-Tate number of the cyclotomic character is 1.) Finally, we call a − cocharacter µ (ρ, i): G H over Q a Hodge-Tate cocharacter, if it is conjugate HT m → ℓ to µ (ρ, i) b in H(K). HT K In any of the aboveb conditions on ρ, if it holds for one ξ then it holds for all ξ (use [DMOS82, Prop. I.3.1]). Whenever ρ is Hodge-Tate, a Hodge-Tate cocharacter exists by [Ser79, 1.4] and is shown to be unique (independent of ξ) by a standard § Tannakian argument. Often we only care about the isomorphism class of ρ, in which case only the H(Qℓ)-conjugacy class of a Hodge-Tate cocharacter matters. Lemma 1.11. Let f : H H be a morphism of connected reductive groups over 1 → 2 Q . If ρ : Gal(K/K) H (Q ) is a Hodge-Tate representation then f ρ is also ℓ → 1 ℓ ◦ Hodge-Tate with µ (f ρ, i)= f µ (ρ, i) for all i : Q K. HT ◦ ◦ HT ℓ → Proof. This is obvious by considering ξ f ρ for any faithful algebraicb representation ◦ ◦ ξ : H GL . 2 → N We return to the global setup. A representation ρ : Γ H(Q ) is said to be → ℓ totally odd if ρ is odd for every v . It is crystalline/semistable/de |Gal(F v/Fv) ∈ V∞ Rham/Hodge-Tate if ρ is crystalline/semistable/de Rham/Hodge-Tate for |Gal(F v/Fv) every place v above ℓ.

Definition 1.12. Let H be a real reductive group. Let KH be a maximal com- 0 pact subgroup of H(R). Put KH := (KH ) Z(H)(R). Let ξ be an irreducible algebraic representation of H C. An irreducible unitary representation π of H(R) ⊗R is said to be cohomological for ξe (or ξ-cohomological) if there exists i 0 such that ≥ Hi(Lie H(C), K , π ξ) = 0. (The definition is independent of the choice of K . H ⊗C 6 H The group KH is consistent with K∞ in 7 below.) e § Example 1.13.e Let H,ξ be as in Definition 1.12. Assume that H(R) has discrete series representations. Let Πξ be the set of (irreducible) discrete series representations ∨ which have the same infinitesimal and central characters as ξ . Then Πξ is a discrete series L-packet, whose L-parameter is going to be denoted by φ : W LH. ξ R → 18 Arno Kret, Sug Woo Shin

Then there are a Borel subgroup B H and a maximal torus T B such that ⊂ ⊂ λξ +ρ φξ(z)=(z/z) , where λξ is the B-dominant highest weight of ξ, and ρ is the half b b b b sum of B-positive roots. Every member of Πξ is ξ-cohomological. More precisely i b 1 ′ H (Lie H(C), KH, π ξ) = 0 exactly when i = 2 dimR H(R)/KH, in which case the b ⊗ 6 cohomology is of dimension [KH Z(H)(R): KH ], cf. Remark 7.2 below. e Definition 1.14. Consider a complex L-parametere φ: WC H. For a suitable max- → imal torus T H, one can describe φ as z µ (z)µ (z) for µ ,µ X (T ) with ⊂ 7→ 1 2 1 2 ∈ ∗ C µ µ X (T ). Write Ω b for the Weyl group of T in Hb. We define µ (φ) 1 − 2 ∈ ∗ H Hodge b b b b to be µ1 viewed as an element of X∗(T )C/ΩH . (When µ1 happens to be integral, b b b i.e. in X∗(T ), then we may also view µHodge(φ) as a conjugacy class of cocharacters b Gm H over C.) Given H,ξ as in the preceding example, define → b b µ (ξ) := µ (φ ). Hodge Hodge ξ|WC So if B, T are as before, then µ (ξ) = λ + ρ 1 X (T ) X (T ) up to the Hodge ξ ∈ 2 ∗ ⊂ ∗ C b ΩH -action. b b b b Let f : H H be a morphism of connected reductive groups over R whose 1 → 2 image is normal in H2 such that f has abelian kernel and cokernel. (Later we will consider the dual of the mapping GL Sp GSp ). Denote by f : H H the 1 × 2n → 2n 2 → 1 dual morphism. We choose maximal tori Ti Hi for i =1, 2 such that f(T2) T1. ⊂ b b b⊂ If φ2 : WR H2 is an L-parameter then obviously → b b b b b f(µ (φ )) = µ (f φ ). (1.1) Hodge 2|WC Hodge ◦ 2|WC

Lemma 1.15. With theb above notation, let π2 be ab member of the L-packet for φ2.

Then the pullback of π2 via f decomposes as a ﬁnite direct sum of irreducible representations of H (R), and all of them lie in the L-packet for f φ . 1 ◦ 2 Proof. This is property (iv) of the Langlands correspondenceb for real groups on page 125 of [Lan89].

2 Arthur parameters for symplectic groups

♭ Assume π is a cuspidal automorphic representation of Sp2n(AF ), such that

• π♭ is cohomological for an irreducible algebraic representation ξ♭ = ξ♭ ⊗v∈V∞ v of Sp2n,F ⊗C,

• ♭ there is an auxiliary ﬁnite place vSt such that the local representation πvSt is

the Steinberg representation of Sp2n(FvSt ). Galois representations for general symplectic groups 19

In this section we apply the construction of Galois representations [Art13,Shi11] to ♭ π to obtain a morphism ρ ♭ : Γ GO (Q ) and then lift ρ ♭ to a representation π → 2n+1 ℓ π ρ ♭ : Γ GSpin (Q ). π → 2n+1 ℓ Let us briefly recall the notion of (formal) Arthur parameters as introduced in [fArt13]. We will concentrate on the discrete and generic case as this is all we need (after Corollary 2.2 below); refer to loc. cit. for the general case. Here genericity means that no nontrivial representation of SU2(R) appears in the global parameter. For any N Z let θ be the involution on (all the) general linear groups GL , ∈ ≥1 N,F defined by θ(x) = J tx−1J where J is the N N-matrix with 1’s on its anti- N N N × diagonal, and 0’s on all its other entries. A generic discrete Arthur parameter for the group Sp is a finite unordered collection of pairs (m , τ ) r , where 2n,F { i i }i=1 • m ,r 1 are positive integers, such that 2n +1= r m , i ≥ i=1 i • P for each i, τi is a unitary cuspidal automorphic representation of GLmi (AF ) such that τ θ τ , i ≃ i

• the τi are mutually non-isomorphic, and

• each τi is of orthogonal type, and the product of the central characters of the

τi is trivial.

We write formally ψ = ⊞r τ for the Arthur parameter (m , τ ) r . The parameter i=1 i { i i }i=1 ψ is said to be simple if r = 1. The representation Πψ is deﬁned to be the isobaric ⊞r sum i=1τi; it is a self-dual automorphic representation of GL2n+1(AF ).

Exploiting the fact that Sp2n is a twisted endoscopic group for GL2n+1, Arthur attaches [Art13, Thm. 2.2.1] to π♭ a discrete Arthur parameter ψ. Let π♯ denote the corresponding isobaric automorphic representation of GL (A ) as in [Art13, 1.3]. 2n+1 F § (If ψ is generic, which will be verified soon, then ψ has the form as in the preceding ♭ paragraph.) For each F -place v, the representation πv belongs to the local Arthur packet Π(ψv) defined by ψ localized at v. This packet Π(ψv) satisfies the character relation ( [Art13, Thm. 2.2.1])

Tr (A π♯ (f )) = Tr τ(f Sp2n ), (2.1) θ ◦ v v v τ∈XΠ(ψv)

Sp2n,F for all pairs of functions fv (GL2n+1(Fv)), fv (Sp2n(Fv)) such that Sp2n,F ∈ H ∈ H fv is a Langlands-Shelstad-Kottwitz transfer of fv. Here Aθ is an intertwin- ♯ 2 ing operator from πv to its θ-twist such that Aθ is the identity map. (The precise normalization is not recalled as it does not matter to us.)

♯ Lemma 2.1. The component πvSt is the Steinberg representation of GL2n+1(FvSt ). 20 Arno Kret, Sug Woo Shin

Proof. This follows from [MS, Prop. 8.2].

♭ Corollary 2.2. The Arthur parameter ψ of π is simple (i.e. ψ = τ1 is cuspidal) and generic.

Proof. Lemma 2.1 implies in particular that ψvSt is a generic parameter which is irreducible as a representation of the local Langlands group WF SU2(R). Hence vSt × the global parameter ψ is simple and generic.

♯ ∼ Denote by π the cuspidal automorphic representation τ1 = Πψ. Let Aθ : ΠΨ θ 2 → ΠΨ the canonical intertwining operator such that Aθ is the identity and Aθ preserves the Whittaker model (see [Art13, sect. 2.1] and [KS99, sect. 5.3] for this normalization). Write η for the L-morphism LSp LGL extending the standard 2n,Fv → 2n+1,Fv representation Sp = SO (C) GL (C) such that η is the identity map 2n 2n+1 → 2n+1 |WFv onto WFv . c ♭ 6 ♯ Lemma 2.3. Let v be a finite F -place where πv is unramified . Then πv is unramified ♯ as well. Let φ ♯ : W SU (R) GL (C) be the Langlands parameter of π . Let πv Fv 2 2n+1 v × → ♭ φ ♭ : WF SU2(R) SO2n+1(C) be the Langlands parameter of π . Then η φ ♭ = πv v × → v ◦ πv φ ♯ . πv Proof. The morphism η∗ : unr(GL (F )) unr(Sp (F )) is surjective because H 2n+1 v → H 2n v the restriction of finite dimensional characters of GL2n+1 to SO2n+1 generate the space spanned by finite dimensional characters of SO2n+1. The lemma now follows Sp2n from (2.1) and the twisted fundamental lemma (telling us that one can take fv = ∗ η (fv) in (2.1)).

♯ The existence of the Galois representation ρπ♯ attached to π follows from [HT01, Thm. VII.1.9], which builds on earlier work by Clozel and Kottwitz. (The local hypothesis in that theorem is satisfied by Lemma 2.1. However this lemma is un- necessary for the existence of ρπ♯ if we appeal to the main result of [Shi11].) The theorem of [HT01] is stated over imaginary CM fields but can be easily adapted to the case over totally real fields, cf. [CHT08, Prop. 4.3.1] and its proof. (Also see [BLGGT14, Thm. 2.1.1] for the general statement incorporating later develop- ments such as the local-global compatibility at v ℓ, which we do not need.) We adopt | the convention in terms of L-algebraic representations as in [BG14, Conj. 5.16] un- like the references just mentioned, in which C-algebraic representations are used (cf. [BG14, 5.3, 8.1]). § § 6 We should mention that in the group Sp2n(Fv), not all hyperspecial subgroups are conjugate ♭ (in contrast to GSp2n(Fv ), where all hyperspecial groups are conjugate). When we say that πv is unramified, we mean that there exists a hyperspecial subgroup for which the representation has an invariant vector. Galois representations for general symplectic groups 21

To state the Hodge-theoretic property at ℓ precisely, we introduce some notation L based on 1. At each y we have a real L-parameter φ ♭ : W Sp arising ∞ ξy Fy 2n ♭ § ∈V → from ξy. The parameter is L-algebraic as well as C-algebraic. Via the embedding

.y : F ֒ C we may identify the algebraic closure F y with C so that WF y = WC → ♭ As explained in Deﬁnition 1.14, we have µHodge(ξy) := µHodge(φξ♭ W ), a conjugacy y | F y class of cocharacters G SO (C). m → 2n+1 Theorem 2.4. There exists an irreducible Galois representation

ρ ♭ = ρ ♭ : Γ SO (Q ), π π ,ι → 2n+1 ℓ ♭ unique up to SO2n+1(Qℓ)-conjugation, attached to π (and ι) such that the following hold.

♭ (i) Let v be a ﬁnite place of F not dividing ℓ. If πv is unramiﬁed then

(ρ ♭ W )ss ιφ ♭ , π | Fv ≃ πv ♭ where φ ♭ is the unramified L-parameter of π , and ( ) is the semisimpli- πv v ss ♭ · fication. For general π , the parameter ιφ ♭ is isomorphic to the Frobenius- v πv 7 semisimplification of the parameter associated to ρ ♭ . π |ΓFv

(ii) Let v be a ﬁnite F -place such that v ∤ ℓ where πv is unramiﬁed. Then ρπ♭,v is

unramiﬁed at v, and for all eigenvalues α of std(ρπ♭ (Frobv))ss and all embed- .dings Q ֒ C we have α =1 → | |

(iii) For every v ℓ, the representation ρ ♭ is potentially semistable. For each y : | π ,v F ֒ C such that ιy induces v, we have → ♭ µHT(ρπ♭,v, ιy)= ιµHodge(ξy).

(iv) For every v ℓ, the Frobenius semisimpliﬁcation of the Weil-Deligne represen- | tation attached to the de Rham representation ρπ♭,v is isomorphic to the Weil- ♭ Deligne representation attached to πv under the local Langlands correspondence.

(v) If π is unramiﬁed at v ℓ, then ρ ♭ is crystalline. If π has a non-zero Iwahori v | π ,v v ﬁxed vector at v ℓ, then ρ ♭ is semistable. | π ,v

(vi) The representation ρπ♭ is totally odd.

♭ (vii) If π is essentially square-integrable at v ∤ then std ρ ♭ is irreducible. v ∞ ◦ π 7 This is equivalent to saying that ι(std φ ♭ ) is isomorphic to the Frobenius-semisimpliﬁcation ◦ πv of the Langlands parameter associated with (std ρ ♭ ) in view of Proposition B.1. ◦ π |ΓFv 22 Arno Kret, Sug Woo Shin

♭ Remark 2.5. We need the ramiﬁed case of (i) only when πv is the Steinberg representation.

♯ ♯ Proof. The representation π is regular algebraic [Clo90, Def. 3.12], and so π∞ is cohomological for an irreducible algebraic representation. By Theorem 1.2 of [Shi11]

(in view of Remark 7.6 in loc. cit.) there exists a Galois representation ρπ♯ : Γ ♯ ♭ → GL2n+1(Qℓ) that satisﬁes properties (i)-(iii),(v) with π in place of π ; property (iv) is established by Caraiani [Car14, Thm. 1.1]. (The reader may also refer to the statement of [BLGGT14, Thm. 2.1.1].) Strictly speaking, the normalization there is n−1 diﬀerent from here, so one has to twist the Galois representation there by the 2 -th power of the cyclotomic character. In particular ρπ♯ is self-dual. ♯ By Lemma 2.1, πvSt is the Steinberg representation and by Taylor–Yoshida [TY07] the representation ρ ♯ : Γ GL (Q ) is irreducible. The determinant of this π → 2n+1 ℓ representation is trivial, since in the Arthur parameter (m , τ ) r , the product of { i i }i=1 the central characters of the τi is trivial. Together with the self-duality of ρπ♯ , we see that ρ ♯ factors through a representation ρ ♭ : Γ SO (Q ) via the standard π π → 2n+1 ℓ .(( embedding SO ֒ GL (after a conjugation by an element of GL (Q 2n+1 → 2n+1 2n+1 ℓ We know the uniqueness of ρπ♭ from Proposition B.1 (and Chebotarev density).

Properties (i)-(v) for ρπ♭ follow from those for ρπ♯ . Part (vi) is deduced from Lemma 1.9 and the main theorem of [Tay12]. Lastly (vii) is [TY07, Cor. B].

For the rest of this section, let π be a cuspidal ξ-cohomological automorphic representation of GSp2n(AF ) for an irreducible algebraic representation ξ of GSp2n,F ⊗C.

Lemma 2.6 (Labesse-Schwermer, Clozel). There exists a cuspidal automorphic Sp2n(AF )- subrepresentation π♭ contained in π.

Proof. This follows from the main theorem of Labesse-Schwermer [LS19].

Lemma 2.7. Suppose that π is a twist of the Steinberg representation at a ﬁnite place.

Then πv is essentially tempered at all places v.

Proof. Let π♭ be as in the previous lemma. We know that π♭ is the Steinberg representation at a ﬁnite place and ξ♭-cohomological, where ξ♭ is the restriction of ξ ♯ to Sp2n,F ⊗C (which is still irreducible). Let π be the self-dual cuspidal automorphic ♯ representation of GL2n+1(AF ) as above. Note also that π is C-algebraic (and regular); this is checked using the explicit description of the archimedean L-parameters. Hence

• π♯ is tempered at all v by Clozel’s purity lemma [Clo90, Lem. 4.9], v |∞ • π♯ is tempered at all v ∤ by [Shi11, Cor. 1.3] and (quadratic) automorphic v ∞ base change. Galois representations for general symplectic groups 23

(Since π♯ is self-dual, if a local component is tempered up to a character twist ♭ then it is already tempered.) Hence πv is a tempered representation of Sp2n(Fv), cf. [Art13, Thm. 1.5.1]. This implies that π∞ itself is essentially tempered. (Indeed, after twisting by a character, one can assume that π∞ restricts to a unitary tempered representation on Sp (F ) Z(F ), which is of finite index in GSp (F ). Then 2n v × v 2n v temperedness is tested by whether the matrix coefficient (twisted by a character so as 2+ε to be unitary on Z(Fv)) belongs to L (GSp2n(Fv)/Z(Fv)). This is straightforward to deduce from the same property of matrix coefficient for its restriction to Sp (F ) 2n v × Z(Fv).)

Corollary 2.8. π∞ belongs to the discrete series L-packet Πξ.

Proof. By [BW00, Thm. III.5.1], Πξ coincides with the set of essentially tempered ξ- cohomological representations. Since π∞ is essentially tempered and ξ-cohomological, the corollary follows.

3 Zariski connectedness of image

Let ρπ♭ be the Galois representation from Section 2. The goal of this section is to prove that ρπ♭ has connected image in the sense deﬁned below. The proof of this fact relies on the existence of the regular unipotent element in the image thanks to assumption (St).

Definition 3.1. Let G0 be a connected reductive group over Qℓ. A representation r : Γ G (Q ) is said to have connected image if the image of r has connected → 0 ℓ Zariski closure in G0. (The definition applies to any topological group in place of Γ.) We say that r : Γ G (Q ) is G -irreducible if its image is not contained in any → 0 ℓ 0 proper parabolic subgroup of G0, and we say that r is strongly G0-irreducible if r Γ′ ′ | is G0-irreducible for every open finite index subgroup Γ of Γ. In case G0 = GLn, we often leave out the reference to the group G0.

When G = GL , a representation r : Γ GL (Q ) is strongly irreducible if it 0 n → n ℓ is irreducible and has connected image in PGLn (Lemma 4.8). Typical examples of irreducible representations that are not strongly irreducible are Artin representations and the 2-dimensional ℓ-adic representation arising from a CM elliptic curve over Q. The lemma below is extracted from arguments of [ST06, p. 675]. (One can always enlarge L to satisfy the ﬁrst two conditions in the lemma.)

′ ′ Lemma 3.2. Let L/L /Qℓ be two ﬁnite extensions in Qℓ. Write ΓL′ := Gal(Qℓ/L ) and Γ := Gal(Q /L). Let ρ: Γ ′ SO (Q ) be a semi-stable representation. Let L ℓ L → 2n+1 ℓ H SO be the Zariski closure of the image of ρ. Assume that ⊂ 2n+1,Qℓ 24 Arno Kret, Sug Woo Shin

• H is deﬁned over L,

• the image of ρ is contained in SO2n+1(L), and

• the Weil-Deligne representation (the functor WD is deﬁned as in [Ber13, p. 12]) WD (B (std ρ))ΓL (3.1) st ⊗Qℓ ◦ has a nilpotent operator Nreg that is regular in Lie(SO2n+1(Qℓ)).

Then the group H(Qℓ) contains a regular unipotent element of SO2n+1(Qℓ).

Proof. The underlying space of the (φ, N)-module (B (std ρ))ΓL is naturally st ⊗Qℓ ◦ a vector space over the maximal subﬁeld L L that is unramiﬁed over Q . We 0 ⊂ ℓ consider the functor

Ψ: Rep (H) Rep (G ), r WD (B (r ρ))ΓL L → L a 7→ st ⊗Qℓ ◦ |Ga where, if V is an L-vector space equipped with the structure of a Weil-Deligne representation for Γ ′ , we write V for the unique representation of the additive L |Ga group ρ: G GL(V ) such that Lie (ρ)(1) is equal to the nilpotent operator of the a → Weil-Deligne representation V .

Let ωH and ωGa be the standard ﬁbre functors of the categories RepL(H) and Rep (G ). The functor Ψ is an exact faithful L-linear -functor. The composition L a ⊗ ω Ψ is therefore a ﬁbre functor of Rep (G)[DMOS82, Sect. II.3]. By Theorem Ga ◦ L 3.2 of [loc. cit.] we have the equivalence of categories,

Fibre functors of Rep (H) ∼ Category of H-torsors over L, η Hom⊗(η,ω ). L → 7→ H

⊗ ⊗ We obtain a morphism Aut (ωG ) Aut (ωH ) , i.e. an algebraic morphism a Qℓ → Qℓ Ψ∗ : Q H(Q ), well-deﬁned up to conjugation. Write N = Ψ∗(1) H(Q ). For ℓ → ℓ ∈ ℓ every linear representation r of H in a ﬁnite dimensional L-vector space the element Lie (Ψ∗)(1) is conjugate to the nilpotent operator of the Weil-Deligne representation attached to (B (r ρ))ΓL . Taking r = std, we see that Lie (Ψ∗)(1) and N are st ⊗Qℓ ◦ reg GL (Q )-conjugate by Assumption (3.1). Thus N = Ψ∗(1) H(Q ) is a regular 2n+1 ℓ ∈ ℓ unipotent element of SO2n+1(Qℓ).

Proposition 3.3. Let H be a semisimple subgroup of SO containing a regular 2n+1,Qℓ unipotent element N SO . Then either H is the full group SO , H is ∈ 2n+1,Qℓ 2n+1,Qℓ the principal PGL in SO , or n =3 and H is the simple exceptional group 2,Qℓ 2n+1,Qℓ G over Q (i.e. H is the automorphism group of the octonion algebra O Q ). 2 ℓ ⊗ ℓ Galois representations for general symplectic groups 25

Proof. Let H0 H denote the identity component. Since there are no non-trivial ⊂ morphisms from G to the ﬁnite group π (H), we must have N H0. Therefore H0 a 0 ∈ is either PGL , G , orSO as in the theorem, by [TZ13, Thm. 1.4] classifying 2,Qℓ 2 2n+1,Qℓ connected semisimple subgroups of SO containing regular unipotent elements. 2n+1,Qℓ From [SS97a] it follows that in each of the 3 cases the subgroup H0 SO ⊂ 2n+1,Qℓ is a maximal closed subgroup (Theorem B, (i.a), (iv.a) and (iv.e) of [loc. cit.]).8 In particular H0 = H.

The following lemma will be used in the proof of Proposition 3.5 below.

Lemma 3.4 (Liebeck–Testerman [LT04, Lem. 2.1]). Let G a semisimple connected algebraic group over an algebraically closed ﬁeld. If X is a connected G-irreducible subgroup of G, then X is semisimple, and the centralizer of X in G is ﬁnite.

Proposition 3.5. Assume n 3. The Zariski closure of the subgroup ρ ♭ (Γ) ≥ π ⊂ SO2n+1(Qℓ) is either PGL2, G2, or SO2n+1. (The embedding of PGL2 is induced by the symmetric 2n-th power representation of GL2. The group G2 occurs only when n =3 and embeds in SO7 via an irreducible self-dual 7-dimensional representation.)

Proof. Write H SO for the Zariski closure of ρ ♭ (Γ). We claim that H contains ⊂ 2n+1 π a regular unipotent element of GL2n+1 (thus also of SO2n+1).

To prove this, we distinguish cases. Let us ﬁrst assume vSt ∤ ℓ. Let NvSt be the

♭ unipotent operator of the Weil–Deligne representation attached to ρπ ,vSt so that it 1 1 corresponds to the image of (1, ( 0 1 )) under ιφπ♭ by Theorem 2.4. Then NvSt is vSt ♭ ♯ regular unipotent in GL2n+1 since πvSt and thus πvSt is Steinberg by Lemma 2.1. The attached Weil-Deligne representation has the property that a positive power of NvSt lies in the image of ρ ♭ . This completes the proof if v ∤ ℓ. If v ℓ, we take a ﬁnite π ,vSt St St| extension E/F such that ρ := ρ ♭ is semistable at places above v . By part π ,vSt |ΓE St (iv) of Theorem 2.4, the last assumption of Lemma 3.2 is satisﬁed. (To satisfy the

♭ ﬁrst two, take L to be large enough.) Applying Lemma 3.2, we see that ρπ ,vSt (ΓE), thus also H, contains a regular unipotent element of GL2n+1. We have shown that H SO contains a regular unipotent element. On ⊂ 2n+1 the other hand, H is an irreducible subgroup of SO by Theorem 2.4. By 2n+1,Qℓ Lemma 3.4, H0 is a semisimple group and hence H as well. By Proposition 3.3, H is 0 one of the groups SO2n+1, G2 or PGL2 over Qℓ. Thus H = H and the proposition follows. 8In fact it is not completely clear whether [SS97a, Thm. B] classiﬁes maximal closed subgroups or maximal connected closed subgroups; see footnote 9 in the proof of [KS20, Prop. 5.2]. Nevertheless, 0 we can still show that H = H in the case at hand as in that proof, using the fact that PGL2,Qℓ and G2 have no outer automorphism, which implies that the conjugation action by elements of H on H0 gives inner automorphisms of H0. 26 Arno Kret, Sug Woo Shin

4 Weak acceptability and connected image

A classical theorem for Galois representations states that if r ,r : Γ GL (Q ) are 1 2 → m ℓ two semisimple representations which are locally conjugate, then they are conjugate. (Recall that every representation is assumed to be continuous by the convention of this paper.) This is a consequence of the Brauer-Nesbitt theorem combined with the Chebotarev density theorem. In this section we investigate analogous statements when GLm(Qℓ) is replaced by a more general (not necessarily connected) reductive group. We show that in many cases the implication still holds if one assumes that one of the two representations has Zariski connected image.

Deﬁnition 4.1. A (possibly disconnected) reductive group G over Qℓ is said to be weakly acceptable if for every proﬁnite group ∆ and any two locally conjugate semisimple continuous representations r ,r : ∆ G(Q ) there exists an open sub- 1 2 → ℓ group ∆′ ∆ such that r and r restricted to ∆′ are conjugate. ⊂ 1 2

Lemma 4.2. Let r1,r2 : ∆ GLm(Qℓ) be two representations such that for some × → ﬁnite subgroup µ of Q we have for each σ ∆ that Tr r (σ)= ζTr r (σ) for some ℓ ∈ 1 2 ζ = ζ µ. Then Tr r (σ) = Tr r (σ) for all σ in some open subgroup ∆′ of ∆. σ ∈ 1 2 Proof. Choose some open ideal I Z such that (1 ζ)m / I for all ζ µ (Q ) ⊂ ℓ − ∈ ∈ t ℓ with ζ = 1. Take U ∆ the set of σ such that for i = 1, 2 we have for all σ U 6 ⊂ ∈ that Tr r (σ) m mod I. Then U is open by continuity of the representations r . i ≡ i Let σ U. The traces Tr r (σ) and Tr r (σ) agree up to an element ζ µ (Q ). ∈ 1 2 σ ∈ t ℓ Reducing modulo I we get m ζ m. This is only possible if ζ = 1. For the ≡ σ σ unit element e ∆ we have Tr r (e) = m (i = 1, 2), consequently U is an open ∈ i neighborhood of e ∆. We may now take ∆′ an open subgroup of ∆ contained in ∈ U.

Lemma 4.3. Assume [µ ֌ H ։ G] is a central extension of reductive groups over

Qℓ, where we assume additionally that µ is ﬁnite. Then the group G is weakly acceptable if and only if H is weakly acceptable.

Proof. “ ”. Assume that H is weakly acceptable. Let ∆ be a proﬁnite group and ⇐ r ,r : ∆ G(Q ) locally conjugate. Choose U H(Q ) an open subgroup with 1 2 → ℓ ⊂ ℓ U µ = 1 . Write U G(Q ) for the image of U in G(Q ). The map H ։ G ∩ { } ⊂ ℓ ℓ induces a bijection U ∼ U. Consider ∆′ := r−1(U) r−1(U) ∆. Using the → 1 ∩ 2 ⊂ isomorphism U ∼ U we obtain lifted representations r , r : ∆′ H(Q ). By the → 1 2 → ℓ local G-conjugacy of r ,r , there exists for each σ ∆′ some element h = h 1 2 ∈ σ ∈ H(Q ) and a z = z µ(Q ) such that e e ℓ σ ∈ ℓ r (σ) = zhr (σ) h−1 H(Q ) (4.1) 2 ss 1 ss ∈ ℓ e e Galois representations for general symplectic groups 27

′′ ′ We claim that in an open subgroup ∆ of ∆ the elements r2(σ) and r1(σ) are in fact H-conjugate for every σ ∆′′ (as opposed to conjugate up to µ H). ∈ ⊂ Choose a faithful representation ϕ: H GL , and writee ϕ as a directe sum of → N irreducible representations ϕ = m ϕ . By Schur’s lemma we have ϕ (µ) µ where ⊕i=1 i i ⊂ t µ are the t-th roots of unity for some t Z . In particular we have the identity t ∈ ≥1 Tr ϕ (r (σ)) = ζTr ϕ (r (σ)) Q , i 2 i 1 ∈ ℓ where ζ := ϕi(z) with z as ine (4.1). Consequentlye there exists by Lemma 4.2 some open subgroup ∆ ∆ such that on ∆ we have Tr ϕ r (σ) = Tr ϕ r (σ). Consider i ⊂ i i 1 i 2 an open subgroup ∆′ ∆ such that Tr ϕ r (σ) = 0 for all σ ∆′ . Now let i ⊂ i i 1 6 ∈ i σ ∆′′ := m ∆′ and let (z, h) be as in Equation (4.1e). By applyinge Tr ϕ to (4.1) ∈ ∩i=1 i i we ﬁnd e 0 = Tr ϕ r (σ) = Tr ϕ r (σ)= ϕ (z)Tr r (σ) Q , 6 i 1 i 2 i 1 ∈ ℓ so ϕi(z) = 1 for all i =1, 2,...,m, and hence z = 1. This proves the claim. e e e The group H is weakly acceptable. Hence the representations r1 ∆′′′ and r2 ∆′′′ ′′′ ′′ | | are conjugate by some h H(Qℓ) on some open subgroup ∆ ∆ . Since H(Qℓ) ∼ ∈ ⊂ ⊃ U U G(Q ) the representations r ′′′ and r ′′′ are conjugatee by the imagee of → ⊂ ℓ 1|∆ 2|∆ h in G(Qℓ). “ ”. Let r ,r : ∆ H(Q ) be two locally conjugate semisimple continuous ⇒ 1 2 → ℓ representations. Their projections r1, r2 are locally conjugate in G(Qℓ). Hence there ′ −1 exists an open ∆ ∆ such that r2 ∆′ = g(r1 ∆′ )g . Lift g to g H(Qℓ) and −1⊂ | | ∈ −1 replace r2 by gr2g . Since µ is central we obtain the character χ(σ) := r1(σ)r2(σ) ′ of ∆ . This character has ﬁnite order. Hence r1 and r2 agree on thee open subgroup ker(χ) ∆′. e e ⊂

Lemma 4.4. Let G/Qℓ be a reductive group with center ZG for which there exists a semisimple subgroup G G such that Z G G is surjective. 1 ⊂ G × 1 →

1. The group G is weakly acceptable if and only if its adjoint group Gad is weakly acceptable.

2. If G is connected and Gad is a product of copies of the groups PGLn, SO2n+1

and PSp2n, then G is weakly acceptable.

3. If G is GPin2n or GO2n, then G is weakly acceptable.

der Remark 4.5. Note that for connected groups G, we may take G1 = G and the condition of the lemma is always satisﬁed. For non-connected G the condition is not empty: One may consider the semi-direct product G of GL with 1 where 1 2 {± } − acts by g g−1,t. In this case there is no semisimple subgroup G G such that 7→ 1 ⊂ 28 Arno Kret, Sug Woo Shin

Z G G is surjective (rk(Z G )=0+1 < 2 = rk(G)). For applications in G × 1 → G × 1 this paper, we only need connected G, so the reader may choose to restrict to this case. However we state a more general version for potential applications elsewhere.

Proof. (1). The statement follows by applying the previous lemma to the natural surjections Z G G and G G . Both maps have ﬁnite kernels as Z is G × 1 → 1 → 1,ad G1 ﬁnite and contains Z G . Therefore G is weakly acceptable if and only if G is G ∩ 1 1 weakly acceptable if and only if G1,ad is weakly acceptable. Finally G1 G induces ∼ ⊂ an isomorphism G1,ad Gad. → der (2). If G is connected we can take G1 = G and use acceptability of the groups

GLn, SO2n+1 and Sp2n; the latter two cases follow from Proposition B.1.

(3). Since O2n is acceptable by Proposition B.1, the group PO2n is weakly acceptable by Lemma 4.3. For G = GPin2n (resp. GO2n) we take G1 := Pin2n (resp. O ), and we have the surjection Z G ։ G. From the surjection G ։ PO we 2n G × 1 1 2n ﬁnd that G1 is weakly acceptable, and then G is also weakly acceptable by (i).

Proposition 4.6. Let G be a reductive group with center Z, cocenter D and µ Z ⊂ the kernel of the natural morphism Z D. Assume that G is weakly acceptable. Let → ∆ be a proﬁnite group and r1,r2 : ∆ G(Qℓ) be two locally conjugate continuous → 9 semisimple representations where we assume that r1 has Zariski connected image modulo Z. Then:

1. We have r = χ gr g−1 where g G(Q ) and χ: ∆ µ(Q ) is some character. 2 · 1 ∈ ℓ → ℓ 2. If the image of r in G(Q ) is Zariski connected modulo a subgroup Z Z 1 ℓ 0 ⊂ with Z µ = 1 , then we have χ =1 in (1). 0 ∩ { } 3. Assume the existence of σ ∆ with the following property: the images of 0 ∈ r (σ ) and gr (σ )g−1 in G/µ(Q ) are not G0(Q )-conjugate for any g 1 0 1 0 ℓ ℓ ∈ G (Q ) G0 (Q ) Then χr and r are G0(Q )-conjugate. ad ℓ \ ad ℓ 1 2 ℓ

Example 4.7. Consider the case G = PGLm. By (1) any two locally conjugate projective Galois representations r1,r2 are conjugate on an open subgroup of ∆ (of index at most m). If furthermore one of the two representations has connected image, then r1 and r2 are PGLm(Qℓ)-conjugate by (2). Similarly, two locally conjugate

GSpin2n+1-valued Galois representations are conjugate on an open subgroup of index

2, since µ is a group of order 2 when G = GSpin2n+1.

Proof. (1). Since G is weakly acceptable, there exists some g G(Q ) and an open ∈ ℓ subgroup ∆′ ∆ such that r = gr g−1 on ∆′. Let ϕ: G/Z GL be a faithful ⊂ 2 1 → N 9Recall that a morphism of finite type affine algebraic groups over a field is closed, in particular there is no difference in taking the Zariski closure before or after reducing modulo Z. Galois representations for general symplectic groups 29 representation and choose x GL (Q ) such that ϕr (σ)= xϕr (σ)x−1 GL (Q ) ∈ N ℓ 1 2 ∈ N ℓ for all σ ∆. For σ ∆′ we obtain ∈ ∈ ϕr (σ)= xϕr (σ)x−1 = xϕ(gr (σ)g−1)x−1 GL (Q ). 1 2 1 ∈ N ℓ

−1 −1 Taking the Zariski closure we have ϕ(i) = xϕ(gig )x GLN for all i in the ′ ∈ Zariski closure I of the image of r1(∆ ). Since I/Z is connected, we obtain ϕr1(σ)= xϕ(gr (σ)g−1)x−1 GL (Q ) for all σ ∆. Earlier we found ϕr (σ)= xϕr (σ)x−1 1 ∈ N ℓ ∈ 1 2 ∈ GL (Q ) for all σ ∆. Thus ϕr (σ)= ϕ(gr (σ)g−1) GL (Q ) for all σ ∆. We N ℓ ∈ 2 1 ∈ N ℓ ∈ obtain that r (σ) = χ(σ) gr (σ)g−1 G(Q ) for some character χ: ∆ Z(Q ). 2 · 1 ∈ ℓ → ℓ By mapping down to the cocenter D of G and using local conjugacy, we see that χ has image in µ.

(2). Repeat the same proof, but now take ϕ a faithful representation of G/Z0 (as opposed to G/Z). Then we find a character χ as above, whose image is in µ and Z0, and therefore trivial. −1 (3). By (1) we have r2 = gχr1g for some g Gad(Qℓ). By local conjugacy at −1 ∈ σ0, we have that r2(σ0)= χ(σ0)gr1(σ0)g and r1(σ0) are Gad(Qℓ)-conjugate. By the assumption, this implies g G0 (Q ). Hence r and χr are G0 (Q )-conjugate. ∈ ad ℓ 2 1 ad ℓ Lemma 4.8. Let ∆ be a profinite group. Let r : ∆ GL(V ) be a representation on a → × finite dimensional Q -vector space V and let χ: ∆ Q be a nontrivial character. ℓ → ℓ 1. If r r χ then r is not strongly irreducible. ≃ ⊗ 2. If r is irreducible and has connected image in PGL(V ), then r is strongly irreducible.

Proof. (1) We may assume r is irreducible. Since det(r) = det(r χ) = det(r)χdim V , ⊗ the character χ has ﬁnite order. Take ∆′ := ker χ, which is an open normal subgroup of ∆. It is enough to show that we have the following inequality

∆ dim Hom∆′ (r, r) = dim Hom∆(r, Ind∆′ r) > 1.

∆ fv We have the tautological ∆-equivariant embedding r ֒ Ind∆′ r given by v (γ χ χ → 7→ 7→ ֒ r(γ)v). Deﬁne fv (γ) := χ(γ)fv(γ). Then v fv gives another embedding r ∆ 7→ → Ind ′ r that is not a scalar multiple of v f . ∆ 7→ v (2) This follows from the fact that the Zariski closure of the image of r in PGL(V ) does not change upon restriction to an open normal subgroup ∆′ ∆ due to con- ⊂ nectedness. Since (strong) irreducibility only depends on the Zariski closure of image in PGL(V ), the lemma follows. 30 Arno Kret, Sug Woo Shin

The preceding lemma leads to the following proposition, which will be employed when studying the spinor norm of GSpin2n+1-valued Galois representations and when proving an automorphic multiplicity one result. Consider r : ∆ GL(V ) and × → χ : ∆ Q as in the setup of Lemma 4.8. Let r t m r be a decomposition into → ℓ ≃ ⊕i=1 i i mutually non-isomorphic irreducible ∆-representations with multiplicities m Z . i ∈ >0 Proposition 4.9. Assume that

• dim r t are mutually distinct. { i}i=1 • im(r) has connected Zariski closure in PGL(V ).

Then each of the following is true.

1. If r r∨ χ and if χ′ r r is a one-dimensional ∆-submodule then χ′ = χ. ≃ ⊗ ⊂ ⊗ 2. If r r χ then χ =1. ≃ ⊗ Proof. (1) The ﬁrst condition says m r m r∨ χ. Since dim(r ) are distinct, ⊕i i i ≃ ⊕i i i ⊗ i r r∨ χ for all i. Again by the dimension assumption, χ′ r r for some i ≃ i ⊗ ⊂ i ⊗ i i. (If χ′ r r for i = j then there is a nonzero ∆-equivariant map from r∨ to ⊂ i ⊗ j 6 i r χ′−1, which must be an isomorphism by irreducibility; however this violates the j ⊗ dimension assumption.) Moreover if r has connected image in PGL then so does ri

(because im(ri) is the image of im(r) by the projection map). Hence we are reduced to the case when r is irreducible. Now suppose r is irreducible. Then χ′ r r implies r r∨ χ′−1. We are ⊂ ⊗ ≃ ⊗ assuming r r∨ χ, so we have r r (χ′χ−1). If χ′ = χ then part (1) of Lemma ≃ ⊗ ≃ ⊗ 6 ′ 4.8 says r ′ is reducible for some open normal subgroup ∆ ∆, contradicting part |∆ ⊂ (2) of the same lemma. Therefore χ′ = χ. (2) Arguing similarly as in the proof of (1), we reduce to the case of irreducible r and deduce that χ = 1 by using Lemma 4.8.

5 GSpin-valued Galois Representations

In this section we study the notion of local conjugacy for the group GSpin2n+1(Qℓ). In general it is not expected that local conjugacy implies (global) conjugacy of Galois representations: In the paper [Lar94a, proof of Prop. 3.10] Larsen constructs a certain ﬁnite group ∆ (in his text it is called Γ), which is a double cover of the (non-simple)

Mathieu group M10 in the 10-th alternating group A10. More precisely, he realizes M SO (Q ) by looking at the standard representation of A GL (Q ). Then 10 ⊂ 9 ℓ 10 ⊂ 10 ℓ ∆ is the inverse image of M10 in Spin9(Qℓ). Let us just assume that ∆ can be realized as a Galois group s: Γ ։ ∆. The group ∆ comes with a map φ : ∆ Spin (Q ), 1 → 9 ℓ Galois representations for general symplectic groups 31 and η is the composition ∆ ։ M ։ M /A Z/2Z ֌ Spin (Q ). He defines 10 10 6 ≃ 9 ℓ φ (x) := η(x)φ (x). We may define r := φ s and r := φ s. Then the argument of 2 1 1 1 ◦ 2 2 ◦ Larsen shows that φ (σ) and φ (σ) are Spin (Q ) conjugate for every σ Γ, while φ 1 2 9 ℓ ∈ 1 and φ2 are not Spin9(Qℓ)-conjugate. The maps φi cannot be GSpin9(Qℓ)-conjugate: × If we would have φ = gφ g−1 for some g GSpin (Q ) then we can find a z Q , 2 1 ∈ 9 ℓ ∈ ℓ such that h = gz Spin (Q ) and hφ h−1 = gφ g−1 = φ which contradicts Larsen’s ∈ 9 ℓ 1 1 2 conclusion. Thus, assuming the inverse Galois problem holds over F for ∆, (r1,r2) is a pair of locally conjugate, but non-conjugate Galois representations. In [Lar94a] Larsen explains that counterexamples may be constructed for all Spin (Q ) with m 8 (for m = 8, see [Lar96, Prop. 2.5])10. Proposition 4.6 shows m ℓ ≥ that any two locally conjugate GSpin2n+1(Qℓ)-valued Galois representations are conjugate up to a quadratic character twist (see also Example 4.7).

Lemma 5.1. Let ρ: Γ GSpin2n+1(Qℓ) be a semisimple representation that contains → × a regular unipotent element in the Zariski closure of its image. If χ: Γ Q is a → ℓ character such that spin(ρ) spin(ρ) χ then χ =1. ≃ ⊗ Proof. This follows from part (2) of Proposition 4.9 applied to r = spin(ρ). For this we need to check the assumptions there; the only nontrivial part is the dimension hypothesis, which we verify as follows. Write im(ρ) for the Zariski closure of im(ρ) in GSpin2n+1. The image of im(ρ) in SO2n+1 is either PGL2, SO2n+1, or G2 (last case only when n = 3) by Proposition 3.3. Hence we have either

(i) SL im(ρ) GL , 2 ⊂ ⊂ 2 (ii) PGL im(ρ) PGL G , 2 ⊂ ⊂ 2 × m (iii) Spin im(ρ) GSpin , or 2n+1 ⊂ ⊂ 2n+1 (iv) G im(ρ) G G (the last case is possible only when n = 3). 2 ⊂ ⊂ 2 × m In each case we explain the dimension hypothesis of Proposition 4.9 in detail when im(ρ) is equal to GL , PGL G , GSpin , or G G , respectively, as the 2 2 × m 2n+1 2 × m argument is essentially the same in general. Case (iii) is obvious. Cases (i) and (ii) follow from the fact that GL or PGL G -representations are determined by the 2 2 × m dimension if the central character has weight 1 (namely the central Gm acts through the identity map). In case (iv), the underlying spin representation has dimension 8, and it decomposes as the direct sum of a one-dimensional representation with an irreducible 7-dimensional representation of G2, whereas the Gm factor acts by weight 1. So the assumption still holds.

10 In a private communication, Chenevier informed us that the group Spin7(Qℓ) is also not acceptable. 32 Arno Kret, Sug Woo Shin

Proposition 5.2. Let r ,r : Γ GSpin (Q ) be two semisimple representations 1 2 → 2n+1 ℓ that are unramiﬁed at almost all places and locally conjugate. Assume the image of r1 contains a regular unipotent element in the Zariski closure of its image. Then r1 and r2 are conjugate.

Proof. As in the proof of Lemma 5.1, the image of r1 in SO2n+1 has connected

Zariski closure. Recall from Lemma 4.4 that GSpin2n+1 is weakly acceptable. By Proposition 4.6 (1) we may assume that r2 = r1χ, with χ a character taking values in the center of Spin2n+1(Qℓ). By Lemma 5.1 we have χ = 1.

6 The trace formula with ﬁxed central character

In this section we recall the general setup for the trace formula with fixed central character.11 For later use, we prove some instances of the Langlands functoriality for general reductive groups under the assumptions analogous to (St) at a finite place and cohomological at infinite places. (One might try to prove such a result by directly applying the L2-Lefschetz formula by [Art89] and [GKM97] but the formula only tells us 0 = 0 when F = Q. This is why we need a version with fixed central 6 character.) Let G be a connected reductive group over a number field F with center Z. Write 0 1 AZ for the maximal Q-split torus in ResF/QZ and set AZ,∞ := AZ(R) . Write G(AF ) for the subgroup of G(A ) as on p.11 of [Art81] so that G(A ) = G(A )1 A . F F F × Z,∞ Consider a closed subgroup X Z(A ) which contains A such that Z(F )X is ⊂ F Z,∞ closed in Z(AF ) (then Z(F )X is always cocompact in Z(AF )) and a continuous character χ : (X Z(F )) X C×. Such a pair (X, χ) is called a central character ∩ \ → datum. In what follows we need to choose Haar measures consistently for various groups, but we will suppress these choices as this is quite standard. For instance the same

Haar measures on G(AF ) and X have to be chosen for each term in the identity of Lemma 6.1 below. × Let v be a place of F , and Xv a closed subgroup of Z(Fv). Let χv : Xv C be a −1 → smooth character. Write (G(Fv), χv ) for the space of smooth compactly supported H −1 functions on G(Fv) which transform under Xv via χv ; if v is archimedean, we also require functions in (G(F ), χ−1) to be K -ﬁnite for a maximal compact subgroup H v v v Kv of G(Fv). (We ﬁx such a Kv in this section, and use the same Kv to compute relative Lie algebra cohomology. In the main case of interest, the choice of Kv is made in 7.) Given a semisimple element γ G(F ) and an admissible representation π § v ∈ v v 11More details are to be available in [KSZ] and Dalal’s thesis. Compare with Section 1 of [Rog83], Sections 2 and 3 of [Art02], or Section 3.1 of [Art13]. Galois representations for general symplectic groups 33

of G(Fv) with central character χv on Xv, the orbital integral and trace character for f (G(F ), χ−1) are deﬁned as follows. Below I denotes the connected v ∈ H v v γv centralizer of γv in G.

−1 Oγv (fv) := fv(x γvx)dx, ZIγv (Fv)\G(Fv ) Tr (f π ) = Tr π (f ) := Tr f (g)π (g)dg . v| v v v v v ZG(Fv )/Xv Note that the trace is well defined since the operator f (g)π (g)dg is of G(Fv)/Xv v v finite rank if v is finite and is of trace class if v is infinite.R For our purpose, we henceforth assume the following. • X = X∞X for an open compact subgroup X∞ Z(A∞) and X = Z(F ), ∞ ⊂ F ∞ ∞ • χ = v χv with χv = 1 at every finite place v, One definesQ the adelic Hecke algebra (G(A ), χ−1) as well as orbital integrals H F and trace characters by taking a product over the local case considered above.

Write Γell,X(G) for the set of X-orbits of elliptic conjugacy classes in G(F ). Let L2 (G(F ) G(A )) denote the space of functions on G(F ) G(A ) transforming disc,χ \ F \ F under X by χ and square-integrable on G(F ) G(A )1/X G(A )1. Write (G) \ F ∩ F Aχ for the set of isomorphism classes of cuspidal automorphic representations of G(AF ) whose central characters restricted to X are χ. (In particular such representations are G(A )-submodules of L2 (G(F ) G(A )).) F disc,χ \ F For f (G(A ), χ−1) we define invariant distributions T G and T G by ∈ H F ell,χ disc,χ T G (f) := ι(γ)−1vol(I (F ) I (A )/X)O (f), ell,χ γ \ γ F γ γ∈ΓXell,X(G) T G (f) := Tr (f L2 (G(F ) G(A ))). disc,χ | disc,χ \ F G Similarly Tcusp,χ is defined by taking trace on the space of square-integrable cusp- forms. We omit the G from the notation when clear from the context. In general we do not expect that Tell,χ(f) = Tdisc,χ(f) (unless G/Z is anisotropic over F ); the equality should hold only after adding more terms on both sides. However we do have Tell,χ(f) = Tdisc,χ(f) if f satisfies some local hypotheses; this is often referred to as the simple trace formula.

We also need to consider the central character datum (X0, χ0) with X0 := AZ,∞ and χ := χ . The quotient X Z(F ) X/X is compact as it is closed in 0 |AZ,∞ ∩ \ 0 Z(F ) Z(A )/A , which is compact. We have a natural surjection (G(A ), χ−1) \ F Z,∞ H F 0 → (G(A ), χ−1) given by H F χ f g f (g) := f (zg)χ(z)dz . 0 7→ 7→ 0 0 Zz∈X∩Z(F )\X/X0 34 Arno Kret, Sug Woo Shin

z Translating the function f0 by z, deﬁne a function f0 (g)= f0(zg). Then

1 χ T G (f z)χ(z)dz = T G (f ), ⋆ ell, disc . volume(X Z(F ) X/X ) ⋆,χ0 0 ⋆,χ 0 ∈{ } ∩ \ 0 Zz∈X∩Z(F )\X/X0 From here on we assume that F is totally real and that G(F∞) admits discrete series representations (for Lefschetz functions to be nontrivial). Let ξ be an irreducible algebraic representation of (Res G) C G F , F/Q ⊗Q ≃ ⊗F ∞ also thought of as a continuous representation of G(F∞) on a complex vector space. Denote by χ : Z(F ) C× the restriction of ξ∨ to Z(F ). Write f = f G ξ ∞ → ∞ ξ ξ ∈ (G(F ), χ−1) for a Lefschetz function (a.k.a. Euler-Poincaréfunction) associated H ∞ ξ with ξ such that Tr π∞(fξ) computes the Euler-Poincarécharacteristic for the relative Lie algebra cohomology of π ξ for every irreducible admissible representation ∞ ⊗ π∞ of G(F∞) with central character χξ. See Appendix A for details. Analogously we have the notion of Lefschetz functions at finite places as recalled in Appendix A.

The following simple trace formula is standard for X = AG,∞ and some other choices of X (such as X = Z(G(AF ))), but we want the result to be more ﬂexible.

Our proof reduces to the case that X = AG,∞. Lemma 6.1. Consider the central character datum (Z(F ), χ). Assume that f ∞ vSt ∈ (G(F )) is a (truncated) Lefschetz function and that f (G(F ), χ−1) is a H vSt ∞ ∈ H ∞ cuspidal function. Then

G G G Tell,χ(f)= Tdisc,χ(f)= Tcusp,χ(f).

−1 Proof. Consider f0 = v f0,v (G(AF ), χ0 ), where f0,v := fv at all finite places ∈ H χ v, and f is a cuspidal function such that f (as defined above) and f have the 0,∞ Q 0,∞ ∞ same trace against every tempered representation of G(F∞). The existence of such an f0,∞ is guaranteed by the trace Paley–Wiener theorem. By assumption fvSt = f0,vSt is strongly cuspidal (Lemma A.7). Hence the simple trace formula [Art88b, Cor. 7.3, 7.4] implies that G G G Tell,χ0 (f0)= Tdisc,χ0 (f0)= Tcusp,χ0 (f0). We deduce the lemma by averaging over Z(F )/Z(F ) G(F )1 against χ, noting ∞ ∞ ∩ ∞ that the average of f0,∞, namely the function g 1 f0,∞(zg)χ(z)dz, 7→ Z(F∞)/Z(F∞)∩G(F∞) recovers f∞ up to a nonzero constant. R Now we go back to the general central character data and discuss the stabilization for the trace formula with fixed central character under simplifying hypotheses. ∗ ∗ Assume that G is quasi-split over F . Write Σell,X(G ) for the set of X-orbits on the set of F -elliptic stable conjugacy classes in G∗(F ). Define

G∗ ∗ −1 G∗ ∗ −1 STell,χ(f) := τ(G ) ˜ι(γ) SOγ,χ(f), f (G (AF ), χ ), ∗ ∈ H γ∈ΣXell,X(G ) Galois representations for general symplectic groups 35 where ˜ι(γ) is the number of Γ-fixed points in the group of connected components in ∗ G∗ the centralizer of γ in G , and SOγ,χ(f) denotes the stable orbital integral of f at ∗ γ. If G has simply connected derived subgroup (such as Sp2n or GSp2n), we always have ˜ι(γ) = 1. Returning to a general reductive group G, let G∗ denote its quasi-split inner form over F (with a fixed inner twist G∗ G over F ). Since Z is canonically ≃ identified with the center of G∗, we may view (X, χ) as a central character datum for G∗. Let f ∗ (G∗(A ), χ−1) denote a Langlands-Shelstad transfer of f to G∗.12 ∈ H F Such a transfer exists in this fixed-central-character setup: One can lift f via the surjection (G(A )) (G(A ), χ−1) given by φ (g φ(gz)χ(z)dz), apply H F → H F 7→ 7→ X the transfer from (G(A )) to (G∗(A )) (the transfer to quasi-split inner forms H F H F R is due to Waldspurger), and then take the image under the similar surjection down to (G∗(A ), χ−1). H F Let vSt be a finite place of F .

G G∗ ∗ Lemma 6.2. Assume that fvSt is a Lefschetz function. Then Tell,χ(f) = STell,χ(f ).

Proof. Since fvSt is stabilizing (Lemma A.7), this follows from [Lab99, Thm. 4.3.4] (specialized to L = G, H = G∗, θ = 1; note that the“(G∗,H)-essentiel” condition there is vacuous).

Let S0 S∞ vSt be a finite subset, and S a finite set of places containing S0 ⊂ ∪{ }∗ ∪∗ S∞. We assume that G is unramified away from S and fix a reductive model for G over [1/S]. (See [ST16, Prop. 8.1] for the existence of such a model.) OF Let G∗ G over F be an inner twist which is trivialized at each place v / S (i.e. ≃ ∈ 0 up to an inner automorphism the inner twist descends to an isomorphism over Fv). In particular G is unramified outside S, and fix a reductive model for G over [1/S] as OF well. By abuse of notation we still write G and G∗ for reductive models. The notion of unramified representations is taken relative to the hyperspecial subgroups given ∗ by these integral models. The inner twist determines an isomorphism GFv GFv ∗ ≃ for v / S0, well defined up to inner automorphisms of GFv . When v / S we further ∈ ∗ ∈ have an isomorphism G GO . We fix these isomorphisms and use them to OFv ≃ Fv transport representations between G∗ and G. We prove weak transfers of a certain class of automorphic representations between G∗ and G.

Proposition 6.3. Assume that the adjoint group of G is nontrivial and simple over ♮ FvSt . Consider π and π as follows: 12The transfer for χ−1-equivariant functions is deﬁned in terms of the usual transfer factors. Its existence is implied by the existence of the usual Langlands-Shelstad transfer (for compactly supported functions). For details, see [Xu16, 3.3] for instance. 36 Arno Kret, Sug Woo Shin

∗ • π is a cuspidal automorphic representation of G (AF ) such that

– π is unramiﬁed at every place apart from S,

– πvSt is an unramiﬁed character twist of the Steinberg representation,

– π∞ is ξ-cohomological.

♮ • π is a cuspidal automorphic representation of G(AF ) such that

– π♮ is unramiﬁed at every place apart from S,

♮ – πvSt is an unramiﬁed character twist of the Steinberg representation, ♮ – π∞ is ξ-cohomological.

1. Suppose that ξ has regular highest weight. Then for each π (resp. π♮) as above, there exists π♮ (resp. π) as above such that π♮ π at every ﬁnite place v / v ≃ v ∈ S v . ∪{ St} 2. For each π as above, suppose that

• π is not one dimensional, • For every τ (G∗) such that τ S πS, if τ is ξ-cohomological τ is ∈Aχ ≃ ∞ vSt an unramiﬁed twist of the Steinberg representation then τ∞ is a discrete series representation.

Then there exists π♮ as above such that π♮ π at every ﬁnite place v / v ≃ v ∈ S v . Moreover the converse is true with G in place of G∗ and the roles ∪{ St} of π and π♮ switched.

Remark 6.4. The adjoint group of G is assumed to be simple over FvSt in order to apply Proposition A.2. Without the assumption we may have a mix of trivial and

Steinberg representations (up to unramiﬁed twists) for πvSt corresponding to simple factors of G at vSt.

Remark 6.5. Corollary 2.8 tells us that the condition in (2) for the transfer from G∗ ∗ to G is satisﬁed by G = GSp2n. Later we will see in Corollary 8.4 that the same is also true for a certain inner form of GSp2n.

Proof. We will only explain how to go from π to π♮ as the opposite direction is proved by the exact same argument. Let f = f be such that f unr(G(F )) v v v ∈ H v for ﬁnite places v / S, f is a Lefschetz function at v , and f is a Lefschetz vSt Q St ∞ ∈ ∗ ∗ ∗ function for ξ. Then we can choose the transfer f = v fv such that fv = fv for ∗ ∗ all v / S∞ vSt , f is a Lefschetz function, and f is a Lefschetz function for ∈ ∪{ } vSt ∞Q Galois representations for general symplectic groups 37

∗ ∗ ξ. We know from Lemmas A.4 and A.11 that fvSt (resp. f∞) and fvSt (resp. f∞) are associated up to a nonzero constant. Hence cf and f ∗ are associated for some c C×. The preceding two lemmas imply that ∈

∗ ∗ T G (f ∗) = STG (f ∗)= c T G (f). cusp,χ ell,χ · cusp,χ By linear independence of characters, we have

∗ ♮ ♮ m(τ)Tr (fS τS)= c m(τ )Tr (fS τS). ∗ | · | τ∈Aχ(G ) τ♮∈Aχ(G) S S τX≃π τ♮,SX≃πS

Let us prove (1). Choose f = f ∗ at finite places v in S but outside v S v v { St} ∪ 0 ∪ S such that Tr (f ∗ π ) > 0. (This is vacuous if there is no such v.) At infinite ∞ v | v places, as soon as Tr (f ∗ τ ) =0 at v , the regularity condition on ξ implies that v | v 6 |∞ τ is a discrete series representations and that Tr (f ∗ τ )=( 1)q(G) by Vogan– v v | v − Zuckerman’s classification of unitary cohomological representations. At v = vSt, ∗ whenever Tr (f τv ) = 0 (which is true for τ = π), the unitary representation τv vSt | St 6 St is either an unramified twist of the trivial or Steinberg representation by Lemma A.1.

If GvSt is anisotropic modulo center then the Steinberg representation is the trivial representation. In case GvSt is isotropic modulo center, if τvSt were one-dimensional then the global representation τ would be one-dimensional by a well known strong approximation argument (e.g. [KST20, Lem. 6.2] for details), implying that τ∞ cannot be tempered. All in all, for all τ as above such that Tr (f ∗ τ ) = 0, we see that τ is an S| S 6 vSt unramified twist of the Steinberg representation and that Tr (f ∗ τ ) has the same S| S sign. Moreover τ = π contributes nontrivially to the left sum by our assumption. ♮ ♮,S Therefore the right hand side is nonzero, i.e. there exists π χ(G) such that π ♮,S ♮ ♮ ∈A ≃ π and m(π )Tr (fS πS) = 0. The nonvanishing of trace confirms the conditions on ♮ | 6 π at vSt and by the same argument as above. ∞ ∗ Now we prove (2). Make the same choice of fv = fv at finite places v in S. Since one-dimensional representations of G(Fv′ ) are excluded by assumption, the condition Tr (f ∗ τ ) = 0 implies that τ is an unramified twist of the Steinberg vSt | vSt 6 vSt representation. By the assumption τ∞ is then a discrete series representation. Hence the nonzero contributions from τ on the left hand side all has the same sign. Thereby we deduce the existence of π♮ as in (1).

As before G∗ is a quasi-split group over a totally real field F . Let E/F be a finite cyclic extension of totally real fields such that [E : F ] is a prime. Write θ ˜∗ ∗ for a generator of the group Gal(E/F ). Set G := ResE/F (GE), which is equipped with a natural action of θ defined over F . Let S be a finite set of places of F such 38 Arno Kret, Sug Woo Shin that the group G∗ and the extension E/F are unramified outside S, so that G˜∗ is ∗ also unramified apart from S. Fix a reductive model of G over F [1/S] as before. O ∗ By base change followed by ResE/F , this gives rise to a reductive model of G˜ over [1/S]. The models determine hyperspecial subgroups of G∗ and G˜∗ away from S, OF thus also unramified Hecke algebras and unramified representations of G∗ and G˜∗. For each place v of F put E := E F . Canonically E E with w run- v ⊗F v v ≃ w|v w ning over places of E above v. Let BC : Irrunr(G∗(F )) Irrunr(G∗(E )) denote E/F v → Q v the base change map for unramified representations. Writing BC∗ : unr(G∗(E )) E/F H v → unr(G∗(F )) for the base change morphism of unramified Hecke algebras, we have H v from Satake theory that

Tr (BC∗ (f ) π ) = Tr(f BC (π )). (6.1) E/F v | v v | E/F v Write ξ = ξ as a representation of G(F ) G(F ). For each inﬁnite ⊗v|∞ v ∞ ≃ v|∞ v place v of F , deﬁne ξ := ξ as a representation of G(E ) G(E ), E,v ⊗w|v v Q v ≃ w|v w where w runs over places of E dividing v (here the two isomorphisms are canonical). Q Set ξ := ξ . ⊗v|∞ E,v ∗ Proposition 6.6. Let π be a cuspidal automorphic representation of G (AF ) such that

• π is unramified at all finite places apart from a finite set S, • πvSt is an unramified character twist of the Steinberg representation,

• π∞ is ξ-cohomological. Suppose that either ξ has regular highest weight or that the condition for π in Proposi- ∗ tion 6.3.(2) is satisﬁed. (This is always true for G = GSp2n, cf. Remark 6.5.) Then ∗ there exists a cuspidal automorphic representation πE of G (AE) such that

• πE is unramified at all finite places apart from S, • πE,vSt is an unramified character twist of the Steinberg representation,

• πE,∞ is ξE-cohomological, and moreover π BC (π ) at every ﬁnite place v / S. E,v ≃ E/F v ∈ Proof. We will be brief as our proposition and its proof are very similar to those in Labesse’s book [Lab99, 4.6], and also as the proof just mimics the argument for § Proposition 6.3 in the twisted case. In particular we leave the reader to ﬁnd further details about the twisted trace formula for base change in loc. cit. Strictly speaking one has to incorporate the central character datum (X, χ) above in Labesse’s argument, but this is done exactly as in the untwisted case. Galois representations for general symplectic groups 39

We begin by setting up some notation. Take K˜ to be a sufficiently small open compact subgroup of G∗(A∞) such that N : Z(A ) Z(A ) maps Z(E) K˜ E E/F E → F ∩ into Z(F ) K. Let X˜ := Z(E) K˜ , redefine X to be N (X˜) and restrict χ to this ∩ ∩ E/F subgroup,χ ˜ := χ N .) Let ξ˜ denote the representation ξ of G∗ (E C)= E/F ⊗E֒→C 2n ⊗Q ◦ .(G∗(F C) (both indexed by F -algebra embedding E ֒ C → E֒→C ⊗Q We choose the test functions Q f˜ = f˜ (G∗(A ), χ˜−1) and f = f (G∗(A ), χ−1) w ∈ H E v ∈ H F w v Y Y ˜ ˜ as follows. If v = vSt then fv = w|v fw and fv are set to be Lefschetz functions as in Appendix A. So f˜ and f are associated up to a nonzero constant by Lemma v v Q A.9. If v is a finite place of F contained in S then choose fv to be the characteristic ∗ function on a sufficiently small open compact subgroup Kv of G (Fv) such that πKv = 0. By [Lab99, Prop. 3.3.2] f is a base change transfer of some function v 6 v f˜ = f˜ (G∗(E ), χ˜−1). Given a finite place v / S and each place w of v w|v w ∈ H v ∈ E above it, let f˜ be an arbitrary function in unr(G∗(E , χ˜−1)). The image of f˜ Q w w v unr ∗ H in (G (Fv)) under the base change map is denoted by fv. At infinite places let ˜ H ˜ ˜ f∞ = w|∞ fw be the twisted Lefschetz function determined by ξ and f∞ = v|∞ fv the usual Lefschetz function for ξ. Again f˜ and f are associated up to a nonzero Q ∞ ∞ Q constant by Lemma A.12. By construction f˜ and cf are associated for some c C×. G˜∗θ G˜∗θ ∈ We write Tcusp,χ˜ and Tell,χ˜ for the cuspidal and elliptic expansions in the base- change twisted trace formula, which are defined analogously as their untwisted ana- G˜∗θ L ˜∗ logues. (For instance Tell,χ˜ is defined as in Te on p.98 of [Lab99] with L = G θ but making the obvious adjustment to account for the central character as earlier in this section.) Just like the trace formula for G and f, the twisted trace formula for G˜ and f˜ as well as its stabilization simplifies greatly exactly as in Lemmas 6.1 and 6.2 in light of Lemmas A.9 and A.12. So we have

∗ ∗ ∗ ∗ T G˜ θ (f˜)= T G˜ θ(f˜)= c STG (f)= c T G (f). cusp,χ˜ ell,χ˜ · ell,χ · cusp,χ By linear independence of characters and the character identity (6.1), we have

θ m˜ (˜τ)Tr (f˜S τ˜S)= c m(τ)Tr (fS τS), ∗ | ∗ | τ˜∈Aχ˜(G (AE)) τ∈Aχ(G (AF )) θ S S τS ≃πS τ˜ ≃τ,˜ τ˜ X≃BCE/F (π ) X where Tr θ denotes the θ-twisted trace (for a suitable intertwining operator for the θ- twist), andm ˜ (˜τ) denotes the relative multiplicity ofτ ˜ as deﬁned on p.106 of [Lab99]. The right hand side is nonzero as in the proof of Proposition 6.3. Therefore there exists π :=τ ˜ (G∗(A )) contributing nontrivially to the left hand side. By E ∈ Aχ E construction of f˜ such a πE satisﬁes all the desired properties. 40 Arno Kret, Sug Woo Shin

7 Cohomology of certain Shimura varieties of abelian type

In this section we construct a Shimura datum and then state the outcome of the Langlands-Kottwitz method on the formula computing the trace of the Frobenius and Hecke operators in the case of good reduction.

We first construct our Shimura datum (ResF/QG,X). The group G/F is a certain ∗ inner form of the quasi-split group G := GSp2n,F . We recall the classification of such inner forms, and then define our G in terms of this classification. The inner twists 1 of GSp2n,F are parametrized by the cohomology H (F, PSp2n). Kottwitz defines in [Kot86, Thm. 1.2] for each F -place v a morphism of pointed sets

α : H1(F , PSp ) π (Z(Spin (C))Γv )D Z/2Z, v v 2n → 0 2n+1 ≃ where ( )D denotes the Pontryagin dual. If v is ﬁnite, then α is an isomorphism. If · v v is inﬁnite, then [Kot86, (1.2.2)] tells us that

ker(α ) = im[H1(F , Sp ) H1(F , PSp )], v v 2n → v 2n

im(α ) = ker[π (Z(Spin (C))Γv )D π (Z(Spin (C))D)]. v 0 2n+1 → 0 2n+1 1 Thus αv is surjective, with trivial kernel as H (R, Sp2n) vanishes by [PR94, Chap. 2]. However α is not a bijection (when n 2). In fact α−1(1) classiﬁes unitary v ≥ v groups associated to Hermitian forms over the Hamiltonian quaternion algebra over

Fv with signature (a, b) with a + b = n modulo the identiﬁcation as inner twists between signatures (a, b) and (b, a). (See [Tai19, 3.1.1] for an explicit computation 1 −1 n of H (Fv, PSp2n).) So αv (1) has cardinality 2 + 1. There is a unique nontrivial ⌊ ⌋ cmpt inner twist of GSp2n,Fv (up to isomorphism), to be denoted by GSp2n,Fv , such that cmpt GSp2n,Fv is compact modulo center. It comes from a deﬁnite Hermitian form. By [Kot86, Prop. 2.6] we have an exact sequence

ker1(F, PSp ) ֌ H1(F, PSp ) H1(F , PSp ) α Z/2Z, 2n 2n → v 2n → v M where α sends (c ) H1(F,G(A Q)) to α (c ). By [Kot84b, Lem. 4.3.1] we v ∈ F ⊗F v v v have ker1(F, PSp ) = 1. We conclude 2n P

H1(F, PSp )= (x ) H1(F , PSp ) α(x )=0 Z/2Z . (7.1) 2n v ∈ v 2n | v ∈ ( v v ) M X

In particular there exists an inner twist G of GSp2n,F such that: Galois representations for general symplectic groups 41

• For the infinite places y v the group G is isomorphic to GSpcmpt . ∈V∞\{ ∞} Fy 2n,Fy For y = v∞ we have GFy = GSp2n,Fy . (Recall: v∞ is the infinite F -place corresponding to the embedding of F into C that we fixed in the Notation section.) • If [F : Q] is odd, we take GA∞ GSp2n,A∞ . F ≃ F

• ∞,v ∞,v If [F : Q] is even we fix a finite F -place vSt, and take GA St GSp2n,A St . F ≃ F The form GFv then has to be the unique nontrivial inner form of GSp . St 2n,FvSt More concretely G can be defined itself as a similitude group but we do not need it in this paper.

Let S be the Deligne torus ResC/RGm. Over the real numbers the group (ResF/QG)R decomposes into the product G F Fy. Let In be the n n-identity matrix y∈V∞ ⊗ × and A be the n n-matrix with all entries 0, except those on the anti-diagonal, n × Q where we put 1. Let h : S (Res G) be the morphism given by 0 → F/Q R

aIn bAn S(R) G(F QR), a+bi , 1,..., 1 (G F Fy)(R), (7.2) → ⊗ 7→ −bAn aIn v∞ ∈ ⊗ y∈V∞ Y for all R-algebras R (the non-trivial component corresponds to the non-compact place v ). We let X be the (Res G)(R)-conjugacy class of h . This set X can ∞ ∈V∞ F/Q 0 more familiarly be described as Siegel double half space Hn, i.e. the n(n+1)/2-dimensional space consisting of complex symmetric n n-matrices with definite (positive × or negative) imaginary part. Let us explain how the bijection X H is obtained. ≃ n The group GSp2n(R) acts transitively on Hn via fractional linear transformations: A B −1 A B ( C D )Z := (AZ + B)(CZ + D) , ( C D ) GSp (R), Z H [Sie64] (in these ∈ 2n ∈ n formulas the matrices A,B,C,D,Z are all of size n n). The place v∞ induces a ։ × surjection (ResF/QG)(R) GSp2n(R) and via this surjection we let (ResF/QG)(R) act on H . The stabilizer K := stab (iI ) of the point iI H has n ∞ (ResF/QG)(R) n n ∈ n the form K∞ := Kv G(Fv), where Kv is the stabilizer of iIn in ∞ × v∈V∞\{v∞} ∞ GSp (F ). Similarly, the stabilizer of h X is equal to K . Thus there is an 2n v∞ Q 0 ∈ ∞ isomorphism X H under which h corresponds to iI . It is a routine verification ≃ n 0 n that Deligne’s axioms (2.1.1.1), (2.1.1.2), and (2.1.1.3) for Shimura data [Del79] are satisfied for (ResF/QG, Hn). Since moreover the Dynkin diagram of the group Gad is of type C, it follows from Deligne [Del79, Prop. 2.3.10] that (ResF/QG, Hn) is of abelian type.

V∞ We write µ = (µy)y∈V∞ X∗(T ) for the cocharacter of ResF/QG such that ∈ xIn 0 µ = 1 if y = v and µ (x)=( 0 In ). Then µ is conjugate to the holomorphic y 6 ∞ v∞ part of h C. An element σ Gal(Q/Q) sends µ = (µ ) to σ(µ)=(µ ). Thus 0 ⊗ ∈ y σ◦y the conjugacy class of µ is ﬁxed by σ if and only if σ Gal(Q/F ). Therefore ∈ 42 Arno Kret, Sug Woo Shin

the reflex field of (Res G, H ) is F (embedded in C via v ). For K G(A∞) F/Q n ∞ ⊂ F a sufficiently small compact open subgroup, write ShK/F for the corresponding

Shimura variety. In case F = Q the datum (ResF/QG, Hn) is the classical Siegel

datum (of PEL type) and ShK are the usual non-compact Siegel modular varieties. If [F : Q] > 1 so that G is anisotropic modulo center, then it follows from the Baily-

Borel compactiﬁcation [BB64, Thm. 1] that ShK is projective. Whenever [F : Q] > 1

the datum (ResF/QG, Hn) is not of PEL type. If moreover n = 1 then the ShK have dimension 1 and they are usually referred to as Shimura curves, which have been extensively studied in the literature. Let ξ = ξ be an irreducible algebraic representation of (Res G) C = ⊗y|∞ y F/Q ×Q y|∞ G(F y) with each F y canonically identiﬁed with C. The central character ωξy of ξ has the form z zwy for some integer w Z. Q y 7→ y ∈ Lemma 7.1. If there exists a ξ-cohomological discrete automorphic representation π

of G(AF ) then wy has the same value for every inﬁnite place y of F .

Proof. Under the assumption, the central character ω : F × A× C× is an (L- π \ F → )algebraic Hecke character. Hence ω = ω w for a ﬁnite Hecke character ω and π 0|·| 0 an integer w by Weil [Wei56]. It follows that w = wy for every inﬁnite place y.

In light of the lemma we henceforth make the hypothesis as follows, which implies × that ξ restricted to the center Z(F )= F of (ResF/QG)(Q) = GSp2n(F ) is the w-th

power of the norm character NF/Q.

(cent) wy is independent of the inﬁnite place y of F (and denoted by w).

Following [Car86b, 2.1] (especially paragraph 2.1.4) we construct an ℓ-adic sheaf ∞ on ShK for each sufficiently small open compact subgroup K of G(AF ) from the ℓ-adic representation ιξ = ξ Q . For simplicity we write for the ℓ-adic sheaf ⊗C,ι ℓ Lιξ (omitting K). It is worth pointing out that the construction relies on the fact that ξ is trivial on Z(F ) K for small enough K. (For a fixed open compact subgroup ∩ K , we see that Z(F ) K × is mapped to 1 under N . So ξ is trivial 0 ∩ 0 ⊂ OF {± } F/Q on Z(F ) K for every K contained in some subgroup of K of index at most 2.) ∩ 0 Similarly, we write for the (complex) local system on Sh (C) attached to ξ. Lξ K Without loss of generality we assume throughout that K decomposes into a product K = K where K G(F ) is a compact open subgroup. We call v∤∞ v v ⊂ v (G,K) unramified at a finite F -place v if the group K is hyperspecial in G(F ). If Q v v so, fix a smooth reductive model G of G over such that G( ) = K . We Fv OFv OFv v say that (G,K) is unramified at a rational prime p if it is unramified at all F -places v above p. For the moment, we do not assume these conditions on (G,K). Galois representations for general symplectic groups 43

We ﬁx an algebraic closure Q C of F with respect to v : F ֒ C. Our primary ⊂ ∞ → interest lies in the ℓ-adic ´etale cohomology with compact support

n(n+1) H (Sh , ) := ( 1)i[Hi (Sh , )] (7.3) c K Lιξ − c K,Q Lιξ i=0 X as a virtual (G(A∞)//K) Γ-module. Let us make the word “virtual”, and the HQℓ F × notation [ ] more precise. As in the book of Harris-Taylor, if Ω is a topological group · that is locally profinite [HT01, p. 23], we let Groth(Ω) be the abelian group of formal, possibly infinite, sums Π∈Irr(Ω) nΠΠ, where Irr(Ω) is the set of isomorphism classes of admissible representations of Ω in Q -vector spaces. Given an admissible P ℓ Ω-representation Π, it defines an element [Π] Groth(Ω) (bottom of page 23 of ∈ [HT01]). In particular we may take Ω = Γ, G(A∞) and Γ G(A∞), or, by proceeding F × F similarly, take Ω equal to a Hecke algebra times a group, such as (G(A∞)//K) Γ. Qℓ F H ×∞ Taking a direct limit over sufficiently small open compact subgroups K of G(AF ), we similarly obtain a G(A∞) Γ-module with admissible G(A∞)-action and a virtual F × F G(A∞) Γ-module F × n(n+1) Hi (Sh, ) := lim Hi (Sh , ), H (Sh, ) := ( 1)i[Hi (Sh, )]. c Lιξ ét K,Q Lιξ c Lιξ − c Lιξ −→K i=0 X Let us introduce some more cohomology spaces to be used mainly for the F = Q case in the next section. For K as above and for each i Z , we write Hi (Sh , ), ∈ ≥0 0 K Lξ Hi (Sh , ), and IHi(Sh , ) for the cuspidal, L2, and ℓ-adic intersection coho- (2) K Lξ K Lιξ mology of Sh , respectively. (The first two spaces are as in [Fal83, 6], considering K § Sh as a complex manifold with respect to F ֒ C corresponding to v . The last one K → ∞ is the cohomology of the intermediate extension of to the Bailey-Borel compact- Lιξ ification [Mor08, Sect. 2.2].) By taking direct limits over K, we obtain admissible ∞ i i i G(AF )-representations H0(Sh, ξ), H(2)(Sh, ξ), and IH (Sh, ιξ). The last one is L ∞ L L equipped with commuting actions of G(AF ) and Γ. We are going to apply the Langlands-Kottwitz method to relate the action of Frobenius elements of Γ at primes of good reduction to the Hecke action on the compact support cohomology. In the PEL case of type A or C, Kottwitz worked it out in [Kot90, Kot92b] including the stabilization. As we are dealing with Shimura varieties of abelian type, we import the result from [KSZ]. In fact the stabilization step is very simple under hypothesis (St). Suppose that (G,K) is unramified at a prime p. Let p be a finite F -place above p with residue field k(p). We fix an isomorphism ι : C Q such that ι v : F ֒ Q p ≃ p p ∞ → p induces the place p of F . Let us introduce some more notation.

• r := [k(p): F ] Z . p ∈ ≥1 44 Arno Kret, Sug Woo Shin

• For j Z denote by Q j the unramiﬁed extension of Q of degree j, and Z j ∈ ≥1 p p p its integer ring.

• Write F := F Q , F := F Q j , and := Z j . p ⊗Q p p,j ⊗Q p OFp,j OF ⊗Z p

• Let σ be the automorphism of Fp,j induced by the (arithmetic) Frobenius

automorphism on Qpj and the trivial automorphism on F .

• ∞ Let (G(AF )) (resp. (G(Fp)), (G(Fp,j))) be the convolution algebra of H H H ∞ compactly supported smooth complex valued functions on G(AF ) (resp. G(Fp), resp. G(Fp,j)), where the convolution integral is deﬁned by the Haar measure giving the group K (resp. K resp. G( )) measure 1. p OFp,j • We write unr(G(F )) (resp. unr(G(F ))) for the spherical Hecke algebra, H p H p,j i.e. the algebra of K (resp. G( ))-bi-invariant functions in (G(F )) (resp. p OFp,j H p (G(F ))). H p,j • Deﬁne φ unr(G(F )) to be the characteristic function of the double coset j ∈ H p,rj G( ) µ(p−1) G( ) in G(F ). OFp,rj · · OFp,rj p,rj • N := ΠG(F∞) π (G(F )/Z(F )) =2n−1 2=2n. ∞ | ξ |·| 0 ∞ ∞ | · • ep(τ ξ) := ∞ ( 1)i dim Hi(Lie G(F ),K ; τ ξ) for an irreducible ∞ ⊗ i=0 − ∞ ∞ ∞ ⊗ admissible representation τ of G(F ). P ∞ ∞ • (X, χ) is the central character datum given as X := (Z(A∞) K) Z(F ) and F ∩ × ∞ χ := χ (extended from Z(F ) to X trivially on Z(A∞) K). ξ ∞ F ∩

Remark 7.2. We point out that K∞ is a subgroup of index π0(G(F∞)/Z(F∞)) ′ | | in some K∞ which is a product of Z(F∞) and a maximal compact subgroup of ′ G(F∞) and that ep(τ∞ ξ) is defined in terms of K∞, not K∞. When τ∞ belongs ⊗ G to the discrete series L-packet Πξ , the representation τ∞ decomposes into a direct sum of irreducible (Lie G(F∞),K∞)-modules, the number of which is equal to π (G(F )/Z(F )) , cf. paragraph below Lemma 3.2 in [Kot92a]13. So ep(τ ξ)= | 0 ∞ ∞ | ∞ ⊗ ( 1)n(n+1)/2 π (G(F )/Z(F )) =( 1)n(n+1)/22 for τ ΠG. − | 0 ∞ ∞ | − ∞ ∈ ξ ∞,p ∞p p Let f (G(AF )//K ). We would like to present a stabilized trace formula ∈ H ∞,p computing the action of f and Frobp on H (Sh , ). c K Lιξ Let (G) denote the set of representatives for isomorphism classes of (G,K)-un- Eell ramified elliptic endoscopic triples of G. For each (H,s,η ) (G) we make a fixed 0 ∈Eell 13The discussion there is correct in our setting, but it can be false for non-discrete series representations (which are allowed in that paper). For instance when ξ and π∞ are the trivial representation in the notation of [Kot92a], it is obvious that τ∞ cannot decompose further even if π (G(F∞)/Z(F∞)) > 1. | 0 | Galois representations for general symplectic groups 45 choice of an L-morphism η : LH LG extending η (which exists since the derived → 0 group of GSp is simply connected). We recall the definition of ι(G,H) Q and 2n ∈ hH = hH,∞,phH hH (H(A )) only for the principal endoscopic triple (H,s,η)= p ∞ ∈ H F (G∗, 1, id), which is all we need. (For other endoscopic triples, the reader is referred to (7.3), second display on p.180 and second display on p.186 of [Kot90]; this is adapted to a little more general setup as ours in [KSZ].) We have ι(G,G∗) = 1 and hG∗ = hG∗,∞,phG∗ hG∗ (G(A ), χ−1) given as follows. p ∞ ∈ H F • G∗,∞,p ∗ ∞,p ∞,p h (G (AF )) is an endoscopic transfer of the function f to the ∈ H ∗ inner form G∗ (hG ,∞,p can be made invariant under Z(A∞,p) Kp by averaging F ∩ over it),

• hG∗ unr(G∗(F )) is the base change transfer of φ unr(G(F )) down p ∈ H p j ∈ H p,rj to unr(G∗(F )) = unr(G(F )), H p H p ∗ ∗ • G ∗ −1 G −1 ∗ h∞ (G (F∞), χ )) := Πξ τ ∈ΠG fτ∞ , i.e. the average of pseudo- ∈ H | | ∞ ξ G∗ coefficients for the discrete series L-packetP Πξ . As shown by [Kot92a, Lem. 3.2], hG∗ is N −1 times the Euler-Poincaréfunction for ξ (defined in 6 but ∞ ∞ § using K∞ of this section):

∗ Tr π (hG )= N −1ep(π ξ), π Irr(G∗(F )). (7.4) ∞ ∞ ∞ ∞ ⊗ ∞ ∈ ∞ The main result of [KSZ] is the following (which works for every Shimura variety of abelian type when p = 2), where the starting point is Kisin’s proof of the Langlands- 6 Rapoport conjecture for all Shimura varieties of abelian type [Kis17]. When F = Q it was already shown by [Kot90, Kot92b].

Theorem 7.3. Let f ∞ (G(A∞)//K). Suppose that p and p are as above such that ∈ H F • (G,K) is unramiﬁed at p,

• f ∞ = f ∞,pf with f ∞,p (G(A∞p)//Kp) and f = 1 . p ∈ H F p Kp ∞ Then there exists a positive integer j0 (depending on ξ, f , p, p) such that for all j j we have ≥ 0

ι−1Tr (ιf ∞Frobj, H (Sh , )) = ι(G,H)STH (hH ). (7.5) p c K Lιξ ell,χ H∈EXell(G) 8 Galois representations in the cohomology

Let π be a cuspidal ξ-cohomological automorphic representation of GSp2n(AF ) satisfying condition (St), and ﬁx the place vSt in that condition. Deﬁne an inner form G 46 Arno Kret, Sug Woo Shin

of GSp2n as in 7; when [F : Q] is even, we take vSt in that definition to be the fixed §♮ place vSt. Let π be a transfer of π to G(AF ) via Proposition 6.3 (which applies thanks ♮,∞ ∞ ♮ to Remark 6.5) so that at the unramified places π and π are isomorphic, πvSt ♮ is an unramified twist of the Steinberg representation, and π∞ is ξ-cohomological. The aim of this section is to compute a certain π♮,∞-isotypical component of the cohomology of the Shimura variety Sh attached to (G,X). Let A(π♮) be the set of (isomorphism classes of) cuspidal automorphic representations τ of G(A ) such that τ π♮ δ for an unramified character δ of the F vSt ≃ vSt ⊗ group G(F ), τ ∞,vSt π♮,∞,vSt and τ is ξ-cohomological. Let K G(A∞) be a vSt ≃ ∞ ⊂ F sufficiently small decomposed compact open subgroup such that π♮,∞ has a non-zero

K-invariant vector. Let Sbad be the set of prime numbers p for which either p = 2, the group ResF/QG is ramified or Kp = v|p Kv is not hyperspecial. Let Hi (Sh, ) denote the semisimplification of Hi (Sh, ) as a G(A∞) Γ- c Lιξ ss Q c Lιξ F × module. Likewise Hi (Sh , ) means the semisimplification as an (G(A∞)//K) c K,Q Lιξ ss H F × Γ-module. For each τ A(π♮) (or any irreducible admissible representation of ∞ ∈∞ ∞ G(AF )), we define the τ -isotypic part of an admissible G(AF )-module R on a C-vector space as ∞ ∞ R[τ ] := Hom ∞ (τ , R), (8.1) G(AF ) ∞ which is finite-dimensional. If the underlying space of R is over Qℓ, we define R[τ ] ∞ ∞ ∞ using ιτ in (8.1). If R has an action of Γ commuting with G(AF ), then R[τ ] i ∞ inherits the Γ-action. As a primary example, Hc(Sh, ιξ)ss[τ ] is a finite-dimensional L ∞ K i representation of Γ, which is isomorphic to HomH(G(A∞)//K)(ι(τ ) , H (Sh , ιξ)ss) F c K,Q L for each K such that (τ ∞)K = 0 since the isomorphism class of (τ ∞)K and that of 6 τ determine each other. In particular the representation is finite-dimensional and unramified at almost all places. Consider two representations τ, τ ′ A(π♮) equivalent and write τ τ ′ if τ ∞ ∈ ∼ ≃ τ ′,∞. Define a virtual Galois representation

n(n+1) ρshim = ρshim(π♮) := ( 1)n(n+1)/2 ( 1)i[Hi (Sh, ) [τ ∞]], (8.2) 2 2 − − c Lιξ ss ♮ i=0 τ∈AX(π )/∼ X in the Grothendieck group of the category of continuous representations of Γ on

finite dimensional Qℓ-vector spaces which are unramified at almost all places. We shim compute in this section ρ2 at almost all F -places not dividing a prime in Sbad. Define a rational number

♮ def n(n+1)/2 −1 a(π ) = ( 1) N∞ m(τ) ep(τ∞ ξ), (8.3) − ♮ · ⊗ τ∈XA(π ) where m(τ) is the multiplicity of τ in the discrete automorphic spectrum of G. Galois representations for general symplectic groups 47

Recall the integer w Z determined by ξ as in condition (cent) of the last section. ∈ ∞ In the following, the subscript ss always means the semisimpliﬁcation as a G(AF )- ∞ module, or as a G(AF ) Γ-module if there is a Γ-action. As in [Fal83, 6], we have ∞ × § natural G(AF )-equivariant maps of C-vector spaces

Hi (Sh, ) Hi (Sh, ) Hi(Sh, ), (8.4) c Lξ → (2) Lξ → Lξ where Hi (Sh, ) and Hi(Sh, ) denote the topological cohomologies, which are c Lξ Lξ isomorphic to the ℓ-adic cohomologies Hi (Sh, ) and Hi(Sh, ) via ι. c Lιξ Lιξ Lemma 8.1. For each τ A(π♮) the following hold. ∈ 1. For every i Z , the maps of (8.4) induce isomorphisms ∈ ≥0 Hi (Sh, )[τ ∞] ∼ Hi (Sh, )[τ ∞] ∼ Hi(Sh, )[τ ∞]. c Lξ → (2) Lξ → Lξ Moreover dim Hi (Sh, ) [τ ∞] = dim Hi (Sh, )[τ ∞]. (Namely all subquotients c Lξ ss c Lξ of Hi (Sh, ) isomorphic to τ ∞ appear as subspaces.) c Lξ 2. For every ﬁnite place x = v of F , if τ is unramiﬁed at x then τ sim w/2 is 6 St x| | tempered and unitary.

Proof. (1) If F = Q the isomorphisms are clear by compactness of Sh . In that 6 K case, Hi(Sh, ) is semisimple as a G(A∞)-module since the same is true for the Lξ F L2-automorphic spectrum (which is entirely cuspidal since G is anisotropic modulo center over F ), via Matsushima’s formula. Suppose that F = Q from now. By Poincaréduality, it suffices to check that (8.4) i ∞ ∼ i ∞ induces an isomorphism H(2)(Sh, ξ)[τ ] H (Sh, ξ)[τ ]. To set things up, let L → L S S Sbad be a finite set of F -places including all infinite places. Then K is a product ⊃ S KS S S of hyperspecial subgroups, and (τ ) = 0. The category of (G(AF )//K )-modules 6 H S S is semisimple and irreducible objects are one-dimensional. For a (G(AF )//K )- S H module R, define the τ -part of R to be the maximal subspace Rτ S such that Rτ S is isomorphic to a self-direct sum of τ S . Then it suffices to prove that (8.4) induces

i ∼ i H (Sh , ) S H (Sh , ) S , i 0. (8.5) (2) K Lξ τ → K Lξ τ ≥ Indeed, the desired isomorphism of (1) will follow from the preceding formula by taking the functor HomG(F )(τS\V , ). The latter assertion of (1) is also implied S\V∞ ∞ · by (8.5) as we now explain. Again by Poincar´eduality, we can reduce to showing it for the ordinary cohomology rather than the compact support cohomology. Then it i suﬃces to verify that H (Sh , ) S is a semisimple G(F )-module. This follows K Lξ τ S\V∞ from (8.5) since Hi (Sh , ) is a semisimple G(A∞)-module by comparison with (2) K Lξ F the L2-discrete automorphic spectrum via Borel–Casselman’s theorem. 48 Arno Kret, Sug Woo Shin

It remains to obtain (8.5). We apply Franke’s spectral sequence in the notation ∞ of [Wal97, Cor., p.151] (cf. [Fra98, Thm. 19]). The spectral sequence, which is G(AF )- equivariant, stays valid when restricted to the τ S-part. We claim that the τ S-part is zero in any parabolic induction of an automorphic representation on a proper Levi ♭ subgroup of GSp2n(A). It suﬃces to check the analogous claim with τ (as in Lemma

2.6) and Sp2n in place of τ and GSp2n. The latter claim follows from Arthur’s main result [Art13, Thm. 1.5.2], Corollary 2.2, and the strong multiplicity one theorem for general linear groups. As a consequence of the claim, the direct summand of p,q S E1 over (w,P ) in [Wal97, p.151] has vanishing τ -part unless P = G and w = 1. For w = 1, we see from the ﬁrst line on p.151 of loc. cit. that there is a nonzero p,q S contribution to E1 for a unique p (independent of q). Therefore the τ -part of the spectral sequence degenerates at E1 and yields an isomorphism, in the notation there (writing i for p + q),

i 1 G ∼ i 1 G H (m ,K ; (G,µ ) E ) S H (m ,K ; (G,µ ) E ) S , i 0. R A2 λ ⊗C λ τ → R A λ ⊗C λ τ ≥ This map is the natural one induced by (G,µ ) (G,µ ), and translates into A2 λ ⊂ A λ (8.5) in our notation via Borel’s theorem [Wal97, p.143] on the right hand side. The proof of (1) is complete. (2) Let ω : F × A× C× denote the common central character of τ, π♮, and π. \ F → (The central characters are the same as they are equal at almost all places.) Since F is totally real, ω = ω a for a ﬁnite character ω and a suitable a C. Since 0|·| 0 ∈ π♮ is ξ-cohomological, we must have a = w. Then π♮ sim w/2 has unitary central ∞ − x| | character and is essentially tempered by Lemma 2.7. We are done since π♮ τ . x ≃ x Proposition 8.2. For almost all ﬁnite F -places v not dividing a prime number in

Sbad and all suﬃciently large integers j, we have

shim j jn(n+1)/4 j Tr ρ (Frob )= q Tr (i ♮ (spin(ιφ ♮ )))(Frob ). 2 v v a(π ) πv v

shim Moreover the virtual representation ρ2 is a true representation.

Proof. We imitate arguments from [Kot92a]. We consider a function f on G(AF ) of the form f = f f f ∞,vSt , where the components are chosen in the following ∞ ⊗ vSt ⊗ way:

• −1 We let f∞ be N∞ times an Euler-Poincar´efunction for ξ (and K∞) on G(F∞) so that

∞ Tr τ (f )= N −1ep(τ ξ)= N −1 ( 1)i dim Hi(g,K ; τ ξ). (8.6) ∞ ∞ ∞ ∞ ⊗ ∞ − ∞ ∞ ⊗ i=0 X Galois representations for general symplectic groups 49

∞,v • We pick a Hecke operator f ∞,vSt = (G(A St )//KvSt ) such v∈V / ∞∪{vSt} ∈ H F that for all automorphic representations τ of G(A ) with τ ∞,K = 0 and Q F 6 Tr τ (f ) = 0 we have ∞ ∞ 6 1, if τ ∞,vSt π♮,∞,vSt Tr τ ∞,vSt (f ∞,vSt )= ≃ (8.7) (0, otherwise.

This is possible since there are only ﬁnitely many such τ (one of which is π♮).

• We let fvSt be a Lefschetz function from Equation (A.4) of Appendix A.

There exists a ﬁnite set of primes Σ containing places over 2 such that f ∞ decomposes f ∞ = f f ∞Σ with Σ ⊗ • f (G(F )//K ), Σ ∈ H Σ Σ ∞,Σ ∞Σ • f = 1 Σ (G(A )), K ∈ H F • KΣ is a product of hyperspecial subgroups, and

• if v , v are F -places above the same rational prime, then v Σ if and only if 1 2 1 ∈ v Σ. 2 ∈ In the rest of the proof we consider only finite places v such that v / Σ, v ∤ ℓ. ∈ Fix such a place and call it p as in 7. We also fix an isomorphism ι : C Q such § p ≃ p that ι v : F ֒ Q induces p. Write p := p . Then K = K is a hyperspecial p ∞ → p |Q p v|p v subgroup of G(F ), and f = f = 1 . Thus (G,K) is unramified at p in p p v|p v v|p Kv Q the sense of 7 so that the Langlands–Kottwitz method applies. § Q Q The stabilized Langlands-Kottwitz formula (Theorem 7.3) simplifies as

∗ ∗ ι−1Tr (ιf ∞Frobj, H (Sh , )) = STG (hG ), (8.8) p c K Lιξ ell,χ essentially for the same reason as in the proof of Lemma 6.2. Indeed fvSt is a stabi- ∗ H lizing function (Lemma A.7), thus unless H = G , the stable orbital integrals of hvSt vanish as they equal κ-orbital integrals of fvSt with κ = 1, which are zero (Deﬁnition 6 G∗ A.6), up to a nonzero constant via the Langlands–Shelstad transfer. Note that hvSt can be chosen to be a Lefschetz function thanks to Lemma A.4. To extract spectral information, we will compare (8.8) with the stabilized trace formula of 6. We choose the following test functions f˜ and f˜ on G(F ) and G(F ). § p ∞ p ∞ • ˜ G∗ ∗ fp := hp via the identiﬁcation G(Fp)= G (Fp),

• ˜ −1 f∞ is N∞ times an Euler-Poincar´efunction associated to ξ. 50 Arno Kret, Sug Woo Shin

G∗ G∗ ˜ ˜ ∗ Then hp and h∞ are transfers of fp and f∞ from G to G , respectively. This is trivial at p and Lemma A.11 at . By construction, hG∗,∞,p is a transfer of f ∞,p. ∞ Therefore Lemmas 6.1 and 6.2 tell us that

G ∞,p ˜ ˜ G∗ G∗ Tcusp,χ(f fpf∞) = STell,χ(f ). (8.9)

By (8.8) and (8.9), we have

ι−1Tr (ιf ∞Frobj, H (Sh , )) = m(τ)Tr τ ∞,p(f ∞,p)Tr τ (f˜ )Tr τ (f˜ ). p c K Lιξ p p ∞ ∞ τ∈AcuspX,χ(G) (8.10) The summand on the right hand side vanishes unless Tr τ (f ) = 0, τ is un- ∞ ∞ 6 p ramiﬁed, and Tr τ ∞,p(f ∞,p) = 0. The latter two imply nonvanishing of Tr τ ∞(f ∞), 6 ∞,vSt ∞,vSt ∞,vSt thus Tr τ (f ) = 0 and Tr τvSt (fvSt ) = 0. By the choice of function f (see 6 6 ♮ (8.7)) and fvSt , it follows that τ contributes nontrivially in (8.10) only if τ A(π ). ∗ ∈ In this case, we have τ π♮ in particular. Recall f˜ = hG . Applying (8.6) and p ≃ p p p (8.7), we identify the right hand side of (8.10) with

−1 ♮ G∗ n(n+1)/2 ♮ ♮ G∗ N∞ m(τ)ep(τ∞ ξ)Tr πp(hp )=( 1) a(π )Tr πp(hp ). (8.11) ♮ ⊗ − τ∈XA(π ) ∞ j By the choice of f the left hand side of (8.10) is equal to the trace of Frobv on ( 1)n(n+1)/2ρshim. To sum up, − 2 j shim ♮ ♮ G∗ Tr (Frobp, ρ2 )= a(π )ιTr πp(hp ). (8.12)

To analyze the right hand side, consider the cocharacter µ =(µ ) from 7. The y y∈V∞ § irreducible representation of Res\F/QG GSpin (C) of highest weight µ ≃ y∈V∞ 2n+1 is spin on the v -component and trivial on the other components (since µ = 1 if ∞ Q y y = v ). This extends uniquely to a representation r of the L-group of (Res G) 6 ∞ µ F/Q Fp as in [Kot84a, Lem. 2.1.2] since the conjugacy class of µ is deﬁned over F (thus also

Fp). The Satake parameter of π π (with respect to G(F ) G(F )) p ≃ v|p v p ≃ v|p v \ \ belongs to the Frobp-coset of (ResF/QQG)Qp , identiﬁed with ResF/QG viaQιp, in the L-group. Then the v∞-component of the Satake parameter of πp comes from the ιp Satake parameter of πp since F ֒ C Qp induces p. With this observation, we → ≃ G ♮ apply [Kot84a, (2.2.1)] (his πϕ, G, n, F , φp correspond to our πp, (ResF/QG)Qp , rj, G∗ Qprj , hp ) to obtain

♮ G∗ jn(n+1)/4 j Tr π (h )= qp Tr (spin(φ ♮ ))(Frob ). (8.13) p p πp p

(See p.656 and the second last paragraph on p.662 of [Kot92a] for a similar computation.) This ﬁnishes the proof of the ﬁrst assertion. Galois representations for general symplectic groups 51

shim It remains to show that ρ2 is not just a virtual representation but a true shim representation. To this end we will show that ρ2 is concentrated in the middle i degree n(n + 1)/2. Let i 0. There are natural maps from Hc(Sh, ιξ) to each i i ≥ L of IH (Sh, ιξ) and H(2)(Sh, ιξ). Compatibility with Hecke correspondences implies L ∞ L that both maps are G(AF )-equivariant; the ﬁrst map is moreover equivariant for ∞ the action of Γ (which commutes with the G(AF )-action). By Zucker’s conjecture, 2 i which is proved in the articles [Loo88, LR91, SS90], the L -cohomology H(2)(Sh, ξ) i L is naturally isomorphic to the intersection cohomology IH (Sh, ιξ) via ι so that we ∞ 14 L have a G(AF )-equivariant commutative diagram

i / i H (Sh, ιξ) / IH (Sh, ιξ) . c L ❙❙ L ❙❙❙❙ ❙❙❙ ∼ ❙❙❙❙ ❙❙) ιHi (Sh, ) (2) Lξ The diagram together with Lemma 8.1 yields a Γ-equivariant isomorphism

Hi (Sh, )[τ ∞] IHi(Sh, )[τ ∞], c Lιξ ≃ Lιξ the semisimplification of which is isomorphic to Hi (Sh, ) [τ ∞]. c Lιξ ss In view of condition (cent), the intersection complex defined by ξ is pure of weight i ∞ i ∞ w. Hence the action of Frobp on IH (Sh, )[τ ], thus also on H (Sh, ) [τ ], is − Lιξ c Lιξ ss pure of weight w + i for every p as above, cf. the proof of [Mor10, Rem. 7.2.5]. In − particular there is no cancellation between different i in (8.2). On the other hand, w/2 ♮ w/2 part (2) of Lemma 8.1 (with x = p) implies that τp sim = πp sim is tempered | | | shim| and unitary. Then the first part of the proposition implies that ρ2 (Frobp)ss has eigenvalues whose absolute values equal the w + n(n+1) -th power of (# /p)1/2. − 2 OF We conclude that IHi(Sh, ) [τ ∞] = 0 unless i = n(n + 1)/2. The proof of the Lιξ ss proposition is complete.

Remark 8.3. The last paragraph in the above proof simpliﬁes signiﬁcantly if F = Q, 6 in which case ShK is proper thus the compact support cohomology coincides with

14 ∞ The G(AF )-equivariance is clear for the horizontal map, which is induced by the map from the extension-by-zero of ιξ to the Baily-Borel compactification to the intermediate extension thereof. For the vertical mapL it can be checked as follows (we thank Sophie Morel for the explanation). i i From the proof of Zucker’s conjecture we know that IH (Sh, ιξ) and H(2)(Sh, ιξ) are represented by two complexes of sheaves on the Baily-Borel compactificationL which are quasi-isomorphicL and become canonically isomorphic to ιξ when restricted to the original Shimura variety. The Hecke correspondences extend to the twoL complexes (so as to define Hecke actions on the intersection and L2-cohomologies) and coincide if restricted to the original Shimura variety. Now the point is that the Hecke correspondences extend uniquely for the intersection complex as follows from [Mor08, Lem. 5.1.3], so the same is true for the L2-complex via Zucker’s conjecture. Therefore the Hecke actions on the two cohomologies are identified via the isomorphism of Zucker’s conjecture. 52 Arno Kret, Sug Woo Shin the intersection cohomology. See the first paragraph in the proof of Lemma 8.1 for another simplification. When F = Q, as an alternative to studying various cohomology spaces, one can also argue by reducing to the F = Q case as follows. Consider 6 many real quadratic extensions E/Q, find a base-change automorphic representation

πE of GSp2n(AE) by Proposition 6.6, and apply the so-called patching lemma [Sor] C ′ n to patch spin(ρπE ) to obtain a Galois representation ρ2 : ΓQ GL2 (Qℓ). The ′ → argument in the next section can be adapted with ρ2 in place of ρ2 to construct ρC :Γ GSpin (Q ). π Q → 2n+1 ℓ Corollary 8.4. If τ A(π♮) then τ belongs to the discrete series L-packet ΠG(F∞). ∈ ∞ ξ Proof. Since τ∞ is ξ-cohomological and unitary, τ∞ appears in the Vogan–Zuckerman classification of [VZ84]. The preceding proof shows that Hi (Sh , )[τ ∞] is nonzero (2) K Lξ only for i = n(n + 1)/2. Hence Hi(Lie G(F ),K′ ; τ K ξ) does not vanish exactly ∞ ∞ ∞ ⊗ for i = n(n + 1)/2. This implies that τ∞ must be a discrete series representation through [VZ84, Thm. 5.5] (namely we must have l k in the notation there). Thus ⊂ τ ΠG(F∞). ∞ ∈ ξ ♮ ∞ ′ ♮ ′ G(F∞) Corollary 8.5. If τ A(π ) then τ τ∞ A(π ) for all τ∞ Πξ and m(τ) = ∞ ′ ∈ ♮ ∈ ∈ m(τ τ∞). Moreover, a(π ) is a positive integer. Remark 8.6. A similar argument appears in [Kot92a, Lem. 4.2]. Proof. We work with the trace formula with fixed central character, with respect to the datum (X, χ), where X = Z(AF ) and χ is the central character of π. (We diverge from the convention of 7 only in this proof.) Let G(F )1 be as in the § vSt paragraph above (A.4), and η : G(F )/G(F )1 C× the unique character such vSt vSt → that πvSt is the η-twist of the Steinberg representation; in particular the restriction ∞,vSt ′ of η to Z(FvSt ) equals χ. Consider test functions f = f fvSt,ηfτ∞ and f = ∞,v St ′ f fvSt,ηfτ∞ , where ∞,v • f ∞,vSt (G(A St ), (χ∞,vSt )−1), ∈ H F −1 G • fv ,η (G(Fv , χ )) is the function f at vSt as in Corollary A.8, and St ∈ H St vSt Lef,η ′ • ′ fτ∞ , fτ∞ are pseudo-coefficients for τ∞, τ∞, respectively.

′ By [Kot92a, Lem. 3.1], fτ∞ and fτ∞ have the same stable orbital integrals. Moreover fvSt,η is stabilizing by Corollary A.8. The simple trace formula and its stabilization (Lemmas 6.1 and 6.2) imply that, by arguing similarly as in the proof of Proposition 6.3,

m(σ)Tr (f σ )Tr (f σ )= m(σ)Tr (f σ )Tr (f ′ σ ), (8.14) vSt,η| vSt τ∞ | ∞ vSt,η| vSt τ∞ | ∞ σ σ X X where both sums run over the set of σ (G) such that ∈Aχ Galois representations for general symplectic groups 53

• σ∞,vSt τ ∞,vSt , ≃ • σ τ , by Corollary A.8 (since σ cannot be one-dimensional, as in the vSt ≃ vSt vSt proof of Proposition 6.3), and

• σ ΠG(F∞). (Since Tr (f σ ) = 0, the inﬁnitesimal character of σ is the ∞ ∈ ξ τ∞ | ∞ 6 ∞ same as that of τ∞. By [SR99, Thm. 1.8], we have that σ∞ is ξ-cohomological. Thus σ A(π♮), to which Corollary 8.4 applies.) ∈

By the last condition, Tr (fτ∞ σ∞) equals 1 if σ∞ τ∞ and 0 otherwise; the obvious ′ | ≃ analogue also holds with τ∞ in place of τ∞. Therefore, we have the equality

∞ ′ m(τ)= m(τ τ∞).

It remains to prove the second assertion. With A(π♮)/ as in (8.2), we apply ∼ Corollary 8.4 to rewrite a(π♮) deﬁned in (8.3) as

♮ n(n+1)/2 −1 ∞ ′ ′ a(π )=( 1) N∞ m(τ τ∞) ep(τ∞ ξ). − ♮ · ⊗ τ∈A(π )/∼ τ ′ ∈ΠG(F∞) X ∞ Xξ The inner sum is equal to

( 1)n(n+1)/2 2 ΠG(F∞) m(τ) − · ·| ξ |· n G(F∞) n−1 by what we just proved as well as Remark 7.2. Since N∞ =2 and Πξ =2 , ♮ ♮ | ♮ | we conclude that a(π )= ♮ m(τ) Z . (Note a(π ) m(π ) > 0.) τ∈A(π )/∼ ∈ >0 ≥ For a Z and m PZ let i : GL GL denote the block diagonal ∈ ≥1 ∈ ≥1 a m → am embedding.

shim ♮ Corollary 8.7. Let ρ2 be as above. For almost all finite F -places v where π is shim n(n+1)/4 unramified and, ρ (Frobv)ss qv ia(π♮)(spin(ιφ ♮ ))(Frobv) in GLa(π♮)2n . 2 ∼ · πv shim n(n+1)/4 Proof. Write γ = ρ (Frob ) and γ = qv i ♮ (spin(ιφ ♮ ))(Frob ). By 1 2 v ss 2 a(π ) πv v j j · Proposition 8.2 we get Tr (γ1) =Tr(γ2) for j sufficiently large. Consequently γ1 and

γ2 are GLa(π♮)2n (Qℓ)-conjugate.

9 Construction of the GSpin-valued representation

Let π be a cuspidal ξ-cohomological automorphic representation of GSp2n(AF ) satisfying (St). Denote by ωπ its central character. We showed in Corollary 8.7 that there ♮ n shim exists a a(π )2 -dimensional Galois representation ρ2 : Γ GLa(π♮)2n (Qℓ). In this shim → section we show that the representation ρ2 factors through the composition

spin GSpin (Q ) GL n (Q ) GL ♮ n (Q ), 2n+1 ℓ → 2 ℓ → a(π )2 ℓ 54 Arno Kret, Sug Woo Shin

to a representation ρC : Γ GSpin (Q ). π → 2n+1 ℓ Denote by Sbad the ﬁnite set of rational primes p such that either p = 2 or p

is ramiﬁed in F or πv is ramiﬁed at a place v of F above p. (As we commented in introduction, it shouldn’t be necessary to include p = 2 in view of [KMP16].) In the theorem below the superscript C designates the C-normalization in the sense of [BG14]. Statement (ii)’ of the theorem will be upgraded in the next section to include all v which are not above S ℓ . bad ∪{ } Theorem 9.1. Let π be a cuspidal ξ-cohomological automorphic representation of

GSp2n(AF ) satisfying the conditions (St) and (L-coh). Let ℓ be a prime number and ι: C ∼ Q a ﬁeld isomorphism. Then exists a representation → ℓ ρC = ρC : Γ GSpin (Q ), π π,ι → 2n+1 ℓ

unique up to conjugation by GSpin2n+1(Qℓ), satisfying (iii.b), (iv) and (v) of The- C orem A with ρπ in place of ρπ, as well as the following (which are slightly modiﬁed from (i), (ii), (iii.a) of Theorem A due to a diﬀerent normalization).

♭ (i’) For each automorphic Sp2n(AF )-subrepresentation π of π, its associated Ga- C lois representation ρ ♭ is isomorphic to ρ composed with the projection GSpin (Q ) π π 2n+1 ℓ → SO2n+1(Qℓ). The composition

[ ρC ]: Γ GSpin (Q ) GL (Q ) N ◦ π → 2n+1 ℓ → 1 ℓ corresponds to ω −n(n+1)/2 by global class field theory. π|·| (ii’) There exists a finite set of prime numbers S with S S such that for ⊃ bad all finite F -places v which are not above S ℓ , ρC is unramified and the ∪{ } π,v element ρC (Frob ) is conjugate to the Satake parameter of ιπ sim −n(n+1)/4 π v ss v| | in GSpin2n+1(Qℓ).

C n(n+1) 15 (iii.a’) µHT(ρπ,v, ιy)= ιµHodge(ξy)+ 4 sim. (See Deﬁnitions 1.10 and 1.14.)

Proof. We have the automorphic representation π of GSp2n(AF ). Consider

• π♭ π a cuspidal automorphic Sp (A )-subrepresentation from Lemma 2.6; ⊂ 2n F ♮ • π a transfer of π to the group G(AF ) from Proposition 6.3;

• ρ ♭ : Γ SO (Q ) the Galois representation from Theorem 2.4; π → 2n+1 ℓ 15 C Even though µHT(ρπ,v, ιy) and ιµHodge(ξy) are only conjugacy classes of (possibly half-integral) n(n+1) cocharacters, the equality makes sense since 4 sim is a central (possibly half-integral) cocharacter. Galois representations for general symplectic groups 55

• ρπ♭ a lift of ρπ♭ to the group GSpin2n+1(Qℓ). To prove the existence of this lift, consider for s Z the group GSpin(s) (Q ) of g GSpin (Q ) such that ∈ ≥1 2n+1 ℓ ∈ 2n+1 ℓ f(g)s = 1. Using the exact sequence µ ֌ GSpin(s) ։ SO and the N 2s 2n+1 2n+1 vanishing of H2(Γ, Q/Z) we see that such a lift indeed exists for s suﬃciently large and divisible.

ia(π♮)spin • ρ : Γ GSpin (Q ) GL ♮ n (Q ) the composition of ρ ♭ with the 1 → 2n+1 ℓ → a(π )2 ℓ π a(π♮)-th power of the spin representation. f shim • ρ = ρ : Γ GL ♮ n (Q ) a semi-simple representative of the virtual 2 2 → a(π )2 ℓ representation introduced in (8.2) (see also Proposition 8.2);

(see also Figure 1 in the introduction). We have

• For all ﬁnite F -places v away from S,

(ρ (Frob ) ) ρ (Frob ) PGL ♮ n (Q ) 1 v ss ∼ 2 v ss ∈ a(π )2 ℓ by Proposition 8.2;

• The representation ρ : Γ PGL ♮ n (Q ) has connected image. This is true 1 → a(π )2 ℓ because ρ1 factors over SO2n+1(Qℓ), has a regular unipotent in its image, and hence has connected image by Proposition 3.5.

By Proposition 4.6(1) and Example 4.7 there exists a g GLa(π♮)2n (Qℓ) and a × ∈ character χ: Γ Q , such that → ℓ −1 ρ = χgρ g : Γ GL ♮ n (Q ). 2 1 → a(π )2 ℓ −1 Without loss of generality, we replace ρ2 with g ρ2g to assume that ρ2 = χρ1. By construction, the representation ρ1 has image inside GSpin2n+1(Qℓ) GLa(π♮)2n (Qℓ), shim ⊂ and consequently the representation ρ2 has image in GSpin2n+1(Qℓ) as well. Thus shim ρ2 induces a representation

ρC : Γ GSpin (Q ) π → 2n+1 ℓ C C such that ρ2 = ia(π♮)spin ρπ . We now compute ρπ at Frobenius conjugacy classes. C ◦ The uniqueness of ρπ will then follow from Proposition 5.2 and the fact that since C qρπ is conjugate to ρπ♭ it has a regular unipotent element of GSpin2n+1 in its image. C We have the semisimple elements ρπ (Frobv)ss and ιφπv (Frobv) in GSpin2n+1(Qℓ). At this point we know that

C n(n+1)/4 • spin(ρ (Frob ) ) and qv spin(ιφ (Frob )) are conjugate in GL n (Q ) π v ss · πv v 2 ℓ by Proposition 8.2. 56 Arno Kret, Sug Woo Shin

• q(ρC (Frob ) ) q(ιφ (Frob )) GL (Q ). π v ss ∼ πv v ∈ 2n+1 ℓ We claim that additionally we have

(spin(ρC (Frob ) )) = (qn(n+1)/4spin(ιφ (Frob ))). (9.1) N π v ss N v πv v C n(n+1)/4 Once the claim is veriﬁed, by Lemma 1.1, ρ (Frob ) is conjugate to qv π v ss · ιφ (Frob ) in GSpin (Q ) since spin, std, is fundamental set of represen- πv v 2n+1 ℓ { N} tations of GSpin2n+1 (table above Lemma 1.1). This proves statement (ii)’. Let us prove the claim. Possibly after conjugation by an element of GL2n (Qℓ), the image of spin ρ lies in GO n (resp. GSp n ) if n(n + 1)/2 is even (resp. odd), for some ◦ π 2 2 symmetric (resp. symplectic) pairing on the underlying 2n-dimensional space. Again

n by the same lemma, we may assume that spin(ιφπv (Frobv)) also belongs to GO2

(resp. GSp2n ). Hereafter we let the central characters ωπ or ωπv also denote the corresponding Galois characters via class ﬁeld theory. For almost all v we have the following isomorphisms.

(spin((ρC ) ))∨ n(n+1)/4 (spin(ιφ ))∨ n(n+1)/4 (spin(ιφ∨ )) π |Γv ss ≃ |·| · πv ≃ |·| · πv n(n+1)/4 spin(ιφ ) ω−1 n(n+1)/2(spin((ρC ) )) ω−1, ≃ |·| · πv ⊗ πv ≃ |·| π |Γv ss ⊗ πv where φ∨ := φ ω by definition. With this definition, the second isomorphism πv πv ⊗ πv is directly verified using the equality (φ )= ω (from functoriality of the Satake N πv πv -isomorphism with respect to the central embedding G ֒ GSp ) and the isomor m → 2n phism spin(ιφ ) ∼ spin(ιφ )∨ sim(φ )−1 induced from the pairing , that πv → πv ⊗ πv h· ·i defines GO2n (resp. GSp2n ). The above isomorphisms imply that

spin(ρC ) spin(ρC )∨ −n(n+1)/2ω . (9.2) π ≃ π ⊗|·| π By Lemma 0.1 we have

spin(ρC ) spin(ρC )∨ (spin(ρC )) π ≃ π ⊗N π From Lemma 5.1 we deduce

(spin(ρC )) = −n(n+1)/2ω . N π |·| π This proves the second part of (i’). Evaluating at unramified places, we obtain (9.1), finishing the proof of the claim. We show statement (i)’. It only remains to check the first part. By Theorem 2.4 and the preceding proof of (ii)’, we have for almost all unramified places v that

ιφπ (Frobv) ιφ ♭ (Frobv) ρ ♭ (Frobv)ss, v ∼ πv ∼ π Galois representations for general symplectic groups 57 where we also used that the Satake parameter of the restricted representation π♭ v ⊂ πv is equal to the composition of the Satake parameter of πv with the natural sur- C jection GSpin2n+1(C) SO2n+1(C) (cf. [Xu18, Lem. 5.2]). Hence ρπ (Frobv)ss → C ∼ ρπ♭ (Frobv)ss for almost all F -places v. Consequently std ρπ std ρπ♭ = ρπ♯ . By C ◦ ∼ ◦ Proposition B.1 the representation ρπ is SO2n+1(Qℓ)-conjugate to the representation

ρπ♭ . This proves the first part of (i)’. ։ We prove statement (v). The composition of ρπ with GSpin2n+1(Qℓ) SO2n+1(Qℓ) is ρπ♭ . Hence statement (v) follows from Proposition 3.5. i The Galois representation in the cohomology H (SK, )ss is potentially semistable C L by Kisin [Kis02, Thm. 3.2]. The representation ρπ appears in this cohomology and is therefore potentially semistable as well (this uses that semisimplification preserves potential semistability). This proves the first assertion of statement (iii). To verify (iii.a)’, let µ (ρC , i): G GSpin be the Hodge-Tate cochar- HT π m,Qℓ 2n+1,Qℓ C → acter of ρ (Definition 1.10). Similarly, we let µ (ρ ♭ , i): G GO be the π HT π m,Qℓ 2n+1,Qℓ C → Hodge-Tate cocharacter of ρπ♭ . We need to check that µHT(ρπ,v, ιy)= ιµHodge(ξy)+ n(n+1) 4 sim. In fact it is enough to check the equalities C qµHT(ρπ,v, ιy) = qιµHodge(ξy), (9.3) µ (ρC , ιy)) = ι (µ (ξ )) , (9.4) N ◦ HT π,v N ◦ Hodge y namely after applying the natural surjection q′ : GSpin ։ GO = SO 2n+1 2n+1 2n+1 × GL given by (q, ), since q′ induces an injection on the set of conjugacy classes 1 N of cocharacters. (The map X (T ) X (T ) induced by q′ is injective since ∗ GSpin → ∗ GO T T is an isogeny, so the map is still injective after taking quotients by the GSpin → GO common Weyl group.) Let y : F C be an embedding, such that ιy induces the place C→ ♭ v. It is easy to see that qµHT(ρπ,v, ιy)= µHT(ρπ♭,v, ιy) and qµHodge(ξy) = µHodge(ξy) from (1.1) and Lemmas 1.15) and 1.11. Thus (9.3) follows from Theorem 2.4 (iii).

The proof of (9.4) is similar: we have the following equalities in X∗(Gm)= Z. µ (ρC , ιy)) = µ ( ρC , ιy)) N ◦ HT π,v HT N ◦ π,v = ιµ (ω )+ n(n+1) = ι(µ (ξ )+ n(n+1) sim), Hodge πy 2 N ◦ Hodge y 4 where the Hodge cocharacter of ωπ at y is denoted by µHodge(ωπy ). Indeed, the ﬁrst, second, and third equalities follow from Lemma 1.11, part (i’) of the current theorem

(just proved above), and the fact that φπy = φωπ . The last fact comes from the N◦ ξy description of the central character of an L-packet; see condition (ii) in [Lan89, 3]. § The third equality also uses the easy observation that sim = 2 under the canonical N◦ identiﬁcation X∗(Gm)= Z. We prove statement (iii.b). So we assume that v ℓ and that K is either hyper- | ℓ special or contains an Iwahori subgroup of GSp (F Q ). We use the following 2n ⊗ ℓ proposition of Conrad: 58 Arno Kret, Sug Woo Shin

Proposition 9.2 (Conrad). Consider a representation σ : Γ GO (Q ) satis- v → 2n+1 ℓ fying a basic p-adic Hodge theory property P de Rham, crystalline, semistable . ∈{ } There exists a lift σ′ : Γ GSpin (Q ) that satisﬁes P if σ admits a lift σ′′ : Γ v → 2n+1 ℓ v → GSpin2n+1(Qℓ) which is Hodge-Tate.

Proof. (see also Wintenberger [Win95]). Combine Proposition 6.5 with Corollary 6.7 of Conrad’s article [Con13]. (Conrad has general statements for central extensions of algebraic groups of the form [Z ֌ H′ ։ H]; we specialized his case to our setting.)

′ −n(n+1)/2 Write ωπ := ωπ . Let rec( ) denote the ℓ-adic Galois representation |·| ×· corresponding to a Hecke character of AF via global class ﬁeld theory. The product ′ rec(ω )ρ ♭ : Γ GO (Q ) = GL (Q ) SO (Q ) is the Galois representa- π π → 2n+1 ℓ 1 ℓ × 2n+1 ℓ tion corresponding to the automorphic representation ω′ π♭ of A× Sp (A ), cf. π ⊗ F × 2n F Theorem 2.4. By comparing Frobenius elements at the unramiﬁed places, and using ′ C ′ Chebotarev and Brauer-Nesbitt, the representations std q ρπ and std rec(ωπ)ρπ♭ ′ C ′ ◦ ◦ ◦ are isomorphic. Hence q ρ and rec(ω )ρ ♭ are GO (Q )-conjugate by Proposi- ◦ π π π 2n+1 ℓ tion B.1. We make two observations:

• C ′ We saw that ρπ is potentially semistable. In particular rec(ωπ)ρπ♭ has a lift C which is a Hodge-Tate representation (namely ρπ is such a lift).

• 16 Under the assumption of (iii.b), ρπ♭ is known to be crystalline (resp. semistable )

by (iv) and (v)) of Theorem 2.4). We have Kℓ Z(AF )= Z( F ) if Kℓ contains ∩ ′ O an Iwahori subgroup and thus the character rec(ωπ) is crystalline at all v ℓ. ′ b | Therefore rec(ωπ)ρπ♭ is crystalline (resp. semistable) as well.

′ By Proposition 9.2 and Theorem 2.4.(v), the local representation rec(ω )ρ ♭ π π |Γv ^′ has a lift rv := rec(ωπ)ρπ♭ : Γv GSpin2n+1(Qℓ) which is crystalline/semistable. C → ′ C Since r and ρ are both lifts of rec(ω )ρ ♭ , the representations r and ρ v π |Γv π π |Γv v π |Γv differ by a quadratic character χv. Hence statement (iii.b) follows. ′ We prove statement (vi). Let ρ : Γ GSpin2n+1(Qℓ) be semisimple and such that ′ → C ′ for almost all F -places v where ρ and ρπ are unramified, ρ (Frobv)ss is conjugate C C to ρπ (Frobv)ss. We know that ρπ has a regular unipotent element in its image (it ։ suffices to check this after composing with GSpin2n+1(Qℓ) SO2n+1(Qℓ), in which case it follows from Proposition 3.5). Hence Proposition 5.2 implies that the two representation are conjugate.

16 If πv has a vector ﬁxed under the standard Iwahori subgroup I of GSp2n(Fv), then the restricted ♭ representation πv has a ﬁxed vector under some Iwahori subgroup of Sp2n(Fv), which may be not conjugate to I Sp (Fv), but the Galois representation ρ ♭ is still semi-stable at v. ∩ 2n π Galois representations for general symplectic groups 59

Statement (iv) reduces to checking that ρ ♭ : Γ GO (Q ) is totally odd π → 2n+1 ℓ since the covering map GSpin GO induces an isomorphism on the level 2n+1 → 2n+1 of Lie algebras and the adjoint action of GSpin2n+1 factors through that of GO2n+1.

We already know from Theorem 2.4.(vi) that ρπ♭ is totally odd, so the proof is complete.

10 Compatibility at unramiﬁed places

Let π be an automorphic representation of GSp2n(AF ), satisfying the same conditions as in Section 9. In this section we identify the representation ρπ,v from Theorem 9.1 at all places v not above S ℓ . bad ∪{ }

Proposition 10.1. Let v be a ﬁnite F -place such that p := v Q does not lie in Sbad C C | ∪ ℓ . Then ρ is unramiﬁed at v. Moreover ρ (Frob ) is conjugate to ιφ −n(n+1)/4 (Frob ). { } π π v ss πv|sim| v ♮ Proof. Let π be a transfer of π to the inner form G(AF ) of GSp2n(AF ) (Proposition ♮ 6.3). Let B(π ) be the set of cuspidal automorphic representations τ of G(AF ) such that

• τvSt and πvSt are isomorphic up to a twist by an unramiﬁed character,

• τ ∞,vSt,v and π♮,∞,vSt,v are isomorphic,

• τv is unramiﬁed,

• τ∞ is ξ-cohomological.

To compare with the definition of A(π♮), notice that the condition at v is different. We define an equivalence relation on the set B(π♮) by declaring that τ τ if ≈ 1 ≈ 2 and only if τ2 A(τ1) (hence, τ1 τ2 if and only if τ1,v τ2,v). Define a (true) ∈ shim≈ ≃ representation of Γ (see (8.2) for ρ2 (τ), cf. Proposition 8.2 and Corollary 8.7)

shim shim ρ3 := ρ2 (τ)= ia(τ) ρ2(τ). (10.1) ♮ ♮ ◦ τ∈BX(π )/≈ τ∈BX(π )/≈ ♮ Recall the definition of a(τ) from (8.3). Define b(π ) := τ∈B(π♮)/≈ a(τ). Since ρ (π♮) and ρ (τ) have the same Frobenius trace at almost all places for 2 2 P τ B(π♮), we deduce that ρ (π♮) ρ (τ). Hence ∈ 2 ≃ 2 shim ♮ ρ i ♮ ρ (π ). 3 ≃ b(π ) ◦ 2 We adapt the argument of Proposition 8.2 to the slightly different setting here. Consider the function f on G(A ) of the form f = f f 1 f ∞,vSt,v, where F ∞ ⊗ vSt ⊗ Kv ⊗ 60 Arno Kret, Sug Woo Shin

∞,vSt,v f∞ and fvSt are as in the proof of that proposition, and f is such that, for all automorphic representations τ of G(A ) with τ ∞,K = 0 and Tr τ (f ) = 0, we have F 6 ∞ ∞ 6 (observe the slight diﬀerence between Equations (8.7) and (10.2) at the place v):

1 if τ ∞,vSt,v π♮,∞,vSt,v Tr τ ∞,vSt,v(f ∞,vSt,v)= ≃ (10.2) (0 otherwise.

Arguing as in the paragraph between (8.11) and (8.12), but with B(π♮), b(π♮), and shim ♮ ♮ shim ρ3 in place of A(π ), a(π ), and ρ2 , we obtain

j shim jn(n+1)/4 j Tr (Frobv, ρ3 )= a(τ)Tr τv(fj)= a(τ)qv Tr (spin(ιφτv ))(Frobv), ♮ ♮ τ∈BX(π )/≈ τ∈BX(π )/≈ where the last equality comes from (8.13). Thus the proof boils down to the next ♮ j lemma (Lemma 10.2). Indeed the lemma and the last equality imply that Tr ρ2(π )(Frobv) jn(n+1)/4 j is equal to qv Tr (spin(ιφ ♮ ))(Frob ). As in the proof of Theorem 9.1, since πv v ♮ C C n(n+1)/4 ρ (π ) = spin ρ by construction, the semisimple parts of ρ (Frob ) and q ιφ ♮ (Frob ) 2 π π v v πv v ◦ n(n+1)/4 are conjugate. (Note that q φ ♮ (Frob )= φ −n(n+1)/4 (Frob ).) v πv v πv|sim| v

Lemma 10.2. With the above notation, if τ B(π♮) then τ π♮ . ∈ v ≃ v ∗ ∗ ♮ Proof. Let τ and π denote transfers of τ and π from G to GSp2n via part (2) of Proposition 6.3; the assumption there is satisfied by Corollary 8.4 (in fact we can ∗ ∗ ∗ ♮ just take π to be π). In particular τx τx and πx πx at all finite places x where ♮ ≃ ≃ τx and πx are unramified). In particular this is true for x = v, so it suffices to show ∗ ∗ that τv πv . ≃ ∗ ∗ By [Xu18, Thm. 1.8] we see that τ and π belong to global L-packets Π1 = Π and Π = Π (as constructed in that paper), respectively, such that ⊗x 1,x 2 ⊗x 2,x Π = Π ω for a quadratic Hecke character ω : F × A× 1 (which is lifted 1 2 ⊗ \ F → {± } to a character of GSp2n(AF ) via the similitude character). Since each local L-packet has at most one unramified representation by [Xu18, Prop. 4.4.(3)] we see that for ∗ ∗ ♮ ♮ almost all finite places x, we have τx πx ωx. Hence πx πx ωx. (Recall that ♮ ≃ ⊗ ≃ ⊗ τx πx by the initial assumption at almost all x.) This implies through (ii’) of ≃ C C Theorem 9.1 that spin(ρπ ) spin(ρπ ) ω (viewing ω as a Galois character via class C ≃ ⊗ field theory). Since ρπ has a regular unipotent element in its image, it follows that ∗ ∗ ω = 1 (by Lemma 5.1). Hence the unramified representations τv and πv belong to the same local L-packet, implying that τ ∗ π∗ as desired. v ≃ v We are now ready to collect everything and prove the main result.

Theorem 10.3. Theorem A is true. Galois representations for general symplectic groups 61

Proof. Let us caution the reader that the normalization for π changes. Namely given π as in Theorem A (so that π sim n(n+1)/4 is ξ-cohomological), the preceding | | C n(n+1)/4 discussions in Sections 9 and 10 apply to π := π sim . We thus deﬁne ρπ := C | | ρπC . Theorem 9.1 proves (iii.b), (iv) and (v) of Theorem A. By Proposition 10.1 we also have (ii). Statements (i) and (iii.a) follow from Theorem 9.1.(i’) and 9.1.(iii.a’), with evident changes due to the diﬀerent normalization. It remains to verify (iii.c) and (vi) of Theorem A. Item (vi) follows from Proposition 5.2 since ρπ contains a regular unipotent element in its image (see the proof of Proposition 3.5). Item

(iii.c) is about the crystallineness of ρπ. By deﬁnition, it is enough to prove that C spin(ρπC ) is crystalline, and hence it is enough to prove that the compact support cohomology of the Shimura variety Sh with coeﬃcients in is crystalline. In K Lξ case F = Q the variety Sh is projective, and the crystallineness follows from the 6 K thesis of Tom Lovering [Lov17]. If F = Q, then the Shimura variety SK is the classical Siegel space and in this case the representation was shown to be crystalline by Faltings-Chai [FC90, VI.6]. Alternatively we may reduce to this case using base change (Proposition 6.6) to a real quadratic extension F/Q in which ℓ is completely split.

Remark 10.4. In the introduction we claimed that we prove [BG14, Conj. 5.16] up to Frobenius semisimpliﬁcation in the case at hand. Our Theorem A covers everything except the veriﬁcation that the GSpin2n+1(Qℓ)-conjugacy class of ρπ(cv) coincides with the one predicted by loc. cit. (Our theorem only shows that ρπ(cv) is odd.) Let α GSpin (C) be as in [BG14, Conj. 5.16], and let ρ = ρ with ι : C Q . v ∈ 2n+1 π π,ι ≃ ℓ Let us sketch the argument that ρπ(cv) is conjugate to ι(αv). By Lemma 1.1, it is enough to prove the conjugacy in GL1 (under spinor norm), SO2n+1, and GL2n

(under spin). The conjugacy in GL1 is checked using the second part of Theorem

A (i). The conjugacy in SO2n+1 follows from the uniqueness of odd conjugacy class in SO (or in any adjoint group, cf. [Gro, 2], since both conjugacy classes are 2n+1 § odd in SO2n+1 (in view of Lemma 1.9 for ρπ(cv); by direct computation for ι(αv)).

Then spin(ρπ(cv)) and spin(ι(αv)) are conjugate in PGL2n ; moreover they are (up to conjugation) images of the diagonal matrix in GL2n with each of 1 and 1 appearing n−1 − exactly 2 times. Since ρ (c ), spin(ι(α ) GL n (Q ) are elements of order 2 whose π v v ∈ 2 ℓ images in PGL n−1 are determined as such, we conclude that ρ (c ) spin(ι(α ) 2 π v ∼ v (with eigenvalues 1 and 1, with multiplicities 2n−1 each). We are done with showing − that ρ (c ) ι(α ) in GSpin (Q ). π v ∼ v 2n+1 ℓ 62 Arno Kret, Sug Woo Shin

11 Galois representations for the exceptional group G2

As an application of our main theorems we realize some instances of the global

Langlands correspondence for G2 in the cohomology of Siegel modular varieties of genus 3 via theta correspondence, following the strategy of Gross–Savin [GS98]. In particular the constructed Galois representations will be motivic17 and come in compatible families as such. We work over F = Q (as opposed to a general totally real ﬁeld) mainly because this is the case in [GS98].

More precisely we are writing G2 for the split simple group of type G2 defined c over Z. Denote by G2 the inner form of G2 over Q which is split at all finite places c and such that G2(R) is compact. The dual group of G2 is G2(C) and fits in the diagram η & & PGL (C) / G (C) / SO (C) / GL (C) 2 2 _ 7 _ 7 _ ζ / / Spin (C) / SO8(C) / GL8(C) 7 8 spin such that G (C)=SO (C) Spin (C). The subgroup PGL (C) is given by a choice 2 7 ∩ 7 2 of a regular unipotent element of SO8(C). See [GS98, pp.169-170] for details. Note that the spin representation of Spin7 is orthogonal and thus factors through SO8. The 8-dimensional representation G (C) ֒ GL (C) decomposes into 1-dimensional 2 → 8 and 7-dimensional irreducible pieces. The former is the trivial representation. The latter factors through SO7(C). Evidently all this is true with Qℓ in place of C. c The (exceptional) theta lift from each of G2 and G2 to PGSp6 uses the fact c that (G2, PGSp6) and (G2, PGSp6) are dual reductive pairs in groups of type E7 c [GRS97, GS98]. In this section we concentrate on the case of G2, only commenting c on the case of G2 at the end. Every irreducible admissible representation of G2(R) is finite-dimensional, and both (C-)cohomological and L-cohomological since the half sum of all positive roots of G2 is integral. Note that an automorphic representation

π of PGSp6(A) is the same as an automorphic representation of GSp6(A) with trivial central character, so we will use them interchangeably. For such a π the subgroup

ρπ(Γ) of GSpin7(Qℓ) is contained in Spin7(Qℓ) by (i) of Theorem A.

c Theorem 11.1. Let σ be an automorphic representation of G2(A). Assume that • σ admits a theta lift to a cuspidal automorphic representation π on PGSp6(A). • σvSt is the Steinberg representation at a ﬁnite place vSt.

17In this paper “motivic” means that it appears in the ´etale cohomology of a smooth quasi- projective variety over a number ﬁeld. Galois representations for general symplectic groups 63

Then for each prime ℓ and ι : C Q , there exists a representation ρ = ρ :Γ ≃ ℓ σ σ,ι → G2(Qℓ) such that

1. For every finite place v = ℓ where σ is unramified, ρ is unramified at v. 6 σ Moreover (ρ ) ιφ as unramified L-parameters for G . σ|WQv ss ≃ σv 2 ∨ 18 2. ρσℓ is de Rham with µHT(ρσℓ )= ιµHodge(σ∞).

3. If σℓ is unramiﬁed then ρσℓ is crystalline.

4. ζ ρ ρ . ◦ σ ≃ π Before starting the proof, we recall the basic properties of the theta lift π. We see from [GS98, 4 Prop. 3.1, 4 Prop. 3.19] that π is the Steinberg representation, § § vSt and that πv is unramified whenever σv is unramified at a finite place v and the unramified L-parameters are related via

ζφ φ . (11.1) σv ≃ πv

Furthermore π∞ is an L-algebraic discrete series representation whose parameter can be explicitly described in terms of σ ( 3 Cor. 3.9 of loc. cit.). §

Proof. We apply Theorem A to the text above π to obtain a representation ρπ : Γ Spin (Q ). Note that spin ρ is a semisimple representation by construction. → 7 ℓ ◦ π Since the image of ρπ♭ contains a regular unipotent element of SO7(Qℓ), it follows that ρπ(Γ) contains a regular unipotent of Spin7(Qℓ) and also that of SO8(Qℓ). (The representation spin: Spin GL 3 induces an inclusion Spin ֒ SO , under which 7 → 2 7 → 8 a regular unipotent maps to a regular unipotent.) By (11.1) and the Chebotarev density theorem, the image of ρπ is locally contained in G2 in the terminology of Gross-Savin, so [GS98, 2 Cor. 2.4] implies that ρ (Γ) is contained in G (Q ) (given § π 2 ℓ as SO (Q ) Spin (Q ) for a suitable choice of the embedding SO ֒ SO ; here 7 ℓ ∩ 7 ℓ 7 → 8 . spin : Spin ֒ SO is ﬁxed). Hence we have ρ satisfying (4), namely that ρ ζ ρ 7 → 8 σ π ≃ ◦ σ Assertions (1)-(3) of the theorem follow from Theorem A and the fact that the set of Weyl group orbits on the maximal torus (resp. on the cocharacter group of a maximal torus) for G2 maps injectively onto that for Spin7. (The latter can be checked explicitly.)

c Example 11.2. There is a unique automorphic representation σ of G2(A) unramiﬁed outside 5 such that σ5 is the Steinberg representation and σ∞ is the trivial representation [GS98, 1 Prop. 7.12]. Proposition 5.8 in 5 of loc. cit. (via a computer § § calculation due to Lansky and Pollack) tells us that σ admits a nontrivial cuspidal

18 ∨ Note that σ∞ is ξ-cohomological for ξ = σ∞. 64 Arno Kret, Sug Woo Shin

theta lift to PGSp6(A). Proposition 5.5 in the same section gives another example of nontrivial theta lift but we will not consider it here.

We conﬁrm the prediction of Gross–Savin that a rank 7 motive whose motivic

Galois group is G2 is realized in the middle degree cohomology of a Siegel modular variety of genus 3.

Corollary 11.3. Let σ be as in Example 11.2. Write π for its theta lift. Then ρσ has Zariski dense image in G (Q ). Moreover spin ρ is isomorphic to the direct sum of 2 ℓ ◦ π η ρσ and the trivial representation. In particular η ρσ and the trivial representation ◦ ∞ 6 ◦ appear in the π -isotypic part in Hc (Sh, Qℓ)(3), where Sh is the tower of Siegel modular varieties for GSp6 and (3) denotes the Tate twist (i.e. the cube power of the cyclotomic character).

Proof. If the image is not dense in G (Q ) then the proof of [GS98, 2 Prop. 2.3] 2 ℓ § shows that the Zariski closure of ρπ(Γ) is PGL2. However we see from the explicit computation of Hecke operators at 2 and 3 on σ carried out by Lansky–Pollack [LP02, p.45, Table V] that the Satake parameters at 2 and 3 do not come from 19 PGL2. We conclude that ρσ(Γ) is dense in G2(Qℓ). The second assertion of the corollary is clear from (4) of Theorem 11.1 and the construction of ρπ.

∞ 6 Remark 11.4. Corollary 12.4 implies that the π -isotypic part in Hc (Sh, Qℓ)(3) is 8-dimensional, consisting of η ρ and the trivial representation without multiplicity. ◦ σ Remark 11.5. The Tate conjecture predicts the existence of an algebraic cycle on Sh which should give rise to the trivial representation in the corollary. Gross and Savin suggest that it should come from a Hilbert modular subvariety of Sh for a totally real cubic extension of Q. See [GS98, 6] for details. § Remark 11.6. Ginzburg–Rallis–Soudry [GRS97, Thm. B] showed that every globally generic automorphic representation σ of G2(A) admits a theta lift to PGSp6(A). (The result is valid over every number ﬁeld.) Using this, Khare–Larsen–Savin [KLS10] established instances of global Langlands correspondence for G2(A), the analogue of c Theorem 11.1 with G2 in place of G2, under a suitable local hypothesis. (See Section 6 of their paper for the hypothesis. They prescribe a special kind of supercuspidal representation instead of the Steinberg representation.) In our notation, their ρσ is constructed inside the SO7-valued representation ρπ♭ , where the point is to show 20 that the image is contained and Zariski dense in G2 [KLS10, Cor. 9.5]. 19 The authors check [LP02, 4.3] that the Satake parameters do not come from SL2 via the map SL (C) G (C) induced by a§ regular unipotent element. But the latter map factors through the 2 → 2 projection SL2 PGL2. 20Thereby they→ give an aﬃrmative answer to Serre’s question on the motivic Galois group of Galois representations for general symplectic groups 65

12 Automorphic multiplicity

In this section we prove multiplicity one results for automorphic representations of

GSp2n(AF ) and the inner form G(AF ) under consideration. For GSp2n we deduce this from Bin Xu’s multiplicity formula using the following property (cf. Lemma 5.1) × ω : Γ Q : ρ ρ ω ω =1 ∀ → ℓ π ≈ π ⊗ ⇒ of the associated Galois representations ρπ. The result is then transferred to G(AF ) via the trace formula ( 6). This is standard except when the highest weight of ξ is not § regular: In that case we need the input from Shimura varieties that the automorphic representations of G(AF ) of interest are concentrated in the middle degree. Theorem 12.1. Let n 2. Let π be a cuspidal automorphic representation of GSp (A ) ≥ 2n F such that conditions (St) and (L-Coh) hold. The automorphic multiplicity m(π) of π is equal to 1, i.e. Theorem B is true. Proof. By Xu [Xu18, Prop. 1.7] we have the formula

m(π)= m(π♭) Y (π)/α( ) , | Sφ | where m(π♭) = 1 in our case by Arthur [Art13, Thm. 1.5.2]. The group Y (π) is equal to the set of characters ω : GSp (A ) C× which are trivial on GSp (F )A×Sp (A ) 2n F → 2n F 2n F ⊂ GSp2n(AF ) and are such that π π ω. The deﬁnition of the subgroup α( φ) of Y (π) ≃ ⊗ S × is not important for us: We claim that Y (π) =1. Let ω Y (π) and let χ: Γ Q ∈ → ℓ be the corresponding character via class ﬁeld theory. We get ρ ρ χ from The- π ≃ π ⊗ orem A, (ii) and (vi). By Lemma 5.1, it follows that χ = 1 and hence ω is trivial as well.

We now prove the analogue of Theorem 12.1 for an inner form of GSp2n,F . Since the analogue of results by Arthur and Xu has not been worked out yet for non- quasi-split inner forms,21 we only obtain a partial result. Let G be an inner form of GSp in the construction of Shimura varieties, cf. 7. 2n,F § Theorem 12.2. Let π be a cuspidal automorphic representation of G(AF ) satisfying conditions (L-coh) and (St). Then the automorphic multiplicity of π is equal to 1.

∗ ∗ ∗ ∗ G G −1 q(G∞) G G G −1 q(G∞) G Proof. Let f∞ := Πξ ( 1) fLef,∞ and f∞ := Πξ ( 1) fLef,∞, where G G∗ | | − | | − ∗ fLef,∞ and fLef,∞ denote the Lefschetz functions on G(F∞) and G (F∞) as in Equa- tion (A.6)), respectively. At the place vSt, we consider the truncated Lefschetz func- G G∗ G q(G ) G tions f and f as introduced in (A.4), and put f := ( 1) vSt f and Lef,vSt Lef,vSt vSt − Lef,vSt type G2, since it is well known that ρπ♭ appears in the cohomology of a unitary PEL-type Shimura variety after a quadratic base change, along the way to proving a result on the inverse Galois problem. Sometimes this contribution of [KLS10] is overlooked in the literature. 21 Certain inner forms of Sp2n,F and special orthogonal groups have been treated by Ta¨ıbi [Tai19]. 66 Arno Kret, Sug Woo Shin

∗ ∗ q(G ) ∗ ∞,vSt G vSt G ∞,vSt ∗ fvSt := ( 1) fLef,vSt . Let f (G(AF )) be arbitrary. Let π (resp. π ) − ∈ H ∗ G G∗ be a discrete automorphic representation of G (resp. G ). Note that fv and fv are associated at v vSt, by Lemmas A.4 and A.11 (with the choice of Haar mea- ∈{ ∞} ∗ ∗ ∞,vSt G G ∞,vSt G G sures and transfer factors explained there). Hence f fvSt f∞ and f fvSt f∞ are associated. Implicitly we are choosing the local transfer factor at each place v to be 22 identically equal to the sign e(Gv) (whenever nonzero) so that the global transfer factor always equals one (whenever nonzero). Comparing the trace formulas for G and G∗ and arguing as in the proof of Propo- sition 6.3, we obtain

∗ ∗ ∞,vSt G G ∗ ∗ ∞,vSt G G m(τ)Tr τ(f fvSt f∞)= m(τ )Tr τ (f fvSt f∞ ), τ τ ∗ X X ∗ where τ (resp. τ ) runs over discrete automorphic representations of G(AF ) (resp. ∗ G (AF )), and m( ) denotes the automorphic multiplicity as usual. By Lemma A.13 G G∗ ∗ the functions fvSt (resp. fvSt ) have non-zero trace against πvSt (resp. πvSt ) only if πvSt ∗ (resp. πvSt ) is an unramiﬁed twist of the Steinberg representation. Therefore

∗ ∗ ΠG −1 m(τ)( 1)q(G∞)ep(τ ξ)= ΠG −1 m(τ ∗)( 1)q(G∞)ep(τ ∗ ξ), | ξ | − ∞ ⊗ | ξ | − ∞ ⊗ τ τ ∗ X X (12.1) G G∗ where Πξ and Πξ are discrete L-packets. In Equation (12.1), each sum ranges over the set of ξ-cohomological discrete automorphic representations which are Steinberg (up to unramiﬁed twist) at v and isomorphic to πvSt,∞ away from v and . St St ∞ For the quasi-split group G∗, Corollary 2.8 shows that any τ ∗ in Equation (12.1) ∗ G∗ is tempered at inﬁnity, thus τ∞ is a discrete series (and belongs to Πξ ). However Corollary 2.8 does not imply the analogue for τ. For τ∞, Corollary 8.4 implies that q(G∞) τ∞ must be a discrete series representation. Hence ep(τ∞ ξ)=( 1) and ∗ ⊗ − ep(τ ∗ ξ)=( 1)q(G∞) above, allowing us to simplify (12.1) to ∞ ⊗ − ∗ ΠG −1 m(τ)= ΠG −1 m(τ ∗), (12.2) | ξ | | ξ | τ τ ∗ X X where each sum runs over discrete automorphic representations which are Steinberg

(up to unramified twist at vSt), belong to the specified discrete L-packet at infinity, and isomorphic to πvSt,∞ away from v and . St ∞ At this point it is useful to show the following lemma by the trace formula argument.

∞ ′ ∗ Lemma 12.3. Suppose that F = Q. We have m(τ) = m(τ τ∞) and m(τ ) = ∗,∞ ∗,′ ′ ∗,′ 6 G G∗ m(τ τ∞ ), where τ∞, τ∞ are arbitrary members of Πξ or Πξ . 22In the special case of inner forms, the deﬁnition of transfer factors in [LS87] simpliﬁes to yield a nonzero constant, independent of conjugacy classes, whenever two conjugacy classes are matching. Galois representations for general symplectic groups 67

∞ ′ Proof. The assertion that m(τ)= m(τ τ∞) was already shown in Corollary 8.5. (We have τ A(π) in the notation there.) The same argument for G∗ proves m(τ ∗) = ∈ m(τ ∗,∞τ ∗,′), with the only change occurring at . Namely, instead of appealing to ∞ ∞ Corollary 2.8, we know that the analogue of (8.14) for G∗ receives contributions only from those representations whose components at belong to ΠG∗ , by Corollary 2.8. ∞ ξ (We already made this observation in the discussion above (12.2).)

We continue the proof of Theorem 12.2. The representation π as in the statement of the theorem appears in the left sum of (12.2). So both sides are positive in that equation. In particular there exists π∗ contributing to the right hand side such that m(π∗) > 0. Any other τ ∗ in the sum is isomorphic to π∗ at all finite places away ∗ ∗ from vSt. We also know that τvSt and πvSt differ by an unramified character and that ∗ ∗ τ∞ and π∞ belong to the same L-packet. Now we claim that τ π . To see vSt ≃ vSt this, we apply [Xu18, Thm. 1.8] to deduce that τ ∗ and π∗ belong to the same global L-packet as in that paper, by the same argument as in the proof of Lemma 10.2.

Since the local L-packet for GSp2n(FvSt ) of (any character twist of) the Steinberg representation is a singleton by [Xu18, Prop. 4.4, Thm. 4.6], the claim follows.23 We have shown that the right hand side of (12.2) may be summed over τ ∗ which ∗ G∗ ∗ is isomorphic to π away from inﬁnity and belongs to Πξ at inﬁnity. Each τ has automorphic multiplicity one by Theorem 12.1 and Lemma 12.3. (Without the lemma, some m(τ ∗) could be zero.) So (12.2) comes down to

m(τ)= ΠG . | ξ | τ X Recall that the sum runs over τ such that τ ∞,vSt π∞,vSt , τ ΠG, and τ ≃ ∞ ∈ ξ vSt ≃ π ǫ for an unramiﬁed character ǫ. By Lemma 12.3, the contribution from τ with vSt ⊗ τ ∞ π∞ is already ΠG m(π), and the other τ (if any) contributes nonnegatively. ≃ | ξ | We conclude that m(π) = 1. (Moreover, all τ in the sum with m(τ) > 0 should be isomorphic to π at vSt; this gets used in the next corollary.)

Corollary 12.4. The integer a(π♮) in Corollary 8.5 (deﬁned in (8.3)) is equal to one.

Proof. Consider π := π♮ −n(n+1)/4, which satisﬁes (St) and (L-coh). Applying the |·| last paragraph in the proof of the preceding theorem to π, we see that for every ♮ ♮ G ♮ τ A(π ), we have an isomorphism τv π . Since τ∞ Π for every τ A(π ) ∈ St ≃ vSt ∈ ξ ∈ 23 Lemma 2.1 implies that the L-parameter WFv SU (R) SO n (C) for the Steinberg St × 2 → 2 +1 representation of Sp2n(FvSt ) restricts to the principal representation of SU2(R) in SO2n+1(C). A lift of this to a GSpin2n+1(C)-valued parameter is attached to the Steinberg representation of GSp2n(FvSt ) in Xu’s construction. This is enough to imply that the group φ˜ in [Xu18, Prop. 4.4] is trivial. S 68 Arno Kret, Sug Woo Shin by Corollary 8.4, it follows from Theorem 12.2, Lemma 12.3, and Remark 7.2 that

♮ n(n+1)/2 −1 G −1 a(π )=( 1) N∞ m(τ) ep(τ∞ ξ)= Πξ m(τ)=1. − ♮ · ⊗ | | τ τ∈A(π ) τ∞≃π∞, τ ∈ΠG X X∞ ξ

13 Meromorphic continuation of the Spin L-function

Let π be a cuspidal automorphic representation of GSp2n(AF ) unramified away from a finite set of places S. The partial spin L-function for π away from S is by definition

S 1 L (s, π, spin) := −s . det(1 qv spin(φπv (Frobv))) vY∈ /S − Various analytic properties of this function would be accessible if the Langlands functoriality conjecture for the L-morphism spin were known. However we are far from it when n 3. In particular no results have been known about the meromorphic ≥ (or analytic) continuation of LS(s, π, spin) when n 6 (see introduction for some ≥ results when 2 n 5). ≤ ≤ The aim of this final section is to establish Theorem C on meromorphic continuation for L-algebraic π under hypotheses (St), (L-coh), and (spin-REG) by applying a potential automorphy theorem, namely Theorem A of [BLGGT14]. Let Sbad be as in Theorem A. A character µ :Γ GL (M ) is considered totally of sign ǫ 1 → 1 π,λ ∈ {± } if µ(cv) = ǫ for all infinite places v of F . More commonly a character totally of sign +1 or 1 is said to be totally odd or even, respectively. For each Q-embedding − -w : F ֒ C define the cocharacter µ (φ ) : GL (C) GSpin (C) by re → Hodge πw 1 → 2n+1 stricting the L-parameter φ : W LSp to W = GL (C) (via w). Condition πw Fw → 2n C 1 (L-coh) ensures that µHodge(φπw ) is an algebraic cocharacter.

Proposition 13.1. Suppose that π satisﬁes (St), (L-coh), and (spin-REG). There exist a number ﬁeld Mπ and a representation

R :Γ GL n (M ) π,λ → 2 π,λ for each ﬁnite place λ of Mπ such that the following hold. (Write ℓ for the rational prime below λ.)

1. At each place v of F not above S ℓ , we have bad ∪{ } char(R (Frob )) = char(spin(ιφ (Frob ))) M [X]. π,λ v πv v ∈ π Galois representations for general symplectic groups 69

2. R is de Rham for every v ℓ. Moreover it is crystalline if π is unramiﬁed π,λ|Γv | v and v / S . ∈ bad =(For each v ℓ and each w : F ֒ C such that ιw induces v, we have µ (R , ιw .3 | → HT π,λ|Γv ι(spin µ (φ )). In particular µ (R , ιw) is a regular cocharacter for ◦ Hodge πw HT π,λ|Γv each w.

4. Rπ,λ is irreducible.

5. Rπ,λ maps into GSp2n (M π,λ) (resp. GO2n (M π,λ)) for a suitable nondegenerate alternating (resp. symmetric) pairing on the underlying 2n-dimensional space over M if n(n + 1)/2 is odd (resp. even). The multiplier character µ :Γ π,λ λ → GL (M ) (so that R R∨ µ ) is totally of sign ( 1)n(n+1)/2. 1 π,λ π,λ ≃ π,λ ⊗ λ − Remark 13.2. Parts (1)-(3) of the lemma imply that the family R is a weakly { π,λ} compatible system of λ-adic Galois representations, cf. [BLGGT14, 5.1]. §

Remark 13.3. Although we do not need it, we can choose Mπ such that Rπ,λ is valued in GL2n (Mπ,λ) for every λ. Concretely we may take Mπ to be the field of definition for the π∞-isotypic part in the compact support Betti cohomology with Q-coefficients (with respect to the local system arising from ξ). When the coefficients ∞ are extended to Mπ,λ the π -isotypic part becomes a λ-adic representation of Γ via

´etale cohomology, which is isomorphic to a single copy of Rπ,λ if the coeﬃcients are further extended to M π,λ by Corollary 12.4.

Proof. The ﬁrst three assertions are immediate from Theorem 9.1. For the fourth assertion, observe that the image of ρπ in SO2n+1 is either PGL2, G2 (n = 3) or

SO2n+1, due to the Steinberg hypothesis (and local global compatibility at vSt). We ﬁrst exclude the case G (n = 3). In this case spin decomposes as a direct 2 7|G2 sum of the trivial representation and an odd-dimensional self-dual representation.

In particular, up to twist the weight 0 occurs in ρπ with multiplicity at least 2. This contradicts however the regularity in item (3). We now exclude the case PGL2 in a similar way. Write ψ : SL Spin for the principal SL , and decompose r := 2 → 2n+1 2 spin ψ = t r into irreducible representations r of SL (cf. [Gro00, Prop. 6.1]). ◦ i=1 i i 2 The dimension of the representations r all have the same parity: dim(r ) dim(r ) L i i ≡ j mod 2 for all i, j. If the dimension of the ri is odd, the weight 0 appears in them, and in the even case, the weight 1 appears in each ri. As n > 2, r is reducible, so (up to twist) either the weight 0 or the weight 1 appears with multiplicity t 2 in ≥ ρπ. Thus, if the Galois representation has image in these smaller groups, it can’t be regular, which contradicts (3). In particular statement (4) follows. 70 Arno Kret, Sug Woo Shin

The ﬁrst part of (5) is clear from Lemma 0.1. Let us check the second part of (5). Consider the diagram

ρπ / spin / Γ ▼ / GSpin2n+1(M π,λ) / GSp2n (M π,λ) or GO2n (M π,λ) . ▼▼▼ ❤ ▼ µλ ❤❤❤ ▼▼▼ ❤❤❤❤ ▼▼ N ❤❤❤❤ ▼▼ ❤❤❤❤ sim & t❤❤❤ GL1(M π,λ)

The outer triangle commutes by the definition of µλ. The right triangle also commutes by Lemma 0.1. Hence µ = ρ . By (L-coh) we know that ̟ n(n+1)/4 λ N ◦ π πv|·| is the inverse of the central character of some irreducible algebraic representation × × ξ at each infinite place v. Hence ̟ n(n+1)/4 : R C is the w-th power map, v πv|·| → where w Z is as in (cent) of 7 (in particular w is independent of v). Hence ∈ n(n+1)/2 § ̟ n(n+1)/4 = ̟ corresponds via class field theory and ι to an even Galois π|·| π|·| character of Γ (regardless of the parity of w). On the other hand, µλ corresponds to

̟π via class ﬁeld theory and ι in the L-normalization, cf. Theorem A. We conclude that µ (c )=( 1)n(n+1)/2 for each v . λ,v v − |∞ Now that we proved the lemma, Theorem A of [BLGGT14] implies

Corollary 13.4. Theorem C is true.

Proof. The conditions of the theorem in loc. cit. are veriﬁed by the above lemma with the following additional observation: the characters µλ forms a weakly compatible system since they are associated to the central character of π (which is an algebraic Hecke character).

Remark 13.5. For the lack of precise local Langlands correspondence for general symplectic groups (however note that Bin Xu established a slightly weaker version in [Xu18]), we cannot extend the partial spin L-function for π to a complete L- function by ﬁlling in the bad places. However the method of proof for the corollary yields a ﬁnite alternating product of completed L-functions made out of Π as in Theorem C, by an argument based on Brauer induction theorem as in the proof of [HSBT10, Thm. 4.2]; this alternating product should be equal to the completed spin L-function as this is indeed true away from S.

A Lefschetz functions

We collect some results on Lefschetz functions that are used in the text. In this appendix, F is a characteristic-zero non-archimedean local field of residue characteristic p > 0, except in Lemma A.13 at the end, where F is global. We collect Galois representations for general symplectic groups 71 the required results from the literature and prove some lemmas to deal with small technical difficulties (non-compact center, twisted group). To help the readers we clarify some terminology. There are three names for the function whose trace computes the Euler-Poincarécharacteristic or the Lefschetz number of the group cohomology (resp. Lie algebra cohomology) for a given reductive p-adic (resp. real) group: they are called Euler-Poincaréfunctions, Kottwitz functions, or Lefschetz functions. In the real case one can consider twisted coefficients by local systems. There are small differences between the three functions. Euler-Poincaréand Lefschetz functions are considered on either p-adic or real groups, and can be described in terms of pseudo-coefficients for certain discrete series representations. The functions may not be unique but their orbital integrals are well defined. A Kottwitz function mainly refers to a particular function ona p-adic group and gives pseudo-coefficients for the Steinberg representations. It is not just characterized by their traces but can be given by an explicit formula, cf. [Kot88, 2]. A § generalization of Kottwitz functions on a p-adic group is given in [SS97b] but we will not need it in this paper. We recall a result of Casselman and Kottwitz. Let G be a connected reductive group and write q(G) for the F -rank of the derived subgroup of G.

Proposition A.1. Suppose that the center of G is anisotropic over F . Then there G exists a locally constant compactly supported function fLef = fLef on G(F ) such that

(i) Assume that the adjoint group of G is simple. For all irreducible admissible

unitary representations τ of G(F ) the trace Tr τ(fLef) vanishes unless τ is either the trivial representation or the Steinberg representation, in which case the trace is equal to 1 or ( 1)q(G) respectively. (If q(G)=0 then the trivial − and Steinberg representations coincide.)

(ii) Let γ G(F ) be a semisimple element, and I the neutral component of its ∈ centralizer; we give I(F ) G(F ) the Euler-Poincar´emeasure. Then the orbital \ integral −1 Oγ (fLef)= fLef(g γg)dg ZI(F )\G(F ) is non-zero if and only if I(F ) has a compact center, in which case Oγ (fLef)= 1.

Proof. Kottwitz constructed in [Kot88, Sect. 2] Lefschetz functions fLef on G(F ). These are functions such that

∞ Tr τ(f )= ( 1)i dim Hi (G(F ), τ), (A.1) Lef − cts i=0 X 72 Arno Kret, Sug Woo Shin for all irreducible representations τ of G(F ). The cohomology in Equation (A.1) is the continuous cohomology, i.e. the derived functors of τ τ G(F ) on the category 7→ of smooth representations of ﬁnite length. By the computation of the cohomology i i spaces Hcts(G(F ), τ) in [BW00, Thm. XI.3.9], for unitary τ, the spaces Hcts(G(F ), τ) vanish unless i = 0 and τ is the trivial representation, or i = q(G) and τ is the q(G) Steinberg representation, in which case dim Hcts (G(F ), τ) = 1.

We now consider the group GL2n+1 equipped with the involution θ deﬁned by g J tg−1J with J the (2n +1) (2n + 1) matrix with all entries 0 except those on 7→ × the antidiagonal, where we put 1. Deﬁne GL+ := GL ⋉ θ . In [BLS96] Borel, 2n+1 2n+1 h i Labesse, and Schwermer introduced twisted Lefschetz functions. Below we will cite mostly from the more recent article of Chenevier–Clozel [CC09, Sect. 3]. The results we need from them are proven in a general twisted setup but specialize to our case as follows.

GL2n+1,θ Proposition A.2. There exists a function fLef with compact support in the subset + GL2n+1(F )θ of GL2n+1(F ), such that

+ GL2n+1,θ (i) For all unitary admissible representations τ of GL2n+1(F ) the trace Tr τ(fLef ) vanishes, unless τ is either the trivial representation or the Steinberg representation, in which case trace is equal to 1 or 2( 1)n respectively; − (ii) Let γθ G(F )θ GL+ (F ) be a semisimple element, and I the neutral ∈ ∈ 2n+1 γθ component of the centralizer of γθ; we give I (F ) G(F ) the Euler-Poincar´e γθ \ measure [CC09, p.30] (cf. [Kot88, Sect. 1]). Then the twisted orbital integral

GL2n+1,θ GL2n+1,θ −1 Oγθ(fLef )= fLef (g γθ(g))dg, ZIγθ(F )\G(F ) is non-zero if and only if Iγθ(F ) has a compact center. Moreover in case Iγθ(F ) GL2n+1,θ has compact center, the integral Oγθ(fLef ) is equal to 1. Proof. We deduce this from Proposition 3.8 and Corollary 3.10 of [CC09]. In fact, [CC09] states that TrSt(f GL2n+1(F )θ)=( 1)q(G)P (1)/ε(θ). In our case one computes Lef − q(G)=2n, ε(θ)=( 1)n and P (1) = 2, hence the statement of our proposition. −

It is well known (cf. [Art13, 1.2] or [Wal10, 1.8]) that Sp2n,F is a θ-twisted en- Sp2n(F ) doscopic group of GL2n+1,F . Let fLef be the Lefschetz function on Sp2n(F ) from Proposition A.1. We introduce the notion of associated functions. Let G0 be a re- ′ ′ ductive group over F . Suppose that G0 is an endoscopic group for G0 (i.e. G0 is part of an endoscopic datum for G ). We say that the functions f (G′ (F )) and 0 0 ∈ H 0 f ′ (H′ (F )) are associated if they have matching orbital integrals in the sense 0 ∈ H 0 of [KS99, (5.5.1)]. The same deﬁnition carries over to the archimedean and adelic setup. Galois representations for general symplectic groups 73

× ×2 −1 Sp2n GL2n+1,θ Lemma A.3. Deﬁne C = F /F . Then the functions (fv ,C f ) are | | St · Lef associated.

Proof of Lemma A.3. We show that for each γ Sp (F ) strongly regular, semisim- ∈ 2n ple and elliptic, we have

Sp2n GL2n+1,θ SOγ (fLef )= ∆(γ, δ)SOδθ(CfLef ) (A.2) Xδ where δ ranges over a set of representatives of the twisted conjugacy classes in Sp2n GL2n+1(F ) which are associated to γ. Note that the stable orbital integral SOγ(fLef ) is equal to the number of conjugacy classes in the stable conjugacy class of γ. Our ﬁrst claim (1) is that ∆(γ, δ) = 1 for all elliptic δ associated to γ. Assuming that this claim is true, the right hand side of Equation (A.2) is the stable twisted orbital GL2n+1,θ integral SOδθ(CfLef ), which equals, up to multiplication by C, the number of twisted conjugacy classes in the stable twisted conjugacy class of δ. Our second claim Sp2n GL2n+1,θ is that SOγ (fLef ) = SOδθ(CfLef ). Clearly the lemma follows from claims (1) and (2). Let us now check the two claims. We begin with claim (1). For this we use the formulas in the article of Waldspurger [Wal10, Sect. 10]. The factors ∆(γ, δ) are complicated and involve many notation to introduce properly, for which we do not have the space to introduce. Thus we use, without introducing, the notation from [Wal10]. By [Wal10, Prop. 1.10]

∆ e(y, x)= χ(ηxDPI (1)PI− ( 1)) sgn (Ci). (A.3) Sp2n,G Fi/F±i − −∗ i∈YI e − Since our conjugacy class is elliptic we have H = Sp2n, the group H is trivial and the sets I−, I−∗ are empty. A priori, χ is any quadratic character of F ×, and each choice defines a different isomorphism class of twisted endoscopic data. The choice affects the L-morphism:

η : LH = SO (C) W LG(C)=GL (C) W , (g,w) (χ(w)g,w). χ 2n+1 × F → 2n+1 × F 7−→ L Since the Steinberg L-parameter for GL2n+1(C), WF SU2(R) GL2n+1(C), is 2n × → trivial on WF (and Sym on SU2(R)), it factors through ηχ only for χ = 1 (and not for non-trivial χ). Consequently the transfer factor in Equation (A.3) is equal to 1, and claim (1) is true. We check claim (2). In Sections 1.3 and 1.4 of [Wal10], Waldspurger describes the (strongly) regular, semisimple (stable) conjugacy classes of Sp2n(F ) in terms of F -algebras. More precisely, a regular semisimple conjugacy class in Sp2n(F ) is given by data (F , F ,c , x ,I), where I is a ﬁnite index set; for each i I, F is a i ±i i i ∈ ±i 74 Arno Kret, Sug Woo Shin

ﬁnite extension of F ; for each i I, F is a commutative F -algebra of dimension ∈ i ±i 2; we have i∈I [Fi : F ]=2n; τi is the non-trivial automorphism of Fi/F±i; for each i I, we have an an element c F × such that τ (c ) = c ; x F × such ∈ P i ∈ i i i − i i ∈ i that xiτi(xi) = 1; to the data (Fi, F±i,ci, xi,I), Waldspurger attaches a conjugacy class [Wal10, Eq. (1)] and this conjugacy class is required to be regular.

The data (Fi, F±i,ci, xi,I) should be taken up to the following equivalence relation: The index set I is up to isomorphism; the triples (Fi, F±i, xi) are up to × isomorphism; the element ci Fi is given up to multiplication by the norm group × ∈ NFi/F±i (Fi ). The stable conjugacy class is obtained from (Fi, F±i,ci, xi,I) by forgetting the elements ci [Wal10, Sect. 1.4], and keeping only (Fi, F±i, xi,I). According to [Wal10, Sect. 1.3] a strongly regular, semisimple twisted conjugacy 24 class in GL2n+1 is given by data (Li, L±i,yi,yD,J) where a ﬁnite index set J; for each i J, L±i is a ﬁnite extension of F ; for each i J, Li is a commutative ∈ ∈ × L±i-algebra of dimension 2 over L±i; for each i J, an element yi Li ; 2n +1= × ∈ ∈ i∈J [Li : F ]; yD an element of F ; to the data (Li, L±i,yi,yD,J) Waldspurger attaches a twisted conjugacy class [Wal10, p.45] and this class is required to be P strongly regular.

The data (Li, L±i,yi,J) should be taken up to the following equivalence relation:

(Li, L±i,J) are under the same equivalence relation as before for the symplectic × group; the elements yi are determined up to multiplication by NLi/L±i (Li ); yD is ×2 determined up to squares F . The stable conjugacy class of (Li, L±i,J,yi,yD) is × obtained by taking yi up to L±i and forgetting the element yD.

By [Wal10, Sect. 1.9] the stable (twisted) conjugacy classes (Li, L±i, xi,J) and

(Fi, F±i,yi,I) correspond if and only if (Li, L±i, xi,J)=(Fi, F±i,yi,I).

The conjugacy class γ (resp. δ) is elliptic if the algebra Fi (resp Li) is a ﬁeld. Let’s assume that this is the case (otherwise there is nothing to prove, because the Sp2n GL2n+1,θ equation SOγ(fLef ) = STOδ(CfLef ) reduces to 0 = 0). By the description above, to reﬁne the stable conjugacy class (Fi, F±i, xi,I) to a conjugacy class, is to give elements c F ×/N F × such that τ (c ) = c . For each i there are 2 i ∈ i Fi/F±i i i i − i choices for this, so we get 2#I conjugacy classes inside the stable conjugacy class.

To reﬁne the stable twisted conjugacy class (L±i, Li,yi,J) to a twisted conju- × × ։ × × gacy class, is to lift yi under the mapping Li /NLi/L±i (Li ) Li /L±i. There are #L× /N (L×) = 2 choices for such a lift for each i J. We get 2#J choices to ±i Li/L±i i ∈ reﬁne the collection (y ) . Finally we also have to choose an element y F ×/F ×2. i i∈J D ∈ In total we have F ×/F ×2 2#J twisted conjugacy classes inside the stable twisted | | conjugacy class. Thus if we take C = F ×/F ×2 −1 the lemma follows. | |

24 To avoid a collision of notation in our exposition, we write L±i, Li, yi, J, yD where Waldspurger writes F±i, Fi, xi, I, xD. Galois representations for general symplectic groups 75

We also need Lefschetz functions on the group GSp2n(F ) and its nontrivial inner form G. Unfortunately Proposition A.1 does not apply since the center of GSp2n(F ) is not compact. We follow Labesse to construct Kottwitz functions generally for an arbitrary reductive group G over F . Let A denote the maximal split torus in the center of G. Deﬁne ν : G(F ) X∗(A) R as in [Lab99, 3.9] and put G(F )1 := ker ν. → ⊗Z Consider the exact sequence

1 A G ς G′ := G/A 1 → → → → and a Lefschetz function f ′ (G′) as in Proposition A.1. The pullback f ′ ς is Lef ∈ H Lef ◦ a function on G(F ). Multiplying with the characteristic function 1G(F )1 , we obtain

G ′ f = f := 1 1 (f ς) (G(F )). (A.4) Lef Lef G(F ) · Lef ◦ ∈ H Let G∗ denote a quasi-split inner form of G. Suppose that the Haar measures on G′(F ) and (G∗)′(F ) are chosen compatibly in the sense of [Kot88, p.631]. Given a Haar measure on A(F ) this determines Haar measures on G(F ) and G∗(F ). Let us normalize the transfer factor between G and G∗ to be (whenever nonzero) the Kottwitz sign e(G), which is equal to ( 1)q(G∗)−q(G) by [Kot83, pp.296-297]. − Lemma A.4. With the above choices, ( 1)q(G)f G and ( 1)q(G∗)f G∗ are associated. − Lef − Lef Remark A.5. Our lemma speciﬁes the constant in [Lab99, Prop. 3.9.2] when θ =1 and H = G∗.

Proof. The proof is quickly reduced to the case when the center is anisotropic, the point being that G(F )1 and G∗(F )1 are invariant under conjugation. From Proposi- G G tion A.1 we deduce that SOγ (fLef) is zero if γ is non-elliptic and equals the number of G(F )-conjugacy classes in the stable conjugacy class of γ if γ is elliptic. Since the same is true for G∗ it is enough to show that the number of conjugacy classes in a stable conjugacy class is the same for γ and γ∗ when they are strongly regular and have matching stable conjugacy classes. This follows from the p-adic case in the proof of [Kot88, Thm. 1].

Deﬁnition A.6 ([Lab99, Def. 3.8.1, 3.8.2]). Let φ (G(F )). We say that φ is ∈ H cuspidal if the orbital integrals of φ vanish on all regular non-elliptic semisimple elements, and strongly cuspidal if the orbital integrals of φ vanish on all non-elliptic elements and if the trace of φ is zero on all constituents of induced representations from unitary representations on proper parabolic subgroups. The function φ is said to be stabilizing if φ is cuspidal and if the κ-orbital integrals of φ vanish on all semisimple elements for all nontrivial κ. 76 Arno Kret, Sug Woo Shin

Lemma A.7. The function f G is strongly cuspidal and stabilizing. If Tr π(f G ) =0 Lef Lef 6 for an irreducible unitary representation π of G(F ) then π is an unramiﬁed character twist of either the trivial or the Steinberg representation.

G Proof. The ﬁrst assertion is [Lab99, Prop. 3.9.1]. By construction fLef is constant 1 G 1 G on Z(F ) G(F ) , which is compact. Since Tr π(fLef) is the sum of Tr π (fLef) over ∩ 1 1 1 G irreducible constituents π of π 1 , we may choose π such that Tr π (f ) = 0. |G(F ) Lef 6 The nonvanishing implies that π1 has trivial central character on Z(F ) G(F )1. Let ∩ ω denote the central character of π. Then ω is trivial on the subgroup A(F ) π π ∩ G(F )1 A(F ), and hence induces a morphism ⊂ ω : A(F )/(A(F ) G(F )1) C×. π|A(F ) ∩ → We have a short exact sequence

1 G(F )1 G(F ) G(F )/G(F )1 1 → → → → 1 ∗ ∗ and G(F )/G(F ) is a lattice of X (A)R via ν. The image of A(F ) in X (A)R, namely A(F )/A(F ) G(F )1, is a sublattice of G(F )/G(F )1 since the map clearly factors ∩ through ν : G(F ) X(A) . Thus A(F )G(F )1 is a subgroup of G(F ) with a ﬁnite → R abelian quotient. We can think of A(F )G(F )1/A(F ) = G(F )1/A(F ) G(F )1 as a ∩ subgroup of G(F )/A(F )= G′(F ) with a ﬁnite abelian quotient as well. Since C× is divisible, we can extend ω to a character ω : G(F )/G(F )1 π|A(F ) → C×. Twisting by ω−1, we may assume that ω is trivial. Let us write f for f G . π|A(F ) Lef Now we compute

Tr π(f) = θπ(ag)f(ag)dadg Zg∈G(F )/A(F ) Za∈A(F ) 1 = vol(A(F ) G(F ) ) θπ(g)f(g)dg ∩ 1 1 ZG(F ) /A(F )∩G(F ) 1 ′ = vol(A(F ) G(F ) ) θπ(g)f (g)dg, ∩ 1 1 ZG(F ) /A(F )∩G(F ) where π is viewed as a representation of G′(F ) (since ω = 1), in which π|A(F ) G(F )1/A(F ) G(F )1 is a finite-index subgroup, and where f ′ is a Lefschetz function ∩ for G′(F ). Notice that the integral after the second equality is well defined as f and π are invariant under A(F ) G(F )1. ∩ Write X for the finite set of characters of G(F )/A(F )G(F )1 = G′(F )/G(F )1. They are unramified characters of G′(F ). Then

1 −1 ′ Tr π(f) = vol(A(F ) G(F ) ) X χ(g)θπ(g)f (g)dg ∩ | | ′ χ∈X G (F ) X Z Galois representations for general symplectic groups 77

= vol(A(F ) G(F )1) X −1 Tr (π χ)(f ′). ∩ | | ⊗ χ∈X X Thus if Tr π(f) = 0 then π χ is the trivial or Steinberg representation of G′(F ). 6 ⊗ We conclude that π is the trivial or Steinberg representation of G(F ) up to an unramiﬁed twist.

When the center of G is not compact, it is also convenient to work with functions with ﬁxed central character. We adopt the notation and convention from 6. § We record a result in the following, where a representation is said to be essentially unitary if its character twist is unitary.

× Corollary A.8. Let η : G(F ) C be a character, and write ω := η A(F ). Deﬁne a G −1→ G −1 G/A | function fLef,η (G(F ),ω ) by fLef,η(g) := η (g)fLef (g), where g G(F )/A(F ) ∈ H G ∈ denotes the image of g under the quotient map. Then fLef,η has the following properties:

G 1. fLef,η is stabilizing (in particular cuspidal).

2. Let π be an irreducible essentially unitary G(F )-representation whose central character on A(F ) coincides with ω. Then Tr (f G , π)=0 unless π Lef,η ∈ η, St η . { G(F ) ⊗ } 3. f G (zg)= η−1(g)f G (g) for all z Z (F ) and g G(F ). Lef,η Lef,η ∈ G ∈ Proof. Let us write G := G/A in this proof. The second point quickly follows from the analogue for f G on G(F ) = G(F )/A(F ). Let us show the ﬁrst point. Let g Lef ∈ G(F ) be a semisimple element, and g G(F ) its image. Write I (resp. I) for the ∈ connected centralizer of g in G (resp. g in G). Then the natural map I I induces → I/A I, as in the proof of [Kot82, Lem. 3.1(1)]. Equipping quotient measures on I ≃ and G with respect to those on A, I, and G, we have

G G G Og(fLef,η)= η(g)Og(fLef,1)= η(g)Og(fLef). (A.5)

G G This implies that fLef,η is cuspidal since fLef is cuspidal. κ G It remains to verify that the κ-orbital integrals Og (fLef,η) vanish for each semisimple g and κ = 1. The ﬁrst observation is that I ։ I induces a canonical isomorphism 6 K(I/F ) ∼ K(I/F ) in the notation of [Kot86, 4]. To see this, one can dually check → § that the induced map E(I/F ) E(I/F ) (using the notation therein) is an iso- → morphism. This follows from the following commutative diagram with exact rows, coming from the long exact sequence associated with 0 A I I 0, since → → → → 78 Arno Kret, Sug Woo Shin

E(I/F ) and E(I/F ) are the kernels of the ﬁrst and second vertical maps, respectively. (We have used Hilbert 90 for the split torus A.)

0 / H1(F,I) / H1(F, I) / H2(F, A)

0 / H1(F,G) / H1(F, G) / H2(F, A)

The second observation is that the map G ։ G induces a surjection from the set of conjugacy classes in the stable conjugacy class of g onto the analogous set for g, where the cardinality of ﬁbers remains constant, as explained on pp. 611–612 of [Hai09]. κ G From these two observations combined with (A.5), we deduce that Og (fLef,η)=0 κ G if and only if Og (fLef) = 0 for each κ = 1 (viewed in either K(I/F ) or K(I/F ) via G 6 κ G the isomorphism above). Since fLef is stabilizing, Og (fLef) vanishes, therefore so does κ G Og (fLef,η). G/A The last assertion of the corollary boils down to the statement that fLef (gz)= G/A fLef (g) for all g G/A(F ) and z ZG/A(F ). This follows from Harish-Chandra’s ∈ ∈ G/A Plancherel theorem [Wal03, Thm. VIII.1.1 (3)] (applied with f = fLef ) since the G/A right hand side of that theorem does not change when fLef is translated by z; the point is that the only irreducible tempered representation of (G/A)(F ) with G/A nonzero trace against fLef is the Steinberg representation, which has the trivial central character.

Assume from now on that G is a non-split inner form of a quasi-split group G∗ over F . Consider a ﬁnite cyclic extension E/F with θ generating the Galois group. ˜∗ ∗ ∗ Put G := ResE/F G equipped with the evident θ-action. (The case G = GSp2n is used in the main text.)

G˜∗ Lemma A.9. The function fLef is strongly cuspidal and stabilizing on the twisted group G˜∗θ. There exist constants c C× such that the functions cf˜ G∗ and f G˜∗ are ∈ Lef Lef associated (via base change).

Proof. [Lab99, Prop. 3.9.2]. (Take θ = 1, H = G∗ in loc. cit. for the ﬁrst assertion.)

Now we turn our attention to Lefschetz functions (also called Euler-Poincar´e functions) on connected reductive groups G over F = R. We assume that G(R) has discrete series representations. Let ξ be an irreducible algebraic representation of G C. Denote by χ : Z(R) C× the restriction of ξ∨ to the center Z(R). ×R ξ → Galois representations for general symplectic groups 79

Write f = f G (G(R), χ−1) for an Euler-Poincar´efunction associated to ξ, ξ ξ ∈ H ξ characterized by the identity

∞ Tr π(f ) = ep (π ξ) def= ( 1)i dim Hi(Lie G(R),K ; π ξ) ξ K∞ ⊗ − ∞ ⊗ i=0 X for every irreducible admissible representation π of G(R) with central character χξ, where K∞ is a ﬁnite index subgroup in the group generated by the center and a maximal compact subgroup of G(R). (A main example for us is K˜H in Deﬁnition

1.12.) The formula does not determine fξ uniquely but the orbital integrals of fξ are well defined. The function fξ exists by Clozel and Delorme and can be constructed as G the average of pseudo-coefficients as follows. Let Πξ denote the L-packet consisting of (the isomorphism classes of) irreducible discrete series representations π of G(R) whose central character and infinitesimal character are the same as ξ∨. Write f π ∈ (G(R), χ−1) for a pseudo-coefficient of π, and q(G) Z for the real dimension of H ξ ∈ G(R)/K∞. Then fξ can be defined by

f G := ( 1)q(G) f . (A.6) ξ − π π∈ΠG Xξ Let G∗ denote a quasi-split inner form of G over R. Then the same ξ gives rise G∗ to the functions fξ . Note that Deﬁnition A.6 makes sense verbatim when F is archimedean and the function has central character.

Lemma A.10. Any pseudo-coeﬃcient fπ for a discrete series representation π is G G∗ cuspidal. In particular fξ and fξ are cuspidal.

Proof. It suﬃces to check that fπ is cuspidal, or equivalently that the trace of every induced representation from a proper Levi subgroup vanishes against fπ, cf. [Art88a, p.538], but this is true by the construction of pseudo-coeﬃcients. (The cuspidality of fπ also follows from the proof of [Kot92a, Lem. 3.1].)

Let A denote the maximal split torus in the center of G (hence also in G∗). Equip G(R)/A(R) and G∗(R)/A(R) with Euler-Poincarémeasures and A(R) with the Lebesgue measure so that the Haar measures on G(R) and G∗(R) are determined. Define q(G) (resp. q(G∗)) to be the real dimension of the symmetric space associated to the derived subgroup of G(R) (resp. G∗(R)). Normalize the transfer factor between G(R) and G∗(R) to be e(G)=( 1)q(G∗)−q(G). Write ΠG (resp. ΠG∗ ) for the discrete − ξ ξ series L-packet for G(R) (resp. G∗(R)) associated to ξ, cf. Example below Definition 1.12.

Lemma A.11. The functions ( 1)q(G) ΠG −1f G and ( 1)q(G∗) ΠG∗ −1f G∗ are associ- − | ξ | ξ − | ξ | ξ ated. 80 Arno Kret, Sug Woo Shin

Proof. This follows from the computation of stable orbital integrals in [Kot92a, Lem. 3.1]; see also [CL11, Prop. 3.3] when A = 1.

A similar construction works in the base change context, cf. [CL11]. We are only concerned with a particular case that G˜∗ = G∗ G∗ G∗ (the number of × ×···× copies is d Z≥1) and θ is the automorphism (g1, ..., gd) (g2, ..., gd,g1). Write ∈ ∗ −1 7→ ξ˜ := ξ ξ. A function f˜ (G˜ (R), χ ) is said to be a (twisted) Lefschetz ⊗···⊗ ξ ∈ H ξ function for ξ˜ if

∞ i i d d Trπ ˜(f˜)= ( 1) Tr (θ H Lie G(R) ,K ; π ξ ξ − | ∞ ⊗ i=0 X for every irreducible admissible representation π of G(R) with central character χξ. Deﬁnition A.6 carries over to this base change setup as in [Lab99, Def. 3.8.1, 3.8.2].

25 G∗ Lemma A.12. The function fξ˜ is cuspidal. Moreover fξ˜ and cf˜ ξ are associated for some c˜ C×. ∈ Proof. The ﬁrst assertion is [CL11, Prop. 3.3]. The second assertion follows from [CL11, Prop. 3.3, Thm. 4.1]. (The reference assumes that A = 1 but the arguments can be adapted to the case of nontrivial A as in the untwisted setup above).

We end this appendix with a global result. We change notation. Let F be a totally real number ﬁeld and vSt a ﬁnite F -place. Let G be an inner form of either the group

GSp2n,F or the group Sp2n,F .

Lemma A.13. Assume that n> 1. Let π be a non-abelian discrete automorphic rep- G resentation of G(AF ), and assume that Tr πv (f ) =0. Then πv is an unramiﬁed St vSt 6 St twist of the Steinberg representation. e Remark A.14. The lemma is false for n = 1.

Proof. We give the argument only in case G is an inner form of GSp2n,F ; the argu- G ment for inner forms of Sp is similar. By the assumption Tr πv (f ) = 0, πv is 2n,F St vSt 6 St either a twist of the Steinberg representation or a twist of the trivial representation (Lemma A.7). We assume that we are in the second case; after twistinge we may assume that π is the trivial representation. Let G G be the kernel of the fac- vSt 1 ⊂ tor of similitudes. By strong approximation the subset G (F )G (F ) G (A ) is 1 1 vSt ⊂ 1 F dense if G1,F is non-anisotropic. Let us assume this for now. Let f π. Since πv vSt ∈ St 25 1 When H (R, Gsc) = 1 (assuming A = 1), Clozel and Labesse prove that fξ˜ is also stabilizing in [Lab99, Thm. A.1.1]. However we do not use it in the main text but instead appeal to the fact that the (twisted) Lefschetz function at a ﬁnite place is stabilizing. Galois representations for general symplectic groups 81

is the trivial representation, f is invariant under G(FvSt ). Thus f is G1(F )G1(FvSt )- invariant, and hence G1(AF ) invariant. This implies that π is abelian. It remains to check that G1,Fv is non-anisotropic. In the split case, we have G1,Fv Sp2n,F . St St ≃ vSt In the non-split case, the group G is of the following form. Let D/F be the 1,FvSt vSt quaternion algebra, and consider the involution on D deﬁned by x = Tr(x) x where − Tr is the reduced trace. Then G(1, F ) Sp (D) is the group of g GL (D) such vSt ≃ n ∈ n that gA tg = c(g)A for some c(g) F × , where A is the matrix with 1’s on the n n ∈ vSt n anti-diagonal, and 0’s everywhere else [PR94, item (3), p.92]. For n > 1 the group Sp (D) has a strict parabolic subgroup and thus G is not anisotropic. n 1,FvSt

B Conjugacy in the standard representation

Let V be a ﬁnite dimensional Q -vector space and , a non-degenerate, symmetric ℓ h· ·i or skew-symmetric bilinear form on V . Let H GL(V ) be the subgroup of elements ⊂ that preserve , (resp. preserve it up to scalar). Thus, H(Qℓ) is either an orthogonal h· ·i × group or a symplectic group (resp. of similitude). Write sim: H(Q ) Q for the ℓ → ℓ factor of similitude. Note sim is non-trivial only for the groups GO(V, , ) and h· ·i GSp(V, , ). h· ·i The results of this appendix largely overlap with [Wan15, Prop A (and Prop 2.1)], although he works in a slightly diﬀerent setting.

Proposition B.1 (Larsen, cf. proof of [Lar94b, Prop. 2.3, Prop. 2.4]). Let Γ be a topological group. Consider two (continuous) representations φ ,φ : Γ H(Q ) 1 2 → ℓ such that φ is semisimple. Then φ ,φ are H(Q )-conjugated if and only if sim φ = 1 1 2 ℓ ◦ 1 sim φ and std φ , std φ are GL (Q )-conjugate. ◦ 2 ◦ 1 ◦ 2 m ℓ Remark B.2. The conclusion of Proposition B.1 fails in general for the special orthogonal group in even dimension. In odd dimension the group O2n+1(Qℓ) equals 1 SO (Q ) and so the proposition is true for SO (Q ) as well. {± } × 2n+1 ℓ 2n+1 ℓ Proof of Proposition B.1. We consider the group H = GO(V, , ) with , sym- h· ·i h· ·i metric and non-degenerate, the other groups are treated in a similar fashion. Fix a × morphism φ: Γ GO(V, , ) with std φ semisimple. Write χ = sim φ: Γ Q . → h· ·i ◦ ◦ → ℓ Consider the set X (φ) of GO(V, , )-conjugacy classes of morphisms φ′ : Γ χ h· ·i → GO(V, , ) such that std φ′ std φ and χ = sim φ′. We view the space V as h· ·i ◦ ≃ ◦ ◦ a Γ-representation via φ. We have the injection

X (φ) ֌ Isom (V,V ∗ χ)/Aut (V ), [φ′, ρ: std φ′ ∼ std φ] ρ , , χ Γ ⊗ Γ ◦ →Γ ◦ 7→ ∗h· ·i whose image is the set of non-degenerate symmetric pairings taken modulo AutΓ(V ), where the automorphisms σ Aut (V ) act on Isom (V,V ∗ χ) via ρ (σ∗ ) ρ ∈ Γ Γ ⊗ 7→ · ◦ ◦ 82 Arno Kret, Sug Woo Shin

(σ ). Using the pairing , on V we can further identify, · h· ·i Isom (V,V ∗ χ)/Aut (V ) ∼ Aut (V )/Aut (V )-congruence, Γ ⊗ Γ → Γ Γ where by congruence we mean the action τ σtτσ for σ Aut (V ); here the trans- 7→ ∈ Γ pose is deﬁned using , . Since (V,φ) is semisimple we can consider the isotypical h·t ·i di t decomposition V = i=1 Vi , and so by Schur’s lemma EndΓ(V ) = i=1 Mdi (Qℓ). We obtain an embedding L Q t ֌ Xχ(φ) GLdi (Qℓ)/GLdi (Qℓ)-congruence, j=1 Y where two matrices X,Y GL (Q ) are GL (Q )-congruent if there exists a third ∈ di ℓ di ℓ matrix g GL (Q ) such that Y = gtXg. The image of X (φ) in the set on the right ∈ di ℓ χ hand side decomposes along the product and is in each GLdj (Qℓ)-factor equal to the

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