Periods of quaternionic Shimura varieties. I
aa r X i v : . [ m a t h . N T ] O c t PERIODS OF QUATERNIONIC SHIMURA VARIETIES. I.
ATSUSHI ICHINO AND KARTIK PRASANNA
Abstract.
We study “quadratic periods” on quaternionic Shimura varieties and formulate an in-tegral refinement of Shimura’s conjecture regarding Petersson inner products of automorphic formsthat are related by the Jacquet-Langlands correspondence. The main result is that this integralrefinement is implied by another conjecture (Conjecture D below) regarding integrality of theta liftsbetween certain quaternionic unitary groups.
Contents
Introduction 11. Quaternionic Shimura varieties 102. Unitary and quaternionic unitary groups 273. Theta correspondences 334. The Rallis inner product formula and the Jacquet–Langlands correspondence 445. Schwartz functions 536. Explicit form of the Rallis inner product formula 807. The main conjecture on the arithmetic of theta lifts 105Appendix A. Polarized Hodge structures, abelian varieties and complex conjugation 117Appendix B. Metaplectic covers of symplectic similitude groups 119Appendix C. Splittings: the case dim B V = 2 and dim B W = 1 123Appendix D. Splittings for the doubling method: quaternionic unitary groups 157References 171 Introduction
In this paper and its sequels [31], [32], we study periods of automorphic forms on quaternionicShimura varieties. Specifically, the periods that we focus on are the Petersson inner products ofHilbert modular forms and of their Jacquet–Langlands lifts to quaternionic Shimura varieties. Thissubject was pioneered by Shimura who proved many results on algebraicity of ratios of Petersson innerproducts and made a precise general conjecture ([68] Conjecture 5.10) that predicts a large number ofalgebraic relations in the Q -algebra generated by such periods. Shimura’s conjecture was proved byHarris [23] under a technical hypothesis on the local components of the corresponding automorphicrepresentation. This hypothesis was relaxed partly by Yoshida [78], who also used these period relationsto prove a refined conjecture of Shimura ([68] Conj. 5.12, [69] Conj 9.3) on the factorization of Petersson inner products into fundamental periods up to algebraic factors. In later papers [24] [25], Harris hasconsidered the question of generalizing such period relations to the setting of unitary Shimura varieties.Specialized to the case of hermitian spaces of dimension two, these latter results provide more preciseinformation about the fields of rationality of quadratic period ratios of quaternionic modular forms.In this series of papers, we will study the corresponding integrality questions. The simplest inter-esting case is the period ratio h f, f ih g, g i where f is a usual modular form of (even) weight 2 k (for GL over Q ) and trivial central character,and g is its lift to a Shimura curve corresponding to an indefinite quaternion algebra also over Q . Theforms f and g here are assumed to be newforms and to be suitably integrally normalized. In this case,there is a very precise rationality result due to Harris and Kudla [27] which asserts that the ratio abovelies in the field generated by the Hecke eigenvalues of f . As for the more refined integrality question,what is known is the following:(i) In the special case when the weight 2 k equals 2 and f corresponds to an isogeny class of ellipticcurves, it can be shown (see [58], § level-lowering congruences satisfied by the form f . This suggests that such period ratios contain rather deeparithmetic information. The proof in this case follows from combining three geometric ingredi-ents: • The work of Ribet on level-lowering [62] and its extension due to Ribet and Takahashi [63]which depend on a study of the geometry of Shimura curves, especially a description oftheir bad reduction and of the component groups of the N´eron models of their Jacobians. • The Tate conjecture for products of curves over number fields, which was proved by Faltings[15], and which implies that modular elliptic curves are equipped with a uniformization map X → E , with X being a Shimura curve. • A study of the
Manin constant for the map X → E , following [51], [13], [1].(ii) In the more general case of weight 2 k >
2, such geometric arguments are not available. Themain obstruction is that the Tate conjecture is unknown for products of Kuga–Sato varieties.Instead, one may try to use purely automorphic techniques. This is the strategy employed in[58], where we showed (using the theta correspondence and results from Iwasawa theory) thatfor f and g of arbitrary even weight, the ratio h f, f i / h g, g i is integral outside of an explicit finiteset of small primes, and further that it is always divisible by primes at which the form f satisfiescertain level-lowering congruences. The converse divisibility and the more precise relation toTamagawa numbers is also expected to hold in general, but seems harder to prove. This is oneproblem that we hope to eventually address by the methods of this paper.Let us now elaborate a bit on the relation of this problem to the Tate conjecture. As describedabove, the case of weight two forms for GL over Q is relatively simple since one knows by Faltingsthat the Tate conjecture holds for a product of curves. This implies that there exists an algebraic cycleon the product X × X , where X and X are modular and Shimura curves respectively, that at thelevel of cohomology, identifies the f and g -isotypic components of the “motives” H ( X ) and H ( X )respectively. The rationality of the period ratio h f, f i / h g, g i is then a simple consequence of the factthat such a cycle induces an isomorphism of the Hodge–de Rham structures [22] attached to f and g .For forms of higher weight, the Jacquet–Langlands correspondence can similarly be used to produceTate classes on a product W × W where W and W are Kuga–Sato varieties fibered over X and X respectively. However, we are very far from understanding the Tate (or even Hodge) conjecture in thiscase. The case of Hilbert modular forms considered in this paper is even harder: in the simplest setting,namely for forms of parallel weight two and trivial central character, the Tate conjecture predicts the existence of algebraic cycles on products of the form X × ( X × X ), where X , X and X are suitablychosen quaternionic Shimura varieties such that dim( X ) = dim( X ) + dim( X ). Again, these cyclesshould induce isomorphisms of Hodge–de Rham structures H ∗ ( X ) Π ≃ H ∗ ( X ) Π ⊗ H ∗ ( X ) Π that inturn should imply the predicted period relations up to rationality. (Here the subscript Π denotes theΠ f -isotypic component for a fixed automorphic representation Π = Π ∞ ⊗ Π f .) This point of view - atleast the factorization of Hodge structures - occurs explicitly in the work of Oda ([55], [56]). It is worthremarking here that the Tate and Hodge conjectures are only expected to hold rationally in generaland not integrally, and thus by themselves do not predict any statements about integrality of periodratios. Nevertheless, the discussion above suggests that in the setting of arithmetic automorphic formson Shimura varieties, such integral relations do hold and that their proofs lie much deeper than thoseof the corresponding rational relations.With this background, we will outline the main results of this paper. Let F be a totally real numberfield with [ F : Q ] = d , ring of integers O F , class number h F and discriminant D F . Let Π = ⊗ v Π v be an irreducible cuspidal automorphic representation of GL ( A F ) of weight ( k, r ) = ( k , . . . , k d , r ),conductor N and central character ξ Π . We assume that k ≡ k ≡ · · · ≡ k d ≡ r (mod 2) and allthe k i ≥
1. These are thus the automorphic representations associated with classical Hilbert modularforms. (Note that we allow forms of partial or parallel weight one.)For simplicity we will assume that at all finite places v where Π v is ramified, it is either a specialrepresentation with square-free conductor (i.e., an unramified twist of the Steinberg representation) ora ramified principal series representation Ind( χ ⊗ χ ) with χ unramified and χ ramified. We canthus factor the conductor N of Π as N = N s · N ps where N s is the (square-free) product of the conductors at places where Π v is special and N ps is theproduct of the conductors at places where Π v is ramified principal series.Let K Π be the number field generated by the Hecke eigenvalues of Π and O K Π the ring of integersof K Π . We set N Π := N N , k Π := max k i and N (Π) := 2 · h F · D F · N Π · k Π ! , R := O Q [1 /N (Π)] . Let Σ F denote the set of all places of F and Σ ∞ and Σ fin the subsets of infinite and finite placesrespectively. Let Σ Π be the set of places v of F at which Π v is discrete series. Thus, Σ Π equals theunion of Σ Π , ∞ and Σ Π , fin , whereΣ Π , ∞ := Σ Π ∩ Σ ∞ = { v ∈ Σ ∞ : k v ≥ } , Σ Π , fin := Σ Π ∩ Σ fin = { v ∈ Σ fin : ord v ( N s ) > } . For any quaternion algebra B over F , let Σ B denote the set of places of F at which B is ramified.Also set Σ B, ∞ := Σ B ∩ Σ ∞ , Σ B, fin := Σ B ∩ Σ fin . Henceforth we suppose that Σ B ⊂ Σ Π , so that by Jacquet–Langlands [35], Π transfers to an au-tomorphic representation Π B of B × ( A ). To such a pair ( B, Π), we will attach in Sec. 1.4 below acanonical quadratic period invariant q B (Π) ∈ C × /R × . This period invariant is essentially (i.e., up to some factors coming from normalizations of measures)equal to the Petersson inner product of a normalized eigenform f B in Π B . Here we use the assumptionthat N s is square-free to first fix f B up to a scalar. The scalar is then fixed by requiring that f B ATSUSHI ICHINO AND KARTIK PRASANNA correspond to an integrally normalized section of a suitable automorphic vector bundle on the Shimuravariety associated with the algebraic group B × .The goal of this paper and its sequels is to study the relations between the invariants q B (Π) forfixed Π as B varies over all quaternion algebras in Σ Π . The following conjecture is a more preciseversion of [59] Conjecture 4.2 and provides an integral refinement of Shimura’s conjecture on algebraicperiod relations. The reader may consult Sec. 4 of [59] for a discussion of the motivation behind thisformulation. To state the conjecture, let L ( s, Π , ad) denote the adjoint (finite) L -function attached toΠ and let Λ( s, Π , ad) denote the corresponding completed L -function that includes the Γ-factors at theinfinite places. Let us recall the following invariant of Π, which has played a crucial role in the studyof congruences of modular forms (see [29], [30], [75]):(I.1) Λ(Π) := Λ(1 , Π , ad) . Conjecture A.
There exists a function c (Π) : Σ Π → C × /R × , v c v (Π) , such that: (i) c v (Π) lies in R (mod R × ) if v is a finite place, and (ii) for all B with Σ B ⊆ Σ Π , we have q B (Π) = Λ(Π) Q v ∈ Σ B c v (Π) ( in C × /R × ) . Remark 1.
It is easy to see that if it exists, the function c (Π) is uniquely determined as long as | Σ Π | ≥
3. Also, as the notation suggests, the invariants c v (Π) are not invariants of the local representationΠ v but rather are really invariants of the global representation Π. Remark 2.
The conjecture should be viewed as describing period relations between the quadraticperiods q B (Π) as B varies. Indeed, the number of B with Σ B ⊆ Σ Π is 2 | Σ Π |− but the conjecturepredicts that the corresponding invariants q B (Π) can all be described using only | Σ Π | + 1 invariants,namely the L -value Λ(Π) and the additional invariants c v (Π), which are | Σ Π | in number. Remark 3.
For B = M ( F ), the conjecture simply predicts that q M ( F ) (Π) = Λ(Π) in C × /R × . This piece of the conjecture is known to be true. Indeed, it follows from the fact that the integralnormalization of f B in the split case coincides with the q -expansion normalization (see [11], § f ∈ Π and the value of the adjoint L -function at s = 1. (See Prop. 6.6 for instance.)It is natural to ask for an independent description of the invariants c v (Π). Before discussing this,we recall the notion of Eisenstein primes for Π. To any finite place λ of K Π , one can associate (by[67], [9], [12], [65], [5], [74], [71], [37]; see also [4]) an irreducible two dimensional Galois representation ρ Π ,λ : Gal( Q /F ) → GL ( K Π ,λ )that is characterized up to isomorphism by the requirement thattr ρ Π ,λ (Frob v ) = a v (Π)for any finite place v of F that is prime to N · N λ , with a v (Π) being the eigenvalue of the Heckeoperator T v acting on a new-vector in Π v . Choose a model for ρ Π ,λ that takes values in GL ( O K Π ,λ ) and denote by ¯ ρ Π ,λ the semisimplification of the mod λ reduction of ρ Π ,λ . The isomorphism class of¯ ρ Π ,λ is independent of the choice of model of ρ Π ,λ . Let F λ = O K Π /λ be the residue field at λ . Theprime λ is said to be Eisenstein for Π if¯ ρ Π ,λ : Gal( Q /F ) → GL ( F λ )is (absolutely) reducible, and non-Eisenstein otherwise.Let N (Π) Eis be the product of the N λ as λ varies over all the Eisenstein primes for Π. (There areonly finitely many such.) Let ˜ R denote the ring˜ R := R [1 /N (Π) Eis ] = O Q [1 /N (Π) N (Π) Eis ] . The following conjecture characterizes the invariants c v (Π) for finite places v up to Eisenstein primes,relating them to level-lowering congruences for Π. (It is obviously conditional on the truth of ConjectureA.) Conjecture B.
Suppose that v belongs to Σ Π , fin . Let L ⊇ K Π be a number field containing (arepresentative of ) c v (Π) and let ˜ λ be a finite place of L such that (˜ λ, N (Π)) = 1 . Let λ be the place of K Π under ˜ λ and suppose that Π is not Eisenstein at λ . Then v ˜ λ ( c v (Π)) equals the largest integer n such that ρ Π ,λ mod λ n is unramified at v . At the infinite places v , one might hope to have similarly a description of the invariants c v (Π) purelyin terms of the compatible system ρ Π ,λ of two-dimensional Galois representations attached to Π. Inprinciple, to such a system one should be able to attach a motive defined over F , and the c v (Π) shouldbe related to periods of this motive taken with respect to suitable integral structures on the de Rhamand Betti realizations. In practice, the only case in which one can make an unconditional definition iswhen Π satisfies the following conditions:(a) Π is of parallel weight 2, that is k = (2 , . . . , d (= [ F : Q ]) is odd or Σ Π , fin is nonempty.If Π satisfies both (a) and (b) above, it is known (using [5]) that one can associate to Π an abelianvariety A over F (or more precisely, an isogeny class of abelian varieties) such that • dim( A ) = [ K Π : Q ]; • End F ( A ) ⊗ Q ⊃ K Π ; • A has good reduction outside N ; • For any prime λ of K Π lying over a rational prime ℓ , the representation ρ Π ,λ is isomorphic to therepresentation of Gal( Q /F ) on H ( A Q , Q ℓ ) ⊗ K Π ⊗ Q ℓ K Π ,λ .We may pick in the isogeny class above an abelian variety A such that End F ( A ) ⊃ O K Π . Then one canmake a precise conjecture for c v (Π) for v ∈ Σ ∞ in terms of the periods of A . Here, we will be contentto state this conjecture in the case K Π = Q , namely when A is an elliptic curve over F . Let A denotethe N´eron model of A over R F := O F [1 /N (Π)]. Then L := H ( A , Ω A /R F ) is an invertible R F -module.This module can be trivialized by picking a large enough number field K ⊇ F and extending scalarsto the ring R K := O K [1 /N (Π)]. Pick a generator ω for L ⊗ R F R K viewed as an R K -module. Let v ′ be any archimedean place of K extending v , and denote by σ v : F → R the real embedding of F corresponding to v . The class of the integral1(2 πi ) Z A ⊗ σv C ω v ′ ∧ ¯ ω v ′ in C × / ˜ R × can be checked to be independent of the choices above. ATSUSHI ICHINO AND KARTIK PRASANNA
Conjecture C.
Suppose that K Π = Q so that A is an elliptic curve. Then c v (Π) = 1(2 πi ) Z A ⊗ σv C ω v ′ ∧ ¯ ω v ′ in C × / ˜ R × . Remark 4.
One expects that the invariants c v (Π) are transcendental for any infinite place v . Notethat if A is the base change of an elliptic curve defined over a smaller totally real field F ′ (in whichcase Π is the base change of a Hilbert modular form for F ′ ), then there are obvious algebraic relationsbetween the c v (Π). It would be interesting to formulate a converse to this: namely, can one give acriterion for Π to be a base change purely in terms of the Q -algebra generated by the invariants c v (Π)? Remark 5.
It would also be interesting to formulate the conjectures above without inverting N (Π).There are lots of obvious difficulties with primes that are small with respect to the weight as wellas with integral models at primes of bad reduction. In [32], we will extend the conjectures above inthe case F = Q to include primes of bad reduction at which the local component of the automorphicrepresentation Π is ramified principal series . The only Shimura varieties that occur then are Shimuracurves and those associated with definite quaternion algebras over Q . The geometric difficulties withprimes of bad reduction can be dealt with in this case “by hand”.The goal of this first paper is to reformulate Conjecture A in terms of a new conjecture (ConjectureD below) on the arithmetic properties of a theta lift between quaternionic Shimura varieties. Thisreformulation has many advantages since the arithmetic of theta lifts can be studied via a range ofautomorphic techniques including the Rallis inner product formula and period integrals along tori.Moreover, the constructions involved seem to be useful in attacking several other related problemsinvolving algebraic cycles. We will briefly discuss two such applications below.Now we outline the main construction. Let B , B and B be three quaternion algebras in Σ Π suchthat B = B · B in the Brauer group of F . There is then, up to isometry, a unique skew-hermitian B -space ( V, h· , ·i ) such that GU B ( V ) ≃ ( B × × B × ) /F × . Here GU B ( V ) denotes the identity component of the group of quaternionic unitary similitudes of V .For computational purposes, we will need an explicit construction of such a space V . For this, we picka CM extension E/F with E = F + F i , i = u ∈ F × , such that E embeds in B and B . Fix embeddings E ֒ → B , E ֒ → B and write B = E + E j , B = E + E j , where j = J and j = J lie in F . Then there is an embedding of E in B such that B = E + E j , j = J, where J = J J . Let V = B ⊗ E B , viewed as a right E -vector space. In Chapter 2 below, we showthat V can naturally be equipped with a right B -action extending the action of E as well as a B -skewHermitian form h· , ·i such that the quaternionic unitary similitude group GU B ( V ) has the form above.Let W be a one-dimensional B -space equipped with the standard B -hermitian form so thatGU B ( W ) = B × . We wish to study the theta lift Θ : A (GU B ( W )) −→ A (GU B ( V ) ) , where A denotes the space of cuspidal automorphic forms. The pair (U B ( W ) , U B ( V )) is an example ofa classical reductive dual pair. For our applications we need to work with the corresponding similitudegroups. In order to construct the theta lift, one needs to first construct (local) splittings of themetaplectic cover over the subgroup { ( g, h ) ∈ GU B ( V ) × GU B ( W ) : ν ( g ) = ν ( h ) } , that satisfy the product formula. (Here ν denotes the similitude character.) For quaternionic unitarysimilitude groups, this does not seem to be covered in the existing literature. This problem is handledin the appendices under the assumption that u , J and J are chosen such that for every finite place v of F , at least one of u , J , J an J is locally a square. (See Remark 7 below.)The splittings being chosen, the correspondence Θ above can be defined and studied. For anyquaternion algebra B ′ with Σ B ′ ⊆ Σ Π , we let π B ′ denote the unitary representation associated withΠ B ′ . Thus π B ′ = Π B ′ ⊗ k ν B ′ k − r/ , where ν B ′ denotes the reduced norm on B ′ . In Chapter 4, we prove the following theorem regarding Θ(in the case B = M ( F )) which gives an explicit realization of the Jacquet–Langlands correspondence,extending the work of Shimizu [66]. Theorem 1. Θ( π B ) = π B ⊠ π B . Remark 6.
Up to this point in the paper, we make no restrictions on F or Π. However from Chapter5 onwards (and thus in the rest of the introduction), we assume for simplicity the following: • N is prime to 2 D F/ Q , where D F/ Q denotes the different of F/ Q .These assumptions simplify some of the local computations in Chapters 5 and 6, and could be relaxedwith more work.While Theorem 1 is an abstract representation theoretic statement, for our purposes we need tostudy a more explicit theta lift. The Weil representation used to define the theta lift above is realizedon a certain Schwartz space S ( X ). In Chapter 5, we pick an explicit canonical Schwartz function ϕ ∈ S ( X ) with the property that θ ϕ ( f B ) is a scalar multiple of f B ⊠ f B . Thus(I.2) θ ϕ ( f B ) = α ( B , B ) · ( f B × f B ) , for some scalar α ( B , B ) ∈ C × . The scalar α ( B , B ) depends not just on B and B but also onthe other choices made above. However, we will omit these other dependencies in the notation.That α ( B , B ) is nonzero follows from the following explicit version of the Rallis inner productformula, proved in Chapter 6. (The assumption B = M ( F ) in the statement below is made since theproof in the case B = M ( F ) would be somewhat different. See for instance § Theorem 2.
Suppose B = B , or equivalently, B = M ( F ) . Then | α ( B , B ) | · h f B , f B i · h f B , f B i = C · h f, f i · h f B , f B i , where C is an explicit constant (see Thm. 6.7) and f is a Whittaker normalized form in Π (as inRemark 3). The arithmetic properties of α ( B , B ) are of key importance. As such, the choice of measuresneeded to define the invariants q B (Π) requires us to work with a slight modification of α ( B , B ),denoted α ( B , B ), as described in § ATSUSHI ICHINO AND KARTIK PRASANNA α ( B , B ) for which we may work one prime at a time. Thus we fix a prime ℓ not dividing N (Π)and then choose all the data (for example, E , J , J , ϕ ) to be suitably adapted to ℓ . The choices aredescribed in detail in Sec. 7.1. Finally, we come to main conjecture of this paper, which is motivatedby combining Theorem 2 with Conj. A. Conjecture D.
Suppose that B = B and Σ B ∩ Σ B ∩ Σ ∞ = ∅ , that is B and B have no infiniteplaces of ramification in common. Then (i) α ( B , B ) lies in Q × . (ii) α ( B , B ) is integral at all primes above ℓ . (iii) If in addition B and B have no finite places of ramification in common, then α ( B , B ) is aunit at all primes above ℓ . While not immediately apparent, Conjecture D implies Conjecture A. Indeed, in Sec. 7.2, we showthe following.
Theorem 3.
Suppose that Conjecture D is true for all ℓ in some set of primes Ξ . Then ConjectureA holds with R replaced by R [1 /ℓ : ℓ Ξ] . Consequently, if Conjecture D is true for all ℓ ∤ N (Π) , thenConjecture A is true. At this point, the reader may feel a bit underwhelmed since all we seem to have done is reformulateConjecture A in terms of another conjecture that is not visibly easier. However, we believe thatConjecture D provides the correct perspective to attack these fine integrality questions about periodratios, for several reasons. Firstly, it does not require an a priori definition of the invariants c v . Second,it fits into the philosophy that theta lifts have excellent arithmetic properties and is amenable to attackby automorphic methods of various kinds. Lastly, it is usually a very hard problem (in Iwasawa theory,say) to prove divisibilities ; on the other hand, if a quantity is expected to be a unit, then this might beeasier to show, for instance using congruences. Part (iii) of Conjecture D, which states that α ( B , B )is often a unit, has hidden in it a large number of divisibilities that would be very hard to show directly,but that might be more accessible when approached in this way. This is the approach taken in thesequels [31] and [32] where we study Conjecture D and give various applications to periods.As mentioned earlier, the constructions discussed above also have concrete applications to problemsabout algebraic cycles. We mention two articles in progress that rely crucially on this paper: • In [33], we study the Bloch–Beilinson conjecture for Rankin–Selberg L -functions L ( f E , χ, s ), where f is a modular form of weight k and χ is a Hecke character of an imaginary quadratic field E of infinity type ( k ′ ,
0) with k ′ ≥ k . The simplest case is when ( k, k ′ ) = (2 , L -function and provea relation between the p -adic logarithms of such cycles and values of p -adic L -functions. (Allprevious constructions of cycles for such L -functions ([19], [54], [3]) only work in the case k > k ′ .)The key input from this paper is the embeddingGU B ( V ) → GU E ( V )which provides a morphism of Shimura varieties that can be used to construct the relevant cycle. • In [34] we consider the Tate conjecture for products X × X where X and X are the Shimuravarieties associated with two quaternion algebras B and B over a totally real field F that haveidentical ramification at the infinite places of F . As explained earlier, the Jacquet–Langlandscorrespondence gives rise to natural Tate classes on X × X and the Tate conjecture predicts theexistence of algebraic cycles on the product giving rise to these Tate classes. While we cannot asyet show the existence of such cycles, we are able to at least give an unconditional construction of the corresponding Hodge classes. Moreover, these Hodge classes are constructed not by comparingperiods but rather by finding a morphism X × X → X into an auxiliary Shimura variety X and constructing Hodge classes on X that restrict nontriviallyto X × X . Thus we reduce the Tate conjecture on X × X to the Hodge conjecture on X whichshould in principle be an easier problem. The relation with the current paper is that X × X and X may be viewed as the Shimura varieties associated with certain skew-hermitian B spaces,with B = B · B .Finally, we give a brief outline of the contents of each chapter. In Chapter 1 we recall the theoryof automorphic vector bundles on quaternionic Shimura varieties and define the canonical quadraticperiod invariants q B (Π). In Chapter 2, we give the key constructions involving quaternionic skew-hermitian forms. Chapter 3 discusses the general theory of the theta correspondence as well as thespecial case of quaternionic dual pairs, while Chapter 4 establishes the general form of the Rallis innerproduct formula in our situation and proves that the theta lift we are considering agrees with theJacquet–Langlands correspondence. In Chapter 5, we pick explicit Schwartz functions, which are thenused in Chapter 6 to compute the precise form of the Rallis inner product formula in our setting. InChapter 7 we first discuss all the choices involved in formulating the main conjecture, Conjecture Dabove, and then show that it implies Conjecture A. Appendix A is strictly not necessary but is usefulin motivating some constructions in Chapter 1. The results from Appendix B on metaplectic coversof symplectic similitude groups are used in the computations in Appendix C. Appendices C and D areinvoked in Chapters 3, 4 and 5, and contain the construction of the relevant splittings, on which moreis said in the remark below. Remark 7.
The problem of constructing the required splittings and checking various compatibilitiesinvolving them turns out to be rather nontrivial and occupies the lengthy Appendices C and D. For isometry groups , these can be handled using the doubling method as in Kudla [39, § v ) s Kudla ,v : U B ( V )( F v ) × U B ( W )( F v ) → C (1) that satisfy the product formula : Y v s Kudla ,v ( γ ) = 1for γ ∈ U B ( V )( F ) × U B ( W )( F ). The problem is really to extend these splittings to the groups { ( g, h ) ∈ GU B ( V ) ( F v ) × GU B ( W )( F v ) : ν ( g ) = ν ( h ) } in such a way that they still satisfy the product formula. A similar problem for the dual pairs consistingof the unitary similitude groups of a hermitian E -space V and a skew-hermitian E -space W can besolved using the fact that V ⊗ E W can be considered as a skew-hermitian E -space, and the group { ( g, h ) ∈ GU E ( V ) × GU E ( W ) : ν ( g ) = ν ( h ) } (almost) embeds in U E ( V ⊗ E W ). This fails when working with B -spaces since B is non-commutativeand the tensor product construction is not available. To circumvent this problem, we first constructby hand, splittings s v : { ( g, h ) ∈ GU B ( V ) ( F v ) × GU B ( W )( F v ) : ν ( g ) = ν ( h ) } → C (1) in Appendix C and check that they satisfy several natural properties including the product formula(Proposition C.20). This suffices to construct the theta lift Θ. In order to prove the Rallis innerproduct formula, we need to check a further compatibility between our splittings s v and the splittings s Kudla ,v , in the context of the doubling method. This is accomplished in Lemma D.4 in Appendix D. Acknowledgments:
During the preparation of this article, A.I. was partially supported by JSPSKAKENHI Grant Numbers 22740021, 26287003, and K.P. was partially supported by NSF grantsDMS 0854900, DMS 1160720, a grant from the Simons Foundation (
Quaternionic Shimura varieties
Shimura varieties.
Shimura varieties and canonical models.
We recall quickly the general theory of Shimura va-rieties and their canonical models [10]. Let S := Res C / R G m denote the Deligne torus. There is anequivalence of categories R -Hodge structures ↔ R -vector spaces with an algebraic action of S , described as follows. Suppose that V is an R -vector space equipped with a pure Hodge structure ofweight n . Thus we have a decomposition of V C := V ⊗ R C : V C = M p + q = n V pq , where V pq = V qp . Define an action h of C × on V C by h ( z ) v = z − p ¯ z − q v for v ∈ V pq . Since h ( z ) commutes with complex conjugation, it is obtained by extension of scalars from an auto-morphism of V defined over R . This gives a map on real points h : S ( R ) = C × → GL( V )( R ), thatcomes from an algebraic map S → GL( V ).A Shimura datum is a pair ( G, X ) consisting of a reductive algebraic group G over Q and a G ( R )-conjugacy class X of homomorphisms h : S → G R satisfying the following conditions:(i) For h in X , the Hodge structure on the Lie algebra g of G R given by Ad ◦ h is of type (0 ,
0) +( − ,
1) + (1 , − h to G m, R ⊂ S is trivial.)(ii) For h in X , Ad h ( i ) is a Cartan involution on G ad R , where G ad is the adjoint group of G .(iii) G ad has no factor defined over Q whose real points form a compact group.These conditions imply that X has the natural structure of a disjoint union of Hermitian symmetricdomains. The group G ( R ) acts on X on the left by( g · h )( z ) = g · h ( z ) · g − . To agree with our geometric intuition, we will sometimes write τ h (or simply τ ) for h in X .Let A and A f denote respectively the ring of ad`eles and finite ad`eles of Q . Let K be an opencompact subgroup of G ( A f ). The Shimura variety associated to ( G, X, K ) is the quotientSh K ( G, X ) = G ( Q ) \ X × G ( A f ) / K . For K small enough, this has the natural structure of a smooth variety over C . The inverse limitSh( G, X ) = lim ←− K Sh K ( G, X )is a pro-algebraic variety that has a canonical model over a number field E ( G, X ), the reflex field ofthe Shimura datum (
G, X ). In particular, each Sh K ( G, X ) has a canonical model over E ( G, X ).We recall the definition of E ( G, X ). This field is defined to be the field of definition of the conjugacyclass of co-characters µ h : G m, C → S C → G C , where the first map is z ( z,
1) and the second is the one induced by h . Let us say more preciselywhat this means. For any subfield k of C , let M ( k ) denote the set of G ( k )-conjugacy classes ofhomomorphisms G m,k → G k . Then the inclusion Q ֒ → C gives a bijection between M ( Q ) and M ( C ).This gives a natural action of Gal( Q / Q ) on M ( C ). The reflex field E ( G, X ) is then the fixed field ofthe subgroup { σ ∈ Gal( Q / Q ) : σM X = M X } where M X is the conjugacy class of µ h for any h ∈ X .1.1.2. Automorphic vector bundles.
We recall the basics of the theory of automorphic vector bundlesfollowing [20], [21], [17]. First, to any µ : G m, C → G C as above one can associate a filtration Filt( µ ) ofRep C ( G C ). This is the functor which assigns to every complex representation ( V, ρ ) of G C the filteredvector space ( V, F · µ ) where F · µ is the filtration on V corresponding to ρ ◦ µ ; that is, F pµ V = ⊕ i ≥ p V iµ ,where V iµ is the subspace of V on which G m ( C ) acts via z z i . In particular, one obtains a filtrationon g C via the adjoint representation of G ( C ). Let P µ be the subgroup of G C that preserves the filtration F · µ in every representation ( V, ρ ). Then P µ is a parabolic subgroup of G C that contains the image of µ and has Lie algebra F µ g C . The unipotent radical R u P µ of P µ has Lie algebra F µ g C and is the subgroupthat acts as the identity on Gr · µ ( V ) in every representation ( V, ρ ). The centralizer Z ( µ ) of µ in G C isa Levi subgroup of P µ , isomorphic to P µ /R u P µ . Thus the composite map¯ µ : G m, C → P µ → P µ /R u P µ is a central homomorphism. Then Filt( µ ) equals Filt( µ ′ ) if and only if P µ = P µ ′ and ¯ µ = ¯ µ ′ .Let ˇ X denote the compact dual Hermitian symmetric space to X . As a set, it may be defined as theset of filtrations of Rep C ( G C ) that are G ( C )-conjugate to Filt( µ h ). Equivalently, it may be describedas the set of equivalence classes [( P, µ )] of pairs where P is a parabolic in G C and µ : G m, C → P isa co-character such that ( P, µ ) is G ( C )-conjugate to ( P µ h , µ h ) for some (and therefore every) h ∈ X .Here we say that ( P, µ ) is equivalent to ( P ′ , µ ′ ) if P = P ′ and ¯ µ = ¯ µ ′ . Note that if ( P, µ ) is conjugateto (
P, µ ′ ), then ¯ µ = ¯ µ ′ . Indeed, if g − ( P, µ ) g = ( P, µ ′ ), then g ∈ N G C ( P ) = P . Write g = ℓu , with ℓ ∈ Z ( µ ) and u ∈ R u P , we see that µ ′ = g − µg = u − µu, so that ¯ µ ′ = ¯ µ as claimed. Thus in a given conjugacy class of pairs ( P, µ ), the homomorphism ¯ µ is determined entirely by P . Conversely, for any pair ( P, µ ) in the conjugacy class of ( P µ h , µ h ), theparabolic P must equal P µ so that µ determines P . It follows from this discussion that the naturalmap G ( C ) × ˇ X → ˇ X, ( g, [( P, µ )]) [ g ( P, µ ) g − ]makes ˇ X into a homogeneous space for G ( C ) and the choice of any basepoint [( P, µ )] gives a bijection G ( C ) /P ≃ ˇ X . Further, there is a unique way to make ˇ X into a complex algebraic variety such thatthis map is an isomorphism of complex varieties for any choice of base point. Moreover, the map ξ : M X → ˇ X, µ [( P µ , µ )]is surjective and ˇ X has the natural structure of a variety over E ( G, X ) such that the map ξ isAut( C /E ( G, X ))-equivariant. When we wish to emphasize the rational structure of ˇ X , we will writeˇ X C instead of ˇ X .There is a natural embedding (the Borel embedding) β : X ֒ → ˇ X, h [( P h , µ h )] , where henceforth we write P h for P µ h . Let ˇ V be a G C -vector bundle on ˇ X . The action of G ( C ) on ˇ X extends the G ( R ) action on X . Thus V := ˇ V| X is a G ( R )-vector bundle on X . For an open compact subgroup K of G ( A f ), define V K = G ( Q ) \V × G ( A f ) / K , which we view as fibered over Sh K ( G, X ). In order that this define a vector bundle on Sh K ( G, X ), weneed to assume that ˇ V satisfies the following condition:(1.1) The action of G C on ˇ V factors through G c C .Here G c = G/Z s , where Z s is the largest subtorus of the center Z G of G that is split over R butthat has no subtorus split over Q . Assuming (1.1), for sufficiently small K , V K is a vector bundle onSh K ( G, X ). If ˇ V is defined over E ⊇ E ( G, X ), then V K has a canonical model over E as well. Remark . The reader may keep in mind the following example which occurs in this paper. Let G = Res F/ Q GL , with F a totally real field. Then Z G = Res F/ Q G m and Z s = ker(N F/ Q : Z G → G m ) . We now recall the relation between sections of the bundle V K and automorphic forms on G ( A ). Thisrequires the choice of a base point h ∈ X . Let K h be the stabilizer in G ( R ) of h . Let k h denote theLie algebra of K h and consider the decomposition of g C with respect to the action of Ad ◦ h : g C = p + h ⊕ k h, C ⊕ p − h . Here p + h = g − , C , p − h = g , − C and k h, C = g , C for the Hodge decomposition on g C induced by Ad ◦ h .Thus p ± h correspond to the holomorphic and antiholomorphic tangent spaces of X at h . Then P h is the parabolic subgroup of G ( C ) with Lie algebra k h, C ⊕ p − h . The choice of h gives identifications X = G ( R ) /K h , ˇ X = G ( C ) /P h and the Borel embedding is given by the natural map G ( R ) /K h ֒ → G ( C ) /P h . Let V h denote the fiber of V at h ; equivalently this is the fiber of the bundle ˇ V at β ( h ) ∈ ˇ X . Thiscomes equipped with a natural action of K h , denoted ρ V h . Let ε h denote the map G ( A ) → Sh K ( G, X ) = G ( Q ) \ X × G ( A f ) / K , g = ( g ∞ , g f ) [( g ∞ ( h ) , g f )] . Then there is a canonical isomorphism ε ∗ h ( V K ) ≃ G ( A ) × V h , via which sections of V K can be identified with suitable functions from G ( A ) into V h . This gives acanonical injective map Lift h : Γ (Sh K ( G, X ) , V ) → C ∞ ( G ( Q ) \ G ( A ) / K , V h )whose image is the subspace A ( G, K , V , h ) consisting of ϕ ∈ C ∞ ( G ( Q ) \ G ( A ) / K , V h ) satisfying:(i) ϕ ( gk ) = ρ V h ( k ) − ϕ ( g ), for g ∈ G ( A ) and k ∈ K h ;(ii) Y · ϕ = 0 for all Y ∈ p − h ;(iii) ϕ is slowly increasing, K h -finite and Z ( g C )-finite, where Z ( g C ) is the center of the universalenveloping algebra of g C .Let us make explicit the map Lift h . Fix some τ = τ h ∈ X and let s be a section of V K . For any g f ∈ G ( A f ), there is a canonical identification V τ ≃ V K , [ τ,g f ] where [ τ, g f ] denotes the class of ( τ, g f ) in Sh K ( G, X ). Let g = ( g ∞ , g f ) ∈ G ( A ) = G ( R ) × G ( A f ). Thesection s gives an element s ([ g ∞ τ, g f ]) ∈ V g ∞ τ . However, the element g ∞ induces an isomorphism t g ∞ : V τ ≃ V g ∞ τ . The map Lift h ( s ) : G ( A ) → V τ is then defined by sending g t − g ∞ s ([ g ∞ τ, g f ]) . Remark . The subgroup P h of G C acts on the fiber ˇ V β ( h ) at the point β ( h ). This gives an equivalenceof categories G C -vector bundles on ˇ X ←→ complex representations of P h . The functor in the opposite direction sends a representation (
V, ρ ) of P h to the vector bundle G C × ρ V = ( G C × V ) / { ( gp, v ) ∼ ( g, ρ ( p ) v ) , p ∈ P h } , which fibers over G C /P h in the obvious way. Sections of this vector bundle can be identified withfunctions f : G ( C ) → V, f ( gp ) = ρ ( p ) − f ( g ) . Example . This example will serve to normalize our conventions. Let G = GL , Q and X the G ( R )-conjugacy class containing h : S → G R , a + bi (cid:18) a b − b a (cid:19) . Identify X with h ± , the union of the upper and lower half planes; h is identified with the point i .Then E ( G, X ) = Q and ˇ X ≃ P Q = G/P , where P is the Borel subgroup (of upper triangular matrices)stabilizing ∞ , for the standard action of G on P . We will fix the isomorphism ˇ X ≃ P Q such that themap h ± = X β ֒ → ˇ X C is the identity map. For k ≡ r mod 2, let ˇ V k,r be the homogeneous G C -bundle on ˇ X C correspondingto the character χ k,r : P C → C × , (cid:18) a ∗ d (cid:19) a k det( · ) r − k of P C . Note that abstractly ˇ V k,r ≃ O ( − k ) though the G C -action depends on r as well. For any h ∈ X ,we write ρ k,r for the corresponding representation of K h . The representation ρ k,r of K h = R × · SO ( R )is the character given by z · κ θ z − r e − ikθ , κ θ = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) . For more general h , the character ρ k,r is given by composing the above character with the isomorphism K h ≃ K h given by x α − xα for any α ∈ G ( R ) such that αh α − = h . The correspondingautomorphic line bundle V k,r, K is defined over Q and is the usual bundle of modular forms of weight k and level K . We can make this more explicit as follows.The connected hermitian space h + carries a natural family of (polarized) elliptic curves, the fiberover τ ∈ h + being the elliptic curve A τ = C / ( Z τ + Z ). Let ω be the sheaf of relative one-forms; itis a line bundle on h + and there is a canonical isomorphism β ∗ ˇ V k,r | h + ≃ ω k . This gives a canonicaltrivialization Triv h : V h ≃ C for all h ∈ h + , namely the map sending dz ⊗ k to 1, where z is thecoordinate on C = Lie( A τ ). Thus any section ϕ of V k,r, K on Sh K ( G, X ) gives rise (via Triv h ◦ Lift h )to a function ϕ h : GL ( A ) → C , h ∈ h + , such that ϕ h ( gκ ) = ρ k,r ( κ ) − ϕ h ( g ) for all κ ∈ K h . In particular, for z · κ θ ∈ K h , we have ϕ h ( g · z · κ θ ) = ϕ h ( g ) · z r · e ikθ . Finally, there is a unique modular form f of weight k on h + such that for all h ∈ h + , we have ϕ h ( g ) = f ( g ∞ ( τ h )) j ( g ∞ , τ h ) − k det( g ) r − k , where g = g Q ( g K g ∞ ) with g Q ∈ G ( Q ), g K ∈ K and g ∞ ∈ G ( R ) + . (Here G ( R ) + denotes the topologicalidentity component of G ( R ).) Example . Let G = B × , where B is a non-split indefinite quaternion algebra over Q . Then E ( G, X ) = Q and ˇ X is a form of P ; in fact it is a Severi–Brauer variety associated to the classof B in the Brauer group of Q . The variety ˇ X (over C ) carries the line bundles O ( k ) but only for k even do these descend to line bundles over Q . Indeed, the canonical bundle on ˇ X has degree −
2, so O ( −
2) descends. On the other hand, O (1) does not descend since if it did, by Riemann–Roch it wouldadmit a section whose zero locus is a rational point. Nevertheless, for any σ ∈ Aut( C / Q ), the linebundle L := O (1) on ˇ X C satisfies σ ∗ L ≃ L , so its field of definition is Q .1.1.3. Integral models.
We assume in this section that the Shimura variety (
G, X ) is of abelian type.Let O denote the ring of integers of E ( G, X ) and λ | ℓ a prime of O . We assume that we are givena reductive group G over Z ( ℓ ) such that G Q = G . Let G = G , Z ℓ and K ℓ = G ( Z ℓ ). Then K ℓ is ahyperspecial (maximal compact) subgroup of G ( Q ℓ ). Suppose that K is an open compact subgroup of G ( A f ) of the form K ℓ · K ℓ , with K ℓ as above and K ℓ a subgroup of G ( A ℓf ), where A ℓf denotes the finiteideles whose component at ℓ is 1. Then Sh K ( G, X ) admits a natural integral model S K ,λ ( G, X ) over O ( λ ) . More precisely, if one fixes K ℓ and allows K ℓ to vary, then Kisin [38] shows that the projectivesystem lim ←− Sh K ℓ K ℓ ( G, X ) admits a canonical model S K ℓ ,λ ( G, X ) over O ( λ ) , which is characterized bya certain extension property. We will also need integral models of automorphic vector bundles onSh K ( G, X ). In the abelian case, these will be constructed in the thesis of Lovering [48], and we nowsummarize the relevant results.Recall that the compact dual ˇ X is naturally defined over E ( G, X ). In addition, ˇ X has a naturalmodel ˇ X over O ( λ ) whose A -valued points for any O ( λ ) -algebra A are in bijection with equivalenceclasses of pairs ( P, µ ) consisting of a parabolic subgroup P of G ,A and a cocharacter µ : G m,A → P ,where ( P, µ ) ∼ ( P ′ , µ ′ ) if P = P ′ and ¯ µ = ¯ µ ′ . The data needed to define integral models of automorphicvector bundles consists of the following: • A finite extension L of E ( G, X ) and a G L -equivariant vector bundle ˇ V on ˇ X L . The correspondingautomorphic vector bundle V K on Sh K ( G, X ) has a canonical model over L . • A prime λ of O L ; we write λ for the induced prime of O as well. • A G -equivariant vector bundle ˇ V λ on ˇ X O L, ( λ ) which extends the G L -equivariant vector bundle ˇ V on ˇ X L .To this data, one can associate (by the results of [48]) in a functorial way a vector bundle V K ,λ over S K ,λ ( G, X ) ⊗ O ( λ ) O L, ( λ ) which extends V K . Likewise, if one fixes K ℓ and varies K ℓ , one gets a vectorbundle V K ℓ ,λ over S K ℓ ,λ ( G, X ) ⊗ O ( λ ) O L, ( λ ) . If f : ˇ V λ → ˇ V λ is a map of G -equivariant vector bundlesover ˇ X O L, ( λ ) , there are natural associated maps f K : V K ,λ → V K ,λ and f K ℓ : V K ℓ ,λ → V K ℓ ,λ .1.1.3.1. Models over O L [ N ]. Suppose now that we are given a reductive group G over Z [ N ] such that G , Q = G and that K is of the form Q ℓ K ℓ , where K ℓ = G ( Z ℓ ) for all ℓ not dividing N , so that K ℓ ishyperspecial for such ℓ . Then the integral models of Sh K ( G, X ) for varying ℓ (not dividing N ) patchtogether to give a canonical model S K , O [ N ] ( G, X ) over O [ N ].The compact dual ˇ X has a natural model ˇ X over O [ N ] as well. If we are given moreover: • A finite extension L of E ( G, X ) and a G L -equivariant vector bundle ˇ V on ˇ X L . • A G -equivariant vector bundle ˇ V on ˇ X O L [ N ] which extends the G L -equivariant vector bundle ˇ V on ˇ X L .Then the integral models V K ,λ (as λ varies over the primes of O L not dividing N ) patch together togive an integral model V K , O L [ N ] over O L [ N ]. Automorphic vector bundles on quaternionic Shimura varieties.
In this section, we re-view the connection between automorphic forms on the multiplicative group of a quaternion algebraover a totally real field and sections of automorphic vector bundles on the corresponding Shimuravariety. We will also define canonical metrics on such bundles.
Remark . Everything in this section goes through verbatim even in the case that the quaternionalgebra B is totally definite , even though this does not strictly speaking give a Shimura variety in thesense of § F be a totally real field and B a quaternion algebra over F . Let G B denote the Q -algebraicgroup Res F/ Q ( B × ). Thus for any Q -algebra R , the R -valued points of G B are given by G B ( R ) = ( B ⊗ Q R ) × . Let Σ B denote the set of places of F at which B is ramified.We fix for the moment some choice of isomorphisms B ⊗ F,σ R ≃ M ( R ) , for σ ∈ Σ ∞ r Σ B ;(1.2) B ⊗ F,σ R ≃ H , for σ ∈ Σ B, ∞ , (1.3)where H is the subalgebra (cid:26)(cid:18) α β − ¯ β ¯ α (cid:19) : α, β ∈ C (cid:27) of M ( C ). (Later we will fix these isomorphisms more carefully.) The choice of isomorphisms abovegives us identifications G B ( R ) ≃ Y σ ∈ Σ ∞ r Σ B GL ( R ) × Y σ ∈ Σ B, ∞ H × and G B ( C ) ≃ Y σ ∈ Σ ∞ GL ( C ) . Let X B be the G B ( R )-conjugacy class of homomorphisms S → G B, R containing h : S → G B, R , h := Y σ h ,σ , h ,σ ( z ) = ( z, if σ ∈ Σ ∞ r Σ B ;1 , if σ ∈ Σ B, ∞ , where we identify C with a subring of M ( R ) (see remark below.) Denote by ˇ X B the correspondingcompact dual hermitian symmetric space. The choice of isomorphisms (1.2) and (1.3) above gives riseto an identification ˇ X B = ( P C ) d B and X B = ( h ± ) d B , with d B being the number of infinite places of F where B is split. Remark . (Choices) We embed C in M ( R ) by identifying a + bi with the matrix (cid:18) a b − b a (cid:19) In addition, we identify the homomorphism S → GL , R , a + bi a + bi = (cid:18) a b − b a (cid:19) . with the element i ∈ h . Note that this is opposite to the usual choice made by Shimura. Shimurawould identify i ∈ h with the map a + bi (cid:18) a − bb a (cid:19) . Hermitian forms.
For σ ∈ Σ ∞ r Σ B , let V σ, R denote the vector space R of column vectorsviewed as a left M ( R )-module. Let h : C × = S ( R ) → ( B ⊗ F,σ R ) × = GL ( R ) be any homomorphismthat is GL ( R )-conjugate to h ,σ . Then we can write(1.4) V σ, C = V σ, R ⊗ R C = V − , σ,h ⊕ V , − σ,h , where the decomposition on the right corresponds to the C -subspaces on which h ( z ) ⊗ ⊗ z and 1 ⊗ ¯ z respectively. The bilinear form(1.5) ( x, y ) t x (cid:18) −
11 0 (cid:19) y on V σ, R is almost GL ( R )-invariant: ( gx, gy ) = det( g ) · ( x, y ) . Further, it satisfies the following conditions:(i) ( x, y ) = − ( y, x ).(ii) ( h ( i ) x, h ( i ) y ) = ( x, y ).(iii) The form ( x, h ( i ) y ) is symmetric. (This follows formally from (i) and (ii).) Further, it is positivedefinite if h is GL ( R ) + -conjugate to h . (Otherwise it is negative definite.) Remark . Let τ be the unique point on the complex upper half plane fixed by K h . The bilinearform above equals πi λ τ where λ τ is the Weil pairing on H ( E τ ) given in the ordered basis { τ, } .The composite map V σ, R → V σ ⊗ R C → V − , σ,h is an R -linear isomorphism; via this isomorphism one gets a skew-smmetric bilinear form on V − , σ,h ,which is the negative of the imaginary part of a (necessarily unique) hermitian form H h on V − , σ,h defined by identifying V − , σ,h with V σ, R and setting H h ( x, y ) = ( x, h ( i ) y ) − i ( x, y ) = ( x, iy ) − i ( x, y ) . Remark . The form H h is linear in the first variable and conjugate linear in the second variable.If we denote the form (1.5) above by ˜ B , then H h agrees with the positive definite form 2 · ˜ B h ( i ) ofAppendix A, where:(1.6) ˜ B h ( i ) ( v, w ) = ˜ B C ( v, h ( i ) ¯ w ) . If h is GL ( R ) + -conjugate to h , the form H h is positive definite on account of condition (iii) above.Note that(1.7) H h ( x, x ) = ( x, h ( i ) x ) . The subgroup K h preserves the decomposition (1.4) and the form H h is K h -invariant up to a scalar.In fact, for κ ∈ K h , we have H h ( κx, κy ) = det( κ ) H h ( x, y ) . Moreover, the natural action of GL ( R ) on V σ, C takes V − , σ,h isomorphically onto V − , σ,g · h (recall g · h = ghg − ) and we have H g · h ( gx, gy ) = ( gx, ( gh ( i ) g − ) gy ) = det( g )( x, h ( i ) y ) = det( g ) H h ( x, y ) . We note also that det( V σ, C ) carries a natural bilinear form induced from the C -linear extension of ( · , · ).We equip det( V σ, C ) with the positive definite Hermitian form(1.8) H det ( x, y ) = ( x, ¯ y ) , where the complex conjugation is with respect to the natural real structure coming from det( V σ, R ).This hermitian form satisfies H det ( gx, gy ) = det( g ) · H det ( x, y )for all g ∈ GL ( R ).For σ ∈ Σ B, ∞ , let V σ, C denote the C -vector space C of column vectors viewed as a left M ( C )-module. The form ( x, y ) t x (cid:18) −
11 0 (cid:19) y is almost GL ( C )-invariant: ( gx, gy ) = det( g ) · ( x, y ) . Let L be the R -linear operator L ( x ) := (cid:18) − (cid:19) ¯ x. The operator L is the analog in this case of the operator w h ( i ) ¯ w in (1.6) and the operator x h ( i ) x in (1.7) above. Define a hermitian form H on V σ, C by(1.9) H ( x, y ) = ( x, Ly ) = t x ¯ y. Note that L commutes with the left action of H , hence the form H is H × -invariant up to a scalar,which is also obvious from the formula above. More precisely, for g ∈ H × , we have H ( gx, gy ) = ν ( g ) H ( x, y )where ν is the reduced norm.Let ρ σ,k,r denote the representation V σ,k,r = Sym k ( V σ, C ) ⊗ det( V σ, C ) r − k of GL ( C ). Note that the central character of ρ σ,k,r is z z r .1.2.2. Hermitian metrics on automorphic vector bundles.
Let ( k, r ) be a multi-index of integers with k = ( k σ ) σ ∈ Σ ∞ such that k σ ≡ r (mod 2) for all σ ∈ Σ ∞ . We assume that k σ ≥ B is split at σ and that k σ ≥ B is ramified at σ .Let ρ k,r = ⊗ σ ρ σ,k σ ,r be the representation of G B ( C ) on V k,r = O σ V σ,k σ ,r . This gives rise to a G B ( C )-homogeneous vector bundle ˇ V ρ k,r on ˇ X B :ˇ V ρ k,r = ˇ X B × V k,r , where the G B ( C ) action is: g · ( x, v ) = ( gx, gv ) . By restriction one gets a G B ( R )-homogeneous vector bundle V ρ k,r on X B . Further, the latter admitsa unique G B ( R )-equivariant sub-bundle V k,r corresponding to the K h -subrepresentation ρ k,r,h on V k,r,h = O σ ∈ Σ ∞ r Σ B (cid:16) ( V − , σ,h ) ⊗ k σ ⊗ det( V σ, C ) ⊗ r − kσ (cid:17) O σ ∈ Σ B, ∞ V σ,k σ ,r Let X + B denote the connected component of X B containing h . Note that for h ∈ X + B , the K h -representation above carries a natural positive definite hermitian metric h· , ·i h obtained from the her-mitian metrics in (1.7), (1.8) and (1.9) above. This gives a metric on V k,r that is almost G B ( R )-equivariant; in fact, one has h gx, gy i g · h = ν ( g ) r h x, y i h for g ∈ G B ( R ) and x, y ∈ V k,r,h . Now consider the vector bundle V k,r | X + B × G B ( A f ) on X + B × G B ( A f ).We equip this with the hermitian metric that assigns to the fiber V k,r,h × { g f } over ( h, g f ) ∈ X + B × G B ( A f ) the metric defined above on V k,r,h multiplied by the factor k ν ( g f ) k r . (Here ν ( g f ) ∈ A × F and k · k denotes the idelic norm.) Recall thatSh K ( G B , X B ) = G B ( Q ) \ X B × G B ( A f ) / K = G B ( Q ) + \ X + B × G B ( A f ) / K and V k,r, K = G B ( Q ) \V k,r × G B ( A f ) / K = G B ( Q ) + \V k,r | X + B × G B ( A f ) / K , where G B ( Q ) + = G B ( R ) + ∩ G B ( Q ). Proposition 1.9.
The metric on V k,r | X + B × G B ( A f ) above descends to a (positive definite hermitian)metric on the vector bundle V k,r, K over Sh K ( G B , X B ) .Proof. Let ( h, g f ) and ( h ′ , g ′ f ) be two elements of X + B × G B ( A f ) whose classes in Sh K ( G B , X B ) areequal. Then there exist elements γ ∈ G B ( Q ) + and κ ∈ K such that( h ′ , g ′ f ) = γ ( h, g f ) κ = ( γ · h, γ f g f κ ) . Here γ f is γ viewed as an element of G B ( A f ). We need to check that the bijection V k,r,h × { g f } → V k,r,h ′ × { g ′ f } = V k,r,γ · h × { γ f g f κ } given by ( v, g f ) ( γv, γg f κ ) is metric preserving. But h γv , γv i γ · h · k ν ( γ f g f κ ) k r = Y σ ∈ Σ ∞ σ ( ν ( γ )) r · h v , v i h · k ν ( γ ) f k r k ν ( g f ) k r = h v , v i h · k ν ( g f ) k r , using the product formula and the fact that k ν ( κ ) k = 1. (cid:3) We will need to work with the dual vector bundle V ∨ k,r . This is motivated by observing that in thecase of GL ( Q ), the bundle V ρ k,r corresponds to the relative homology of the universal elliptic curveand the sub-bundle V k,r corresponds to its relative Lie algebra. The line bundle of usual modularforms corresponds to the bundle of relative differentials, which is why we need to replace V k,r by itsdual. We begin by making the following completely elementary remark, which we nevertheless statecarefully to avoid any confusion. Remark . If ρ is a representation of a group G on a finite-dimensional complex vector space V ,then ρ ∨ is defined by ρ ∨ ( g )( L ) = L ◦ ρ ( g − )for L ∈ V ∨ = Hom( V, C ) and g ∈ G . Thus for the tautological pairing( · , · ) : V ∨ × V → C , ( L, v ) = L ( v ) , we have ( ρ ∨ ( g − ) L, v ) = (
L, ρ ( g ) v ) . Suppose V is equipped with a non-degenerate hermitian pairing h· , ·i that is linear in the first variableand conjugate linear in the second variable, and such that h gv, gw i = χ ( g ) h v, w i for some character χ : G → C × . Since h· , ·i is non-degenerate, it induces a conjugate linear isomorphism V ≃ V ∨ , w L w , L w ( v ) = h v, w i . Composing the inverse of this isomorphism with the canonical isomorphism V ≃ V ∨∨ gives a conjugatelinear isomorphism V ∨ ≃ ( V ∨ ) ∨ , which one may view as a hermitian form on V ∨ . Explicitly thisisomorphism sends L w to the linear functional eval w ∈ ( V ∨ ) ∨ , so that for any L ∈ V ∨ , we have h L, L w i = L ( w ) . Note that gL w ( v ) = L w ( g − v ) = h g − v, w i = χ ( g ) − h v, gw i = χ ( g ) − L gw ( v ) , so that gL w = χ ( g ) − L gw . For any L ∈ V ∨ , we have h gL, gL w i = h gL, χ ( g ) − L w i = χ ( g ) − h gL, L gw i = χ ( g ) − ( gL )( gw ) = χ ( g ) − L ( w )= χ ( g ) − h L, L w i , so for any L , L ∈ V ∨ , we have h gL , gL i = χ ( g ) − h L , L i .From the remark above, it is clear that for x, y ∈ V ∨ k,r,h and g ∈ G B ( R ), we have h gx, gy i = ν ( g ) − r h x, y i . Thus we take on V ∨ k,r | X + B × G B ( A f ) the metric which on V ∨ k,r,h × { g f } is k ν ( g f ) k − r times the inducedmetric on V ∨ k,r,h . This descends to a positive definite hermitian metric hh· , ·ii on V ∨ k,r, K . (See Prop. 1.9above.) Definition 1.11.
A holomorphic automorphic form of weight ( k, r ) and level K on G B is a holomorphic section s of the bundle V ∨ k,r, K on Sh K ( G B , X B ). Let ˜ K ⊇ K be any open compact subgroup of G B ( A f )such that hh s ( x ) , s ( x ) ii descends to a function on Sh ˜ K ( G B , X B ). Then the Petersson norm of the section s (normalized with respect to ˜ K ) is defined to be the integral hh s, s ii ˜ K := Z Sh ˜ K ( G B ,X B ) hh s ( x ) , s ( x ) ii dµ x where dµ x is the measure on Sh ˜ K ( G B , X B ) defined in Sec. 6.1.2. Remark . Defn. 1.11 above has the advantage that it does not depend on any choice of base point.In practice though, one usually needs to pick a base point to make any computation at all, and so weshall now discuss the translation between these two points of view.Pick a base point h ∈ X + B . Via the isomorphism Lift h , the space of holomorphic automorphic forms s as above is identified with the space of functions A ( G B , K , V ∨ k,r , h ). An element F : G B ( Q ) \ G B ( A ) / K → V ∨ k,r,h in A ( G B , K , V ∨ k,r , h ) satisfies in particular the condition(1.10) F ( gκ h ) = ρ ∨ k,r,h ( κ h ) − F ( g ) , for all κ h ∈ K h . Henceforth we will fix a character ξ of F × \ A × F which satisfies ξ ( z · z ∞ ) = N( z ∞ ) r · ξ ( z )for z ∈ A × F , z ∞ ∈ A × F, ∞ , and assume that the section s satisfies the following invariance under thecenter Z G B ( A f ) = A × F,f :(1.11) s ( x · α ) = ξ ( α ) · s ( x ) . This enables us to take ˜ K containing the maximal open compact subgroup of Z G B ( A f ), and implies thatthe corresponding function F above satisfies the following invariance property: for α ∈ A × F = Z G B ( A ),we have F ( g · α ) = ξ ( α ) · F ( g )and h F ( g · α ) , F ( g · α ) i = k α k r · h F ( g ) , F ( g ) i . Proposition 1.13.
Suppose
Lift h ( s ) = F . Let K denote any maximal compact subgroup of G B ( A f ) containing ˜ K . Then hh s, s ii ˜ K = 2 | Σ ∞ r Σ B | · h F · [ K : ˜ K ] · h F, F i h , where h F, F i h = Z [ G B ] h F ( g ) , F ( g ) i h · k ν ( g ) k − r dg. Here and henceforth we write [ G B ] for G B ( Q ) Z G B ( A ) \ G B ( A ). Also, dg denotes the standardmeasure on [ G B ] which is defined in § Proof.
Recall that if g = ( g ∞ , g f ), we have F ( g ) = g − ∞ s [( g ∞ · h, g f )] , where we view s [( g ∞ · h, g f )] as an element in V ∨ k,r,g ∞ · h . Now h F ( g ) , F ( g ) i h = ν ( g ∞ ) r h s [( g ∞ · h, g f )] , s [( g ∞ · h, g f )] i g ∞ · h = ν ( g ∞ ) r k ν ( g f ) k r · hh s [( g ∞ · h, g f )] , s [( g ∞ · h, g f )] ii = k ν ( g ) k r hh s [( g ∞ · h, g f )] , s [( g ∞ · h, g f )] ii . The proposition follows from this and the comparison of measures in Lemma 6.3. (cid:3)
Next, we simplify further to scalar valued forms. For κ = ( z σ e iθ σ ) σ ∈ Σ ∞ ∈ ( C × ) d , let κ h be theelement of K h ⊂ G B ( R ) defined by: κ h,σ = ( h σ ( z σ e iθ σ ) , if σ ∈ Σ ∞ r Σ B ; z σ e iθ σ , if σ ∈ Σ B, ∞ , where for σ ∈ Σ B, ∞ , we view z σ e iθ σ as an element in C × ⊂ H × ≃ ( B ⊗ F,σ R ) × via (1.3). The equation(1.10) can be rewritten as F ( gκ h ) = Y σ ∈ Σ ∞ z rσ · Y σ ∈ Σ ∞ r Σ B e ik σ θ σ · O σ ∈ Σ B, ∞ ρ ∨ σ,k σ ,r ( e − iθ σ ) F ( g ) . For σ ∈ Σ ∞ r Σ B , let v σ,k σ be any nonzero vector in the one-dimensional C -vector space( V − , σ,h ) ⊗ k σ ⊗ det( V σ, C ) ⊗ r − kσ , so that(1.12) ρ σ,k σ ,r ( κ h,σ ) · v σ,k σ = z rσ e ik σ θ σ · v σ,k σ . For σ ∈ Σ B, ∞ , let v σ,k σ ∈ V σ,k σ ,r be any nonzero vector such that the condition (1.12) is satisfied forall κ ∈ ( C × ) d . Such a vector is well-defined up to scaling.Set v k = ⊗ σ ∈ Σ ∞ v σ,k σ ∈ V k,r,h . Define φ F ( g ) = ( F ( g ) , v k ) . Then φ F ( g ) satisfies(1.13) φ F ( gκ h ) = Y σ ∈ Σ ∞ z rσ e ik σ θ σ · φ F ( g )and(1.14) φ F ( αg ) = ξ ( α ) φ F ( g ) , for α ∈ Z G B ( A ) = A × F . Proposition 1.14.
The map F φ F is injective.Proof. This follows immediately from the fact that V k,r,h is irreducible as a module over G = Q σ ∈ Σ B, ∞ ( B ⊗ F,σ R ) × . Indeed, given any w ∈ V k,r,h , there exist elements κ i ∈ G and α i ∈ C such that w = X i α i ρ ( κ i ) v k , where ρ denotes the natural action of G on V k,r,h . Then( F ( g ) , w ) = X i α i ( F ( g ) , ρ ( κ i ) v k ) = X i α i ( ρ ∨ ( κ i ) − F ( g ) , v k )= X i α i ( F ( gκ i ) , v k ) = X i α i φ F ( gκ i ) . Thus if φ F is identically zero, then so is F . (cid:3) We will now compare h F, F i to h φ F , φ F i , where h φ F , φ F i = Z [ G B ] φ F ( g ) φ F ( g ) · k ν ( g ) k − r dg. We use the following well known lemma.
Lemma 1.15.
Let K be a compact Lie group and V a (finite dimensional) irreducible complex repre-sentation of K . Let h· , ·i be a nonzero K -invariant hermitian form on V (such a form is unique up toscalar multiples) and denote also by h· , ·i the induced hermitian form on V ∨ . Then for all v , v ∈ V and L , L ∈ V ∨ , we have Z K ( ρ ∨ ( k ) L , v )( ρ ∨ ( k ) L , v ) dk = 1dim( V ) · h v , v ih L , L i , where dk is Haar measure normalized to have total volume .Remark . It is immediate to check that if the form h· , ·i on V is scaled by α ∈ C × , then the form h· , ·i on V ∨ is scaled by ¯ α − , so the right hand side is independent of the choice of h· , ·i . Proposition 1.17. h F, F i h = rank V k,r h v k , v k i h · h φ F , φ F i . Proof.
Let K h denote the maximal compact subgroup of K h . Since V k,r,h is an irreducible represen-tation of K h , using Lemma 1.15 we get h φ F , φ F i = Z [ G B ] φ F ( g ) φ F ( g ) · k ν ( g ) k − r dg = Z K h Z [ G B ] φ F ( g ) φ F ( g ) · k ν ( g ) k − r dg dκ = Z K h Z [ G B ] φ F ( gκ − ) φ F ( gκ − ) · k ν ( gκ − ) k − r dg dκ = Z K h Z [ G B ] ( F ( gκ − ) , v k )( F ( gκ − ) , v k ) · k ν ( g ) k − r dg dκ = Z [ G B ] Z K h ( ρ ∨ ( κ ) F ( g ) , v k )( ρ ∨ ( κ ) F ( g ) , v k ) · k ν ( g ) k − r dκ dg = 1rank V k,r h v k , v k i h Z [ G B ] h F ( g ) , F ( g ) i h k ν ( g ) k − r dg. (cid:3) Rational and integral structures.
Let Π = ⊗ v Π v be an irreducible cuspidal automorphicrepresentation of GL ( A F ) corresponding to a Hilbert modular form of weight ( k, r ), character ξ Π andconductor N = N s · N ps , as in the introduction. Thus the character ξ Π satisfies ξ Π ( z · z ∞ ) = N( z ∞ ) r · ξ Π ( z )for z ∈ A × F and z ∞ ∈ A × F, ∞ . We also let π = ⊗ v π v denote the corresponding unitary representation: π := Π ⊗ k det( · ) k − r/ . Recall that Σ Π denotes the set of all places v of F at which Π v is discrete series. Thus Σ Π containsΣ ∞ but will typically be larger. Let B be any quaternion algebra over F such that Σ B ⊆ Σ Π , where Σ B denotes the set of places v of F where B is ramified. By the Jacquet–Langlands correspondence, thereexists (up to isomorphism) a unique irreducible (cuspidal) automorphic representation Π B ≃ ⊗ v Π B,v of G B ( A ) such that Π B,v ≃ Π v for all v Σ B . Let k B = ( k B,σ ) σ ∈ Σ ∞ be defined by:(1.15) k B,σ = ( k σ , if B is split at σ , k σ − , if B is ramified at σ .Then Π B has weight ( k B , r ) at infinity.Choose a maximal order O B in B . Recall that we have assumed that the conductor N of Π satisfies N = N s · N ps where N s is divisible exactly by those primes at which Π v is discrete series and N ps is divisible exactlyby those primes at which Π v is ramified principal series. Let d B be the (finite part of the) discriminantof B , so that d B divides N s . Then there is a unique integral ideal N B in O F such that N = N B · d B , and we may choose and fix an Eichler order O B ( N B ) in O B of level N B . We will also fix an orientation of this order at the places dividing N ps . By this, we mean a ring homomorphism o : O B ( N B ) → O F / N ps . This choice determines an open compact subgroup K = Q K ℓ of G B ( A f ), namely K ℓ = Q v | ℓ K v wherefor any finite place v of F , we have K v = ker (cid:2) o v : ( O B ( N B ) ⊗ O F O F,v ) × → ( O F,v / N ps O F,v ) × (cid:3) . Here o v is the natural map induced by the orientation o . For all rational primes ℓ such that ( ℓ, N (Π)) =1, the subgroup K ℓ is a hyperspecial maximal compact subgroup of G B ( Q ℓ ).Now, we will assume that B is not totally definite , relegating the case of totally definite B to Remark1.19 at the end of this section. Let ℓ be such that ( ℓ, N (Π)) = 1. Then for each prime λ of E ( G B , X B )dividing such an ℓ , one has (see § S K ,λ = S K ,λ ( G B , X B )of Sh K ( G B , X B ) defined over O E ( G B ,X B ) , ( λ ) .We will now fix more carefully the isomorphism(1.16) φ B : B ⊗ R ≃ Y σ ∈ Σ ∞ r Σ B M ( R ) × Y σ ∈ Σ B, ∞ H . Note that the vector bundles previously denoted by V ρ kB ,r, K and V k B ,r, K actually depend on thechoice of φ B . In this section alone, we will be pedantic and write V φ B ρ kB ,r, K and V φ B k B ,r, K to indicate thedependence on φ B . Let L ⊃ F be a number field such that L splits B . We may assume by enlarging L if necessary that it is Galois over Q . Then L contains E ( G B , X B ). We pick the isomorphism φ B abovesuch that B maps into Q σ ∈ Σ ∞ M ( L ). This data defines an L -rational structure ([20], [52]) on theautomorphic vector bundle V φ B ρ kB ,r, K on Sh K ( G B , X B ) associated to the G B ( R )-homogeneous vectorbundles V φ B ρ kB ,r as well as the sub-bundles V φ B k B ,r, K . To define integral models of these vector bundles,we first pick a rational prime ℓ prime to N (Π) and insist that the isomorphism φ B satisfy(1.17) φ B ( O B ) ⊂ Y σ M ( O L, ( ℓ ) ) , so that φ B gives an isomorphism(1.18) O B ⊗ O L, ( ℓ ) ≃ Y σ M ( O L, ( ℓ ) ) . By the discussion in Sec. 1.1.3, this data defines for all primes λ ′ of L with λ ′ | λ | ℓ , natural integralmodels for the bundles V φ B ρ kB ,r, K and V φ B k B ,r, K over S K ,λ ⊗ O E ( GB,XB ) , ( λ ) O L, ( λ ′ ) . Indeed, the choice of theEichler order O B ( N B ) determines a reductive group G over Z ( ℓ ) such that G , Q = G B ; namely for any Z ( ℓ ) -algebra A , we have G ( A ) = ( O B ( N B ) ⊗ A ) × . Further, the map φ B induces an isomorphism(1.19) G ⊗ O L, ( ℓ ) ≃ Y σ ∈ Σ ∞ GL / O L, ( ℓ ) . This gives an integral model over O L, ( ℓ ) for the compact dual symmetric space and the vector bundleˇ V k B ,r . Via the identification (1.19) above, the integral model ˇ X O L, ( ℓ ) for the compact dual is simplythe conjugacy class of the parabolic subgroup P := Y σ ∈ Σ ∞ r Σ B B × Y σ ∈ Σ B, ∞ GL / O L, ( ℓ ) of G ⊗ O L, ( ℓ ) , where B = (cid:26)(cid:18) ∗ ∗ ∗ (cid:19)(cid:27) ⊂ GL / O L, ( ℓ ) . Thus ˇ X O L, ( ℓ ) is isomorphic to Q σ ∈ Σ ∞ r Σ B P O L, ( ℓ ) , the isomorphism depending on the choice of φ B . Let L = O L, ( ℓ ) with the obvious left action of GL ( O L, ( ℓ ) ). Then the integral model of the vector bundleˇ V k B ,r over ˇ X O L, ( ℓ ) is the vector bundle ˇ V φ B O L, ( ℓ ) corresponding to the representation Y σ ∈ Σ ∞ r Σ B χ k B,σ ,r · O σ ∈ Σ B, ∞ Sym k B,σ ( L ) ⊗ det( L ) r − kB,σ of P . (Recall that χ k,r has been defined in Eg. 1.3.)For all finite places v of F at which B is split, we will fix an isomorphism(1.20) i v : B ⊗ F v ≃ M ( F v )such that for all but finitely many v , we have(1.21) i v : O B ( N B ) ⊗ O F,v ≃ M ( O F,v ) . Let ∆ be a large enough finite set of places of F such that • ∆ contains all the infinite places, and all the finite places v at which Π v is ramified. • For all v ∆, the condition (1.21) holds.For all finite places v not in ∆, we get (using i v ) an identification(1.22) K v ≃ GL ( O F,v ) , H ′ v ≃ H v where H ′ v and H v denote the spherical Hecke algebras on B × v and GL ( F v ) constructed using themaximal compact subgroups K v and GL ( O F,v ) respectively. Let H ′ ∆ = O v ∆ H ′ v , H ∆ = O v ∆ H v . Note that H ′ ∆ acts naturally on the space of sections of V k B ,r, K and we have an identification H ′ ∆ ≃ H ∆ .Also H ∆ acts on ⊗ v ∆ Π v . Let ϕ = ⊗ v ∆ ϕ v be a new-vector in the space ⊗ v ∆ Π v , so that ϕ is aneigenvector for the action of H ∆ . Let Λ Π denote the corresponding character of H ∆ . Proposition 1.18.
There exists up to scaling a unique non-zero section s B of the bundle V φ B k B ,r, K which satisfies the following conditions: • s is an eigenvector for the action of H ′ ∆ and H ′ ∆ acts on it by Λ Π , via the identification H ′ ∆ ≃ H ∆ above. • s satisfies (1.11) for ξ = ξ Π .Proof. Let s be any section of V φ B k B ,r, K . Pick some point h ∈ X B . Let F s,h = Lift h ( s ) and set φ s,h = φ F s,h , notations as in the previous section. By strong multiplicity one, the assignment s φ s,h gives a bijection of the space of sections of V φ B k B ,r, K on which H ′ ∆ acts by Λ Π with the space of functions φ : G B ( Q ) \ G B ( A ) / K → C that satisfy (1.13) and (1.14) and on which H ′ ∆ acts by Λ Π . By the Jacquet–Langlands correspondenceand the uniqueness of newforms [6], this latter space is one-dimensional, generated by a nonzero element φ . If s B is such that φ s B ,h = φ , then s B is our required section. (cid:3) Let us enlarge L if necessary so that E Π ⊂ L where E Π is the field generated by the Hecke eigenvaluesof Π. By [23] Prop. 2.2.4, the section s B of Prop. 1.18 can be chosen to be L -rational. Further, for λ ′ | λ | ℓ as above, the integral model of V φ B k B ,r, K over S K ,λ ⊗ O E ( GB,XB ) , ( λ ) O L, ( λ ′ ) defines an O L, ( λ ′ ) -lattice M λ ′ in H (Sh K ( G B , X B ) /L , V φ B k B ,r, K ) . Fixing ℓ , choose s B (by suitably scaling) such that for all λ ′ | λ | ℓ , it is a generator for the rankone O L, ( λ ′ ) -lattice M λ ′ ∩ Ls B . We will say that the section s B is ℓ -normalized. Remark . In this remark we deal with the case of totally definite B . Pick φ B satisfying (1.17),(1.18) above with an appropriate choice of L . Then X B = { h } , and sections s of V φ B k B ,r, K are identifiedwith functions F : G B ( Q ) \ G B ( A ) / K → V k B ,r = O σ ∈ Σ ∞ V σ,k B,σ ,r satisfying the appropriate invariance property under the right action of G B ( R ). Then V k B ,r admits anatural L -rational structure as well as a natural O L, ( ℓ ) -submodule: V k B ,r ⊃ V k B ,r ( L ) = O σ ∈ Σ ∞ Sym k B,σ L ⊗ det( L ) r − kB,σ ⊃ V k B ,r ( O L, ( ℓ ) ) = O σ ∈ Σ ∞ Sym k B,σ O L, ( ℓ ) ⊗ det( O L, ( ℓ ) ) r − kB,σ . This gives an L -rational structure and an O L, ( ℓ ) -integral structure on the space of sections of V φ B k B ,r, K ,namely we take sections s which on G B ( A f ) take values in V k B ,r ( L ) and V k B ,r ( O L, ( ℓ ) ) respectively. Wepick isomorphisms i v as above and Prop. 1.18 continues to hold. Finally, we pick s B to be ℓ -normalizedwith respect to the integral structure provided by V k B ,r ( O L, ( ℓ ) ).1.4. Canonical quadratic period invariants.
We can now define the canonical quadratic periodinvariants attached to Π and state the main conjecture relating these invariants. Let B be a quaternionalgebra such that Σ B ⊆ Σ Π . As in the introduction, let R be the ring O Q [1 /N (Π)]. For any rationalprime ℓ prime to N (Π), we define an invariant q B (Π , ℓ ) ∈ C × /R × ( ℓ ) as follows. Let ˜ K ⊇ K be the opencompact subgroup of G B ( A f ) defined by ˜ K = Q v ˜ K v with˜ K v = ( O B ( N B ) ⊗ O F O F,v ) × . Choosing a section s B as above that is ℓ -normalized, define q B (Π , ℓ ) := hh s B , s B ii ˜ K ∈ C × /R × ( ℓ ) , to be the Petersson norm of the section s B as in Defn. 1.11. Proposition 1.20.
The invariant q B (Π , ℓ ) is well defined, in that as an element of C × /R × ( ℓ ) , it doesnot depend on the choices of the number field L , the pair ( O B ( N B ) , o ) consisting of the Eichler order O B ( N B ) and the orientation o : O B ( N B ) → O F / N ps , the isomorphism (1.16) satisfying (1.17) , (1.18) above and the collection of isomorphisms (1.20) .Proof. We will give the argument in the case when B is not totally definite. In the case of a totallydefinite B , a similar (but simpler) argument can be given which we leave to the reader.Independence of the choice of L is clear since we can always replace L by a larger field withoutchanging the choice of s B . Implicitly in the arguments below we may need to make such a fieldextension and we do this without comment. Let us first show that fixed choices of other data, thereis no dependence on the choice of isomorphisms (1.20). Indeed, for all but finitely many v , theisomorphisms i v must satisfy (1.21). Let { i ′ v } be a different set of choices. Then for all but finitelymany v , the isomorphisms i v and i ′ v must differ by conjugation by an element of K v . For such v , theidentifications H ′ v ≃ H v given by i v and i ′ v are the same. This implies that the same choice of s B canbe used if { i v } is replaced by { i ′ v } and the norm hh s B , s B ii ˜ K is unchanged. Next let us look at the dependence on the choice of isomorphism φ B in (1.16), for fixed choices ofother data. Let φ ′ B be a different choice of isomorphism satisfying (1.17). Then φ B and φ ′ B differ byconjugation by an element t ∈ Y σ GL ( L ) ∩ Y σ ∈ Σ ∞ r Σ B GL ( R ) × Y σ ∈ Σ B, ∞ H × that normalizes Q σ M ( O L, ( ℓ ) ). The normalizer of M ( O L, ( ℓ ) ) in GL ( L ) is L × · GL ( O L, ( ℓ ) ), so wemay assume that t lies in Q σ GL ( O L, ( ℓ ) ). Then there is a natural morphism of integral modelsˇ V φ B O L, ( ℓ ) ≃ ˇ V φ ′ B O L, ( ℓ ) which is just given by the (left) action of t on the fibers. This induces an isomorphism between theintegral models of the corresponding automorphic vector bundles that is also given by the action of t on the fibers. (Keep in mind that the G ( Q )-action on the fibers are different, and differ by conjugationby t , so the left action of t on the fibers is indeed a map of bundles.) Thus if s B is an ℓ -normalizedsection of V φ B k B ,r, K , then t · s B is an ℓ -normalized section of V φ ′ B k B ,r, K . Then the inner products hh s B , s B ii ˜ K and hh s ′ B , s ′ B ii ˜ K differ by a power of k ν ( t ) k , which is a unit at ℓ .Finally, we consider dependence on the choice of the pair ( O B ( N B ) , o ). Let ( O B ( N B ) ′ , o ′ ) beanother such pair and let φ ′ B (respectively i ′ v ) denote our choices of isomorphism (1.16) satisfying(1.17) (respectively the isomorphisms (1.20) satisfying (1.21) for all but finitely many v ). Let ussuppose first that the pair ( O B ( N B ) ′ , o ′ ) is conjugate to ( O B ( N B ) , o ) by an element in B × , say O B ( N B ) ′ = b − O B ( N B ) b and o ′ ( x ) = o ( bxb − ). By what we have shown so far we may assume that φ ′ B ( x ) = φ B ( bxb − ) , i ′ v ( x ) = i v ( bxb − ) . The open compact subgroup K ′ of G B ( A f ) determined by the pair ( O B ( N B ) ′ , o ′ ) satisfies K ′ = b − K b .Let us write b = b ∞ · b f where b ∞ and b f denote the infinite and finite components of b respectively,viewed as elements in G B ( A ). There is a natural isomorphism of Shimura varietiesSh K ( G B , X B ) = G B ( Q ) \ X B × G B ( A f ) / K ξ b ≃ G B ( Q ) \ X B × G B ( A ) / K ′ = Sh K ′ ( G B , X B ) , given by ( h, g f ) ( h, g f b f ) . Further, there is a natural isomorphism V φ B k B ,r, K = G B ( Q ) \V φ B k B ,r × G B ( A f ) / K ˜ ξ b ≃ G B ( Q ) \V φ ′ B k B ,r × G B ( A f ) / K ′ = V φ ′ B k B ,r, K ′ covering ξ b , given by(1.23) ( v, g f ) ( φ B ( b ) · v, g f b f ) . Note that if γ is an element in G B ( Q ) then γ · ( v, g f ) = ( φ B ( γ ) · v, γg f ) ( φ B ( b ) φ B ( γ ) · v, γg f b f )= ( φ ′ B ( γ ) φ B ( b ) · v, γg f b f ) = γ · ( φ B ( b ) · v, g f b f ) , so that the assignment in (1.23) does descend to equivalence classes for the G B ( Q )-action. Note alsothat ˜ ξ B is the map on automorphic vector bundles corresponding to a morphism of vector bundles thatextend to the integral models, since these integral models are defined using the triples ( O B ( N B ) , o, φ B )and ( O B ( N B ) ′ , o ′ , φ ′ B ) respectively. Thus ˜ ξ b is an isomorphism at the level of integral models, and sowe may assume that s ′ B = ˜ ξ b ( s B ). But then we see from the definition of the metrics on the vectorbundles V φ B k B ,r, K and V φ ′ B k B ,r, K ′ and the product formula that hh s ′ B , s ′ B ii ˜ K ′ = k ν ( b ∞ ) k − r · k ν ( b f ) k − r · hh s B , s B ii ˜ K = hh s B , s B ii ˜ K . In general, it may not be true that the pairs ( O B ( N B ) , o ) and ( O B ( N B ) ′ , o ′ ) are conjugate by anelement of B × . Nevertheless, we can always find an element β f ∈ B × ( A f ) such that O B ( N B ) ′ = β − f O B ( N B ) β f , o ′ ( x ) = o ( β f xβ − f ) . Let b be an element of B × approximating β f = ( β v ) at ℓ so that O B ( N B ) ′ ⊗ Z ( ℓ ) = ( b − O B ( N B ) b ) ⊗ Z ( ℓ ) . Then we may assume that φ ′ B ( x ) = φ B ( bxb − ) , i ′ v ( x ) = i v ( β v xβ − v ) . The open compact subgroup K ′ satisfies K ′ = β − f K β f . We now run through the same argument asabove, defining ξ b [( h, g f )] = [( h, g f β f )] , ˜ ξ b [( v, g f )] = [( φ B ( b ) · v, g f β f )] . The result follows from observing that k ν ( b ∞ ) k · k ν ( β f ) k , while not necesarily 1, is still an element in R × ( ℓ ) . (cid:3) We can also define an invariant q B (Π) ∈ C × /R × such that the class of q B (Π) in C × /R × ( ℓ ) equals q B (Π , ℓ ). Indeed, pick the isomorphism φ B , the number field L and the maximal order O B in B suchthat φ B ( O B ) ⊂ Y σ M ( O L ) . Choose a pair ( O B ( N B ) , o ) consisting of an Eichler order and an orientation. The constructions in § O L [ N (Π) ]. (See Sec.1.1.3.1.) By enlarging L if need be, we can pick a section s B that is ℓ -normalized at all rational primesthat are prime to N (Π) and then define q B (Π) to equal hh s B , s B ii ˜ K for such a choice of s B . This isan element of C × that maps to q B (Π , ℓ ) under the natural map C × → C × /R × ( ℓ ) for all ℓ such that( ℓ, N (Π)) = 1. Since the map C × /R × → Y ( ℓ,N (Π))=1 C × /R × ( ℓ ) is injective, the class of q B (Π) in C × /R × is well defined. This defines the invariants needed in theformulation of Conjecture A of the introduction.2. Unitary and quaternionic unitary groups In § § § Hermitian and skew-hermitian spaces.
Hermitian spaces.
Let F be a field of characteristic zero and E a quadratic extension of F ,possibly split. Let V be a right E -vector space of dimension n (i.e., a free E -module of rank n ),equipped with a Hermitian form ( · , · ) : V × V → E. Such a form is linear in one variable and antilinear in the other, and we fix any one convention at thispoint. For example, if ( · , · ) is antilinear in the first variable and linear in the second, then:( vα, v ′ β ) = α ρ ( v, v ′ ) β, ( v, v ′ ) = ( v ′ , v ) ρ , where ρ denotes the nontrivial involution of E/F .To such a V , one associates the following invariants: dim( V ) = n and disc( V ) ∈ F × / N E/F E × ,where disc( V ) = ( − n ( n − / det (( v i , v j )) , with { v i } an E -basis for V . Since ( · , · ) is Hermitian, disc( V ) lies in F × and its class in F × / N E/F E × is independent of the choice of basis.Let GU( V ) denote the unitary similitude group of V . (Occasionally, we will write GU E ( V ) forclarity.) This is an algebraic group over F such that for any F -algebra R , we haveGU( V )( R ) := { g ∈ GL( V ⊗ R ) : ( gv, gv ′ ) = ν ( g )( v, v ′ ) for all v, v ′ and ν ( g ) ∈ R × } . If E = F × F , then GU( V ) ≃ GL n × GL . If E is a field, the various possibilities for GU( V ) arediscussed below.2.1.1.1. p -adic local fields. Let F be p -adic. As a Hermitian space, V is determined up to isomorphismby its dimension and discriminant. Further, given any choice of dimension and discriminant, there isa space V with these as its invariants. If dim( V ) is odd, the group GU( V ) is (up to isomorphism)independent of disc( V ) and is quasi-split. If dim( V ) is even, there are two posibilities for GU( V ) upto isomorphism and GU( V ) is quasi-split if and only if disc( V ) = 1.2.1.1.2. Archimedean fields. Let F = R and E = C . Then the form ( · , · ) can be put into the diagonalform (1 , . . . , , − , . . . , −
1) which is called the signature of V ; we say V is of type ( p, q ) if the numberof 1s is p and the number of −
1s is q . Hermitian spaces are classified up to isomorphism by theirsignature (which determines both the dimension and discrminant) and we write GU( p, q ) for theassociated group. The only isomorphisms between these groups are GU( p, q ) ≃ GU( q, p ).2.1.1.3. Number fields. Let
E/F be a quadratic extension of number fields. If V is a Hermitian E -space, then for each place v of F , one gets a local space V v which is a Hermitian space for E v /F v andsuch that for almost all v , the discriminant of V v is 1. The Hasse principle says that V is determinedup to isomorphism by this collection of local spaces. Conversely, suppose we are given for each place v a local space V v (of some fixed dimension n ) such that almost all of the local discriminants are equalto 1. The collection of local discriminants gives an element of A × F / N E/F A × E . Such a collection of localspaces comes from a global space if and only if this element lies in the image of F × , i.e., is trivial inthe quotient A × F /F × N E/F A × E , which has order 2.2.1.2. Skew-Hermitian spaces.
Let
E/F be a quadratic extension as in the beginning of the previoussection. Skew-hermitian E -spaces are defined similarly to hermitian spaces but with the condition( v, v ′ ) = − ( v ′ , v ) ρ . We can go back and forth between hermitian and skew-hermitian spaces simply by multiplying theform by an element in E × of trace zero. Indeed, pick a trace zero element i ∈ E × . If ( · , · ) is skew-hermitian form on V , the product ( · , · ) ′ := i · ( · , · ) is hermitian. The group GU( V ) is the same for both ( · , · ) and ( · , · ) ′ . Thus the classification of skew-hermitian forms (and the corresponding groups)can be deduced from the hermitian case.2.1.3. Quaternionic hermitian spaces.
Let F be a field and B a quaternion algebra over F . Let a a ∗ denote the main involution on B . A B -Hermitian space is a right B -space V equipped with a B -valuedform h· , ·i : V × V → B satisfying h vα, v ′ β i = α ∗ h v, v ′ i β, h v, v ′ i = h v ′ , v i ∗ , for v, v ′ ∈ V and α, β ∈ B .Let GU( V ) denote the unitary similitude group of V . (Sometimes, we write GU B ( V ) for clarity.)This is an algebraic group over F such that for any F -algebra R , we haveGU( V )( R ) := { g ∈ GL( V ⊗ R ) : h gv, gv ′ i = ν ( g ) h v, v ′ i for all v, v ′ and ν ( g ) ∈ R × } . If B is split, there is a unique such space V of any given dimension n over B . The correspondinggroup GU( V ) is identified with GSp(2 n ). If B is nonsplit, the classification of such spaces over p -adicfields and number fields is recalled below.2.1.3.1. p -adic fields. If F is a p -adic field, there is a unique such space of any given dimension, up toisometry. The corresponding group is the unique nontrivial inner form of GSp(2 n ).2.1.3.2. Archimedean fields. If F = R , such spaces are classified by dimension and signature. If thesignature is of type ( p, q ), the associated group is denoted GSp( p, q ). The only isomorphisms betweenthese are GSp( p, q ) ≃ GSp( q, p ).2.1.3.3. Global fields. The Hasse principle holds in this case, so a global B -hermitian space is deter-mined up to isometry by the collection of corresponding local B v -Hermitian spaces. Conversely, givenany collection of B v -hermitian spaces, there is a (unique) B -Hermitian space that gives rise to thislocal collection up to isometry.2.1.4. Quaternionic skew-hermitian spaces.
These are defined similarly to B -hermitian spaces but withthe condition h v, v ′ i = −h v ′ , v i ∗ . To such a space V is associated the invariant det( V ) ∈ F × / ( F × ) as follows. Pick a B -basis { v i } for V and set det( V ) = ν B ( h v i , v j i ) . Here ν B denotes the reduced norm. (Often, we will omit the subscript B when the choice of quaternionalgebra is clear.) The group GU( V ) is defined similarly as above. It is however not connected as analgebraic group. We now recall the classification of such spaces and the associated groups. Note thatif B is split, we can associate to V a quadratic space V † over F of dimension 2 n (where n = dim B ( V ))and GU( V ) ≃ GO( V † ).2.1.4.1. p -adic fields. Let F be p -adic. If B is split, V is determined by dim( V ), det( V ) and the Hasseinvariant of V † . If B is nonsplit, V is determined by dim( V ) and det( V ).2.1.4.2. Archimedean fields. If F = R and B is split, V is determined by the signature of V † . Thegroup GU( V ) is isomorphic to GO( p, q ) where ( p, q ) is the signature. If B is nonsplit, V is determinedjust by n = dim B ( V ). The group GU( V ) is isomorphic to GO ∗ (2 n ). If F = C , then B must be splitand there is a unique skew-hermitian space of any given dimension. Then GU( V ) ≃ GO(2 n, C ). F be a number field. If B is split, then the classification reduces to thatfor quadratic spaces via the assignment V V † . In this case, the Hasse principle holds. If B isnonsplit, then the Hasse principle does not hold. Let Σ B be the set of places v where B is ramifiedand let s = | Σ B | . The space V gives rise to a collection of local spaces and up to isometry there areexactly 2 s − global B -skew-hermitian spaces that give rise to the same set of local spaces. Converselya collection of local B v -skew-hermitian spaces V v arises from a global B -skew-hermitian space V ifand only if there exists a global element d ∈ F × such that det( V v ) = d in F × v / ( F × v ) for all v and foralmost all v , the Hasse invariant of V † v is trivial.2.2. The key constructions.
In this section, we assume that B and B are two quaternion algebrasover a number field F and E/F is a quadratic extension that embeds in both B and B . We willfix embeddings E ֒ → B and E ֒ → B . Via these embeddings, B and B are hermitian spaces over E . Let τ i and ν i be respectively the reduced trace and norm on B i . We think of B and B as right E -vector spaces, the Hermitian form being described below. Write B = E + E j = E + j E, B = E + E j = E + j E, where τ ( j ) = τ ( j ) = 0. We write pr i for the projection B i → E onto the “first coordinate” and ∗ i for the main involution on B i . Then B i is a right Hermitian E -space, the form being given by:( x, y ) i = pr i ( x ∗ i y ) . If x = a + j i b , y = c + j i d , then( x, y ) i = ( a + j i b, c + j i d ) i = a ρ c − J i b ρ d, where J i := − ν i ( j i ) = j i . This form satisfies the relations( xα, yβ ) i = α ρ ( x, y ) i β, for α, β ∈ E, and ( x, y ) i = ( y, x ) ρi . We note that B × i acts naturally on B i by left multiplication, and this action is E -linear. Further,(2.1) ( α x, α y ) i = ν i ( α )( x, y ) i for all α ∈ B i . Thus B × i embeds naturally in GU E ( B i ). In fact, F × \ ( B × i × E × ) ≃ GU E ( B i ) , where E × acts on B i by right multiplication, and we think of F × as a subgroup of B × i × E × via λ ( λ − , λ ).Consider the (right) E -vector space V := B ⊗ E B . Remark . If x ∈ B , y ∈ B , α ∈ E , then by definition,( x ⊗ y ) α = xα ⊗ y = x ⊗ yα. The E -vector space V is equipped with a natural Hermitian form given by the tensor product( · , · ) ⊗ ( · , · ) . We fix a nonzero element i ∈ E of trace 0, and define ( · , · ) on V by(2.2) ( · , · ) := i · ( · , · ) ⊗ ( · , · ) . Clearly, ( · , · ) satisfies ( xα, yβ ) = α ρ ( x, y ) β, for α, β ∈ E, ( x, y ) = − ( y, x ) ρ . Thus ( · , · ) is an E -skew-Hermitian form on V . It will be useful to write down the form ( · , · ) explicitly in terms of coordinates with respect to asuitable E -basis. We pick the following (orthogonal) basis:(2.3) e := 1 ⊗ e := j ⊗ e := 1 ⊗ j e := j ⊗ j . In this basis,( e a + e b + e c + e d, e a ′ + e b ′ + e c ′ + e d ′ ) = i · [ a ρ a ′ − J b ρ b ′ − J c ρ c ′ + J J d ρ d ′ ] . There is a natural map GU E ( B ) × GU E ( B ) → GU E ( V ) , given by the actions of GU E ( B ) and GU E ( B ) on the first and second component respectively of V = B ⊗ E B . The kernel of this map isZ := (cid:8) ([ λ , α ] , [ λ , α ]) : λ i ∈ F × , α i ∈ E × , λ λ α α = 1 (cid:9) . Let B := B · B be the product in the Brauer group over F . Then E embeds in B as well, and wewill fix an embedding E → B . We may write B = E + E j where τ ( j ) = 0 and J := − ν ( j ) = j satisfies J = J J . Here τ and ν are respectively the reduced trace and reduced norm on B . We define a right action of B on V (extending the right E -action on V ) by setting(1 ⊗ · j = j ⊗ j (2.4) ( j ⊗ · j = J (1 ⊗ j )(2.5) (1 ⊗ j ) · j = J ( j ⊗ j ⊗ j ) · j = J J (1 ⊗ j on V to be conjugate E -linear. (It is straightforward to check thatthis gives an action.) Then V is a free rank-2 right B -module. For example, a basis for V as a right B -module is given either by { ⊗ , j ⊗ } or { ⊗ , ⊗ j } . Further, one checks that (equivalently) ( x j , y ) = ( y j , x )(2.8) ( x j , y j ) ρ = − J ( x, y )(2.9)for all x, y ∈ V .We will now show that there is a B -valued skew-Hermitian form h· , ·i on V such thatpr ◦h· , ·i = ( · , · ) . Indeed, define(2.10) h x, y i = ( x, y ) − J · j · ( x j , y ) . It may be checked (using (2.8) and (2.9)) that h x α , y β i = α ∗ h x, y i β , (2.11) h x, y i = −h y, x i ∗ , (2.12)for all α , β ∈ B . For future reference, we write down the matrix of inner products h e i , e j i . h· , ·i e e e e e i ije − J i − ij e − ij − J i e ij J i From the table, we see that det( V ) = ν ( − J u ) = 1 in F × / ( F × ) .Notice that B × and B × act on V by left multiplication on the first and second factor of V = B ⊗ E B respectively. These actions are (right) E -linear and in fact (right) B -linear as is easily checked using(2.4) through (2.7). Further, it follows from (2.1), (2.2), and (2.10) that(2.13) h α i · x, α i · y i = ν i ( α i ) h x, y i for α i ∈ B × i . Clearly, the actions of B × and B × commute, hence one gets an embedding(2.14) F × \ ( B × × B × ) ֒ → GU B ( V ) , the quaternionic unitary group of the B -skew-Hermitian form h· , ·i . (Here we think of F × as embeddedantidiagonally in B × × B × via λ ( λ − , λ ).) Then (2.14) gives an isomorphism F × \ ( B × × B × ) ≃ GU B ( V ) , where GU B ( V ) denotes the identity component of GU B ( V ). Further, one has a commutative diagram F × \ ( B × × B × ) ≃ / / (cid:127) _ (cid:15) (cid:15) GU B ( V ) (cid:127) _ (cid:15) (cid:15) Z \ (GU E ( B ) × GU E ( B )) / / GU E ( V )where the vertical map F × \ ( B × × B × ) ֒ → Z \ (GU E ( B ) × GU E ( B ))is [ b , b ] [[( b , , [( b , , and the vertical map GU B ( V ) ֒ → GU E ( V ) is just the natural inclusion.Let V = Res E/F ( V ), that is V is just V thought of as an F -space, with non-degenerate symplecticform hh v , v ii := 12 tr E/F ( v , v ) . Let X = F e ⊕ F e ⊕ F e ⊕ F e ⊂ V . Since X is maximal isotropic for hh· , ·ii , there exists a unique maximal isotropic subspace Y in V , suchthat V = X ⊕ Y . Let ( e ∗ , . . . , e ∗ ) be an F -basis for Y that is dual to ( e , . . . , e ). We can identify thisbasis precisely: letting i = u ∈ F × , we have(2.15) e ∗ = 1 u e i , e ∗ = − J u e i , e ∗ = − J u e i , e ∗ = 1 Ju e i . The unitary group U E ( V ) at infinite places. This section will not be relevant in this paper. Wesimply record for future use the isomorphism class of the unitary group U E ( V ) at the infinite places v assuming that F v = R and E v = C . The skew-hermitian form is given by i · [ a ρ a ′ − J b ρ b ′ − J c ρ c ′ + Jd ρ d ′ ]Thus we have the following table which summarizes the relation between the ramification of B and B at v and the isomorphism class of U E ( V ). B , B J , J U E ( V )split, split J > , J > , J < , J > , J > , J < , J < , J < , The failure of the Hasse principle.
The constructions above illustrate the failure of the Hasseprinciple for skew-hermitian B -spaces. Indeed, let us fix a B and consider pairs ( B , B ) such thatΣ B ∩ Σ B = Σ , where Σ is some fixed set of places not intersecting Σ B . Let E/F be a quadraticextension that is nonsplit at all the places in Σ B ∪ Σ . Then E embeds in B , B , B , so the constructionsfrom the previous section apply. The various spaces V obtained by taking different choices of B and B are all locally isometric, since all of them have det( V ) = 1 and the Hasse invariant of V † v is independentof V for v Σ B . Since interchanging B and B gives an isometric global space, the number of differentglobal spaces obtained in this way (up to isometry) is exactly 2 s − , where s = | Σ B | .Conversely, suppose that we are given a quaternion division algebra B and a collection of local B v -skew-hermitian spaces V v such that det( V v ) = 1 for all v and the Hasse invariant of V † v (for v Σ B ) is1 for all but finitely many v . Then there are up to isometry 2 s − different global skew-hermitian spacesthat give rise to this collection of local spaces, and all of them may be obtained by the constructionabove, by suitably choosing B , B and i .3. Theta correspondences
Preliminaries.
Weil indices.
Let F be a local field of characteristic not 2 and fix a non-trivial additive character ψ of F . For a non-degenerate symmetric F -bilinear form q , we let γ F ( ψ ◦ q ) ∈ µ denote the Weilindex associated to the character of second degree x ψ ( q ( x, x )) (see [73], [61, Appendix]). When q ( x, y ) = xy for x, y ∈ F , we write γ F ( ψ ) := γ F ( ψ ◦ q ). Put γ F ( a, ψ ) := γ F ( aψ ) γ F ( ψ ) for a ∈ F × , where ( aψ )( x ) := ψ ( ax ) for x ∈ F . Then we have γ F ( ab , ψ ) = γ F ( a, ψ ) ,γ F ( ab, ψ ) = γ F ( a, ψ ) · γ F ( b, ψ ) · ( a, b ) F ,γ F ( a, bψ ) = γ F ( a, ψ ) · ( a, b ) F ,γ F ( a, ψ ) = ( − , a ) F ,γ F ( a, ψ ) = 1 ,γ F ( ψ ) = γ F ( − , ψ ) − ,γ F ( ψ ) = 1for a, b ∈ F × (see [61, p. 367]). Here ( · , · ) F is the quadratic Hilbert symbol of F .Let q be a non-degenerate symmetric F -bilinear form. Let det q ∈ F × / ( F × ) and h F ( q ) ∈ {± } denote the determinant and the Hasse invariant of q . For example, when q ( x, y ) = a x y + · · · + a m x m y m for x = ( x , . . . , x m ), y = ( y , . . . , y m ) ∈ F m , thendet q = Y ≤ i ≤ m a i , h F ( q ) = Y ≤ i Let V be a 2 n -dimensional F -vector space with a non-degenerate symplecticform hh· , ·ii : V × V → F . For maximal isotropic subspaces Y , Y ′ , Y ′′ of V , the Leray invariant q ( Y , Y ′ , Y ′′ ) is a non-degenerate symmetric F -bilinear form defined as follows. (See also Definitions2.4 and 2.10 of [61].)Suppose first that Y , Y ′ , Y ′′ are pairwise transverse. Let P Y be the maximal parabolic subgroup ofSp( V ) stabilizing Y and let N Y be its unipotent radical. By Lemma 2.3 of [61], there exists a unique g ∈ N Y such that Y ′ g = Y ′′ . We write g = (cid:18) b (cid:19) , b ∈ Hom( Y ′ , Y )with respect to the complete polarization V = Y ′ + Y . Then q = q ( Y , Y ′ , Y ′′ ) is a symmetric bilinearform on Y ′ defined by q ( x ′ , y ′ ) := hh x ′ , y ′ b ii . In general, we consider a symplectic space V R := R ⊥ / R , where R := ( Y ∩ Y ′ ) + ( Y ′ ∩ Y ′′ ) + ( Y ′′ ∩ Y ) , and maximal isotropic subspaces Y R := ( Y ∩ R ⊥ ) / R , Y ′ R := ( Y ′ ∩ R ⊥ ) / R , Y ′′ R := ( Y ′′ ∩ R ⊥ ) / R of V R . By Lemma 2.9 of [61], Y R , Y ′ R , Y ′′ R are pairwise transverse. We put q ( Y , Y ′ , Y ′′ ) := q ( Y R , Y ′ R , Y ′′ R ) . By Theorem 2.11 of [61], we have q ( Y g, Y ′ g, Y ′′ g ) = q ( Y , Y ′ , Y ′′ ) for g ∈ Sp( V ).3.2. Weil representation for Mp . Heisenberg group, Heisenberg representations. Let F be a local field of characteristic not 2. Forsimplicity, we assume that F is non-archimedean.Let V be a finite dimensional vector space over F equipped with a non-degenerate symplectic form hh· , ·ii : V × V −→ F. The Heisenberg group H ( V ) is defined by H ( V ) := V ⊕ F as a set, with group law ( v , z ) · ( v , z ) = (cid:18) v + v , z + z + 12 hh v , v ii (cid:19) . The center of H ( V ) is F .Let ψ be a nontrivial additive character of F . Then by the Stone–von Neumann theorem, H ( V )admits a unique (up to isomorphism) irreducible representation ρ ψ on which F acts via ψ . Thisrepresentation can be realized as follows. Fix a complete polarization V = X ⊕ Y , i.e., X and Y are maximal isotropic subspaces of V . We construct a character ψ Y of H ( Y ) = Y ⊕ F bysetting ψ Y ( y, z ) = ψ ( z ) . Define S Y := Ind H ( V ) H ( Y ) ψ Y . i.e. S Y is the space of functions f : H ( V ) → C satisfying(i) f (˜ y ˜ v ) = ψ Y (˜ y ) f (˜ v ) for ˜ y ∈ H ( Y ), ˜ v ∈ H ( V ).(ii) f is smooth i.e. there exists an open compact subgroup (a lattice !) L in V such that f (˜ vℓ ) = f (˜ v ) for all ℓ ∈ L ⊂ V ⊂ H ( V ) . Then H ( V ) acts on S Y on the right naturally. We can identify S Y with S ( X ), the Schwartz space oflocally constant functions with compact support on X , via the restriction of functions to X .3.2.2. Metaplectic group. Let Sp( V ) be the symplectic group of V . Following Weil, we let Sp( V ) acton V on the right. Recall that Sp( V ) acts on H ( V ) by( v, z ) g := ( vg, z ) . Let f Sp( V ) be the unique non-trivial 2-fold central extension of Sp( V ). The metaplectic group Mp( V )is a central extension 1 −→ C −→ Mp( V ) −→ Sp( V ) −→ V ) := f Sp( V ) × {± } C . Lemma 3.1. Any automorphism of Mp( V ) which lifts the identity map of Sp( V ) and which restrictsto the identity map of C must be the identity map of Mp( V ) . Proof. Let p : Mp( V ) → Sp( V ) be the projection. Let f : Mp( V ) → Mp( V ) be such an automorphism.Since p ( f ( g )) · p ( g ) − = 1 for g ∈ Mp( V ), there exists a character κ : Mp( V ) → C such that f ( g ) · g − = κ ( g ). Since f ( z ) = z for z ∈ C , κ is trivial on C , and hence induces a character ofSp( V ). Since [Sp( V ) , Sp( V )] = Sp( V ), this character must be trivial. (cid:3) One can realize Mp( V ) explicitly as follows. Put z Y ( g , g ) = γ F ( 12 ψ ◦ q ( Y , Y g − , Y g ))for g , g ∈ Sp( V ). By Theorem 4.1 of [61], z Y is a 2-cocycle, and the groupMp( V ) Y := Sp( V ) × C with group law ( g , z ) · ( g , z ) = ( g g , z z · z Y ( g , g ))is isomorphic to Mp( V ). Moreover, by Lemma 3.1, this isomorphism is canonical. If there is noconfusion, we identify Mp( V ) Y with Mp( V ).Let V = X ′ ⊕ Y ′ be another complete polarization. Lemma 3.2. We have z Y ′ ( g , g ) = λ ( g g ) λ ( g ) − λ ( g ) − · z Y ( g , g ) , where λ : Sp( V ) → C is given by λ ( g ) := γ F ( 12 ψ ◦ q ( Y , Y ′ g − , Y ′ )) · γ F ( 12 ψ ◦ q ( Y , Y ′ , Y g )) . In particular, the bijection Mp( V ) Y −→ Mp( V ) Y ′ ( g, z ) ( g, z · λ ( g )) is an isomorphism.Proof. See Lemma 4.2 of [39]. (cid:3) Suppose that V = V ⊕ V , where each V i is a non-degenerate symplectic subspace. One can liftthe natural embedding Sp( V ) × Sp( V ) ֒ → Sp( V ) to a homomorphismMp( V ) × Mp( V ) −→ Mp( V ) . If V i = X i ⊕ Y i is a complete polarization and X = X ⊕ X , Y = Y ⊕ Y , then this homomorphism is given byMp( V ) Y × Mp( V ) Y −→ Mp( V ) Y , (( g , z ) , ( g , z )) ( g g , z z )i.e., we have z Y ( g , g ′ ) · z Y ( g , g ′ ) = z Y ( g g , g ′ g ′ )for g i , g ′ i ∈ Sp( V i ) (see Theorem 4.1 of [61]). Let L be a self-dual lattice in V and let K be the stabiliser of L in Sp( V ). If the residual characteristicof F is not 2, then there exists a splitting Mp( V ) (cid:15) (cid:15) K / / < < ②②②②②②②②② Sp( V ) . Moreover, if the residue field of F has at least four elements, then [ K, K ] = K (see Lemma 11.1 of[53]), and hence such a splitting is unique. In the next section, we shall describe this splitting by usingthe Schr¨odinger model.3.2.3. Weil representation, Schr¨odinger model. Recall that ρ ψ is the unique (up to isomorphism) irre-ducible smooth representation of H ( V ) with central character ψ . Let S be the underlying space of ρ ψ .The Weil representation ω ψ of Mp( V ) on S is a smooth representation characterized by the followingproperties: • ρ ψ ( h g ) = ω ψ ( g ) − ρ ψ ( h ) ω ψ ( g ) for all h ∈ H ( V ) and g ∈ Mp( V ). • ω ψ ( z ) = z · id S for all z ∈ C .One can realize ω ψ explicitly as follows. We regard V = F n as the space of row vectors. Fix acomplete polarization V = X ⊕ Y . Choose a basis e , . . . , e n , e ∗ , . . . , e ∗ n of V such that X = F e + · · · + F e n , Y = F e ∗ + · · · + F e ∗ n , hh e i , e ∗ j ii = δ ij . Then we have Sp( V ) = (cid:26) g ∈ GL n ( F ) | g (cid:18) n − n (cid:19) t g = (cid:18) n − n (cid:19)(cid:27) . The Weil representation ω ψ of Mp( V ) Y on the Schwartz space S ( X ) is given as follows: ω ψ (cid:18)(cid:18) a t a − (cid:19) , z (cid:19) ϕ ( x ) = z · | det a | / · ϕ ( xa ) ω ψ (cid:18)(cid:18) n b n (cid:19) , z (cid:19) ϕ ( x ) = z · ψ (cid:18) xb t x (cid:19) · ϕ ( x ) ω ψ (cid:18)(cid:18) n − n (cid:19) , z (cid:19) ϕ ( x ) = z · Z F n ϕ ( y ) ψ ( x t y ) dy. for ϕ ∈ S ( X ), x ∈ X ∼ = F n , a ∈ GL( X ) ∼ = GL n ( F ), b ∈ Hom( X , Y ) ∼ = M n ( F ) with t b = b , and z ∈ C .More generally, for ( g, z ) ∈ Mp( V ) Y with g = (cid:0) a bc d (cid:1) ∈ Sp( V ), we have ω ψ ( g, z ) ϕ ( x ) = z · Z F n / ker( c ) ψ (cid:18) 12 ( xa, xb ) + ( xb, yc ) + 12 ( yc, yd ) (cid:19) ϕ ( xa + yc ) dµ g ( y ) , where ( x, y ) = x t y for row vectors x, y ∈ F n , and the measure dµ g ( y ) on F n / ker( c ) is normalized sothat this operator is unitary (see [40, Proposition 2.3]). In particular, if det c = 0, then ω ψ ( g, ϕ ( x ) is equal to Z F n ψ (cid:18) 12 ( xa, xb ) + ( xb, yc ) + 12 ( yc, yd ) (cid:19) ϕ ( xa + yc ) dµ g ( y )= | det c | − · ψ (cid:18) 12 ( xa, xb ) (cid:19) · Z F n ψ (cid:18) ( xb, y − xa ) + 12 ( y − xa, yc − d − xac − d ) (cid:19) ϕ ( y ) dµ g ( y )= | det c | − · ψ (cid:18) 12 ( xa, x ( ac − d − b )) (cid:19) · Z F n ψ (cid:18) ( y, x ( b − ac − d )) + 12 ( y, yc − d ) (cid:19) ϕ ( y ) dµ g ( y )= | det c | − · ψ (cid:18) 12 ( xa, x t c − ) (cid:19) · Z F n ψ (cid:18) − ( y, x t c − ) + 12 ( y, yc − d ) (cid:19) ϕ ( y ) dµ g ( y )= | det c | − / · ψ (cid:18) 12 ( xac − , x ) (cid:19) · Z F n ψ (cid:18) − ( x t c − , y ) + 12 ( yc − d, y ) (cid:19) ϕ ( y ) dy, where dy is the self-dual Haar measure on F n with respect to the pairing ψ ◦ ( · , · ).If the residual characteristic of F is not 2, let K be the stabilizer of the self-dual lattice o e + · · · + o e n + o e ∗ + · · · + o e ∗ n . Then the splitting K → Mp( V ) Y is given as follows. Assume that ψ is of order zero. Let ϕ ∈ S ( X )be the characteristic function of o e + · · · + o e n ∼ = o n . Since the residual characteristic of F is not 2,we see that ω ψ ( k, ϕ = ϕ for k = (cid:18) a t a − (cid:19) , (cid:18) n b n (cid:19) and (cid:18) n − n (cid:19) , where a ∈ GL n ( o ) and b ∈ M n ( o ). Since these elements generate K , there exists a function s Y : K → C such that ω ψ ( k, ϕ = s Y ( k ) − · ϕ for all k ∈ K. Thus we obtain z Y ( k , k ) = s Y ( k k ) s Y ( k ) − s Y ( k ) − for k , k ∈ K , so that the map k ( k, s Y ( k )) is the desired splitting.3.2.4. Change of models. Suppose V = X ′ ⊕ Y ′ is another polarization of V . Then likewise the rep-resentation ρ ψ can be realized on S Y ′ ≃ S ( X ′ ). We will need an explicit isomorphism between theserepresentations of H ( V ). At the level of the induced representations, this is given by the map S Y ′ → S Y , f ′ ff (˜ v ) = Z Y ∩ Y ′ \ Y f ′ (( y, · ˜ v ) · ψ Y ( y, − dy = Z Y ∩ Y ′ \ Y f ′ (( y, · ˜ v ) dy. For now, we will take any Haar measure on Y to define this. We will fix this more carefully later.Let us now write down this isomorphism in terms of Schwartz spaces. Lemma 3.3. Suppose that ϕ ∈ S ( X ) and ϕ ′ ∈ S ( X ′ ) correspond to f ∈ S Y and f ′ ∈ S Y ′ respectively.Then we have (3.2) ϕ ( x ) = Z Y ∩ Y ′ \ Y ψ (cid:18) hh x ′ , y ′ ii − hh x, y ii (cid:19) ϕ ′ ( x ′ ) dy, where x ′ = x ′ ( x, y ) ∈ X ′ and y ′ = y ′ ( x, y ) ∈ Y ′ are given by x ′ + y ′ = x + y ∈ V . Proof. Let ϕ ′ ∈ S ( X ′ ). Let ( x ′ + y ′ , z ) ∈ H ( V ). We write this as:( x ′ + y ′ , z ) = (cid:18) y ′ , z − hh y ′ , x ′ ii (cid:19) · ( x ′ , . Thus if f ′ ∈ S Y ′ corresponds to ϕ ′ , then f ′ ( x ′ + y ′ , z ) = ψ (cid:18) z − hh y ′ , x ′ ii (cid:19) · ϕ ′ ( x ′ ) . We can rewrite this as: (with v = x ′ + y ′ ) f ′ ( v, z ) = ψ (cid:18) z − hh v, x ′ ii (cid:19) · ϕ ′ ( x ′ ) . Thus f ′ corresponds to f ∈ S Y given by f ( x + y, z ) = Z Y ∩ Y ′ \ Y f ′ (( y , · ( x + y, z )) dy = Z Y ∩ Y ′ \ Y f ′ (cid:18) x + y + y , z + 12 hh y , x ii (cid:19) dy . Thus ϕ ( x ) = Z Y ∩ Y ′ \ Y f ′ (cid:18) x + y , hh y , x ii (cid:19) dy = Z Y ∩ Y ′ \ Y ψ (cid:18) hh y , x ii (cid:19) f ′ ( x + y , dy = Z Y ∩ Y ′ \ Y ψ (cid:18) − hh x + y , x ′ ii + 12 hh y , x ii (cid:19) ϕ ′ ( x ′ ) dy . (cid:3) Thus we obtain an H ( V )-equivariant isomorphism S ( X ′ ) ∼ = S ( X ) defined by the partial Fouriertransform (3.2). Using the characterization of the Weil representation of Mp( V ), one sees that thisisomorphism is also Mp( V )-equivariant.The isomorphism S ( X ′ ) ∼ = S ( X ) is in fact a partial Fourier transform. To see this, using [61, Lemma2.2], we choose a basis e , . . . , e n , e ∗ , . . . , e ∗ n of V such that X = F e + · · · + F e n , X ′ = F e + · · · + F e k + F e ∗ k +1 + · · · + F e ∗ n , Y = F e ∗ + · · · + F e ∗ n , Y ′ = F e ∗ + · · · + F e ∗ k + F e k +1 + · · · + F e n , and hh e i , e ∗ j ii = δ ij , where k = dim( Y ∩ Y ′ ). In particular, we have Y ∩ Y ′ = F e ∗ + · · · + F e ∗ k .Let ϕ ∈ S ( X ) and ϕ ′ ∈ S ( X ′ ) be as in (3.2). We also regard ϕ ′ as a function on F n via the basis e , . . . , e k , e ∗ k +1 , . . . , e ∗ n . Write x + y = x ′ + y ′ with x ∈ X , y ∈ Y , x ′ ∈ X ′ , y ′ ∈ Y ′ . If we write x = x e + · · · + x n e n , y = y e ∗ + · · · + y n e ∗ n with x i , y j ∈ F , then x ′ = x e + · · · + x k e k + y k +1 e ∗ k +1 + · · · + y n e ∗ n ,y ′ = y e ∗ + · · · + y k e ∗ k + x k +1 e k +1 + · · · + x n e n , and hh x, y ii = x y + · · · + x n y n , hh x ′ , y ′ ii = x y + · · · + x k y k − x k +1 y k +1 − · · · − x n y n . Hence we have ϕ ( x ) = Z Y ∩ Y ′ \ Y ψ (cid:18) 12 ( hh x ′ , y ′ ii − hh x, y ii ) (cid:19) ϕ ′ ( x ′ ) dy = Z F n − k ψ ( − x k +1 y k +1 − · · · − x n y n ) ϕ ′ ( x , . . . , x k , y k +1 , . . . , y n ) dy k +1 · · · dy n . Over global fields. In this section, let F be a number field with ring of adeles A . Let V be asymplectic space over F . The global metaplectic group Mp( V ) A is defined as follows.Fix a lattice L in V . For each finite place v , let K v be the stabilizer of L v in Sp( V v ). For almostall v , L v is self-dual and there exists a unique splitting K v ֒ → Mp( V v ), in which case we identify K v with its image in Mp( V v ).For a finite set S of places of F including all archimedean places, we define a central extension1 −→ C −→ Mp( V ) S −→ Y v ∈ S Sp( V v ) −→ V ) S := Y v ∈ S Mp( V v ) ! / ( ( z v ) ∈ Y v ∈ S C | Y v ∈ S z v = 1 ) . Put K S := Q v / ∈ S K v . If S ⊂ S ′ are sufficiently large, the splitting K v ֒ → Mp( V v ) induces an embeddingMp( V ) S × K S ֒ → Mp( V ) S ′ × K S ′ . Then Mp( V ) A is defined by Mp( V ) A := lim −→ S (cid:0) Mp( V ) S × K S (cid:1) . There exists a unique splitting Mp( V ) A (cid:15) (cid:15) Sp( V )( F ) / / qqqqqqqqqq Sp( V )( A )given as follows. Fix a complete polarization V = X ⊕ Y over F . Recall that the metaplectic groupMp( V v ) = Sp( V v ) × C is determined by the 2-cocycle z Y v . Moreover, for almost all v , there exists afunction s Y v : K v → C such that z Y v ( k , k ) = s Y v ( k k ) s Y v ( k ) − s Y v ( k ) − for k , k ∈ K v . Lemma 3.4. Let γ ∈ Sp( V )( F ) . Then we have s Y v ( γ ) = 1 for almost all v .Proof. By the Bruhat decomposition, we may write γ = p wp with some p i = (cid:18) a i b it a − i (cid:19) , w = k n − k − k n − k , where a i ∈ GL n ( F ) and b i ∈ M n ( F ). By Theorem 4.1 of [61], we have z Y v ( p , g ) = z Y v ( g, p ) = 1for all v and g ∈ Sp( V v ), so that( p wp , 1) = ( p , · ( w, · ( p , 1) in Mp( V v ) . On the other hand, for almost all v , we have p i ∈ K v and ω ψ v ( p i , ϕ v = ϕ v , ω ψ v ( w, ϕ v = ϕ v , where ψ v is a non-trivial character of F v of order zero and ϕ v is the characteristic function of o nv . Thuswe obtain ω ψ v ( γ, ϕ v = ω ψ v ( p , ω ψ v ( w, ω ψ v ( p , ϕ v = ϕ v for almost all v . (cid:3) For γ ∈ Sp( V )( F ), let ( γ, 1) be the element in Q v Mp( V v ) such that ( γ, v = ( γ, 1) for all v . ByLemma 3.4, we have ( γ, v ∈ K v for almost all v . Hence, if S is a sufficiently large finite set of placesof F , then ( γ, 1) maps to an element i ( γ ) in Mp( V ) S × K S . Lemma 3.5. The map i : Sp( V )( F ) −→ Mp( V ) A is a homomorphism.Proof. Let γ , γ ∈ Sp( V )( F ). For each v , we have( γ , v · ( γ , v = ( γ γ , z Y v ( γ , γ )) in Mp( V v ) . Choose a finite set S of places of F such that γ , γ ∈ K v , s Y v ( γ ) = s Y v ( γ ) = s Y v ( γ γ ) = 1for v / ∈ S . Then we have z Y v ( γ , γ ) = 1for v / ∈ S . Moreover, by the product formula for the Weil index, we have Y v ∈ S z Y v ( γ , γ ) = 1 . Hence the image of ( γ , · ( γ , 1) in Mp( V ) S × K S is equal to i ( γ γ ). (cid:3) Fix a non-trivial additive character ψ of A /F . We have the global Weil representation ω ψ of Mp( V ) A on the Schwartz space S ( X ( A )). For each ϕ ∈ S ( X ( A )), the associated theta function on Mp( V ) A isdefined by Θ ϕ ( g ) := X x ∈ X ω ψ ( g ) ϕ ( x ) . Then Θ ϕ is a left Sp( V )( F )-invariant slowly increasing smooth function on Mp( V ) A .3.3. Reductive dual pairs. In this section, we consider the reductive dual pair (GU( V ) , GU( W )),where V is a skew-Hermitian right B -space of dimension two and W is a Hermitian left B -space ofdimension one. Reductive dual pairs; examples. Recall that in § B -space V = B ⊗ E B with the skew-Hermitian form h· , ·i : V × V → B . Let W = B be the 1-dimensional Hermitian left B -space with the Hermitian form h· , ·i : W × W → B defined by h x, y i = xy ∗ . These forms satisfy that h v α , v ′ β i = α ∗ h v, v ′ i β h v ′ , v i = −h v, v ′ i ∗ h α w, β w ′ i = α h w, w ′ i β ∗ h w ′ , w i = h w, w ′ i ∗ for v, v ′ ∈ V , w, w ′ ∈ W and α , β ∈ B . We let GL( V ) act on V on the left and let GL( W ) act on W on the right. Let GU( V ) and GU( W ) be the similitude groups of V and W with the similitudecharacters ν : GU( V ) → F × and ν : GU( W ) → F × respectively:GU( V ) := { g ∈ GL( V ) | h gv, gv ′ i = ν ( g ) · h v, v ′ i for all v, v ′ ∈ V } , GU( W ) := { g ∈ GL( W ) | h wg, w ′ g i = ν ( g ) · h w, w ′ i for all w, w ′ ∈ W } . Let U( V ) := ker ν and U( W ) := ker ν be the unitary groups of V and W respectively.Put V := V ⊗ B W. Then V is an F -space equipped with a symplectic form hh· , ·ii := 12 tr B/F ( h· , ·i ⊗ h· , ·i ∗ ) . If we identify Res B/F ( V ) with V via the map v v ⊗ 1, then the associated symplectic form on V isgiven by hh· , ·ii = 12 tr B/F h· , ·i = 12 tr E/F ( · , · ) , where ( · , · ) = pr ◦h· , ·i is the associated E -skew-Hermitian form. We let GL( V ) × GL( W ) act on V onthe right: ( v ⊗ w ) · ( g, h ) := g − v ⊗ wh. This gives a natural homomorphismG(U( V ) × U( W )) −→ Sp( V ) , where G(U( V ) × U( W )) := { ( g, h ) ∈ GU( V ) × GU( W ) | ν ( g ) = ν ( h ) } . Splittings. Recall that V = e E + e E + e E + e E. Let V = X + Y be the complete polarization given by X = F e + F e + F e + F e , Y = F e ∗ + F e ∗ + F e ∗ + F e ∗ . We first suppose that F is a local field. In Appendix C, we define a function s : G(U( V ) × U( W )) −→ C such that z Y ( g , g ) = s ( g g ) s ( g ) − s ( g ) − , so that the map ι : G(U( V ) × U( W )) −→ Mp( V ) Y g ( g, s ( g )) is a homomorphism. Thus we have a commutative diagramMp( V ) (cid:15) (cid:15) G(U( V ) × U( W )) ι ♥♥♥♥♥♥♥♥♥♥♥♥ / / Sp( V ) . If V = X ′ + Y ′ is another complete polarization, we choose g ∈ Sp( V ) such that Y ′ = Y g and definea function s ′ : G(U( V ) × U( W )) −→ C by s ′ ( g ) = s ( g ) · z Y ( g , gg − ) · z Y ( g, g − )= s ( g ) · z Y ( g gg − , g ) − · z Y ( g , g ) . By Lemma 3.2, we have z Y ′ ( g , g ) = s ′ ( g g ) s ′ ( g ) − s ′ ( g ) − , so that the map G(U( V ) × U( W )) −→ Mp( V ) Y ′ g ( g, s ′ ( g ))is a homomorphism.We next suppose that F is a number field. For each place v of F , we have defined a function s v : G(U( V v ) × U( W v )) −→ C with associated homomorphism ι v : G(U( V v ) × U( W v )) −→ Mp( V v ) . Lemma 3.6. The homomorphisms ι v induce a homomorphism ι : G(U( V ) × U( W )) ( A ) −→ Mp( V ) A . Moreover, the diagram G(U( V ) × U( W )) ( F ) (cid:15) (cid:15) (cid:31) (cid:127) / / G(U( V ) × U( W )) ( A ) ι (cid:15) (cid:15) Sp( V )( F ) i / / Mp( V ) A is commutative.Proof. Recall that, for almost all v , K v is the maximal compact subgroup of Sp( V v ) and s Y v : K v → C is the function which defines the splitting K v ֒ → Mp( V v ). Put K v := G(U( V v ) × U( W v )) ∩ K v . Then K v is a maximal compact subgroup of G(U( V v ) × U( W v )) for almost all v . By Lemma C.19,we have s v | K v = s Y v | K v for almost all v . Hence, for g = ( g v ) ∈ G(U( V ) × U( W )) ( A ), the element ( ι v ( g v )) ∈ Q v Mp( V v ) mapsto an element ι ( g ) in Mp( V ) S × K S if S is sufficiently large. This proves the first assertion. Let γ ∈ G(U( V ) × U( W )) ( F ). By Proposition C.20, we have Y v s v ( γ ) = 1 . Hence, if S is sufficiently large, the image of ( ι v ( γ )) in Mp( V ) S × K S is equal to that of ( γ, (cid:3) Weil representation for the above dual reductive pair. If F is local, we get the Weil representation ω ψ ◦ ι of G(U( V ) × U( W )) on S ( X ), where ω ψ is the Weil representation of Mp( V ) Y and ι : G(U( V ) × U( W )) → Mp( V ) Y is the above homomorphism. Similarly, if F is global, we get the global Weilrepresentation ω ψ ◦ ι of G(U( V ) × U( W )) ( A ) on S ( X ( A )). If there is no confusion, we suppress ι fromthe notation.4. The Rallis inner product formula and the Jacquet–Langlands correspondence Setup. Let F be a number field and B a quaternion algebra over F . As in Appendix D, weconsider the following spaces: • V = B ⊗ E B is the 2-dimensional right skew-hermitian B -space. • W = B is the 1-dimensional left hermitian B -space. • W (cid:3) = W ⊕ W is the 2-dimensional left hermitian B -space. • V = V ⊗ B W is the 8-dimensional symplectic F -space. • V (cid:3) = V ⊗ B W (cid:3) = V ⊕ V is the 16-dimensional F -space. • W (cid:3) = W ▽ ⊕ W △ is the complete polarization over B . • V = X ⊕ Y is the complete polarization over F . • V v = X ′ v ⊕ Y ′ v is the complete polarization over F v . • V (cid:3) = V ▽ ⊕ V △ is the complete polarization over F . • V (cid:3) = X (cid:3) ⊕ Y (cid:3) is the complete polarization over F . • V (cid:3) v = X ′ (cid:3) v ⊕ Y ′ (cid:3) v is the complete polarization over F v .We have a natural map ι : G(U( W ) × U( W )) −→ GU( W (cid:3) )and a see-saw diagram GU( W (cid:3) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ G(U( V ) × U( V )) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ G(U( W ) × U( W )) GU( V ) . Partial Fourier transform. Fix a non-trivial character ψ = ⊗ v ψ v of A /F . Recall that e , . . . , e is a basis of X over F . For each place v of F , this basis and the self-dual measure on F v with respectto ψ v define a Haar measure dx v on X v . Then the product measure dx = Q v dx v is the Tamagawameasure on X ( A ). We define a hermitian inner product h· , ·i on S ( X ( A )) by h ϕ , ϕ i := Z X ( A ) ϕ ( x ) ϕ ( x ) dx. Recall that V (cid:3) = V ▽ ⊕ V △ = X (cid:3) ⊕ Y (cid:3) . We define a partial Fourier transform S ( X (cid:3) ( A )) −→ S ( V ▽ ( A )) ϕ ˆ ϕ by ˆ ϕ ( u ) = Z ( V △ ∩ Y (cid:3) \ V △ )( A ) ϕ ( x ) ψ (cid:18) 12 ( hh x, y ii − hh u, v ii ) (cid:19) dv, where we write u + v = x + y with u ∈ V ▽ ( A ), v ∈ V △ ( A ), x ∈ X (cid:3) ( A ), y ∈ Y (cid:3) ( A ), and dv is theTamagawa measure. Lemma 4.1 ([47, p. 182, (13)]) . If ϕ = ϕ ⊗ ¯ ϕ ∈ S ( X (cid:3) ( A )) with ϕ i ∈ S ( X ( A )) , then we have ˆ ϕ (0) = h ϕ , ϕ i . Proof. We include a proof for convenience. Since V △ ∩ Y (cid:3) = Y △ , we haveˆ ϕ ( u ) = Z X △ ( A ) ϕ ( x ) ψ (cid:18) 12 ( hh x, y ii − hh u, v ii ) (cid:19) dv. We write v = ( v , v ) , u = ( u , − u ) , u = x + y with v , x ∈ X ( A ) and y ∈ Y ( A ), so that x = ( v + x , v − x ) , y = ( y , − y ) . We have hh x, y ii = hh v + x , y ii − hh v − x , − y ii = 2 hh v , y ii , hh u, v ii = 2 hh u , v ii = 2 hh y , v ii , and hence ˆ ϕ ( u ) = Z X ( A ) ϕ ( v + x , v − x ) ψ (2 hh v , y ii ) dv , where dv is the Tamagawa measure on X ( A ). In particular, we haveˆ ϕ (0) = Z X ( A ) ϕ ( v , v ) dv . (cid:3) For each place v of F , we define a hermitian inner product h· , ·i on S ( X v ) with respect to the Haarmeasure dx v on X v given above. Fix a Haar measure on X ′ v and define a hermitian inner product h· , ·i on S ( X ′ v ) similarly. For ϕ ′ ∈ S ( X ′ v ), we define its partial Fourier transform ϕ ∈ S ( X v ) by ϕ ( x ) = Z Y v ∩ Y ′ v \ Y v ϕ ′ ( x ′ ) ψ v (cid:18) 12 ( hh x ′ , y ′ ii − hh x, y ii ) (cid:19) dy, where we write x + y = x ′ + y ′ with x ∈ X v , y ∈ Y v , x ′ ∈ X ′ v , y ′ ∈ Y ′ v , and we normalize a Haarmeasure dy so that h ϕ , ϕ i = h ϕ ′ , ϕ ′ i holds for ϕ ′ , ϕ ′ ∈ S ( X ′ v ) and their partial Fourier transforms ϕ , ϕ ∈ S ( X v ).4.1.2. Weil representations. Fix a place v of F and suppress the subscript v from the notation. InAppendices C, D, we have defined the maps • ˆ s : G(U( V ) × U( W (cid:3) )) → C such that z V △ = ∂ ˆ s , • s : GU( V ) × GU( W ) → C such that z Y = ∂s , • s ′ : GU( V ) × GU( W ) → C such that z Y ′ = ∂s ′ .Let ω ψ and ω (cid:3) ψ be the Weil representations of Mp( V ) and Mp( V (cid:3) ) with respect to ψ , respectively.Composing these with ˆ s , s , s ′ , we obtain: • a representation ω (cid:3) ψ of G(U( V ) × U( W (cid:3) )) on S ( V ▽ ), • a representation ω ψ of G(U( V ) × U( W )) on S ( X ), • a representation ω ψ of G(U( V ) × U( W )) on S ( X ′ ).By § D.4, the partial Fourier transform S ( V ▽ ) ∼ = S ( X (cid:3) ) = S ( X ) ⊗ S ( X )induces an isomorphism ω (cid:3) ψ ◦ (id V ⊗ ι ) ∼ = ω ψ ⊗ ¯ ω ψ as representations of G(U( V ) × U( W ) × U( W )). By definition, the partial Fourier transform S ( X ′ ) ∼ = S ( X ) is G(U( V ) × U( W ))-equivariant.4.2. The Jacquet–Langlands–Shimizu correspondence. Let F be a number field and B a quater-nion algebra over F . We assume that B is division. Set G = GU( W ) , H = GU( V ) , H = GU( V ) ,G = U( W ) , H = U( V ) , H = U( V ) . Recall that G ∼ = B × and 1 −→ F × i −→ B × × B × −→ H −→ , where B and B are quaternion algebras over F such that B · B = B in the Brauer group and i ( z ) = ( z, z − ).Put ( A × ) + = ν ( G ( A )) ∩ ν ( H ( A )), G ( A ) + = { g ∈ G ( A ) | ν ( g ) ∈ ( A × ) + } , H ( A ) + = { h ∈ H ( A ) | ν ( h ) ∈ ( A × ) + } . For each place v of F , we define ( F × v ) + , G + v , ( H v ) + similarly. We have ( F × v ) + = F × v if v is eitherfinite or complex. If v is real, then we have( F × v ) + = ( R × if B v , B ,v , B ,v are split, R × + otherwise.We have ( A × ) + = Q ′ v ( F × v ) + , G ( A ) + = Q ′ v G + v , and H ( A ) + = Q ′ v ( H v ) + .Let π be an irreducible unitary cuspidal automorphic representation of GL ( A ). We assume thatits Jacquet–Langlands transfers π B , π B , π B to B × ( A ), B × ( A ), B × ( A ) exist. We regard π B and π B ⊠ π B as irreducible unitary automorphic representations of G ( A ) and H ( A ) respectively.We define a theta distribution Θ : S ( X ( A )) → C byΘ( ϕ ) = X x ∈ X ( F ) ϕ ( x )for ϕ ∈ S ( X ( A )). Let ϕ ∈ S ( X ( A )) and f ∈ π B . For h ∈ H ( A ) + , choose g ∈ G ( A ) + such that ν ( g ) = ν ( h ) and put(4.1) θ ϕ ( f )( h ) := Z G ( F ) \ G ( A ) Θ( ω ψ ( g gh ) ϕ ) f ( g g ) dg . Here dg = Q v dg ,v is the Tamagawa measure on G ( A ) and we may assume that the volume of ahyperspecial maximal compact subgroup of G ,v with respect to dg ,v is 1 for almost all v . Note thatvol( G ( F ) \ G ( A )) = 1. Using Eichler’s norm theorem, one can see that θ ϕ ( f )( γh ) = θ ϕ ( f )( h ) for γ ∈ H ( F ) ∩ H ( A ) + and h ∈ H ( A ) + . Since H ( A ) = H ( F ) H ( A ) + , θ ϕ ( f ) defines an automorphicform on H ( A ). Let Θ( π B ) be the automorphic representation of H ( A ) generated by θ ϕ ( f ) for all ϕ ∈ S ( X ( A )) and f ∈ π B . Lemma 4.2. The automorphic representation Θ( π B ) is cuspidal. Proof. If both B and B are division, then H is anisotropic and the assertion is obvious. Hencewe may assume that either B or B is split. Then there exists a complete polarization V = ˜ X ⊕ ˜ Y over B . As in § C.3, we regard V as a left B -space. Choosing a basis ˜ v , ˜ v ∗ of V such that ˜ X = B ˜ v ,˜ Y = B ˜ v ∗ , h ˜ v , ˜ v ∗ i = 1, we may write H = (cid:26) h ∈ GL ( B ) (cid:12)(cid:12)(cid:12)(cid:12) h (cid:18) − (cid:19) t h ∗ = ν ( h ) · (cid:18) − (cid:19) (cid:27) . Put n ( b ) := (cid:18) b (cid:19) ∈ H for b ∈ F . It remains to show that Z F \ A θ ϕ ( f )( n ( b )) db = 0for all ϕ ∈ S ( X ( A )) and f ∈ π B .Let V = ˜ X ⊕ ˜ Y be another complete polarization over F given by ˜ X = W ⊗ B ˜ X and ˜ Y = W ⊗ B ˜ Y ,where we regard W as a right B -space. As in [39, § ω ψ of G ( A ) × H ( A ) with respect to ψ on S ( ˜ X ( A )) ∼ = S ( W ( A )). Note that˜ ω ψ ( g ) ˜ ϕ ( x ) = ˜ ϕ ( g − x ) , ˜ ω ψ ( n ( b )) ˜ ϕ ( x ) = ψ ( h x, x i b ) ˜ ϕ ( x )for ˜ ϕ ∈ S ( W ( A )), x ∈ W ( A ), g ∈ G ( A ), and b ∈ A . Let ˜ ϕ ∈ S ( W ( A )) be the partial Fouriertransform of ϕ ∈ S ( X ( A )). Then we haveΘ( ω ψ ( g ) ϕ ) = χ ( g ) X x ∈ W ( F ) ˜ ω ψ ( g ) ˜ ϕ ( x )for g ∈ G ( A ) × H ( A ) with some character χ of G ( A ) × H ( A ) trivial on G ( F ) × H ( F ). One cansee that χ ( g ) = χ ( n ( b )) = 1 for g ∈ G ( A ) and b ∈ A . Since W is anisotropic, we have Z F \ A θ ϕ ( f )( n ( b )) db = Z F \ A Z G ( F ) \ G ( A ) X x ∈ W ( F ) ψ ( h x, x i b )˜ ω ψ ( g ) ˜ ϕ ( x ) f ( g ) dg db = Z G ( F ) \ G ( A ) ˜ ω ψ ( g ) ˜ ϕ (0) f ( g ) dg = ˜ ϕ (0) Z G ( F ) \ G ( A ) f ( g ) dg . Since π is cuspidal, the restriction of π B to G ( A ) is orthogonal to the trivial representation of G ( A ),so that this integral vanishes. This completes the proof. (cid:3) Lemma 4.3. The automorphic representation Θ( π B ) is non-zero. The proof of this lemma will be given in § Proposition 4.4. We have Θ( π B ) = π B ⊠ π B . Proof. Since Θ( π B ) is cuspidal and non-zero, the assertion follows from the local theta correspondencefor unramified representations and the strong multiplicity one theorem. (cid:3) The doubling method. Degenerate principal series representations. Set G (cid:3) = GU( W (cid:3) ) , G (cid:3) = U( W (cid:3) ) . Choosing a basis w , w ∗ of W (cid:3) such that W ▽ = B w , W △ = B w ∗ , h w , w ∗ i = 1, we may write G (cid:3) = (cid:26) g ∈ GL ( B ) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) (cid:19) t g ∗ = ν ( g ) · (cid:18) (cid:19) (cid:27) . Let P and P be the Siegel parabolic subgroups of G (cid:3) and G (cid:3) given by P = (cid:26) (cid:18) a ∗ ν · ( a ∗ ) − (cid:19) ∈ G (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) a ∈ B × , ν ∈ F × (cid:27) and P = P ∩ G (cid:3) respectively. Let δ P and δ P denote the modulus characters of P ( A ) and P ( A )respectively. We have δ P (cid:18)(cid:18) a ∗ ν · ( a ∗ ) − (cid:19)(cid:19) = | ν ( a ) | · | ν | − and δ P = δ P | P ( A ) . Put d ( ν ) := (cid:18) ν (cid:19) ∈ P for ν ∈ F × . We fix a maximal compact subgroup K of G (cid:3) ( A ) such that G (cid:3) ( A ) = P ( A ) K and G (cid:3) ( A ) = P ( A ) K , where K = K ∩ G (cid:3) ( A ).For s ∈ C , we consider a degenerate principal series representation I ( s ) := Ind G (cid:3) P ( δ s/ P ) of G (cid:3) ( A )consisting of smooth functions F on G (cid:3) ( A ) which satisfy F (cid:18)(cid:18) a ∗ ν · ( a ∗ ) − (cid:19) g (cid:19) = | ν ( a ) | s + · | ν | − s − · F ( g ) . We define a degenerate principal series representation I ( s ) := Ind G (cid:3) P ( δ s/ P ) of G (cid:3) ( A ) similarly. Thenthe restriction I ( s ) → I ( s ) to G (cid:3) ( A ) as functions is a G (cid:3) ( A )-equivariant isomorphism. For eachplace v of F , we define degenerate principal series representations I v ( s ) and I ,v ( s ) of G (cid:3) v and G (cid:3) ,v similarly.For ϕ ∈ S ( V ▽ ( A )), we define F ϕ ∈ I ( ) by F ϕ ( g ) = | ν ( g ) | − · ( ω (cid:3) ψ ( d ( ν ( g ) − ) g ) ϕ )(0) . One can see that the map ϕ 7→ F ϕ is G(U( V ) × U( W (cid:3) ))( A )-equivariant, where GU( V )( A ) acts triviallyon I ( ).4.3.2. Eisenstein series. For a holomorphic section F s of I ( s ), we define an Eisenstein series E ( F s )on G (cid:3) ( A ) by (the meromorphic continuation of) E ( g, F s ) = X γ ∈ P ( F ) \ G (cid:3) ( F ) F s ( γg ) . For a holomorphic section F ,s of I ( s ), we define an Eisenstein series E ( F ,s ) on G (cid:3) ( A ) similarly. If F s is a holomorphic section of I ( s ), then F s | G (cid:3) ( A ) is a holomorphic section of I ( s ) and E ( F s ) | G (cid:3) ( A ) = E ( F s | G (cid:3) ( A ) ). By [77, Theorem 3.1], E ( F s ) is holomorphic at s = . In particular, the map E : I ( ) −→ A ( G (cid:3) )given by E ( F ) := E ( F s ) | s = is G (cid:3) ( A )-equivariant, where A ( G (cid:3) ) is the space of automorphic formson G (cid:3) ( A ) and F s is the holomorphic section of I ( s ) such that F = F and F s | K is independent of s . Doubling zeta integrals. Let h· , ·i be the invariant hermitian inner product on π B given by h f , f i := Z Z G ( A ) G ( F ) \ G ( A ) f ( g ) f ( g ) dg for f , f ∈ π B . Here Z G is the center of G and dg is the Tamagawa measure on Z G ( A ) \ G ( A ). Notethat vol( Z G ( A ) G ( F ) \ G ( A )) = 2. Fix an isomorphism π B ∼ = ⊗ v π B,v . For each place v of F , we choosean invariant hermitian inner product h· , ·i on π B,v so that h f , f i = Q v h f ,v , f ,v i and h f ,v , f ,v i = 1for almost all v for f = ⊗ v f ,v , f = ⊗ v f ,v ∈ π B . Set G = G(U( W ) × U( W )) = { ( g , g ) ∈ G × G | ν ( g ) = ν ( g ) } . Then the doubling zeta integral of Piatetski-Shapiro and Rallis [57] is given by Z ( F s , f , f ) = Z Z ( A ) G ( F ) \ G ( A ) E ( ι ( g , g ) , F s ) f ( g ) f ( g ) d g for a holomorphic section F s of I ( s ) and f , f ∈ π B . Here Z is the center of G (cid:3) and d g is theTamagawa measure on Z ( A ) \ G ( A ). Note that vol( Z ( A ) G ( F ) \ G ( A )) = 2. For each place v of F , put Z ( F s,v , f ,v , f ,v ) = Z G ,v F s,v ( ι ( g ,v , h π B,v ( g ,v ) f ,v , f ,v i dg ,v for a holomorphic section F s,v of I v ( s ) and f ,v , f ,v ∈ π B,v . Note that, for fixed f ,v and f ,v , thisintegral depends only on the holomorphic section F s,v | G (cid:3) ,v of I ,v ( s ). Lemma 4.5. We have Z ( F s , f , f ) = L S ( s + , π, ad) ζ S ( s + ) ζ S (2 s + 1) · Y v ∈ S Z ( F s,v , f ,v , f ,v ) for a holomorphic section F s = ⊗ v F s,v of I ( s ) and f = ⊗ v f ,v , f = ⊗ v f ,v ∈ π B . Here S is asufficiently large finite set of places of F .Proof. The assertion follows from the doubling method [57]. Indeed, as in [57], [23, § E ( ι ( g , g ) , F s ) and see that only the open G -orbit P \ P G in P \ G (cid:3) contributesto the integral Z ( F s , f , f ). Hence we have Z ( F s , f , f ) = Z Z ( A ) G △ ( F ) \ G ( A ) F s ( ι ( g , g )) f ( g ) f ( g ) d g , where G △ = { ( g, g ) | g ∈ G } . We have F s ( ι ( g , g )) = F s ( ι ( g − g , g , g ) ∈ G . Writing g = g and g ′ = g − g , we have Z ( F s , f , f ) = Z G ( A ) Z Z G ( A ) G ( F ) \ G ( A ) F s ( ι ( g ′ , f ( gg ′ ) f ( g ) dg dg ′ = Z G ( A ) F s ( ι ( g ′ , h π B ( g ′ ) f , f i dg ′ = Y v Z ( F s,v , f ,v , f ,v ) . By [57], we have Z ( F s,v , f ,v , f ,v ) = L ( s + , π v , ad) ζ v ( s + ) ζ v (2 s + 1)for almost all v . This completes the proof. (cid:3) Local zeta integrals. Lemma 4.6. The integral Z ( F v , f ,v , f ,v ) is absolutely convergent for F v ∈ I v ( ) and f ,v , f ,v ∈ π B,v .Proof. If B v is split, then the lemma is proved in [16, Lemma 6.5]. If B v is division, then G ,v iscompact and the assertion is obvious. (cid:3) Lemma 4.7. There exist ϕ v ∈ S ( V ▽ v ) and f ,v , f ,v ∈ π B,v such that Z ( F ϕ v , f ,v , f ,v ) = 0 .Proof. If B v is split, then the lemma is proved in [16, Lemma 6.6]. Assume that B v is division. As in[42, Theorem 3.2.2], [44, Proposition 7.2.1], one can see that there exist F v ∈ I v ( ) and f ,v , f ,v ∈ π B,v such that Z ( F v , f ,v , f ,v ) = 0. On the other hand, by [76, Theorems 1.2, 9.2], the map S ( V ▽ v ) −→ I ,v ( ) ϕ v ϕ v | G (cid:3) ,v is surjective . This yields the lemma. (cid:3) If ϕ v is the partial Fourier transform of ϕ ,v ⊗ ¯ ϕ ,v ∈ S ( X (cid:3) v ) with ϕ i,v ∈ S ( X v ), then we have(4.2) Z ( F ϕ v , f ,v , f ,v ) = Z G ,v h ω ψ ( g ,v ) ϕ ,v , ϕ ,v ih π B,v ( g ,v ) f ,v , f ,v i dg ,v . This will be used later to explicate the Rallis inner product formula.4.4. The Rallis inner product formula. Theta integrals. Recall that G(U( V ) × U( W (cid:3) ))( A ) acts on S ( V ▽ ( A )) via the Weil representation ω (cid:3) ψ . We define a G (cid:3) ( A )-equivariant and H ( A )-invariant map I : S ( V ▽ ( A )) −→ A ( G (cid:3) )as follows. Here A ( G (cid:3) ) is the space of automorphic forms on G (cid:3) ( A ).Let Θ : S ( V ▽ ( A )) → C be the theta distribution given byΘ( ϕ ) = X x ∈ V ▽ ( F ) ϕ ( x )for ϕ ∈ S ( V ▽ ( A )). Let dh be the Haar measure on H ( A ) such that vol( H ( F ) \ H ( A )) = 1.First we assume that either B or B is split. Then the integral(4.3) Z H ( F ) \ H ( A ) Θ( ω (cid:3) ψ ( g h ) ϕ ) dh may not be convergent. Following Yamana [77, § v ∈ Σ B and an element z in theBernstein center of H ,v or the universal enveloping algebra of the complexified Lie algebra of H ,v .Then the integral I ( g , ϕ ) := Z H ( F ) \ H ( A ) Θ( ω (cid:3) ψ ( g h )( z · ϕ )) dh is absolutely convergent for all g ∈ G (cid:3) ( A ) and ϕ ∈ S ( V ▽ ( A )), and defines an automorphic formon G (cid:3) ( A ). Note that I ( g , ϕ ) = (4.3) if the right-hand side is absolutely convergent for all g . Inparticular, I ( g , ϕ ) does not depend on choice of v and z . Next we assume that both B and B are division. Then H ( F ) \ H ( A ) is compact. For ϕ ∈ S ( V ▽ ( A )), we define an automorphic form I ( ϕ ) on G (cid:3) ( A ) by I ( g , ϕ ) := Z H ( F ) \ H ( A ) Θ( ω (cid:3) ψ ( g h )( z · ϕ )) dh , where we write z for the identity operator for uniformity.Similarly, we define a G (cid:3) ( A )-equivariant and H ( A )-invariant map I : S ( V ▽ ( A )) −→ A ( G (cid:3) )by I ( g , ϕ ) := Z H ( F ) \ H ( A ) Θ( ω (cid:3) ψ ( g h )( z · ϕ )) dh , where dh is the Tamagawa measure on H ( A ). Note that vol( H ( F ) \ H ( A )) = 2. Lemma 4.8. We have I = 2 · I. Proof. The lemma follows from [41, Proposition 4.2] with slight modifications. We include a proof forconvenience. For each place v / ∈ Σ B , we consider the space Hom H ,v ( S ( V ▽ v ) , C ) with the natural actionof H ,v \ H ,v . Let V † v and ( W (cid:3) v ) † be the 4-dimensional quadratic F v -space and the 4-dimensionalsymplectic F v -space associated to V v and W (cid:3) v respectively. Since dim V † v > dim( W (cid:3) v ) † , we haveHom H ,v ( S ( V ▽ v ) , sgn v ) = { } by [60, p. 399], where sgn v is the non-trivial character of H ,v \ H ,v .Hence H ,v acts trivially on Hom H ,v ( S ( V ▽ v ) , C ). On the other hand, we have H ,v = H ,v for all v ∈ Σ B . Hence H ( A ) acts trivially on Hom H ( A ) ( S ( V ▽ ( A )) , C ), so that I ( g , ϕ ) = Z H ( A ) H ( F ) \ H ( A ) I ( g , ω (cid:3) ψ ( ˙ h ) ϕ ) d ˙ h = Z H ( A ) H ( F ) \ H ( A ) I ( g , ϕ ) d ˙ h = 12 · I ( g , ϕ ) , where d ˙ h is the Haar measure on H ( A ) \ H ( A ) such that vol( H ( A ) H ( F ) \ H ( A )) = . (cid:3) The Siegel–Weil formula. The Siegel–Weil formula [77, Theorem 3.4] due to Yamana says that I ( ϕ ) = E ( F ϕ ) | G (cid:3) ( A ) for ϕ ∈ S ( V ▽ ( A )). Hence, by Lemma 4.8, we have(4.4) I ( ϕ ) = 2 · E ( F ϕ ) | G (cid:3) ( A ) for ϕ ∈ S ( V ▽ ( A )).4.4.3. The Rallis inner product formula. Let Z H be the center of H and dh the Tamagawa measureon Z H ( A ) \ H ( A ). Note that vol( Z H ( A ) H ( F ) \ H ( A )) = 4. Proposition 4.9. Let ϕ = ⊗ v ϕ v ∈ S ( V ▽ ( A )) be the partial Fourier transform of ϕ ⊗ ¯ ϕ ∈ S ( X (cid:3) ( A )) with ϕ i = ⊗ v ϕ i,v ∈ S ( X ( A )) . Let f = ⊗ v f ,v , f = ⊗ v f ,v ∈ π B . Then we have Z Z H ( A ) H ( F ) \ H ( A ) θ ϕ ( f )( h ) · θ ϕ ( f )( h ) dh = 2 · L S (1 , π, ad) ζ S (2) · Y v ∈ S Z ( F ϕ v , f ,v , f ,v ) . Here S is a sufficiently large finite set of places of F . Proof. Put ( F × ) + = F × ∩ ( A × ) + , G ( F ) + = G ( F ) ∩ G ( A ) + , H ( F ) + = H ( F ) ∩ H ( A ) + . Set C = ( A × ) ( F × ) + \ ( A × ) + . Then the similitude characters induce isomorphisms Z G ( A ) G ( A ) G ( F ) + \ G ( A ) + ∼ = C , Z H ( A ) H ( A ) H ( F ) + \ H ( A ) + ∼ = C . Fix cross sections c g c and c h c of G ( A ) + → C and H ( A ) → C respectively. Since G ( A ) = Z ( A ) · G ( F ) · ( G × G )( A ) · { ( g c , g c ) | c ∈ C} , we have Z ( F ϕ,s , f , f ) = 2 Z C Z G ( F ) \ G ( A ) Z G ( F ) \ G ( A ) E ( ι ( g g c , g g c ) , F ϕ,s ) f ( g g c ) f ( g g c ) dg dg dc, where dg , dg are the Tamagawa measures on G ( A ) and dc is the Haar measure on C such thatvol( C ) = 1. For each c ∈ C , put ϕ c = ω (cid:3) ψ ( ι ( g c , g c ) , h c ) ϕ . Since E ( gι ( g c , g c ) , F ϕ ) = E ( g, F ϕ c ), we have Z ( F ϕ , f , f )= 2 Z C Z G ( F ) \ G ( A ) Z G ( F ) \ G ( A ) E ( ι ( g , g ) , F ϕ c ) f ( g g c ) f ( g g c ) dg dg dc = Z C Z G ( F ) \ G ( A ) Z G ( F ) \ G ( A ) I ( ι ( g , g ) , ϕ c ) f ( g g c ) f ( g g c ) dg dg dc = Z C Z G ( F ) \ G ( A ) Z G ( F ) \ G ( A ) Z H ( F ) \ H ( A ) Θ( ω (cid:3) ψ ( ι ( g , g ) h )( z · ϕ c )) × f ( g g c ) f ( g g c ) dh dg dg dc = Z C Z H ( F ) \ H ( A ) Z G ( F ) \ G ( A ) Z G ( F ) \ G ( A ) Θ( ω (cid:3) ψ ( ι ( g , g ) h ) ϕ c ) ∗ z × f ( g g c ) f ( g g c ) dg dg dh dc by the Siegel–Weil formula (4.4). On the other hand, we haveΘ( ω (cid:3) ψ ( ι ( g , g ) h ) ϕ c ) = Θ( ω ψ ( g g c h h c ) ϕ ) · Θ( ω ψ ( g g c h h c ) ϕ ) . Hence we have Z G ( F ) \ G ( A ) Z G ( F ) \ G ( A ) Θ( ω (cid:3) ψ ( ι ( g , g ) h ) ϕ c ) ∗ z · f ( g g c ) f ( g g c ) dg dg = (cid:16) θ ϕ ( f )( h h c ) · θ ϕ ( f )( h h c ) (cid:17) ∗ z . By Lemma 4.2, the function h θ ϕ ( f )( h h c ) · θ ϕ ( f )( h h c ) is integrable over H ( F ) \ H ( A ), sothat Z H ( F ) \ H ( A ) (cid:16) θ ϕ ( f )( h h c ) · θ ϕ ( f )( h h c ) (cid:17) ∗ z dh = Z H ( F ) \ H ( A ) θ ϕ ( f )( h h c ) · θ ϕ ( f )( h h c ) dh and hence Z ( F ϕ , f , f ) = Z C Z H ( F ) \ H ( A ) θ ϕ ( f )( h h c ) · θ ϕ ( f )( h h c ) dh dc. Since H ( A ) = Z H ( A ) · H ( F ) · H ( A ) · { h c | c ∈ C} , this integral is equal to12 Z Z H ( A ) H ( F ) \ H ( A ) θ ϕ ( f )( h ) · θ ϕ ( f )( h ) dh . Now the assertion follows from this and Lemma 4.5. (cid:3) Now Lemma 4.3 follows from Proposition 4.9 and Lemma 4.7.5. Schwartz functions Let F be a number field. Let o be the integer ring of F and d the different of F over Q . Let D bethe discriminant of F . For each finite place v of F , let o v be the integer ring of F v , p v = ̟ v o v themaximal ideal of o v , ̟ v a uniformizer of o v , and q v the cardinality of the residue field o v / p v . Let d v be the non-negative integer such that d ⊗ o o v = ̟ d v v o v . Then we have | D | = Q v ∈ Σ fin q d v v .Let ψ = ⊗ v ψ ,v be the non-trivial character of A Q / Q given by • ψ , ∞ ( x ) = e π √− x for x ∈ R , • ψ ,p ( x ) = e − π √− x for x ∈ Q p .Let ψ = ⊗ v ψ v be the non-trivial character of A /F defined by ψ = ψ ◦ tr F/ Q . We call ψ the standardadditive character of A /F . If v is a real place of F , then ψ v ( x ) = e π √− x for x ∈ F v . If v is a complexplace of F , then ψ v ( x ) = e π √− x +¯ x ) for x ∈ F v , where ¯ x is the complex conjugate of x . If v is afinite place of F , then ψ v is trivial on ̟ − d v v o v but non-trivial on ̟ − d v − v o v . For each place v of F , wedefine a Fourier transform S ( F v ) −→ S ( F v ) φ ˆ φ by ˆ φ ( x ) = Z F v φ ( y ) ψ v ( xy ) dy, where dy is the self-dual Haar measure on F v with respect to ψ v .Let V = B ⊗ E B be the 2-dimensional right skew-hermitian B -space given in § W = B the 1-dimensional left hermitian B -space given in § E = F + F i , B = E + E j , B = E + E j , B = E + E j ,u = i , J = j , J = j , J = j , where J = J J . Let V = V ⊗ B W be the 8-dimensional symplectic F -space. We identify V withRes B/F ( V ) via the map v v ⊗ 1. In § V = X ⊕ Y over F . Let e , . . . , e and e ∗ , . . . , e ∗ be the bases of X and Y , respectively, given by (2.3), (2.15).5.1. Complete polarizations. In Appendix C, we also choose a complete polarization V v = X ′ v ⊕ Y ′ v over F v for each place v of F . Note that in picking the polarization, we use the assumption that forany place v of F , at least one of u , J , J , J is a square in F v . In this subsection, we recall the choiceof this polarization. Later, we will pick a Schwartz function on X ′ v and then transfer it to a Schwartzfunction on X v by a partial Fourier transform. From now on, we fix a place v of F and suppress thesubscript v from the notation. The case u ∈ ( F × ) . Choose t ∈ F × such that u = t . We define an isomorphism i : B → M ( F )of quaternion F -algebras by(5.1) i (1) = (cid:18) (cid:19) , i ( i ) = (cid:18) t − t (cid:19) , i ( j ) = (cid:18) J (cid:19) , i ( ij ) = (cid:18) t − tJ (cid:19) . Put e = 12 + 12 t i , e ′ = 12 j + 12 t ij , e ′′ = 12 J j − tJ ij , e ∗ = 12 − t i , so that i ( e ) = (cid:18) (cid:19) , i ( e ′ ) = (cid:18) (cid:19) , i ( e ′′ ) = (cid:18) (cid:19) , i ( e ∗ ) = (cid:18) (cid:19) . Let W † := eW be the 2-dimensional F -space associated to W equipped with a non-degenerate sym-plectic form h· , ·i † defined by(5.2) h x, y i ∗ = h x, y i † · e ′ for x, y ∈ W † . Then the restriction to W † induces a natural isomorphism GU( W ) ∼ = GSp( W † ). Wehave h e, e i † = h e ′ , e ′ i † = 0 , h e, e ′ i † = 1 , and (cid:20) e · α e ′ · α (cid:21) = i ( α ) · (cid:20) ee ′ (cid:21) for α ∈ B . We take a complete polarization W † = X ⊕ Y given by X = F e, Y = F e ′ . Similarly, let V † := V e be the 4-dimensional F -space associated to V equipped with a non-degeneratesymmetric bilinear form h· , ·i † defined by(5.3) 12 · h x, y i = h x, y i † · e ′′ for x, y ∈ V † . Then the restriction to V † induces a natural isomorphism GU( V ) ∼ = GO( V † ). We takea complete polarization V = X ′ ⊕ Y ′ given by X ′ = V † ⊗ X, Y ′ = V † ⊗ Y. We identify X ′ with V † via the map v v ⊗ e . Put(5.4) v = 2 e e = e + t e ∗ , v ∗ = − t e e ∗ = − t e + 12 e ∗ , v = 2 e e = e − tJ e ∗ , v ∗ = 1 tJ e e ∗ = 12 tJ e + 12 e ∗ , v = − tJ e e ′′ = − tJ e + 12 J e ∗ , v ∗ = − e e ′ = − J e − tJ e ∗ , v = − t e e ′′ = − tJ e − e ∗ , v ∗ = 2 e e ′ = e − tJ e ∗ . Then v , . . . , v and v ∗ , . . . , v ∗ are bases of X ′ and Y ′ , respectively, such that hh v i , v ∗ j ii = δ ij .We may identify the quadratic space V † with the space M ( F ) equipped with a non-degeneratesymmetric bilinear form(5.5) tr( xy ∗ ) = x y − x y − x y + x y for x = ( x x x x ), y = ( y y y y ). Indeed, the basis v , . . . , v of V † gives rise to an isomorphism V † ∼ =M ( F ) of quadratic spaces by v (cid:18) (cid:19) , v (cid:18) (cid:19) , v (cid:18) (cid:19) , v (cid:18) (cid:19) . Under this identification, we have α v = v · i ( α ) ∗ , α v = i ( α ) · v for α i ∈ B i and v ∈ V † ∼ = M ( F ), where i : B → M ( F ) and i : B → M ( F ) are isomorphisms ofquaternion F -algebras given by(5.6) i ( a + b i + c j + d ij ) = (cid:18) a − bt − ( c − dt ) − J ( c + dt ) a + bt (cid:19) , i ( a + b i + c j + d ij ) = (cid:18) a + bt − tJ ( c + dt ) − tJ ( c − dt ) a − bt (cid:19) . The case J ∈ ( F × ) . Choose t ∈ F × such that J = t . We define an isomorphism i : B → M ( F )of quaternion F -algebras by(5.7) i (1) = (cid:18) (cid:19) , i ( i ) = (cid:18) u (cid:19) , i ( j ) = (cid:18) t − t (cid:19) , i ( ij ) = (cid:18) − ttu (cid:19) . Put e = 12 + 12 t j , e ′ = 12 i − t ij , e ′′ = 12 u i + 12 tu ij , e ∗ = 12 − t j , so that i ( e ) = (cid:18) (cid:19) , i ( e ′ ) = (cid:18) (cid:19) , i ( e ′′ ) = (cid:18) (cid:19) , i ( e ∗ ) = (cid:18) (cid:19) . Let W † := eW be the 2-dimensional F -space associated to W equipped with a non-degenerate sym-plectic form h· , ·i † defined by (5.2). We have h e, e i † = h e ′ , e ′ i † = 0 , h e, e ′ i † = 1 , and (cid:20) e · α e ′ · α (cid:21) = i ( α ) · (cid:20) ee ′ (cid:21) for α ∈ B . We take a complete polarization W † = X ⊕ Y given by X = F e, Y = F e ′ . Similarly, let V † := V e be the 4-dimensional F -space associated to V equipped with a non-degeneratesymmetric bilinear form h· , ·i † defined by (5.3). We take a complete polarization V = X ′ ⊕ Y ′ given by X ′ = V † ⊗ X, Y ′ = V † ⊗ Y. We identify X ′ with V † via the map v v ⊗ e . Put˜ v = e e = 12 e + 12 t e , ˜ v ∗ = 2 u e e ′ = e ∗ + t e ∗ , ˜ v = e e ′′ = 12 e ∗ − t e ∗ , ˜ v ∗ = − e e ∗ = − e + 1 t e , ˜ v = e e = 12 e + J t e , ˜ v ∗ = − uJ e e ′ = e ∗ + tJ e ∗ , ˜ v = e e ′′ = − J e ∗ + t e ∗ , ˜ v ∗ = 2 J e e ∗ = 1 J e − t e . Then ˜ v , . . . , ˜ v and ˜ v ∗ , . . . , ˜ v ∗ are bases of X ′ and Y ′ , respectively, such that hh ˜ v i , ˜ v ∗ j ii = δ ij . We need to use two coordinate systems given as follows:5.1.2.1. The case (i). We fix s ∈ F × and define bases v , . . . , v and v ∗ , . . . , v ∗ of X ′ and Y ′ , respec-tively, such that hh v i , v ∗ j ii = δ ij by(5.8) v = ˜ v , v = ˜ v , v = 1 s ˜ v , v = 1 s ˜ v , v ∗ = ˜ v ∗ , v ∗ = ˜ v ∗ , v ∗ = s ˜ v ∗ , v ∗ = s ˜ v ∗ . We may identify the quadratic space V † with the space B equipped with a non-degenerate sym-metric bilinear form − 14 tr B /F ( xy ∗ ) . Indeed, since h ˜ v , ˜ v i † = u , h ˜ v , ˜ v i † = − , h ˜ v , ˜ v i † = − uJ , h ˜ v , ˜ v i † = J , and h ˜ v i , ˜ v j i † = 0 if i = j , the basis ˜ v , . . . , ˜ v of V † gives rise to an isomorphism V † ∼ = B of quadraticspaces by ˜ v i , ˜ v , ˜ v j i , ˜ v j . Under this identification, we have α v = α · v , α v = v · i ( α ) ∗ for α i ∈ B i and v ∈ V † ∼ = B , where i : B → B is an isomorphism of quaternion F -algebras givenby(5.9) i ( α + β j ) = α ρ + tβ ρ J j for α, β ∈ E .5.1.2.2. The case (ii). Assume that J ∈ ( F × ) . We choose t ∈ F × such that J = t and define bases v , . . . , v and v ∗ , . . . , v ∗ of X ′ and Y ′ , respectively, such that hh v i , v ∗ j ii = δ ij by(5.10) v = ˜ v + 1 t ˜ v = 12 e + 12 t e + t t e + 12 t e , v = ˜ v + 1 t ˜ v = 12 e ∗ − t e ∗ + t t e ∗ − t e ∗ , v = ˜ v − t ˜ v = 12 e ∗ + t e ∗ − t t e ∗ − t e ∗ , v = 1 u ˜ v − t u ˜ v = 12 u e − t u e − t tu e + 12 tu e , v ∗ = 12 ˜ v ∗ + t v ∗ = 12 e ∗ + t e ∗ + t t e ∗ + t e ∗ , v ∗ = 12 ˜ v ∗ + t v ∗ = − e + 12 t e − t t e + 12 t e , v ∗ = 12 ˜ v ∗ − t v ∗ = − e − t e + t t e + 12 t e , v ∗ = u v ∗ − t u v ∗ = u e ∗ − t u e ∗ − tu t e ∗ + tu e ∗ . We may identify the quadratic space V † with the space M ( F ) equipped with the non-degeneratesymmetric bilinear form (5.5). Indeed, the basis v , . . . , v of V † gives rise to an isomorphism V † ∼ = M ( F ) of quadratic spaces by v (cid:18) (cid:19) , v (cid:18) (cid:19) , v (cid:18) (cid:19) , v (cid:18) (cid:19) . Under this identification, we have α v = i ( α ) · v , α v = v · i ( α ) ∗ for α i ∈ B i and v ∈ V † ∼ = M ( F ), where i : B → M ( F ) and i : B → M ( F ) are isomorphisms ofquaternion F -algebras given by(5.11) i ( a + b i + c j + d ij ) = (cid:18) a + ct b − dt u ( b + dt ) a − ct (cid:19) , i ( a + b i + c j + d ij ) = (cid:18) a − c tt − u ( b + d tt ) − ( b − d tt ) a + c tt (cid:19) . The case J ∈ ( F × ) or J ∈ ( F × ) . We only consider the case J ∈ ( F × ) ; we switch theroles of B and B in the other case. Choose t ∈ F × such that J = t . We define isomorphisms i : B → M ( F ) and i : B → B of quaternion F -algebras by(5.12) i (1) = (cid:18) (cid:19) , i ( i ) = (cid:18) u (cid:19) , i ( j ) = (cid:18) t − t (cid:19) , i ( ij ) = (cid:18) − t tu (cid:19) , and(5.13) i ( α + β j ) = α + βt j for α, β ∈ E . Put v := 12 e + 12 t e , v ∗ := e ∗ + t e ∗ = 1 u e i − tu e i . Then v , v ∗ is a basis of V over B such that h v , v i = h v ∗ , v ∗ i = 0 , h v , v ∗ i = 1 . Moreover, we have (cid:2) α i · v α i · v ∗ (cid:3) = (cid:2) v v ∗ (cid:3) · i i ( α i )for α i ∈ B i . Here we identify i ( α ) with the scalar matrix i ( α ) · in M ( B ). Let V ′ := V , regardedas a left B -space via α · x ′ := ( x · α ∗ ) ′ , where for an element x ∈ V , we write x ′ for the correspondingelement in V ′ . We have a natural skew-hermitian form h· , ·i ′ on V ′ defined by h x ′ , y ′ i ′ = h x, y i . LetGL( V ′ ) act on V ′ on the right. We may identify GU( V ) with GU( V ′ ) via the isomorphismGL( V ) −→ GL( V ′ ) .g (cid:2) x ′ ( g − · x ) ′ (cid:3) Under this identification, we have (cid:20) v ′ · α i ( v ∗ ) ′ · α i (cid:21) = t ( i i ( α i ) − ) ∗ · (cid:20) v ′ ( v ∗ ) ′ (cid:21) for α i ∈ B i . We take a complete polarization V ′ = X ′ ⊕ Y ′ given by X ′ = B · v ′ , Y ′ = B · ( v ∗ ) ′ . Similarly, let W ′ := W , regarded as a right B -space via x ′ · α := ( α ∗ · x ) ′ . We have a natural hermitianform h· , ·i ′ on W ′ defined by h x ′ , y ′ i ′ = h x, y i . Let GL( W ′ ) act on W ′ on the left. We may identifyGU( W ) with GU( W ′ ) via the isomorphismGL( W ) −→ GL( W ′ ) .g (cid:2) x ′ ( x · g − ) ′ (cid:3) We now consider an F -space V ′ := W ′ ⊗ B V ′ equipped with a non-degenerate symplectic form hh· , ·ii ′ := 12 tr B/F ( h· , ·i ′ ⊗ h· , ·i ′∗ ) . Let GL( V ′ ) act on V ′ on the right. We identify V with V ′ via the map x = x ⊗ y x ′ = y ′ ⊗ x ′ . Thenby Lemma C.10, we may identify GSp( V ) with GSp( V ′ ) via the isomorphismGL( V ) −→ GL( V ′ ) , g [ x ′ ( x · g ) ′ ]which induces a commutative diagramGU( V ) × GU( W ) / / (cid:15) (cid:15) GSp( V ) (cid:15) (cid:15) GU( W ′ ) × GU( V ′ ) / / GSp( V ′ ) . We take a complete polarization V ′ = ( W ′ ⊗ B X ′ ) ⊕ ( W ′ ⊗ B Y ′ ) . Under the identification V = V ′ , this gives a complete polarization V = X ′ ⊕ Y ′ , where X ′ = ( v · B ) ⊗ B W, Y ′ = ( v ∗ · B ) ⊗ B W. We identify X ′ with W via the map w v ⊗ w . We fix s ∈ F × and put(5.14) v = v = 12 e + 12 t e , v ∗ = v ∗ = e ∗ + t e ∗ , v = 1 u vi = 12 e ∗ − t e ∗ , v ∗ = − v ∗ i = − e + 1 t e , v = 1 s vj = 12 s e + t s e , v ∗ = − sJ v ∗ j = s e ∗ + st e ∗ , v = 1 su vij = − J s e ∗ + J st e ∗ , v ∗ = sJ v ∗ ij = sJ e − stJ e . Then v , . . . , v and v ∗ , . . . , v ∗ are bases of X ′ and Y ′ , respectively, such that hh v i , v ∗ j ii = δ ij .5.2. Weil representations. Recall that we have the Weil representation ω ψ of G(U( V ) × U( W ))on S ( X ) obtained from the map s : GU( V ) × GU( W ) → C such that z Y = ∂s given in Appendix C.This Weil representation is unitary with respect to the hermitian inner product h· , ·i on S ( X ) given by h ϕ , ϕ i = Z X ϕ ( x ) ϕ ( x ) dx, where dx = dx · · · dx for x = x e + · · · + x e with the self-dual Haar measure dx i on F with respectto ψ . The map s is defined in terms of another map s ′ : GU( V ) × GU( W ) → C such that z Y ′ = ∂s ′ given in Appendix C, based on [39]. Thus we obtain the Weil representation ω ψ of G(U( V ) × U( W )) on S ( X ′ ) from s ′ , as in [39, § § h· , ·i on S ( X ′ ) given in terms of certain Haar measure on X ′ . In this subsection, wedefine this Haar measure on X ′ and give explicit formulas for the Weil representation on S ( X ′ ). The case u ∈ ( F × ) . Recall that we identified X ′ with V † . We take the self-dual Haar measureon V † with respect to the pairing ( x, y ) ψ ( h x, y i † ). More explicitly, this measure is given by dx = dx · · · dx for x = x v + · · · + x v ∈ X ′ , where v , . . . , v is the basis of X ′ given by (5.4) and dx i is the self-dualHaar measure on F with respect to ψ .We identity GU( W ) ∼ = B × with GL ( F ) via the isomorphism i given by (5.1). Then U( W ) ∼ = SL ( F )acts on S ( X ′ ) by ω ψ (cid:18) a a − (cid:19) ϕ ( x ) = | a | ϕ ( ax ) , a ∈ F × ,ω ψ (cid:18) b (cid:19) ϕ ( x ) = ψ (cid:18) b h x, x i † (cid:19) ϕ ( x ) , b ∈ F,ω ψ (cid:18) − (cid:19) ϕ ( x ) = Z X ′ ϕ ( y ) ψ ( −h x, y i † ) dy. This action extends to an action of G(U( V ) × U( W )) by ω ψ ( g, h ) = ω ψ ( g · d ( ν ) − ) ◦ L ( h ) = L ( h ) ◦ ω ψ ( d ( ν ) − · g )for g ∈ GU( W ) ∼ = GL ( F ) and h ∈ GU( V ) ∼ = GO( V † ) such that ν ( g ) = ν ( h ) =: ν , where d ( ν ) =( ν ) and L ( h ) ϕ ( x ) = | ν | − ϕ ( h − x ) . The case J ∈ ( F × ) . Recall that we identified X ′ with V † . We take the self-dual Haar measureon V † with respect to the pairing ( x, y ) ψ ( h x, y i † ). More explicitly, according the coordinate system,this measure is given as follows:(i) dx = (cid:12)(cid:12)(cid:12)(cid:12) uJ s (cid:12)(cid:12)(cid:12)(cid:12) dx · · · dx for x = x v + · · · + x v ∈ X ′ , where v , . . . , v is the basis of X ′ given by (5.8) and dx i is theself-dual Haar measure on F with respect to ψ .(ii) dx = dx · · · dx for x = x v + · · · + x v ∈ X ′ , where v , . . . , v is the basis of X ′ given by (5.10) and dx i is theself-dual Haar measure on F with respect to ψ .We identity GU( W ) ∼ = B × with GL ( F ) via the isomorphism i given by (5.7). Then U( W ) ∼ = SL ( F )acts on S ( X ′ ) by ω ψ (cid:18) a a − (cid:19) ϕ ( x ) = | a | ϕ ( ax ) , a ∈ F × ,ω ψ (cid:18) b (cid:19) ϕ ( x ) = ψ (cid:18) b h x, x i † (cid:19) ϕ ( x ) , b ∈ F,ω ψ (cid:18) − (cid:19) ϕ ( x ) = γ B Z X ′ ϕ ( y ) ψ ( −h x, y i † ) dy, where γ B = ( B is split, − B is ramified. This action extends to an action of G(U( V ) × U( W )) by ω ψ ( g, h ) = ω ψ ( g · d ( ν ) − ) ◦ L ( h ) = L ( h ) ◦ ω ψ ( d ( ν ) − · g )for g ∈ GU( W ) ∼ = GL ( F ) and h ∈ GU( V ) ∼ = GO( V † ) such that ν ( g ) = ν ( h ) =: ν , where d ( ν ) =( ν ) and L ( h ) ϕ ( x ) = | ν | − ϕ ( h − x ) . The case J ∈ ( F × ) or J ∈ ( F × ) . We only consider the case J ∈ ( F × ) ; we switch theroles of B and B in the other case. Recall that we identified X ′ with W . We take the self-dual Haarmeasure on W with respect to the pairing ( x, y ) ψ ( tr B/F h x, y i ). More explicitly, this measure isgiven by dx = (cid:12)(cid:12)(cid:12)(cid:12) Js u (cid:12)(cid:12)(cid:12)(cid:12) dx · · · dx for x = x v + · · · + x v ∈ X ′ , where v , . . . , v is the basis of X ′ given by (5.14) and dx i is theself-dual Haar measure on F with respect to ψ .We identity GU( V ) ∼ = ( B × × B × ) /F × with the group (cid:26) g ∈ GL ( B ) (cid:12)(cid:12)(cid:12)(cid:12) t g ∗ (cid:18) − (cid:19) g = ν ( g ) (cid:18) − (cid:19) (cid:27) via the map ( α , α ) i ( α ) i ( α ), where i and i are the isomorphisms given by (5.12), (5.13).Then U( V ) acts on S ( X ′ ) via the identification U( V ) ∼ = U( V ′ ) followed by the Weil representationof U( V ′ ) on S ( W ′ ⊗ B X ′ ) given in [39, § V ) acts on S ( X ′ ) by ω ψ (cid:18) a ( a − ) ∗ (cid:19) ϕ ( x ) = | ν ( a ) | − ϕ ( a − x ) , a ∈ B × ,ω ψ (cid:18) b (cid:19) ϕ ( x ) = ψ (cid:18) − b h x, x i (cid:19) ϕ ( x ) , b ∈ F,ω ψ (cid:18) − (cid:19) ϕ ( x ) = γ B Z X ′ ϕ ( y ) ψ (cid:18) − 12 tr B/F h x, y i (cid:19) dy, where γ B = ( B is split, − B is ramified.This action extends to an action of G(U( V ) × U( W )) by ω ψ ( g, h ) = ω ψ ( h · d ( ν ) − ) ◦ R ( g ) = R ( g ) ◦ ω ψ ( d ( ν ) − · h )for g ∈ GU( W ) and h ∈ GU( V ) such that ν ( g ) = ν ( h ) =: ν , where d ( ν ) = ( ν ) and R ( g ) ϕ ( x ) = | ν | ϕ ( xg ) . Partial Fourier transforms. Recall that the partial Fourier transform ϕ ∈ S ( X ) of ϕ ′ ∈ S ( X ′ )is given by ϕ ( x ) = Z Y / Y ∩ Y ′ ϕ ′ ( x ′ ) ψ (cid:18) 12 ( hh x ′ , y ′ ii − hh x, y ii ) (cid:19) dµ Y / Y ∩ Y ′ ( y ) , where for x ∈ X and y ∈ Y , we write x + y = x ′ + y ′ with x ′ = x ′ ( x, y ) ∈ X ′ and y ′ = y ′ ( x, y ) ∈ Y ′ ,and we take the Haar measure µ Y / Y ∩ Y ′ on Y / Y ∩ Y ′ so that the map S ( X ′ ) −→ S ( X ) ϕ ′ ϕ respects the hermitian inner products (given in terms of the Haar measures on X and X ′ given in § V ) × U( W )) on S ( X ) and S ( X ′ ). In this subsection, we explicate the Haarmeasure µ Y / Y ∩ Y ′ and the partial Fourier transform S ( X ′ ) → S ( X ).We write x = x e + · · · + x e ∈ X , y = y e ∗ + · · · + y e ∗ ∈ Y ,x ′ = x ′ v + · · · + x ′ v ∈ X ′ , y ′ = y ′ v ∗ + · · · + y ′ v ∗ ∈ Y ′ , where v , . . . , v and v ∗ , . . . , v ∗ are the bases of X ′ and Y ′ , respectively, given in § dx i , dy j , dx ′ i , dy ′ j be the self-dual Haar measures on F with respect to ψ .5.3.1. The case u ∈ ( F × ) . Recall that v i , v ∗ j are given by (5.4). Note that Y ∩ Y ′ = { } . We definea Haar measure µ Y / Y ∩ Y ′ on Y by dµ Y / Y ∩ Y ′ ( y ) = | u | − dy · · · dy for y = y e ∗ + · · · + y e ∗ . We will see below that the partial Fourier transform with respect to thisHaar measure is an isometry.If x + y = x ′ + y ′ , then we have x ′ = 12 t ( y + tx ) , y ′ = y − tx ,x ′ = − tJ ( y − tJ x ) , y ′ = y + tJ x ,x ′ = J ( y − tJ x ) , y ′ = − tJ ( y + tJ x ) ,x ′ = − ( y + tJx ) , y ′ = − tJ ( y − tJx ) . Namely, putting a = t, a = − tJ , a = − tJ , a = tJ, b = b = 1 , b = b = − tJ, we have x ′ i = b i a i ( y i + a i x i ) , y ′ i = 1 b i ( y i − a i x i ) , so that x ′ i y ′ i − x i y i = x ′ i (cid:18) a i b i x ′ i − a i b i x i (cid:19) − x i (cid:18) a i b i x ′ i − a i x i (cid:19) = 2 a i b i ( x ′ i ) − a i b i x i x ′ i + a i x i . Hence, if ϕ ′ ( x ′ ) = Q i =1 ϕ ′ i ( x ′ i ) with ϕ ′ i ∈ S ( F ), then we have ϕ ( x ) = | u | − Y i =1 ϕ i ( x i ) , where ϕ i ( x i ) = Z F ϕ ′ i ( x ′ i ) ψ (cid:18) 12 ( x ′ i y ′ i − x i y i ) (cid:19) dy i = (cid:12)(cid:12)(cid:12)(cid:12) a i b i (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) a i x i (cid:17) Z F ϕ ′ i ( x ′ i ) ψ (cid:18) a i b i ( x ′ i ) − a i b i x i x ′ i (cid:19) dx ′ i . Since Y i =1 a i b i = 4 u, the partial Fourier transform with respect to µ Y / Y ∩ Y ′ is an isometry.5.3.2. The case J ∈ ( F × ) . v i , v ∗ j are given by (5.8). Note that Y ∩ Y ′ = F v ∗ + F v ∗ . Let µ Y and µ Y ∩ Y ′ be the Haar measures on Y and Y ∩ Y ′ , respectively, defined by dµ Y ( y ) = dy · · · dy , dµ Y ∩ Y ′ ( y ′ ) = dy ′ dy ′ for y = y e ∗ + · · · + y e ∗ and y ′ = y ′ v ∗ + y ′ v ∗ . We define a Haar measure µ Y / Y ∩ Y ′ on Y / Y ∩ Y ′ by µ Y / Y ∩ Y ′ = (cid:12)(cid:12)(cid:12)(cid:12) uJ s J (cid:12)(cid:12)(cid:12)(cid:12) µ Y µ Y ∩ Y ′ . We will see below that the partial Fourier transform with respect to this Haar measure is an isometry.If x + y = x ′ + y ′ , then we have x ′ = x + tx , y ′ = 12 (cid:18) y + 1 t y (cid:19) ,x ′ = y − t y , y ′ = − 12 ( x − tx ) ,x ′ = s (cid:18) x + tJ x (cid:19) , y ′ = 12 s (cid:18) y + J t y (cid:19) ,x ′ = − sJ (cid:18) y − J t y (cid:19) , y ′ = J s (cid:18) x − tJ x (cid:19) , so that x ′ y ′ − x ′ y ′ = x y + x y , x ′ y ′ − x ′ y ′ = x y + x y . Also, we have dx ′ dx ′ dy ′ dy ′ = | J | dx · · · dx , dx ′ dx ′ dy ′ dy ′ = | J | − dy · · · dy . Hence, if ϕ ′ ( x ′ ) = Q i =1 ϕ ′ i ( x ′ i ) with ϕ ′ i ∈ S ( F ), then we have ϕ ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) uJJ s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′ ( x ′ ) ϕ ′ ( x ′ ) Z F Z F ϕ ′ ( x ′ ) ϕ ′ ( x ′ ) ψ ( x ′ y ′ + x ′ y ′ ) dx ′ dx ′ = (cid:12)(cid:12)(cid:12)(cid:12) uJJ s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′ ( x ′ ) ˆ ϕ ′ ( y ′ ) ϕ ′ ( x ′ ) ˆ ϕ ′ ( y ′ ) . In particular, the partial Fourier transform with respect to µ Y / Y ∩ Y ′ is an isometry.5.3.2.2. The case (ii). Recall that v i , v ∗ j are given by (5.10). Note that Y ∩ Y ′ = F v ∗ + F v ∗ . Let µ Y and µ Y ∩ Y ′ be the Haar measures on Y and Y ∩ Y ′ , respectively, defined by dµ Y ( y ) = dy · · · dy , dµ Y ∩ Y ′ ( y ′ ) = dy ′ dy ′ for y = y e ∗ + · · · + y e ∗ and y ′ = y ′ v ∗ + y ′ v ∗ . We define a Haar measure µ Y / Y ∩ Y ′ on Y / Y ∩ Y ′ by µ Y / Y ∩ Y ′ = | uJ | − µ Y µ Y ∩ Y ′ . We will see below that the partial Fourier transform with respect to this Haar measure is an isometry. If x + y = x ′ + y ′ , then we have x ′ = 12 (cid:18) x + t x + tt x + tx (cid:19) , y ′ = 12 (cid:18) y + 1 t y + t t y + 1 t y (cid:19) ,x ′ = 12 (cid:18) y − t y + t t y − t y (cid:19) , y ′ = − (cid:18) x − t x + tt x − tx (cid:19) ,x ′ = 12 (cid:18) y + 1 t y − t t y − t y (cid:19) , y ′ = − (cid:18) x + t x − tt x − tx (cid:19) ,x ′ = u (cid:18) x − t x − tt x + tx (cid:19) , y ′ = 12 u (cid:18) y − t y − t t y + 1 t y (cid:19) , so that x ′ y ′ − x ′ y ′ − x ′ y ′ + x ′ y ′ = x y + x y + x y + x y . Also, we have dx ′ dx ′ dy ′ dy ′ = | uJ | dx · · · dx , dx ′ dx ′ dy ′ dy ′ = | uJ | − dy · · · dy . Hence, if ϕ ′ ( x ′ ) = Q i =1 ϕ ′ i ( x ′ i ) with ϕ ′ i ∈ S ( F ), then we have ϕ ( x ) = | uJ | ϕ ′ ( x ′ ) ϕ ′ ( x ′ ) Z F Z F ϕ ′ ( x ′ ) ϕ ′ ( x ′ ) ψ ( x ′ y ′ + x ′ y ′ ) dx ′ dx ′ = | uJ | ϕ ′ ( x ′ ) ˆ ϕ ′ ( y ′ ) ˆ ϕ ′ ( y ′ ) ϕ ′ ( x ′ ) . In particular, the partial Fourier transform with respect to µ Y / Y ∩ Y ′ is an isometry.5.3.3. The case J ∈ ( F × ) or J ∈ ( F × ) . We only consider the case J ∈ ( F × ) ; we switch the rolesof B and B in the other case. Recall that v i , v ∗ j are given by (5.14). Note that Y ∩ Y ′ = F v ∗ + F v ∗ .Let µ Y and µ Y ∩ Y ′ be the Haar measures on Y and Y ∩ Y ′ , respectively, defined by dµ Y ( y ) = dy · · · dy , dµ Y ∩ Y ′ ( y ′ ) = dy ′ dy ′ for y = y e ∗ + · · · + y e ∗ and y ′ = y ′ v ∗ + y ′ v ∗ . We define a Haar measure µ Y / Y ∩ Y ′ on Y / Y ∩ Y ′ by µ Y / Y ∩ Y ′ = | s u | − µ Y µ Y ∩ Y ′ . We will see below that the partial Fourier transform with respect to this Haar measure is an isometry.If x + y = x ′ + y ′ , then we have x ′ = x + tx , y ′ = 12 (cid:18) y + 1 t y (cid:19) ,x ′ = y − t y , y ′ = − 12 ( x − tx ) ,x ′ = s (cid:18) x + 1 t x (cid:19) , y ′ = 12 s ( y + ty ) ,x ′ = − sJ ( y − ty ) , y ′ = J s (cid:18) x − t x (cid:19) , so that x ′ y ′ − x ′ y ′ = x y + x y , x ′ y ′ − x ′ y ′ = x y + x y . Also, we have dx ′ dx ′ dy ′ dy ′ = | J | dx · · · dx , dx ′ dx ′ dy ′ dy ′ = | J | − dy · · · dy . Hence, if ϕ ′ ( x ′ ) = Q i =1 ϕ ′ i ( x ′ i ) with ϕ ′ i ∈ S ( F ), then we have ϕ ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) J s u (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′ ( x ′ ) ϕ ′ ( x ′ ) Z F Z F ϕ ′ ( x ′ ) ϕ ′ ( x ′ ) ψ ( x ′ y ′ + x ′ y ′ ) dx ′ dx ′ = (cid:12)(cid:12)(cid:12)(cid:12) J s u (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′ ( x ′ ) ˆ ϕ ′ ( y ′ ) ϕ ′ ( x ′ ) ˆ ϕ ′ ( y ′ ) . In particular, the partial Fourier transform with respect to µ Y / Y ∩ Y ′ is an isometry.5.4. Automorphic representations. Suppose that F is a totally real number field. Let π B ∼ = ⊗ v π B,v be an irreducible unitary cuspidal automorphic representation of B × ( A ) satisfying the followingconditions: • For v ∈ Σ fin r Σ B, fin ,(ur) π B,v = Ind( χ v ⊗ µ v ) is a principal series representation, where χ v and µ v are unitary unram-ified; or(rps) π B,v = Ind( χ v ⊗ µ v ) is a principal series representation, where χ v is unitary unramified and µ v is unitary ramified; or(st) π B,v = St ⊗ χ v is a twist of the Steinberg representation, where χ v is unitary unramified. • For v ∈ Σ B, fin ,(1d) π B,v = χ v ◦ ν v is a 1-dimensional representation, where χ v is unitary unramified. • For v ∈ Σ ∞ r Σ B, ∞ ,(ds) π B,v = DS k v is the irreducible unitary (limit of) discrete series representation of weight k v . • For v ∈ Σ B, ∞ ,(fd) π B,v = Sym k v is the irreducible unitary ( k v + 1)-dimensional representation.We assume that π B,v is unramified for all finite places v of F such that F v is ramified or of residualcharacteristic 2. By Proposition 7.1, we may assume that the following conditions (which are relevantto the choice of the polarization V v = X ′ v ⊕ Y ′ v ) are satisfied: • If v / ∈ Σ B , then J ∈ ( F × v ) except in the case (ur). • If v ∈ Σ B , then either J ∈ ( F × v ) or J ∈ ( F × v ) .In fact, Proposition 7.1 enables us to impose more precise ramification conditions as described in thefollowing table: π B B , B E u F J J , J ur ur.p.s. split spl, spl split sq of unit ur integer integerssplit sq of unit ram sq of unit sqs of unitsinert nonsq unit ur unit · sq of int units · sqs of intsramified uniformizer ur sq of unit sqs of unitsrps r.p.s. split spl, spl split sq of unit ur sq of unit sqs of unitsst St split spl, spl split sq of unit ur sq of unit sqs of unitsram, ram inert nonsq unit ur sq of uniform uniforms ∗ 1d St ramified spl, ram inert nonsq unit ur uniform sq of unit, uniformram, spl inert nonsq unit ur uniform uniform, sq of unitds d.s. split spl, spl C negative R positive positive, positiveram, ram C negative R positive negative, negativefd d.s. ramified spl, ram C negative R negative positive, negativeram, spl C negative R negative negative, positive ⋄ All places above 2 fall into the case (ur) with E being split. ⋄ In the case (ur) with E being inert, we need to consider separately the case J ∈ ( F × ) and the case J ∈ ( F × ) or J ∈ ( F × ) .Here π ∼ = ⊗ v π v is the Jacquet–Langlands transfer of π B to GL ( A ). These conditions will be veryuseful in the computation of the partial Fourier transform. From now on, we fix a place v of F andsuppress the subscript v from the notation.5.5. Schwartz functions on X ′ . In this subsection, we pick a Schwartz function ϕ ′ ∈ S ( X ′ ) suchthat h ϕ ′ , ϕ ′ i = 1, together with maximal compact subgroups K , K , K of B × , B × , B × , respectively.Also, we study equivariance properties of ϕ ′ under the action of K and K × K , regarded as subgroupsof GU( W ) ∼ = B × and GU( V ) ∼ = ( B × × B × ) /F × , respectively.We need to introduce some notation. For any set A , let I A denote the characteristic function of A .If F is non-archimedean, then for any positive integer n , we define a subalgebra R n of M ( F ) by R n = (cid:26) (cid:18) a bc d (cid:19) ∈ M ( o ) (cid:12)(cid:12)(cid:12)(cid:12) c ∈ ̟ n o (cid:27) . Note that R is an Iwahori subalgebra of M ( F ). If F = R , then we choose an isomorphism E ∼ = C such that i √− > , i.e., i = | u | √− 1. Put ∂∂z = 12 (cid:18) ∂∂x + 1 i ∂∂y (cid:19) , ∂∂z ρ = 12 (cid:18) ∂∂x − i ∂∂y (cid:19) for z = x + y i . For any integer k , we define a character χ k of C × by χ k ( α ) = (cid:18) α √ αα ρ (cid:19) k . ∗ The ratio J /J is a square of a unit. Put H = (cid:18) − (cid:19) , X = (cid:18) (cid:19) , Y = (cid:18) (cid:19) . The case (ur). E is split and F is unramified. In this case, we have: • F is non-archimedean, • ψ is of order zero, • u = t for some t ∈ o × , • J, J , J ∈ o .We define maximal orders o B , o B , o B in B, B , B , respectively, by o B = i − (M ( o )) , o B = i − (M ( o )) , o B = i − (M ( o )) , where i , i , i are the isomorphisms given by (5.1), (5.6). Put K = o × B , K = o × B , K = o × B . We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with V † ∼ = M ( F ) as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = I M ( o ) , i.e., ϕ ′ ( x ) = I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.4). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ K , k ∈ K , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ).5.5.1.2. The case when E is split and F is ramified. In this case, we have: • F is non-archimedean, • u = t for some t ∈ o × , • J, J , J ∈ ( o × ) .Let d be the non-negative integer such that ψ is trivial on ̟ − d o but non-trivial on ̟ − d − o . We definemaximal orders o B , o B , o B in B, B , B , respectively, by o B = i − (cid:18)(cid:18) ̟ d (cid:19) M ( o ) (cid:18) ̟ − d (cid:19)(cid:19) , o B = i − (M ( o )) , o B = i − (M ( o )) , where i , i , i are the isomorphisms given by (5.1), (5.6). Put K = o × B , K = o × B , K = o × B . We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with V † ∼ = M ( F ) as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = q d · I M ( o ) , i.e., ϕ ′ ( x ) = q d · I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.4). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ K , k ∈ K , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ). E is inert and J ∈ ( F × ) . In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u ∈ o × r ( o × ) , • J = t for some t ∈ o , • J ∈ s o × for some s ∈ o , • J ∈ o .We define maximal orders o B , o B , o B in B, B , B , respectively, by o B = i − (M ( o )) , o B = o + o i + o j s + o ij s , o B = i − ( o B ) , where i , i are the isomorphisms given by (5.7), (5.9). Put K = o × B , K = o × B , K = o × B . We take the complete polarization V = X ′ ⊕ Y ′ as in § X ′ with V † ∼ = B as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = I o B , i.e., ϕ ′ ( x ) = I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.8). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ K , k ∈ K , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ).5.5.1.4. The case when E is inert, and J ∈ ( F × ) or J ∈ ( F × ) . We only consider the case J ∈ ( F × ) ; we switch the roles of B and B in the other case. In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u ∈ o × r ( o × ) , • J = t for some t ∈ o , • J ∈ s o × for some s ∈ o , • J ∈ o .We define maximal orders o B , o B , o B in B, B , B , respectively, by o B = o + o i + o j s + o ij s , o B = i − (M ( o )) , o B = i − ( o B ) , where i , i are the isomorphisms given by (5.12), (5.13). Put K = o × B , K = o × B , K = o × B . We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with W = B as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = I o B , i.e., ϕ ′ ( x ) = I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.14). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ K , k ∈ K , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ). E is ramified. In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u ∈ ̟ o × , • J = t for some t ∈ o × , • J = t for some t ∈ o × , • J ∈ ( o × ) .We define maximal orders o B , o B , o B in B, B , B , respectively, by o B = i − (M ( o )) , o B = i − (M ( o )) , o B = i − (M ( o )) , where i , i , i are the isomorphisms given by (5.7), (5.11). Put K = o × B , K = o × B , K = o × B . We take the complete polarization V = X ′ ⊕ Y ′ as in § X ′ with V † ∼ = M ( F ) as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = I M ( o ) , i.e., ϕ ′ ( x ) = I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.10). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ K , k ∈ K , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ).5.5.2. The case (rps). In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u = t for some t ∈ o × , • J, J , J ∈ ( o × ) .We define maximal orders o B , o B , o B in B, B , B and subalgebras o B,n , o B ,n , o B ,n of B, B , B ,respectively, by o B = i − (M ( o )) , o B = i − (M ( o )) , o B = i − (M ( o )) , o B,n = i − ( R n ) , o B ,n = i − ( R n ) , o B ,n = i − ( R n ) , where i , i , i are the isomorphisms given by (5.1), (5.6). We define orientations o B : o B,n −→ o /̟ n o , o B : o B ,n −→ o /̟ n o , o B : o B ,n −→ o /̟ n o by o B (cid:0) i − (cid:0) a bc d (cid:1)(cid:1) = d mod ̟ n o , o B (cid:0) i − (cid:0) a bc d (cid:1)(cid:1) = d mod ̟ n o , o B (cid:0) i − (cid:0) a bc d (cid:1)(cid:1) = a mod ̟ n o . Put K = o × B , K = o × B , K = o × B , K n = o × B,n , K ,n = o × B ,n , K ,n = o × B ,n . We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with V † ∼ = M ( F ) as in § µ of F × of conductor q n , i.e., trivial on 1+ ̟ n o but non-trivial on 1+ ̟ n − o (resp. o × ) if n > n = 1), we define ϕ ′ = ϕ ′ µ ∈ S ( X ′ ) by ϕ ′ ( x ) = q n +12 ( q − − · I o ( x ) I o ( x ) I ̟ n o ( x ) I o × ( x ) µ ( x ) for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.4). Then we have ω ψ ( k, ( k , k )) ϕ ′ = µ ( k ) − µ ( k ) µ ( k ) − µ ( ν ( k )) ϕ ′ for k ∈ K n , k ∈ K ,n , k ∈ K ,n such that ν ( k ) = ν ( k ) ν ( k ), where µ is the character of R × n (andthose of K n , K ,n , K ,n via i , i , i ) defined by µ ( k ) := µ ( d )for k = (cid:0) a bc d (cid:1) .5.5.3. The case (st). B and B are split. In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u = t for some t ∈ o × , • J, J , J ∈ ( o × ) .We define maximal orders o B , o B , o B in B, B , B and Iwahori subalgebras I , I , I of B, B , B ,respectively, by o B = i − (M ( o )) , o B = i − (M ( o )) , o B = i − (M ( o )) , I = i − ( R ) , I = i − ( R ) , I = i − ( R ) , where i , i , i are the isomorphisms given by (5.1), (5.6). Put K = o × B , K = o × B , K = o × B , I = I × , I = I × , I = I × . We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with V † ∼ = M ( F ) as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ ( x ) = q · I o ( x ) I o ( x ) I p ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.4). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ I , k ∈ I , k ∈ I such that ν ( k ) = ν ( k ) ν ( k ).5.5.3.2. The case when B and B are ramified. In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u ∈ o × r ( o × ) , • J = t for some t ∈ ̟ o × , • J , J ∈ ̟ o × .We define a maximal order o B in B and an Iwahori subalgebra I of B by o B = i − (M ( o )) , I = i − ( R ) , where i is the isomorphism given by (5.7). Let o B and o B be the unique maximal orders in B and B , respectively. Then we have o B = o + o i + o j + o ij , o B = i − ( o B ) , where i is the isomorphism given by (5.9). Put K = o × B , I = I × , K = o × B , K = o × B . Put s = 1. We take the complete polarization V = X ′ ⊕ Y ′ as in § X ′ with V † ∼ = B as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = q · I o B , i.e., ϕ ′ ( x ) = q · I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.8). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ I , k ∈ K , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ).5.5.4. The case (1d). We only consider the case J ∈ ( F × ) ; we switch the roles of B and B in theother case. In this case, we have: • F is non-archimedean, • ψ is of order zero, • ∈ o × , • u ∈ o × r ( o × ) , • J = t for some t ∈ o × , • J, J ∈ ̟ o × .We define a maximal order o B in B and an Iwahori subalgebra I of B by o B = i − (M ( o )) , I = i − (cid:18)(cid:18) ̟ − (cid:19) R (cid:18) ̟ (cid:19)(cid:19) , where i is the isomorphism given by (5.12). Let o B and o B be the unique maximal orders in B and B , respectively. Then we have o B = o + o i + o j + o ij , o B = i − ( o B ) , where i is the isomorphism given by (5.13). Put K = o × B , K = o × B , I = I × , K = o × B . Put s = 1. We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with W = B as in § ϕ ′ ∈ S ( X ′ ) by ϕ ′ = q · I o B , i.e., ϕ ′ ( x ) = q · I o ( x ) I o ( x ) I o ( x ) I o ( x )for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.14). Then we have ω ψ ( k, ( k , k )) ϕ ′ = ϕ ′ for k ∈ K , k ∈ I , k ∈ K such that ν ( k ) = ν ( k ) ν ( k ).5.5.5. The case (ds). B and B are split. In this case, we have: • F = R , • ψ ( x ) = e π √− x , • u < • J = t for some t ∈ F × , • J = s for some s ∈ F × , • J > v = | u | . We take the complete polarization V = X ′ ⊕ Y ′ as in § X ′ with V † ∼ = B as in § k , we define ϕ ′ = ϕ ′ k ∈ S ( X ′ ) by ϕ ′ ( x ) = c − k · ( x − x i ) k · e − π v ( x − ux + x − ux ) for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.8) and c k = k ! | u | k +1 π k . Lemma 5.1. We have h ϕ ′ , ϕ ′ i = 1 and ω ψ ( α, ( α , α )) ϕ ′ = χ k ( α ) − χ k ( α ) χ k ( α ) ϕ ′ for α, α , α ∈ E × such that ν ( α ) = ν ( α ) ν ( α ) .Proof. Recall that the Haar measure on X ′ is given by dx = | u | dx · · · dx . We have | u | Z F ( x − ux ) k e − πv ( x − ux + x − ux ) dx · · · dx = | u | π (cid:16) vπ (cid:17) k Z F ( x + x ) k e − ( x + x + x + x ) dx · · · dx = | u | k +1 π k +2 · (2 π ) Z ∞ Z ∞ r k e − ( r + r ) r dr r dr = | u | k +1 π k Z ∞ Z ∞ r k e − ( r + r ) dr dr = | u | k +1 π k · Γ( k + 1)and hence h ϕ ′ , ϕ ′ i = 1. If we write z = x + x i and z = x + x i , then ϕ ′ ( x ) = c − k · ( z ρ ) k · e − π v ( z z ρ + z z ρ ) , and it is easy to see that ω ψ ( ν , ( α , α )) ϕ ′ = χ k ( α ) χ k ( α ) ϕ ′ for α , α ∈ E × and ν = ν ( α ) ν ( α ). On the other hand, we have ω ψ ( H ) ϕ ′ ( x ) = (cid:18) x ∂∂x + · · · + x ∂∂x (cid:19) ϕ ′ ( x ) ,ω ψ ( X ) ϕ ′ ( x ) = π √− 12 ( ux − x − ux + x ) ϕ ′ ( x ) ,ω ψ ( Y ) ϕ ′ ( x ) = − π √− (cid:18) u ∂ ∂x − ∂ ∂x − u ∂ ∂x + ∂ ∂x (cid:19) ϕ ′ ( x ) , where we identity GU( W ) ∼ = B × with GL ( F ) via the isomorphism i given by (5.7). Thus, noting that ∂ ∂z ∂z ρ = 14 (cid:18) ∂ ∂x − u ∂ ∂x (cid:19) , ∂ ∂z ∂z ρ = 14 (cid:18) ∂ ∂x − u ∂ ∂x (cid:19) , we see that ω ψ ( v − X − v Y ) ϕ ′ = −√− kϕ ′ . This implies that ω ψ ( α, (1 , ϕ ′ = χ k ( α ) − ϕ ′ for α ∈ E since i ( α ) = (cid:18) a bbu a (cid:19) = (cid:18) v (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) v − (cid:19) = exp(( v − X − v Y ) θ )if we write α = a + b i = e √− θ . This completes the proof. (cid:3) B and B are ramified. In this case, we have: • F = R , • ψ ( x ) = e π √− x , • u < • J = t for some t ∈ F × , • J = − s for some s ∈ F × , • J < v = | u | . We take the complete polarization V = X ′ ⊕ Y ′ as in § X ′ with V † ∼ = B as in § k , we define ϕ ′ = ϕ ′ k ∈ S ( X ′ ) by ϕ ′ ( x ) = c − k · ( x − x i ) k · e − π v ( x − ux + x − ux ) for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.8) and c k = k ! | u | k +1 π k . Lemma 5.2. We have h ϕ ′ , ϕ ′ i = 1 and ω ψ ( α, ( α , α )) ϕ ′ = χ k +2 ( α ) − χ k ( α ) χ k ( α ) ϕ ′ for α, α , α ∈ E × such that ν ( α ) = ν ( α ) ν ( α ) .Proof. The proof is the same as that of Lemma 5.1 and we omit the details. Note that, in this case,we have ω ψ ( H ) ϕ ′ ( x ) = (cid:18) x ∂∂x + · · · + x ∂∂x (cid:19) ϕ ′ ( x ) ,ω ψ ( X ) ϕ ′ ( x ) = π √− 12 ( ux − x + ux − x ) ϕ ′ ( x ) ,ω ψ ( Y ) ϕ ′ ( x ) = − π √− (cid:18) u ∂ ∂x − ∂ ∂x + 1 u ∂ ∂x − ∂∂x (cid:19) ϕ ′ ( x ) . (cid:3) The case (fd). We only consider the case J ∈ ( F × ) ; we switch the roles of B and B in theother case. In this case, we have: • F = R , • ψ ( x ) = e π √− x , • u < • J = t for some t ∈ F × , • J = − s for some s ∈ F × , • J < v = | u | . We take the complete polarization V = X ′ ⊕ Y ′ and identify X ′ with W = B as in § k , we define ϕ ′ = ϕ ′ k ∈ S ( X ′ ) by ϕ ′ ( x ) = c − k · (cid:18) x − x i u (cid:19) k · e − πv ( x − u x + x − u x ) for x = x v + · · · + x v , where v , . . . , v is the basis of X ′ given by (5.14) and c k = k ! π k | u | k +1 . Lemma 5.3. We have h ϕ ′ , ϕ ′ i = 1 and ω ψ ( α, ( α , α )) ϕ ′ = χ k ( α ) − χ k +2 ( α ) χ k ( α ) ϕ ′ for α, α , α ∈ E × such that ν ( α ) = ν ( α ) ν ( α ) .Proof. Recall that the Haar measure on X ′ is given by dx = | u | dx · · · dx . We have1 | u | Z F (cid:18) x − u x (cid:19) k e − πv ( x − u x + x − u x ) dx · · · dx = 1 π | u | (cid:18) πv (cid:19) k Z F ( x + x ) k e − ( x + x + x + x ) dx · · · dx = 1 π k +2 | u | k +1 · (2 π ) Z ∞ Z ∞ r k e − ( r + r ) r dr r dr = 1 π k | u | k +1 Z ∞ Z ∞ r k e − ( r + r ) dr dr = 1 π k | u | k +1 · Γ( k + 1)and hence h ϕ ′ , ϕ ′ i = 1. If we write z = x + x i u and z = x + x i u , then ϕ ′ ( x ) = c − k · ( z ρ ) k · e − πv ( z z ρ + z z ρ ) , and it is easy to see that ω ψ ( α, ( ν , α )) ϕ ′ = χ k ( α ) − χ k ( α ) ϕ ′ for α, α ∈ E × and ν = ν ( α ) ν ( α ) − . On the other hand, we have ω ψ ( H ) ϕ ′ ( x ) = − (cid:18) x ∂∂x + · · · + x ∂∂x (cid:19) ϕ ′ ( x ) ,ω ψ ( X ) ϕ ′ ( x ) = 14 π √− (cid:18) ∂ ∂x − u ∂ ∂x + ∂ ∂x − u ∂∂x (cid:19) ϕ ′ ( x ) ,ω ψ ( Y ) ϕ ′ ( x ) = − π √− (cid:18) x − u x + x − u x (cid:19) ϕ ′ ( x ) , where we identity GU( V ) ∼ = ( B × × B × ) /F × with a subgroup of GL ( B ) via the isomorphisms i , i given by (5.12), (5.13). Thus, noting that ∂ ∂z ∂z ρ = 14 (cid:18) ∂ ∂x − u ∂ ∂x (cid:19) , ∂ ∂z ∂z ρ = 14 (cid:18) ∂ ∂x − u ∂ ∂x (cid:19) , we see that ω ψ (2 v − X − − v Y ) ϕ ′ = √− k + 2) ϕ ′ . This implies that ω ψ (1 , ( α, ϕ ′ = χ k +2 ( α ) ϕ ′ for α ∈ E since i ( α ) = (cid:18) a b bu a (cid:19) = (cid:18) − v (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) v − (cid:19) = exp((2 v − X − − v Y ) θ )if we write α = a + b i = e √− θ . This completes the proof. (cid:3) Schwartz functions on X . Let ϕ ∈ S ( X ) be the partial Fourier transform of the Schwartzfunction ϕ ′ ∈ S ( X ′ ) given in § ϕ µ and ϕ k for the partial Fourier transforms of ϕ ′ µ and ϕ ′ k , respectively, to indicate the dependence on a unitary ramified character µ in the case (rps)and on a non-negative integer k in the cases (ds), (fd).) Then we have h ϕ, ϕ i = h ϕ ′ , ϕ ′ i = 1 . Also, since the partial Fourier transform is a G(U( V ) × U( W ))-equivariant map, ϕ satisfies the sameequivariance properties as ϕ ′ . In this subsection, we compute ϕ explicitly.We need to introduce more notation. Put κ = 1 and κ = − J . We define a quadratic F -algebra K by K = F + F j . We write x = e z + e z = x e + x e + x e + x e ∈ X , z i = α i + β i j ∈ K, so that α = x , β = x , α = x , β = 1 J x . Recall that the Weil index γ F ( ψ ) is an 8th root of unity such that Z F φ ( x ) ψ ( x ) dx = γ F ( ψ ) | | − Z F ˆ φ ( x ) ψ (cid:18) − x (cid:19) dx for all φ ∈ S ( F ), where ˆ φ is the Fourier transform of φ with respect to ψ and dx is the self-dualHaar measure on F with respect to ψ . For any non-negative integer n , let H n ( x ) denote the Hermitepolynomial defined by H n ( x ) = ( − n e x d n dx n (cid:16) e − x (cid:17) . The case (ur). E is split and F is unramified. We use the notation of § § ϕ ( x ) = | | − Q i =1 ϕ i ( x i ), where ϕ i ( x i ) = (cid:12)(cid:12)(cid:12)(cid:12) a i b i (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) a i x i (cid:17) Z o ψ (cid:18) a i b i ( x ′ i ) − a i b i x i x ′ i (cid:19) dx ′ i . Lemma 5.4. Assume that ψ is of order zero. Put I ( a, b ) = Z o ψ ( ax + bx ) dx for a, b ∈ F . (i) We have I ( a, b ) = ( I o ( b ) if a ∈ o , ψ ( − b a ) γ F ( aψ ) | a | − I o ( b a ) if a / ∈ p ,where aψ is the non-trivial character of F given by ( aψ )( x ) = ψ ( ax ) . (ii) If F = Q and a ∈ o × , then we have I ( a, b ) = I o × ( ba ) . Proof. If a ∈ o , then we have I ( a, b ) = ˆ I o ( b ) = I o ( b ). If 2 a / ∈ p , then we change the variable x x + y a with y ∈ o to get I ( a, b ) = Z o ψ (cid:18) a (cid:16) x + y a (cid:17) + b (cid:16) x + y a (cid:17)(cid:19) dx = ψ (cid:18) y a + by a (cid:19) Z o ψ ( ax + xy + bx ) dx = ψ (cid:18) y a + by a (cid:19) I ( a, b ) . Assume that 4 a / ∈ p . Then we have I ( a, b ) = ψ ( by a ) I ( a, b ) for all y ∈ o , so that I ( a, b ) = 0 unless b a ∈ o . If b a ∈ o , then we have I ( a, b ) = Z o ψ a (cid:18) x + b a (cid:19) − b a ! dx = ψ (cid:18) − b a (cid:19) Z o ψ ( ax ) dx. On the other hand, by definition, we have Z F φ ( x ) ψ ( ax ) dx = γ F ( aψ ) | a | − Z F ˆ φ ( x ) ψ (cid:18) − x a (cid:19) dx for all φ ∈ S ( F ). Hence we have Z o ψ ( ax ) dx = γ F ( aψ ) | a | − Z o ψ (cid:18) − x a (cid:19) dx = γ F ( aψ ) | a | − . This proves (i).Assume that F = Q and 2 a ∈ o × . As above, we have I ( a, b ) = ψ ( by a ) I ( a, b ) for all y ∈ o , so that I ( a, b ) = 0 unless ba ∈ o . Also, we have I ( a, b ) = ψ (cid:18) − b a (cid:19) Z o ψ ( ax ) dx if ba ∈ o . Changing the variable x x + 1, we have Z o ψ ( ax ) dx = Z o ψ ( ax + 2 ax + a ) dx = ψ ( a ) Z o ψ ( ax ) dx. Since F = Q , ψ is of order zero, and ψ ( a ) = ψ (2 a ) = 1, we must have ψ ( a ) = − 1. Hence we have Z o ψ ( ax ) dx = 0 , so that I ( a, b ) = 0 if ba ∈ o . Assume that ba ∈ o × . Since F = Q , we may write a = y + and b a = z + for some y, z ∈ o . Then we have I ( a, b ) = Z o ψ (cid:18) x (cid:18) y + 12 (cid:19) + 2 x (cid:18) y + 12 (cid:19) (cid:18) z + 12 (cid:19)(cid:19) dx = Z o ψ (cid:18) x + 12 x (cid:19) dx = 1since x ( x + 1) ∈ o for all x ∈ o . This proves (ii). (cid:3) By Lemma 5.4, we have I (cid:18) a i b i , − a i b i x i (cid:19) = I o ( a i b i x i ) if a i b i ∈ o , ψ ( − a i x i ) γ F ( a i ψ ) (cid:12)(cid:12)(cid:12) b i a i (cid:12)(cid:12)(cid:12) I o ( b i x i ) if a i b i / ∈ p ,so that ϕ ( x ) = | | · ψ (cid:18) t x (cid:19) · I o (2 x ) ,ϕ ( x ) = | J | · ψ (cid:18) − tJ x (cid:19) · I o (2 J x ) ,ϕ ( x ) = γ F ( − tJ ψ ) · | J | · ψ (cid:18) tJ x (cid:19) · I o (2 Jx ) ,ϕ ( x ) = γ F ( tJψ ) · | J | · ψ (cid:18) − tJ x (cid:19) · I o (2 Jx ) . We have γ F ( − tJ ψ ) · γ F ( tJψ ) = γ F ( − tJ , ψ ) · γ F (2 tJ, ψ ) · γ F ( 12 ψ ) = γ F ( − J , ψ ) · ( − tJ , tJ ) F · γ F ( − , ψ ) − = γ F ( J , ψ ) · (2 tJ , J ) F . Hence we have ϕ ( x ) = γ F ( J , ψ ) · (2 tJ , J ) F · | | | J | | J |× ψ (cid:18) t x − J x + J x − Jx ) (cid:19) · I o (2 x ) I o (2 J x ) I o (2 Jx ) I o (2 Jx )= γ F ( J , ψ ) · (2 tJ , J ) F · | | | J | | J | · ˜ ϕ κ ( z ) ˜ ϕ κ ( z ) , where ˜ ϕ κ ( z ) = ψ (cid:18) κt K/F ( z ) (cid:19) · I o + o j J (2 κz )for z ∈ K . E is split and F is ramified. We use the notation of § d − is equal to ̟ − d o . By the partial Fourier transform given in § ϕ ( x ) = q d | | − Q i =1 ϕ i ( x i ), where ϕ i ( x i ) = (cid:12)(cid:12)(cid:12)(cid:12) a i b i (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) a i x i (cid:17) Z o ψ (cid:18) a i b i ( x ′ i ) − a i b i x i x ′ i (cid:19) dx ′ i . In particular, we have ϕ i ( x i ) ∈ Z [ q − , µ p ∞ ]for all x i ∈ F , where p is the residual characteristic of F and µ p ∞ is the group of p -power roots ofunity. If further 2 ∈ o × , then we have ϕ i ( x i ) = ψ (cid:16) a i x i (cid:17) · ˆ I o (cid:18) − a i b i x i (cid:19) = q − d · ψ (cid:16) a i x i (cid:17) · I d − ( x i )and hence ϕ ( x ) = q − d · ψ (cid:18) t x − J x − J x + Jx ) (cid:19) · I d − ( x ) I d − ( x ) I d − ( x ) I d − ( x )= q − d · ˜ ϕ κ ( z ) ˜ ϕ κ ( z ) , where ˜ ϕ κ ( z ) = ψ (cid:18) κt K/F ( z ) (cid:19) · I d − + d − j ( z )for z ∈ K .5.6.1.3. The case when E is inert and J ∈ ( F × ) . We use the notation of § § ϕ ( x ) = | J | · I o ( x + tx )ˆ I o (cid:18) − 12 ( x − tx ) (cid:19) I o (cid:18) s (cid:18) x + tJ x (cid:19)(cid:19) ˆ I o (cid:18) J s (cid:18) x − tJ x (cid:19)(cid:19) = | J | · I o ( x ) I o ( sx ) I o (cid:18) stJ x (cid:19) I o ( tx )= | J | · I o + o j t ( z ) I o + o j t ( sz ) . E is inert, and J ∈ ( F × ) or J ∈ ( F × ) . We use the notation of § § ϕ ( x ) = | J | · I o ( x + tx )ˆ I o (cid:18) − 12 ( x − tx ) (cid:19) I o (cid:18) s (cid:18) x + 1 t x (cid:19)(cid:19) ˆ I o (cid:18) J s (cid:18) x − t x (cid:19)(cid:19) = | J | · I o ( x ) I o ( tx ) I o (cid:16) st x (cid:17) I o ( sx )= | J | · I o + o j s ( z ) I o + o j s ( tz ) . E is ramified. We use the notation of § § ϕ ( x ) = q − · I o (cid:18) (cid:18) x + t x + tt x + tx (cid:19)(cid:19) ˆ I o (cid:18) − (cid:18) x − t x + tt x − tx (cid:19)(cid:19) × ˆ I o (cid:18) − (cid:18) x + t x − tt x − tx (cid:19)(cid:19) I o (cid:18) u (cid:18) x − t x − tt x + tx (cid:19)(cid:19) = q − · I o ( x − tx ) I o (cid:18) x − tJ x (cid:19) I o (cid:18) x + t x + tt x + tx (cid:19) I ̟ − o (cid:18) x − t x − tt x + tx (cid:19) = q − · I o ( α − tβ ) I o ( α − tβ ) I o ( α + tβ + t α + tt β ) I ̟ − o ( α + tβ − t α − tt β ) . The case (rps). We use the notation of § § ϕ ( x ) = q n +12 ( q − − Q i =1 ϕ i ( x i ), where ϕ i ( x i ) = ψ (cid:16) a i x i (cid:17) · ˆ I o (cid:18) − a i b i x i (cid:19) = I o ( x i )for i = 1 , ϕ ( x ) = ψ (cid:16) a x (cid:17) · ˆ I ̟ n o (cid:18) − a b x (cid:19) = q − n · ψ (cid:18) − tJ x (cid:19) · I ̟ − n o ( x ) , and ϕ ( x ) = ψ (cid:16) a x (cid:17) · [I o × µ (cid:18) − a b x (cid:19) = ψ (cid:18) tJ x (cid:19) · [I o × µ ( x ) . Since µ is of conductor q n , we have [I o × µ = q − n · g ( µ, ψ ) · I ̟ − n o × µ − , where g ( µ, ψ ) = Z ̟ − n o × µ ( y ) ψ ( y ) dy. Note that | g ( µ, ψ ) | = q n . Hence we have ϕ ( x ) = q − n + ( q − − · g ( µ, ψ ) × ψ (cid:18) t − J x + Jx ) (cid:19) · I o ( x ) I o ( x ) I ̟ − n o ( x ) I ̟ − n o × ( x ) µ ( x ) − = q − n + ( q − − · g ( µ, ψ ) × ψ (cid:18) κ tJ β (cid:19) · I o ( α ) I ̟ − n o × ( β ) µ ( β ) − · ψ (cid:18) κ tJ β (cid:19) · I o ( α ) I ̟ − n o ( β ) . The case (st). B and B are split. We use the notation of § § ϕ ( x ) = q Q i =1 ϕ i ( x i ), where ϕ i ( x i ) = ψ (cid:16) a i x i (cid:17) · ˆ I o (cid:18) − a i b i x i (cid:19) = I o ( x i )for i = 1 , , ϕ ( x ) = ψ (cid:16) a x (cid:17) · ˆ I p (cid:18) − a b x (cid:19) = q − · ψ (cid:18) − tJ x (cid:19) · I ̟ − o ( x ) . Hence we have ϕ ( x ) = q − · ψ (cid:18) − tJ x (cid:19) · I o ( x ) I o ( x ) I ̟ − o ( x ) I o ( x )= q − · I o ( α ) I o ( β ) · ψ (cid:18) κ tJ β (cid:19) · I o ( α ) I ̟ − o ( β ) . B and B are ramified. We use the notation of § § ϕ ( x ) = q − · I o ( x + tx )ˆ I o (cid:18) − 12 ( x − tx ) (cid:19) I o (cid:18) s (cid:18) x + tJ x (cid:19)(cid:19) ˆ I o (cid:18) J s (cid:18) x − tJ x (cid:19)(cid:19) = q − · I o ( x ) I ̟ − o ( x ) I o (cid:18) x + tJ x (cid:19) I ̟ − o (cid:18) x − tJ x (cid:19) = q − · I o ( α ) I ̟ − o ( β ) I o ( α + tβ ) I ̟ − o ( α − tβ ) . The case (1d). We use the notation of § § ϕ ( x ) = q − · I o ( x + tx )ˆ I o (cid:18) − 12 ( x − tx ) (cid:19) I o (cid:18) s (cid:18) x + 1 t x (cid:19)(cid:19) ˆ I o (cid:18) J s (cid:18) x − t x (cid:19)(cid:19) = q − · I o ( x ) I o ( x ) I o (cid:18) x + 1 t x (cid:19) I ̟ − o (cid:18) x − t x (cid:19) = q − · I o ( α ) I o ( α ) I o ( β + tβ ) I ̟ − o ( β − tβ ) . The case (ds). B and B are split. We use the notation of § § ϕ ( x ) = c − k (cid:12)(cid:12)(cid:12)(cid:12) uJ (cid:12)(cid:12)(cid:12)(cid:12) Z ∞−∞ ( x ′ − v √− x ′ ) k e − π ( vx ′ + v x ′ ) e π √− x ′ y ′ dx ′ Z ∞−∞ e − π ( vx ′ + v x ′ ) e π √− x ′ y ′ dx ′ . Lemma 5.5. Let k be a non-negative integer and v a positive real number. Put I ( x, y ) = Z ∞−∞ (cid:18) x + √− v w (cid:19) k e − π ( vx + v w ) e π √− wy dw for x, y ∈ R . Then we have I ( x, y ) = 1 √ k − π k v k − · H k (cid:18) √ πv (cid:18) x − y (cid:19)(cid:19) · e − πv ( x +2 y ) . Proof. We have I ( x, y ) = e − πv x r vπ Z ∞−∞ x + r πv √− w ! k e − w e √ πv √− wy dw = e − πv x r k +1 π k +1 v k − Z ∞−∞ (cid:18)r πv x + √− w (cid:19) k e − ( w −√ πv √− y ) − πvy dw = e − πv x − πvy r k +1 π k +1 v k − Z ∞−∞ (cid:18)r πv x + √− w − √ πvy (cid:19) k e − w dw. Hence the assertion follows from the integral representation of the Hermite polynomial: H k ( x ) = 2 k √ π Z ∞−∞ ( x + √− w ) k e − w dw. (cid:3) By Lemma 5.5, we have ϕ ( x ) = c − k (cid:12)(cid:12)(cid:12)(cid:12) uJ (cid:12)(cid:12)(cid:12)(cid:12) · ( − v √− k √ k − π k v k − · H k (cid:18) √ πv (cid:18) x ′ − y ′ (cid:19)(cid:19) · e − πv ( x ′ +2 y ′ + x ′ +2 y ′ ) = | uJ | ( −√− k k − √ k ! · H k ( √ πvx ) · e − πv ( x + Jx + J x + JJ x ) = | uJ | ( −√− k k − √ k ! · H k ( √ πvα ) · e − πv ( α + Jβ ) · e − πvJ ( α + Jβ ) . B and B are ramified. We use the notation of § § ϕ ( x ) = c − k (cid:12)(cid:12)(cid:12)(cid:12) uJ (cid:12)(cid:12)(cid:12)(cid:12) Z ∞−∞ ( x ′ − v √− x ′ ) k e − π ( vx ′ + v x ′ ) e π √− x ′ y ′ dx ′ Z ∞−∞ e − π ( vx ′ + v x ′ ) e π √− x ′ y ′ dx ′ . By Lemma 5.5, we have ϕ ( x ) = c − k (cid:12)(cid:12)(cid:12)(cid:12) uJ (cid:12)(cid:12)(cid:12)(cid:12) · ( − v √− k √ k − π k v k − · H k (cid:18) √ πv (cid:18) x ′ − y ′ (cid:19)(cid:19) · e − πv ( x ′ +2 y ′ + x ′ +2 y ′ ) = | uJ | ( −√− k k − √ k ! · H k ( √ πvx ) · e − πv ( x + Jx − J x − JJ x ) = | uJ | ( −√− k k − √ k ! · H k ( √ πvα ) · e − πv ( α + Jβ ) · e πvJ ( α + Jβ ) . The case (fd). We use the notation of § § ϕ ( x ) = c − k (cid:12)(cid:12)(cid:12)(cid:12) Ju (cid:12)(cid:12)(cid:12)(cid:12) Z ∞−∞ (cid:18) x ′ + √− v x ′ (cid:19) k e − π ( vx ′ + v x ′ ) e π √− x ′ y ′ dx ′ Z ∞−∞ e − π ( vx ′ + v x ′ ) e π √− x ′ y ′ dx ′ . By Lemma 5.5, we have ϕ ( x ) = c − k (cid:12)(cid:12)(cid:12)(cid:12) Ju (cid:12)(cid:12)(cid:12)(cid:12) · √ k − π k v k − · H k (cid:18) √ πv (cid:18) x ′ − y ′ (cid:19)(cid:19) · e − πv ( x ′ +2 y ′ + x ′ +2 y ′ ) = | uJ | k − √ k ! · H k ( √ πvx ) · e − πv ( x + J x − Jx − JJ x ) = | uJ | k − √ k ! · H k ( √ πvα ) · e − πv ( α − Jβ ) · e − πvJ ( α − Jβ ) . Explicit form of the Rallis inner product formula In this section, we shall explicate the Rallis inner product formula (Proposition 4.9). Measures. In § 4, for any connected reductive algebraic group G over a number field F , we havealways taken the Tamagawa measure on G ( A ), which is a product of Haar measures on G v defined interms of a non-zero invariant differential form of top degree on G over F . However, with respect tothis Haar measure, the volume of a hyperspecial maximal compact subgroup of G v is not necessarily 1for almost all v . For our applications, it is more convenient to take the “standard” measure on G ( A ),which is a product of Haar measures on G v such that the volume of a maximal compact subgroup of G v is 1 for all v . In this subsection, we give a precise definition of the standard measures on A × \ B × ( A )and B ( A ), where B is a quaternion algebra over F , and compare them with the Tamagawa measures.Let F be a number field and ψ the standard additive character of A /F . Let D = D F be thediscriminant of F . We have | D | = Q v ∈ Σ fin q d v v , where d v is the non-negative integer such that ψ v istrivial on ̟ − d v v o v but non-trivial on ̟ − d v − v o v . For each place v of F , let ζ v ( s ) be the local zetafunction of F v defined by ζ v ( s ) = (1 − q − sv ) − if v is finite, π − s Γ( s ) if v is real,2(2 π ) − s Γ( s ) if v is complex.Note that ζ v (1) = ( v is real, π − if v is complex.Let ζ F ( s ) = Q v ∈ Σ fin ζ v ( s ) be the Dedekind zeta function of F . Put ρ F := Res s =1 ζ F ( s ) = 2 r (2 π ) r hR | D | w , where r is the number of the real places of F , r is the number of the complex places of F , h = h F is the class number of F , R = R F is the regulator of F , and w = w F is the number of roots of unityin F . For any connected reductive algebraic group G over F , let τ ( G ) denote the Tamagawa numberof G .From now on, we assume that F is totally real.6.1.1. Measures on A × . For each place v of F , we define a Haar measure d × x Tam v on F × v by d × x Tam v := ζ v (1) · dx v | x v | , where dx v is the self-dual Haar measure on F v with respect to ψ v . Note that: • vol( o v , dx v ) = q − dv v if v is finite, • dx v is the Lebesgue measure if v is real.Then the Tamagawa measure on A × is given by d × x Tam := ρ − F · Y v d × x Tam v . We have τ ( G m ) = 1.On the other hand, we define the standard measure on A × as a product measure d × x := Q v d × x v ,where • d × x v is the Haar measure on F × v such that vol( o × v , d × x v ) = 1 if v is finite, • d × x v = dx v | x v | if v is real. We have(6.1) d × x Tam = | D | − ρ − F · d × x. Measures on B × ( A ) . For each place v of F , we define a Haar measure d × α Tam v on B × v by d × α Tam v := ζ v (1) · d α v | ν ( α v ) | , where d α v is the self-dual Haar measure on B v with respect to the pairing ( α v , β v ) ψ v (tr B v /F v ( α v β v )).Then the Tamagawa measure on B × ( A ) is given by d × α Tam := ρ − F · Y v d × α Tam v . Also, the Tamagawa measure on ( B × / G m )( A ) = B × ( A ) / A × is given by the quotient measure d × α Tam /d × x Tam .We have τ ( B × / G m ) = 2.On the other hand, we define the standard measure on B × ( A ) as a product measure d × α := Q v d × α v , where d × α v is given as follows: • For v ∈ Σ fin r Σ B, fin , fix an isomorphism i v : B v → M ( F v ) of quaternion F v -algebras and let d × α v be the Haar measure on B × v such that vol( i − v (GL ( o v )) , d × α v ) = 1. Since i v is unique upto inner automorphisms, d × α v is independent of the choice of i v . • For v ∈ Σ B, fin , let d × α v be the Haar measure on B × v such that vol( o × B v , d × α v ) = 1, where o B v isthe unique maximal order in B v . • For v ∈ Σ ∞ r Σ B, ∞ , fix an isomorphism i v : B v → M ( F v ) of quaternion F v -algebras and definea Haar measure d × α v on B × v by d × α v = dx v dy v | y v | dz v z v dκ v for α v = i − v (cid:0)(cid:0) x v (cid:1) (cid:0) y v (cid:1) z v κ v (cid:1) with x v ∈ R , y v ∈ R × , z v ∈ R × + , κ v ∈ SO(2), where dx v , dy v , dz v are the Lebesgue measures and dκ v is the Haar measure on SO(2) such that vol(SO(2) , dκ v ) = 1.Since i v is unique up to inner automorphisms, d × α v is independent of the choice of i v . • For v ∈ Σ B, ∞ , let d × α v be the Haar measure on B × v such that vol( B × v /F × v , d × α v /d × x v ) = 1.Also, we define the standard measure on B × ( A ) / A × as the quotient measure d × α /d × x . Lemma 6.1. We have d × α Tam = (2 π ) | Σ ∞ r Σ B, ∞ | · (4 π ) | Σ B, ∞ | · Y v ∈ Σ B, fin ( q v − − · | D | − · ρ − F · ζ F (2) − · d × α . Proof. For each place v of F , let C v be the constant such that d × α Tam v = C v · d × α v . If v ∈ Σ fin r Σ B, fin ,we identify B v with M ( F v ). Then we have vol(M ( o v ) , d α v ) = q − d v v and hence C v = vol(GL ( o v ) , d × α Tam v )= ζ v (1) · | GL ( F q v ) | · vol(1 + M ( p v ) , d α v )= q − d v v · ζ v (2) − . If v ∈ Σ B, fin , then we have vol( o B v , d α v ) = q − d v − v and hence C v = vol( o × B v , d × α Tam v )= ζ v (1) · | F × q v | · vol(1 + p B v , d α v )= q − d v v · ( q v − − · ζ v (2) − . If v ∈ Σ ∞ r Σ B, ∞ , we identify B v with M ( R ). Then d × α Tam v arises from the gauge form on GL ( R )determined (up to sign) by the lattice M ( Z ) in Lie GL ( R ) = M ( R ). Also, the measures dx v dy v | y v | , dz v z v , dκ v in the definition of d × α v arise from the (left invariant) gauge forms determined by the lattices Z (cid:18) (cid:19) + Z (cid:18) (cid:19) , Z (cid:18) (cid:19) , π Z (cid:18) − (cid:19) , respectively. Hence we have C v = 2 π. If v ∈ Σ B, ∞ , we identify B v with H := (cid:26)(cid:18) α β − ¯ β ¯ α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α, β ∈ C (cid:27) . Then d × α Tam v arises from the gauge form on H × determined by the lattice spanned by1 √ (cid:18) (cid:19) , √ (cid:18) √− −√− (cid:19) , √ (cid:18) − (cid:19) , √ (cid:18) √− √− (cid:19) in Lie H × = H . Let d × ˙ α v be the Haar measure on H × / R × which arises from the gauge form determinedby the lattice spanned by12 (cid:18) √− −√− (cid:19) , (cid:18) − (cid:19) , (cid:18) √− √− (cid:19) , so that we have d × α Tam v /d × x v = 2 · d × ˙ α v . Note that this lattice is an integral lattice in Lie H × / R × constructed in [50]. It follows from [50] that vol( H × / R × , d × ˙ α v ) = 2 π and hence C v = vol( H × / R × , d × α Tam v /d × x v ) = 4 π . This completes the proof. (cid:3) Example . Suppose that B is totally definite. Put B ∞ = B ⊗ Q R and fix a maximal compact subgroup K of B × ( A fin ). We can write B × ( A ) = n G i =1 A × B × ( F ) B ×∞ α i K , where { α i ∈ B × ( A fin ) | ≤ i ≤ n } is a (finite) set of representatives for A × B × ( F ) B ×∞ \ B × ( A ) / K . Put mass := vol (cid:18) A × B × ( F ) \ B × ( A ) , d × α d × x (cid:19) = n X i =1 | Γ i | , where Γ i = F × \ ( B × ( F ) ∩ A × fin α i K α − i ). Then it follows from (6.1) and Lemma 6.1 that mass = τ ( B × / G m ) · (4 π ) − d · Y v ∈ Σ B, fin ( q v − · | D | · ζ F (2)= ( − d · − d +1 · Y v ∈ Σ B, fin ( q v − · ζ F ( − , where d = [ F : Q ].Finally, we compare the standard measure on A × \ B × ( A ) with the measure on the Shimura variety.Let ( G, X ) = (Res F/ Q ( B × ) , X B ) be the Shimura datum given in § v ∈ Σ ∞ r Σ B, ∞ , we identify B v with M ( R ). As explained in § X = Y v ∈ Σ ∞ r Σ B, ∞ h ± . Fix an open compact subgroup K of B × ( A fin ) such thatˆ o × ⊂ K, where ˆ o × := Q v ∈ Σ fin o × v ⊂ A × fin . Let Sh K ( G, X ) be the associated Shimura variety:Sh K ( G, X ) = B × ( F ) \ X × B × ( A fin ) /K. Since h ± = GL ( R ) / R × + · SO(2), we have a natural surjective map p : B × ( A ) −→ Sh K ( G, X ) . Recall that in Definition 1.11, we have taken the measure on Sh K ( G, X ) given as follows: • On X , we take the product over v ∈ Σ ∞ r Σ B, ∞ of the GL ( R )-invariant measure dx v dy v | y v | for x v + √− y v ∈ h ± , where dx v , dy v are the Lebesgue measures. This measure is independentof the choice of identification (6.2). • On B × ( A fin ) /K , we take the counting measure. • If B is not totally definite, then o × \ B × ( F ) acts on X × B × ( A fin ) /K properly discontinuously,and we take a natural measure dµ x on Sh K ( G, X ) induced by the product of the above measures. • If B is totally definite, then Sh K ( G, X ) is a finite set, and for any x ∈ Sh K ( G, X ), its stabilizerΓ x in o × \ B × ( F ) is a finite group. We take a measure dµ x on Sh K ( G, X ) given by Z Sh K ( G,X ) φ ( x ) dµ x = X x ∈ Sh K ( G,X ) | Γ x | − φ ( x ) . Lemma 6.3. Let φ be an integrable function on Sh K ( G, X ) such that φ ( x · z ) = φ ( x ) for all x ∈ Sh K ( G, X ) and z ∈ A × . Then we have (6.3) Z Sh K ( G,X ) φ ( x ) dµ x = 2 | Σ ∞ r Σ B, ∞ | · [ K : K ] · h F · Z A × B × ( F ) \ B × ( A ) p ∗ φ ( α ) d × α , where K is any maximal compact subgroup of B × ( A fin ) containing K .Proof. Put F ∞ = F ⊗ Q R and B ∞ = B ⊗ Q R . We can write B × ( A ) = n G i =1 B × ( F ) B ×∞ α i K, where { α i ∈ B × ( A fin ) | ≤ i ≤ n } is a (finite) set of representatives for B × ( F ) B ×∞ \ B × ( A ) /K . Thenwe have Sh K ( G, X ) = n G i =1 Γ i \ X, where Γ i = o × \ ( B × ( F ) ∩ α i K α − i ). For each i , we have a natural commutative diagram B ×∞ α i K p i / / (cid:15) (cid:15) X (cid:15) (cid:15) B × ( F ) B ×∞ α i K p / / Γ i \ X . First we assume that B is not totally definite. Since both sides of (6.3) are proportional, we mayassume that for each i , the restriction of φ to Γ i \ X is of the form φ ( x ) = X γ ∈ Γ i ϕ i ( γx )for some continuous compactly supported function ϕ i on X . Then, noting that Γ i acts on X faithfully,we have Z Γ i \ X φ ( x ) dµ x = Z X ϕ i ( x ) dµ x , where the measure dµ x on X on the right-hand side is as defined above. By the definition of thestandard measure, we have Z X ϕ i ( x ) dµ x = 2 | Σ ∞ r Σ B, ∞ | · vol( K ) − · Z B ×∞ α i K/F ×∞ ˆ o × p ∗ i ϕ i ( α ) d × α . (Here the factor 2 arises from [ R × : R × + ].) Since p ∗ φ ( α ) = φ ( p ( α )) = X γ ∈ Γ i ϕ i ( γp i ( α )) = X γ ∈ Γ i ϕ i ( p i ( γ α )) = X γ ∈ Γ i p ∗ i ϕ i ( γ α )for α ∈ B ×∞ α i K , we have Z B ×∞ α i K/F ×∞ ˆ o × p ∗ i ϕ i ( α ) d × α = Z Γ i \ B ×∞ α i K/F ×∞ ˆ o × p ∗ φ ( α ) d × α . Thus, noting that Γ i \ B ×∞ α i K/F ×∞ ˆ o × = B × ( F ) \ B × ( F ) B ×∞ α i K/F ×∞ ˆ o × , we have Z Γ i \ X φ ( x ) dµ x = 2 | Σ ∞ r Σ B, ∞ | · vol( K ) − · Z B × ( F ) \ B × ( F ) B ×∞ α i K/F ×∞ ˆ o × p ∗ φ ( α ) d × α . Summing over i , we obtain Z Sh K ( G,X ) φ ( x ) dµ x = 2 | Σ ∞ r Σ B, ∞ | · vol( K ) − · Z B × ( F ) \ B × ( A ) /F ×∞ ˆ o × p ∗ φ ( α ) d × α = 2 | Σ ∞ r Σ B, ∞ | · vol( K ) − · vol( F × \ A × /F ×∞ ˆ o × ) · Z A × B × ( F ) \ B × ( A ) p ∗ φ ( α ) d × α . On the other hand, we have vol( K ) = [ K : K ] − for any maximal compact subgroup K of B × ( A fin )containing K , and vol( F × \ A × /F ×∞ ˆ o × ) = h F since the standard measure on A × /F ×∞ ˆ o × is the countingmeasure. This proves (6.3).Next we assume that B is totally definite. Sincevol( B × ( F ) \ B × ( F ) B ×∞ α i K/F ×∞ ˆ o × ) = | Γ i | − · vol( K ) , we have Z B × ( F ) \ B × ( A ) /F ×∞ ˆ o × p ∗ φ ( α ) d × α = vol( K ) · n X i =1 | Γ i | − p ∗ φ ( α i )= vol( K ) · Z Sh K ( G,X ) φ ( x ) dµ x . The rest of the proof is the same as before. (cid:3) Measures on B ( A ) . We recall the exact sequence1 −→ B −→ B × ν −→ G m −→ F . For each place v of F , this induces an exact sequence1 −→ B v −→ B × v ν −→ F × v . We define a Haar measure dg Tam v on B v by requiring that Z B × v φ ( α v ) d × α Tam v = Z ν ( B × v ) ˙ φ ( x v ) d × x Tam v for all φ ∈ L ( B × v ), where ˙ φ ( ν ( α v )) := Z B v φ ( g v α v ) dg Tam v . Note that ν ( B × v ) = F × v unless v ∈ Σ B, ∞ , in which case we have ν ( B × v ) = R × + . Then the Tamagawameasure on B ( A ) is given by dg Tam := Y v dg Tam v . We have τ ( B ) = 1.On the other hand, we define the standard measure on B ( A ) as a product measure dg := Q v dg v ,where dg v is given as follows: • For v ∈ Σ fin r Σ B, fin , fix an isomorphism i v : B v → M ( F v ) of quaternion F v -algebras, whichis unique up to inner automorphisms by elements of GL ( F v ), and let dg v be the Haar measureon B v such that vol( i − v (SL ( o v )) , dg v ) = 1. Noting that there are exactly 2 conjugacy classesof maximal compact subgroups of SL ( F v ), i.e., those of SL ( o v ) and (cid:0) ̟ v (cid:1) SL ( o v ) (cid:0) ̟ − v (cid:1) , wehave vol( i − v ( h v SL ( o v ) h − v ) , dg v ) = vol( i − v (SL ( o v )) , dg v )for h v ∈ GL ( F v ). Hence dg v is independent of the choice of i v . • For v ∈ Σ B, fin , let dg v be the Haar measure on B v such that vol( B v , dg v ) = 1. • For v ∈ Σ ∞ r Σ B, ∞ , fix an isomorphism i v : B v → M ( F v ) of quaternion F v -algebras, which isunique up to inner automorphisms by elements of GL ( F v ), and define a Haar measure dg v on B v by dg v = dx v dy v y v dκ v for g v = i − v (cid:16)(cid:0) x v (cid:1) (cid:0) √ y v √ y v − (cid:1) κ v (cid:17) with x v ∈ R , y v ∈ R × + , κ v ∈ SO(2), where dx v , dy v are theLebesgue measures and dκ v is the Haar measure on SO(2) such that vol(SO(2) , dκ v ) = 1. Thismeasure dg v does not change if we replace i v by Ad( h v ) ◦ i v for h v ∈ SL ( F v ). If we replace i v byAd (cid:0) − (cid:1) ◦ i v , then dg v becomes dx v dy v y v dκ v for g v = i − v (cid:16)(cid:0) − x v (cid:1) (cid:0) √ y v √ y v − (cid:1) κ − v (cid:17) , which is infact equal to the original dg v . Hence dg v is independent of the choice of i v . • For v ∈ Σ B, ∞ , let dg v be the Haar measure on B v such that vol( B v , dg v ) = 1. Lemma 6.4. We have dg Tam = π | Σ ∞ r Σ B, ∞ | · (4 π ) | Σ B, ∞ | · Y v ∈ Σ B, fin ( q v − − · | D | − · ζ F (2) − · dg. Proof. For each place v of F , let C v be the constant such that dg Tam v = C v · dg v . If v ∈ Σ fin r Σ B, fin ,we identify B v with M ( F v ). As in the proof of Lemma 6.1, we have C v = vol(SL ( o v ) , dg Tam v ) = vol(GL ( o v ) , d × α Tam v )vol( o × v , d × x Tam v ) = q − dv v · ζ v (2) − . If v ∈ Σ B, fin , then as in the proof of Lemma 6.1, we have C v = vol( B v , dg Tam v ) = vol( o × B v , d × α Tam v )vol( o × v , d × x Tam v ) = q − dv v · ( q v − − · ζ v (2) − . If v ∈ Σ ∞ r Σ B, ∞ , we identify B v with M ( R ). For α v ∈ GL ( R ) + , we write α v = z v · g v with z v ∈ R × + and g v ∈ SL ( R ). Then we have d × α Tam v = 2 · d × z v dg Tam v , d × α v = d × z v dg v on GL ( R ) + . Since d × α Tam v = 2 π · d × α v as in the proof of Lemma 6.1, we have C v = 12 · π = π. If v ∈ Σ B, ∞ , then we have d × α Tam v = 2 · d × z v dg Tam v for α v = z v · g v with z v ∈ R × + and g v ∈ B v . Hencewe have C v = vol( B v , dg Tam v )= 12 · vol( B × v / R × + , d × α Tam v /d × z v )= vol( B × v / R × , d × α Tam v /d × z v )= 4 π as in the proof of Lemma 6.1. This completes the proof. (cid:3) Example . Suppose that B = M ( F ). Put vol := vol SL ( o ) \ h d , Y v ∈ Σ ∞ dx v dy v y v ! , where d = [ F : Q ]. Since SL ( o ) \ h d ∼ = SL ( F ) \ SL ( A ) /K, where K = Q v ∈ Σ ∞ SO(2) × Q v ∈ Σ fin SL ( o v ), we havevol(SL ( F ) \ SL ( A ) , dg ) = vol · vol {± }\ K, Y v ∈ Σ ∞ dκ v · Y v ∈ Σ fin dg v ! = vol · . On the other hand, it follows from Lemma 6.4 thatvol(SL ( F ) \ SL ( A ) , dg ) = τ ( B ) · π − d · | D | · ζ F (2) = ( − π ) d · ζ F ( − . Hence we have vol = ( − d · d +1 · π d · ζ F ( − . New vectors. In this subsection, we define a 1-dimensional subspace of new vectors in the spaceof an irreducible representation of B × v . For the moment, we fix a place v of F and suppress thesubscript v from the notation. We only consider representations π of B × listed below: • If F is non-archimedean and B is split, then(ur) π = Ind( χ ⊗ µ ) is a principal series representation, where χ and µ are unitary unramified; or(rps) π = Ind( χ ⊗ µ ) is a principal series representation, where χ is unitary unramified and µ isunitary ramified of conductor q n ; or(st) π = St ⊗ χ is a twist of the Steinberg representation, where χ is unitary unramified. • If F is non-archimedean and B is ramified, then(1d) π = χ ◦ ν is a 1-dimensional representation, where χ is unitary unramified. • If F = R and B is split, then(ds) π = DS k is the irreducible unitary (limit of) discrete series representation of weight k . • If F = R and B is ramified, then(fd) π = Sym k is the irreducible unitary ( k + 1)-dimensional representation.If F is non-archimedean, we define a compact subgroup K n of GL ( F ) by K n = (cid:26) (cid:18) a bc d (cid:19) ∈ GL ( o ) (cid:12)(cid:12)(cid:12)(cid:12) c ∈ ̟ n o (cid:27) . Note that I := K is an Iwahori subgroup of GL ( F ). If F = R , we define a character χ k of C × by χ k ( α ) = (cid:18) α √ αα ρ (cid:19) k . The case (ur). Fix an isomorphism i : B → M ( F ). This determines a maximal compactsubgroup K = i − (GL ( o )) of B × . We say that f ∈ π is a new vector with respect to K if π ( k ) f = f for all k ∈ K .6.2.2. The case (rps). Fix an isomorphism i : B → M ( F ). This determines a compact subgroup K n = i − ( K n ) of B × . We define a character µ of K n by µ ( k ) = µ ( d ) for k = i − (cid:0) a bc d (cid:1) . We say that f ∈ π is a new vector with respect to ( K n , µ ) if π ( k ) f = µ ( k ) f for all k ∈ K n .6.2.3. The case (st). Fix an isomorphism i : B → M ( F ). This determines an Iwahori subgroup I = i − ( I ) of B × . We say that f ∈ π is a new vector with respect to I if π ( k ) f = f for all k ∈ I .6.2.4. The case (1d). Let K = o × B be the unique maximal compact subgroup of B × . Then we have π ( k ) f = f for all k ∈ K and f ∈ π . For uniformity, we call any f ∈ π a new vector with respect to K . The cases (ds), (fd). Fix an embedding h : C × ֒ → B × . We say that f ∈ π is a new vector withrespect to h if π ( h ( z )) f = χ k ( z ) f for all z ∈ C × .6.3. An explicit Rallis inner product formula. Suppose that F is global. Let π ∼ = ⊗ v π v be anirreducible unitary cuspidal automorphic representation of GL ( A ) such that for v ∈ Σ fin , • π v = Ind( χ v ⊗ µ v ), where χ v and µ v are unitary unramified; or • π v = Ind( χ v ⊗ µ v ), where χ v is unitary unramified and µ v is unitary ramified of conductor q n v v ;or • π v = St ⊗ χ v , where χ v is unitary unramified,and for v ∈ Σ ∞ , • π v = DS k v , where k v ≥ π v is unramified for all finite places v of F such that F v is ramified or of residualcharacteristic 2. Put Σ π = { v | π v is a discrete series } , Σ π, fin = Σ π ∩ Σ fin , andΣ ′ π, fin := { v ∈ Σ fin | π v is a ramified principal series } . We consider a non-zero vector f = ⊗ v f v ∈ π such that: • for v ∈ Σ fin r (Σ π, fin ∪ Σ ′ π, fin ), f v is a new vector with respect to GL ( o v ); • for v ∈ Σ π, fin , f v is a new vector with respect to the Iwahori subgroup I of GL ( F v ) given in § • for v ∈ Σ ′ π, fin , f v is a new vector with respect to ( K n v , µ v ), where K n v is the compact subgroupof GL ( F v ) given in § µ v is the character of K n v defined by µ v (cid:0) a bc d (cid:1) = µ v ( d ); • for v ∈ Σ ∞ , f v is a new vector with respect to the embedding h v : C × ֒ → GL ( R ) defined by h v ( a + b √− 1) = (cid:0) a b − b a (cid:1) .We normalize such a vector f , which is unique up to scalars, so that W f (cid:18) δ − (cid:19) = e − πd , where W f is the Whittaker function of f defined by W f ( g ) = Z F \ A f (cid:18)(cid:18) x (cid:19) g (cid:19) ψ ( x ) dx with the Tamagawa measure dx on A , δ = ( ̟ d v v ) ∈ A × fin , and d = [ F : Q ] (see also Lemmas 6.8, 6.10,6.12 and (6.4) below). Let h f, f i be the Petersson norm of f defined by h f, f i = Z A × GL ( F ) \ GL ( A ) | f ( g ) | dg, where dg is the standard measure on A × \ GL ( A ). In § Proposition 6.6. We have h f, f i = 2 · Y v ∈ Σ ∞ ( k v − k v +1 π k v +1 · Y v ∈ Σ π, fin ∪ Σ ′ π, fin q v q v + 1 · | D | · L (1 , π, ad) , where L ( s, π, ad) = Q v ∈ Σ fin L ( s, π v , ad) is the adjoint L -function of π . Let B , B , B be quaternion algebras over F such that B = B · B in the Brauer group. We assumethat Σ B = ∅ and Σ B ∪ Σ B ∪ Σ B ⊂ Σ π , i.e., B is division and the Jacquet–Langlands transfers π B , π B , π B of π to B × ( A ), B × ( A ), B × ( A ) exist. Now, we choose a totally imaginary quadratic extension E of F such that E embeds into B , B , B , and write E = F + F i , B = E + E j , B = E + E j , B = E + E j . We also impose the ramification conditions on u, J, J , J in § f B = ⊗ v f B,v ∈ π B , f B = ⊗ v f B ,v ∈ π B , f B = ⊗ v f B ,v ∈ π B such that: • for v ∈ Σ fin r (Σ π, fin ∪ Σ ′ π, fin ), f B,v , f B ,v , f B ,v are new vectors with respect to K , K , K ,respectively, given in § • for v ∈ Σ π, fin r (Σ B, fin ∪ Σ B , fin ∪ Σ B , fin ), f B,v , f B ,v , f B ,v are new vectors with respect to I , I , I , respectively, given in § • for v ∈ Σ B , fin ∩ Σ B , fin , f B,v , f B ,v , f B ,v are new vectors with respect to I , K , K , respectively,given in § • for v ∈ Σ B, fin ∩ Σ B , fin , f B,v , f B ,v , f B ,v are new vectors with respect to K , I , K , respectively,given in § B and B for v ∈ Σ B, fin ∩ Σ B , fin ; • for v ∈ Σ ′ π, fin , f B,v , f B ,v , f B ,v are new vectors with respect to ( K n v , µ v ), ( K ,n v , µ v ), ( K ,n v , µ − v · µ v ◦ ν ), respectively, given in § • for v ∈ Σ ∞ , f B,v , f B ,v , f B ,v are new vectors with respect to the embeddings C × ∼ = E × v ֒ → B × v , C × ∼ = E × v ֒ → B × ,v , C × ∼ = E × v ֒ → B × ,v , respectively, given in §§ f B , f B , f B , which are unique up to scalars. We emphasize that the 1-dimensionalsubspaces of π B , π B , π B spanned by f B , f B , f B , respectively, depend on the choice of E , i , j , j , j . Let h f B , f B i be the Petersson norm of f B defined by h f B , f B i = Z A × B × ( F ) \ B × ( A ) | f B ( g ) | dg, where dg is the standard measure on A × \ B × ( A ). We define h f B , f B i and h f B , f B i similarly.Let ϕ = ⊗ v ϕ v ∈ S ( X ( A )) be the Schwartz function given in § • ϕ v = ϕ µ v for v ∈ Σ ′ π, fin ; • ϕ v = ϕ k v for v ∈ Σ ∞ r (Σ B, ∞ ∪ Σ B , ∞ ∪ Σ B , ∞ ); • ϕ v = ϕ k v − for v ∈ Σ B , ∞ ∩ Σ B , ∞ ; • ϕ v = ϕ k v − for v ∈ Σ B, ∞ ∩ Σ B , ∞ ; we switch the roles of B and B for v ∈ Σ B, ∞ ∩ Σ B , ∞ .In § θ ϕ ( f B ), but for our purposes, we slightly modify its definition:on the right-hand side of (4.1), we take the standard measure on B ( A ) rather than the Tamagawameasure on B ( A ). We regard θ ϕ ( f B ) as an automorphic form on B × ( A ) × B × ( A ). Then it followsfrom the equivariance properties of ϕ that there exists a constant α ( B , B ) ∈ C (once we fix f B , f B , f B ) such that θ ϕ ( f B ) = α ( B , B ) · ( f B × f B ) . Now we state an explicit Rallis inner product formula. Theorem 6.7. We have | α ( B , B ) | · h f B , f B i · h f B , f B i = C · h f, f i · h f B , f B i , where C = | D | · Q v C v with C v = if v ∈ Σ fin r (Σ π, fin ∪ Σ ′ π, fin ) , q v ( q v + 1) if v ∈ Σ π, fin r (Σ B, fin ∪ Σ B , fin ∪ Σ B , fin ) , q v if v ∈ Σ B , fin ∩ Σ B , fin , q v if v ∈ Σ B, fin , q n v − v ( q v − q v + 1) if v ∈ Σ ′ π, fin , k v +2 π k v k v ! if v ∈ Σ ∞ r (Σ B, ∞ ∪ Σ B , ∞ ∪ Σ B , ∞ ) , k v π k v − ( k v − · ( k v − if v ∈ Σ B , ∞ ∩ Σ B , ∞ , k v − π k v − ( k v − · ( k v − if v ∈ Σ B, ∞ .Proof. By Proposition 4.9, we have C B · C B · ( C B ) · | α ( B , B ) | · h f B , f B i · h f B , f B i = 2 · C B · C B · L S (1 , π, ad) ζ SF (2) · h f B , f B i · Y v ∈ S Z v for a sufficiently large finite set S of places of F , where • C B is the constant such that dg Tam = C B · dg , where dg Tam is the Tamagawa measure on A × \ B × ( A ) and dg is the standard measure on A × \ B × ( A ); we define C B and C B similarly; • C B is the constant such that dg Tam1 = C B · dg , where dg Tam1 is the Tamagawa measure on B ( A )and dg is the standard measure on B ( A ); • Z v is the integral defined by Z v = Z B v h ω ψ ( g ,v ) ϕ v , ϕ v ih π B,v ( g ,v ) f B,v , f B,v i dg ,v (cf. (4.2)), where · the hermitian inner product h· , ·i on S ( X v ) is normalized as in § · the invariant hermitian inner product h· , ·i on π B,v is normalized so that h f B,v , f B,v i = 1; · dg ,v is the standard measure on B v .Hence, by (6.1) and Lemmas 6.1, 6.4, we have | α ( B , B ) | · h f B , f B i · h f B , f B i = C ′ · L (1 , π, ad) · h f B , f B i · Y v ∈ S fin ζ v (2) L (1 , π v , ad) · Y v ∈ S Z v , where S fin = S ∩ Σ fin and C ′ = 2 · C B C B · C B · C B · ζ F (2) = 2 · | Σ ∞ r Σ B, ∞ | · (2 π ) −| Σ ∞ r Σ B , ∞ |−| Σ ∞ r Σ B , ∞ | · (4 π ) −| Σ B , ∞ |−| Σ B , ∞ | × Y v ∈ Σ B , fin ( q v − · Y v ∈ Σ B , fin ( q v − · | D | = 2 · | D | · Y v C ′ v with C ′ v = v ∈ Σ fin r (Σ B, fin ∪ Σ B , fin ∪ Σ B , fin ),( q v − if v ∈ Σ B , fin ∩ Σ B , fin , q v − v ∈ Σ B, fin ,(2 π ) − if v ∈ Σ ∞ r (Σ B, ∞ ∪ Σ B , ∞ ∪ Σ B , ∞ ),(8 π ) − if v ∈ Σ B , ∞ ∩ Σ B , ∞ ,(8 π ) − if v ∈ Σ B, ∞ .Moreover, by Proposition 6.6, we have | α ( B , B ) | · h f B , f B i · h f B , f B i = C ′′ · h f, f i · h f B , f B i · Y v ∈ S fin ζ v (2) L (1 , π v , ad) · Y v ∈ S Z v , where C ′′ = | D | · Q v C ′′ v with C ′′ v = v ∈ Σ fin r (Σ π, fin ∪ Σ ′ π, fin ), q v + 1 q v if v ∈ Σ π, fin r (Σ B, fin ∪ Σ B , fin ∪ Σ B , fin ),( q v − ( q v + 1) q v if v ∈ Σ B , fin ∩ Σ B , fin ,( q v − q v + 1) q v if v ∈ Σ B, fin , q v + 1 q v if v ∈ Σ ′ π, fin ,2 k v π k v − ( k v − v ∈ Σ ∞ r (Σ B, ∞ ∪ Σ B , ∞ ∪ Σ B , ∞ ),2 k v − π k v − ( k v − v ∈ Σ B , ∞ ∩ Σ B , ∞ ,2 k v − π k v − ( k v − v ∈ Σ B, ∞ .Now Theorem 6.7 follows from this and Lemmas 6.21, 6.22, 6.23, 6.24, 6.25, 6.26 in § Z v explicitly. (cid:3) Computation of h f, f i . Proposition 6.6 follows from a standard computation of the Rankin–Selberg integral, but we give the details of the proof for the convenience of the reader. We retain thenotation of § n ( x ) = (cid:18) x (cid:19) , t ( y ) = (cid:18) y (cid:19) , w = (cid:18) − (cid:19) . By [36, § § C · h f, f i = 2 ρ F · Res s =1 L S ( s, π × π ∨ ) ζ SF (2) · | D | − · Y v ∈ S k W v k for a sufficiently large finite set S of places of F , where • C is the constant such that dg Tam = C · dg , where dg Tam is the Tamagawa measure on A × \ GL ( A )and dg is the standard measure on A × \ GL ( A ); • the Whittaker function W f of f is decomposed as a product W f = Q v W v , where W v is theWhittaker function of π v with respect to ψ v normalized so that · W v ( t ( ̟ − d v v )) = 1 for v ∈ Σ fin r (Σ π, fin ∪ Σ ′ π, fin ); · W v (1) = 1 for v ∈ Σ π, fin ; · W v (1) = 1 for v ∈ Σ ′ π, fin ; · W v (1) = e − π for v ∈ Σ ∞ ; • k W v k is the integral defined by k W v k = Z F × v | W v ( t ( y v )) | d × y v , where d × y v is the standard measure on F × v .We remark that: • the volume of F × \ A given in [80, Proposition 3.1] is equal to ρ F , • Q v d × y Tam v = | D | − · Q v d × y v .Hence, by (6.1) and Lemma 6.1, we have h f, f i = 2 · (2 π ) − [ F : Q ] · | D | · L (1 , π, ad) · Y v ∈ S fin ζ v (2) ζ v (1) · L (1 , π v , ad) · Y v ∈ S k W v k , where S fin = S ∩ Σ fin . Now Proposition 6.6 follows from this and Lemmas 6.9, 6.11, 6.13, 6.14 below,where we compute the integral k W v k explicitly.For the rest of this subsection, we fix a place v of F and suppress the subscript v from the notation.6.4.1. The case v ∈ Σ fin r (Σ π, fin ∪ Σ ′ π, fin ) . Recall that π = Ind( χ ⊗ µ ), where χ and µ are unitaryunramified. Put α = χ ( ̟ ) and β = µ ( ̟ ). We have L ( s, π, ad) = 1(1 − q − s )(1 − αβ − q − s )(1 − α − βq − s ) . Lemma 6.8. We have W ( t ( ̟ i − d )) = q − i · α i +1 − β i +1 α − β if i ≥ , if i < .Proof. Recall that d is the non-negative integer such that ψ is trivial on ̟ − d o but non-trivial on ̟ − d − o . We define a non-trivial character ψ of F of order zero by ψ ( x ) = ψ ( ̟ − d x ). Let W be theWhittaker function of π with respect to ψ such that • W ( gk ) = W ( g ) for all g ∈ GL ( F ) and k ∈ GL ( o ), • W (1) = 1.Then we have W ( g ) = W ( t ( ̟ d )) g ), so that the assertion follows from the Casselman–Shalika formula[8]. (cid:3) Lemma 6.9. We have k W k = ζ (1) · L (1 , π, ad) ζ (2) . Proof. By Lemma 6.8, we have k W k = ∞ X i =0 | W ( t ( ̟ i − d )) | = 1( α − β )( α − − β − ) · ∞ X i =0 q − i ( α i +1 − β i +1 )( α − i − − β − i − )= 1( α − β )( α − − β − ) · (cid:18) − q − − αβ − − αβ − q − − α − β − α − βq − + 11 − q − (cid:19) = 1 + q − (1 − q − )(1 − αβ − q − )(1 − α − βq − ) . (cid:3) The case v ∈ Σ π, fin . Recall that π = St ⊗ χ , where χ is unitary unramified. Put α = χ ( ̟ ). Wehave L ( s, π, ad) = ζ ( s + 1). Lemma 6.10. We have W ( t ( ̟ i )) = ( q − i · α i if i ≥ , if i < .Proof. We may assume that χ = 1. We recall the exact sequence0 −→ St −→ Ind( | · | ⊗ | · | − ) M −→ −→ , where M : Ind( | · | ⊗ | · | − ) → Ind( | · | − ⊗ | · | ) is the intertwining operator defined by M ( f )( g ) = Z F f ( w n ( x ) g ) dx with the Haar measure dx on F such that vol( o , dx ) = 1. In particular, we haveSt = { f ∈ Ind( | · | ⊗ | · | − ) | M ( f )(1) = 0 } . Also, we have dim C St I = 1 , dim C Ind( | · | ⊗ | · | − ) I = 2 . Let f , f w be the basis of Ind( | · | ⊗ | · | − ) I determined by f | GL ( o ) = I I , f w | GL ( o ) = I IwI . Then f − q − f w is a basis of St I . Indeed, noting that w n ( x ) = (cid:18) − x (cid:19) = (cid:18) x − − x (cid:19) (cid:18) x − (cid:19) , we have M ( f )(1) = ∞ X j =1 Z ̟ − j o × | x | − dx = ∞ X j =1 q − j (1 − q − ) = q − and M ( f w )(1) = Z o dx = 1 . We consider the Jacquet integral W k ( g ) := Z F f k ( w n ( x ) g ) ψ ( x ) dx for k = 1 , w , where we recall that ψ is assumed to be of order zero. We have W k ( t ( y )) = | y | − Z F f k ( w n ( xy − )) ψ ( x ) dx = Z F f k ( w n ( x )) ψ ( xy ) dx. If k = 1, then we have W ( t ( y )) = ∞ X j =1 Z ̟ − j o × | x | − ψ ( xy ) dx = ∞ X j =1 q − j · ˆ I o × ( ̟ − j y ) . Since ˆ I o × ( x ) = − q − if x ∈ o , − q − if x ∈ ̟ − o × ,0 otherwise,we have W ( t ( ̟ i )) = P ij =1 q − j (1 − q − ) + q − ( i +1) · ( − q − ) = q − − q − i − − q − i − if i > q − · ( − q − ) = − q − if i = 0,0 if i < k = w , then we have W w ( t ( y )) = Z o ψ ( xy ) dx = I o ( y ) . Hence, if we put W = W − q − W w , then we have W ( t ( ̟ i )) = ( − q − i − (1 + q − ) if i ≥ i < W = W (1) − · W and the assertion follows. (cid:3) Lemma 6.11. We have k W k = L (1 , π, ad) . Proof. By Lemma 6.10, we have k W k = ∞ X i =0 | W ( t ( ̟ i )) | = ∞ X i =0 q − i = 11 − q − . (cid:3) The case v ∈ Σ ′ π, fin . Recall that π = Ind( χ ⊗ µ ), where χ is unitary unramified and µ is unitaryramified of conductor q n . Put α = χ ( ̟ ). We have L ( s, π, ad) = ζ ( s ). Lemma 6.12. We have W ( t ( ̟ i )) = ( q − i · α i if i ≥ , if i < .Proof. Let f ∈ Ind( χ ⊗ µ ) be the new vector with respect to ( K n , µ ) determined by f | GL ( o ) = I K n µ . We consider the Jacquet integral W ( g ) := Z F f ( w n ( x ) g ) ψ ( x ) dx, where we recall that ψ is assumed to be of order zero. We have W ( t ( y )) = µ ( y ) | y | − Z F f ( w n ( xy − )) ψ ( x ) dx = µ ( y ) | y | Z F f ( w n ( x )) ψ ( xy ) dx. Noting that w n ( x ) = (cid:18) − x (cid:19) = (cid:18) x − − x (cid:19) (cid:18) x − (cid:19) , we have Z F f ( w n ( x )) ψ ( xy ) dx = ∞ X j = n Z ̟ − j o × χ ( x ) − µ ( x ) | x | − ψ ( xy ) dx = ∞ X j = n α j µ ( ̟ ) − j · [I o × µ ( ̟ − j y ) . Since [I o × µ = q − n · g ( µ, ψ ) · I ̟ − n o × µ − , where g ( µ, ψ ) = Z ̟ − n o × µ ( x ) ψ ( x ) dx, we have W ( t ( ̟ i )) = µ ( ̟ ) i q − i · α i + n µ ( ̟ ) − ( i + n ) · q − n · g ( µ, ψ ) · µ ( ̟ ) n = q − i − n · α i + n · g ( µ, ψ )if i ≥ 0, and W ( t ( ̟ i )) = 0 if i < 0. Thus W = W (1) − · W and the assertion follows. (cid:3) Lemma 6.13. We have k W k = L (1 , π, ad) . Proof. By Lemma 6.12, we have k W k = ∞ X i =0 | W ( t ( ̟ i )) | = ∞ X i =0 q − i = 11 − q − . (cid:3) The case v ∈ Σ ∞ . Recall that π = DS k and ψ ( x ) = e π √− x . It is known that(6.4) W ( t ( y )) = ( y k e − πy if y > y < Lemma 6.14. We have k W k = ( k − π ) k . Proof. By (6.4), we have k W k = Z ∞ | W ( t ( y )) | dyy = Z ∞ y k − e − πy dy = Γ( k )(4 π ) k . (cid:3) Matrix coefficients of the Weil representation. Suppose that F is local. In this subsection,we compute the function Φ( g ) := h ω ψ ( g ) ϕ, ϕ i on U( W ) ∼ = B explicitly, where ϕ ∈ S ( X ) is the Schwartz function given in § ϕ is the partialFourier transform of the Schwartz function ϕ ′ ∈ S ( X ′ ) given in § g ) = h ω ψ ( g ) ϕ ′ , ϕ ′ i . Put m ( a ) = (cid:18) a a − (cid:19) , n ( b ) = (cid:18) b (cid:19) for a ∈ F × , b ∈ F .6.5.1. The case (ur). We identify B × with GL ( F ) via: • the isomorphism i given by (5.1) if E is split and F is unramified; • the isomorphism Ad (cid:0) ̟ − d (cid:1) ◦ i , where i is the isomorphism given by (5.1) and ̟ − d o is the inversedifferent, if E is split and F is ramified; • the isomorphism i given by (5.7) if E is inert and J ∈ ( F × ) ; • any fixed isomorphism i : B → M ( F ) such that i ( o B ) = M ( o ), where o B is the maximal orderin B given in § E is inert, and J ∈ ( F × ) or J ∈ ( F × ) ; • the isomorphism i given by (5.7) if E is ramified.Under this identification, we have K = GL ( o ), where K is the maximal compact subgroup of B × given in § Lemma 6.15. We have Φ( m ( a )) = | a | for a ∈ o r { } .Proof. Put φ ( a ) := Z F I o ( ax ) I o ( x ) dx = q − d × ( a ∈ o , | a | − otherwise,where dx is the self-dual Haar measure on F with respect to ψ . Note that d = 0 unless E is split and F is ramified.Assume that E is split and F is unramified. We use the notation of § ω ψ on S ( X ′ ) is given in § m ( a )) = | a | · Z X ′ ϕ ′ ( ax ) ϕ ′ ( x ) dx = | a | · Y i =1 Z F I o ( ax i ) I o ( x i ) dx i = | a | · φ ( a ) . This yields the desired identity.Assume that E is split and F is ramified. We use the notation of § ω ψ on S ( X ′ ) is given in § B × is identified with GL ( F ) via i rather than Ad (cid:0) ̟ − d (cid:1) ◦ i .We haveΦ( m ( a )) = | a | · Z X ′ ϕ ′ ( ax ) ϕ ′ ( x ) dx = q d · | a | · Y i =1 Z F I o ( ax i ) I o ( x i ) dx i = q d · | a | · φ ( a ) . This yields the desired identity. Assume that E is inert and J ∈ ( F × ) . We use the notation of § ω ψ on S ( X ′ ) is given in § m ( a )) = | a | · Z X ′ ϕ ′ ( ax ) ϕ ′ ( x ) dx = | a | · Y i =1 Z F I o ( ax i ) I o ( x i ) dx i = | a | · φ ( a ) . This yields the desired identity.Assume that E is inert and J ∈ ( F × ) ; the case when E is inert and J ∈ ( F × ) is similar. Weuse the notation of § ω ψ on S ( X ′ ) is given in § m ( a )) = Z X ′ ϕ ′ ( x m ( a )) ϕ ′ ( x ) dx = Z M ( F ) ϕ ′ ( x m ( a )) ϕ ′ ( x ) dx = φ ( a ) · φ ( a − ) , where we identify X ′ ∼ = W with M ( F ) via the fixed isomorphism i and normalize the Haar measureon M ( F ) so that vol(M ( o )) = 1. This yields the desired identity.Assume that E is ramified. We use the notation of § ω ψ on S ( X ′ ) is given in § m ( a )) = | a | · Z X ′ ϕ ′ ( ax ) ϕ ′ ( x ) dx = | a | · Y i =1 Z F I o ( ax i ) I o ( x i ) dx i = | a | · φ ( a ) . This yields the desired identity. (cid:3) The case (rps). We identify B × with GL ( F ) via the isomorphism i given by (5.1). Under thisidentification, we have K = GL ( o ), where K is the maximal compact subgroup of B × given in § µ to indicate the dependence of ϕ = ϕ µ on a unitary ramified character µ of conductor q n . Lemma 6.16. We have Φ µ ( n ( b ) m ( a )) = ( µ ( a ) if a ∈ o × and b ∈ o , otherwise.Proof. We use the notation of § ω ψ on S ( X ′ ) is given in § µ ( n ( b ) m ( a )) = | a | · Z X ′ ϕ ′ ( ax ) ϕ ′ ( x ) ψ (cid:18) b h x, x i † (cid:19) dx = q n +1 ( q − − · µ ( a ) · | a | × Z F I o ( ax ) I o ( x ) I o ( ax ) I o ( x ) I ̟ n o ( ax ) I ̟ n o ( x ) I o × ( ax ) I o × ( x ) ψ ( b ( x x − x x )) dx dx dx dx . Since I o × ( ax ) I o × ( x ) = I o × ( a ) I o × ( x ), the above integral is zero unless a ∈ o × , in which case it isequal to Z F I o ( x ) I o ( x ) I ̟ n o ( x ) I o × ( x ) ψ ( b ( x x − x x )) dx dx dx dx = Z F I o ( bx ) I ̟ n o ( x ) dx · Z F I o ( bx ) I o × ( x ) dx = ( q − n (1 − q − ) if b ∈ o ,0 otherwise.This yields the lemma. (cid:3) The case (st). We identify B × with GL ( F ) via: • the isomorphism i given by (5.1) if B and B are split; • the isomorphism i given by (5.7) if B and B are ramified.Under this identification, we have K = GL ( o ), where K is the maximal compact subgroup of B × given in § w = (cid:18) − (cid:19) . Lemma 6.17. We have Φ( m ( ̟ i )) = ( q − i if i ≥ , q i if i ≤ , Φ( m ( ̟ i ) w ) = ( γ B · q − i − if i ≥ , γ B · q i +1 if i < ,where γ B = ( if B is split, − if B is ramified.Proof. For convenience, we write p i = ̟ i o for i ∈ Z . Put φ ( j, k ) := Z F I p j ( x ) I p k ( x ) dx = ( q − j if j ≥ k , q − k if j ≤ k for j, k ∈ Z , where dx is the self-dual Haar measure on F with respect to ψ .Assume that B and B are split. We use the notation of § ω ψ on S ( X ′ ) is given in § ϕ ′ ( x ) = q · I o ( x ) I o ( x ) I p ( x ) I o ( x ) ,ω ψ ( w ) ϕ ′ ( x ) = q − · I o ( x ) I p − ( x ) I o ( x ) I o ( x ) , so that Φ( m ( ̟ i )) = q − i · Z X ′ ϕ ′ ( ̟ i x ) ϕ ′ ( x ) dx = q − i +1 · φ ( − i, · φ ( − i + 1 , , Φ( m ( ̟ i ) w ) = q − i · Z X ′ ω ψ ( w ) ϕ ′ ( ̟ i x ) ϕ ′ ( x ) dx = q − i · φ ( − i, · φ ( − i − , · φ ( − i, . This yields the desired identity.Assume that B and B are ramified. We use the notation of § ω ψ on S ( X ′ ) is given in § ϕ ′ ( x ) = q · I o ( x ) I o ( x ) I o ( x ) I o ( x ) ,ω ψ ( w ) ϕ ′ ( x ) = − q − · I o ( x ) I o ( x ) I p − ( x ) I p − ( x ) , so that Φ( m ( ̟ i )) = q − i · Z X ′ ϕ ′ ( ̟ i x ) ϕ ′ ( x ) dx = q − i · φ ( − i, , Φ( m ( ̟ i ) w ) = q − i · Z X ′ ω ψ ( w ) ϕ ′ ( ̟ i x ) ϕ ′ ( x ) dx = − q − i − · φ ( − i, · φ ( − i − , . This yields the desired identity. (cid:3) 00 ATSUSHI ICHINO AND KARTIK PRASANNA The case (1d). Let K = o × B be the unique maximal compact subgroup of B × . We have B ⊂ K . Lemma 6.18. We have Φ( g ) = 1 for g ∈ B .Proof. We use the notation of § ω ψ ( g ) ϕ ′ = ϕ ′ and hence Φ( g ) = h ϕ ′ , ϕ ′ i = 1 for all g ∈ B . (cid:3) The case (ds). We identify B × with GL ( F ) via the isomorphism i given by (5.7). We writeΦ = Φ k to indicate the dependence of ϕ = ϕ k on a non-negative integer k . Lemma 6.19. We have Φ k ( m ( a )) = (cid:18) a + a − (cid:19) − k − for a > .Proof. Assume that B and B are split. We use the notation of § ω ψ on S ( X ′ ) is given in § k ( m ( a )) = a · Z X ′ ϕ ′ ( ax ) ϕ ′ ( x ) dx = | u | · c − k · a k +2 · Z F ( x − ux ) k · e − π v ( a +1)( x − ux + x − ux ) dx · · · dx = | u | · c − k · a k +2 · v − · (cid:16) π v ( a + 1) (cid:17) − k − · φ ( k ) · φ (0) , where φ ( k ) := Z ∞−∞ Z ∞−∞ ( x + y ) k e − ( x + y ) dx dy with the Lebesgue measures dx , dy on R . Since φ ( k ) = Z π Z ∞ r k e − r r dr dθ = π · Z ∞ r k e − r dr = π · k ! , we haveΦ k ( m ( a )) = | u | · π k k ! | u | k +1 · a k +2 · v − · (cid:16) π v ( a + 1) (cid:17) − k − · π · k ! = (cid:18) a + a − (cid:19) − k − . Assume that B and B are ramified. We use the notation of § ω ψ on S ( X ′ ) is given in § B and B aresplit. (cid:3) The case (fd). We identify C × with a subgroup of B × via the isomorphism E ∼ = C such that i / √− > E ֒ → B . Let φ k be the matrix coefficient of Sym k such that • φ k ( αgβ ) = χ k ( α ) χ k ( β ) φ k ( g ) for α, β ∈ C × and g ∈ B × , • φ k (1) = 1.We write Φ = Φ k to indicate the dependence of ϕ = ϕ k on a non-negative integer k . Lemma 6.20. We have Φ k ( g ) = φ k ( g ) for g ∈ B . Proof. We use the notation of § ω ψ on S ( X ′ ) is given in § x = z + z j s ∈ X ′ ∼ = W = B with z , z ∈ E , then ϕ ′ ( x ) = c − k · ( z ρ ) k · e − πv ( z z ρ + z z ρ ) . Let S k be the subspace of S ( X ′ ) generated by ω ψ ( g ) ϕ ′ for all g ∈ B . Since ω ψ ( g ) ϕ ′ ( z , z ) = ϕ ′ ( z α − z α ρ , z α + z α ρ )for g = α + α j s ∈ B with α , α ∈ E , S k is generated by( z ρ ) i · ( z ρ ) k − i · e − πv ( z z ρ + z z ρ ) for all 0 ≤ i ≤ k . Moreover, the representation of B on S k is isomorphic to the unique irreducible( k + 1)-dimensional representation Sym k | B , so that Φ k is a matrix coefficient of Sym k | B . On theother hand, by Lemma 5.3, we have Φ k ( αgβ ) = χ k ( α ) − χ k ( β ) − Φ k ( g ) for α, β ∈ C and Φ k (1) = h ϕ ′ , ϕ ′ i = 1. Hence we must have Φ k = ¯ φ k | B . (cid:3) Computation of Z v . To finish the proof of Theorem 6.7, it remains to compute the integral Z v .We fix a place v of F and suppress the subscript v from the notation. Recall that Z = Z B Φ( g )Ψ( g ) dg, where • Φ is the function on B given in § • Ψ is the function on B defined byΨ( g ) = h π B ( g ) f B , f B i , where f B ∈ π B is the new vector as in § h· , ·i is the invariant hermitian inner product on π B normalized so that h f B , f B i = 1; • dg is the standard measure on B .6.6.1. The case (ur). In this case, π B = Ind( χ ⊗ µ ), where χ and µ are unitary unramified. We have L ( s, π, ad) = 1(1 − q − s )(1 − γq − s )(1 − γ − q − s ) , where γ = χ ( ̟ ) · µ ( ̟ ) − . Lemma 6.21. We have Z = L (1 , π, ad) ζ (2) . Proof. We retain the notation of §§ K ′ = SL ( o ). Then we have Z = ∞ X i =0 Φ( m ( ̟ i ))Ψ( m ( ̟ i )) vol( K ′ m ( ̟ i ) K ′ ) . By Macdonald’s formula [49], [7], we haveΨ( m ( ̟ i )) = q − i q − · (cid:18) γ i · − γ − q − − γ − + γ − i · − γq − − γ (cid:19) . Also, we see that vol( K ′ m ( ̟ i ) K ′ ) = ( i = 0, q i (1 + q − ) if i ≥ 02 ATSUSHI ICHINO AND KARTIK PRASANNA Combining these with Lemma 6.15, we obtain Z = 1 + ∞ X i =1 q − i · (cid:18) γ i · − γ − q − − γ − + γ − i · − γq − − γ (cid:19) = 1 + γq − − γq − · − γ − q − − γ − + γ − q − − γ − q − · − γq − − γ = (1 + q − )(1 − q − )(1 − γq − )(1 − γ − q − ) . (cid:3) The case (rps). In this case, π B = Ind( χ ⊗ µ ) and Φ = Φ µ , where χ is unitary unramified and µ is unitary ramified of conductor q n . We have L ( s, π, ad) = ζ ( s ). Lemma 6.22. We have Z = 1 q n − ( q − q + 1) · L (1 , π, ad) ζ (2) . Proof. Following [45, Chapter VIII], we shall compute Z explicitly. We retain the notation of §§ K ′ = SL ( o ) and K ′ n = K n ∩ SL ( o ). We take the invariant hermitian inner product h· , ·i on π B defined by h f , f i = Z K f ( k ) f ( k ) dk, where dk is the Haar measure on K such that vol( K ) = 1. Then f B is determined by f B | K = vol( K n ) − · I K n µ . We can define a new vector ˜ f B ∈ π B with respect to ( K n , µ ) by˜ f B ( h ) = Z B Φ( g ) f B ( hg ) dg for h ∈ B × . Since ˜ f B = ˜ f B (1) f B (1) · f B and h f B , f B i = 1, we have Z = h ˜ f B , f B i = vol( K n ) · ˜ f B (1) . We have˜ f B (1) = Z B Φ( g ) f B ( g ) dg = Z K ′ Z F × Z F Φ( n ( b ) m ( a ) k ) · f B ( n ( b ) m ( a ) k ) · | a | − db da dk = vol( K n ) − · Z K ′ n Z F × Z F Φ( n ( b ) m ( a )) µ ( k ) − · χ ( a ) µ ( a ) − | a | µ ( k ) · | a | − db da dk = vol( K n ) − · vol( K ′ n ) · Z F × Z F Φ( n ( b ) m ( a )) · χ ( a ) µ ( a ) − | a | − db da, where • db is the Haar measure on F such that vol( o ) = 1; • da is the Haar measure on F × such that vol( o × ) = 1; • dk is the Haar measure on K ′ such that vol( K ′ ) = 1. By Lemma 6.16, we have Z F × Z F Φ( n ( b ) m ( a )) · χ ( a ) µ ( a ) − | a | − db da = Z o × Z o χ ( a ) | a | − db da = 1 . Hence we have Z = vol( K ′ n ) = 1 q n − ( q + 1) . (cid:3) The case (st). In this case, π B = St ⊗ χ , where χ is unitary unramified. We have L ( s, π, ad) = ζ ( s + 1). Lemma 6.23. (i) If B and B are split, then we have Z = q ( q + 1) · L (1 , π, ad) ζ (2) . (ii) If B and B are ramified, then we have Z = q ( q − ( q + 1) · L (1 , π, ad) ζ (2) . Proof. We retain the notation of §§ I ′ = I ∩ SL ( o ). Let ˜ W = N ( T ) /T be theextended affine Weyl group of GL ( F ), where T = { ( a d ) | a, d ∈ o × } and N ( T ) is the normalizer of T in GL ( F ). Then we have GL ( F ) = G ˜ w ∈ ˜ W I ˜ w I . We can write ˜ W = Ω ⋉ W a with Ω = h ω i and W a = h w , w i , where ω = (cid:18) ̟ (cid:19) , w = (cid:18) (cid:19) , w = (cid:18) ̟ − ̟ (cid:19) . Noting that w = w = 1 and w w = m ( ̟ ), we haveSL ( F ) = G j =0 ∞ G i = −∞ I ′ m ( ̟ i ) w j I ′ and hence Z = X j =0 ∞ X i = −∞ Φ( m ( ̟ i ) w j )Ψ( m ( ̟ i ) w j ) vol( I ′ m ( ̟ i ) w j I ′ ) . Let ℓ be the length function on ˜ W , so that ℓ ( ω ) = 0 and ℓ ( w ) = ℓ ( w ) = 1. By [18, § ω k ˜ w ) = ( − χ ( ̟ )) k · ( − q ) − ℓ ( ˜ w ) for k ∈ Z and ˜ w ∈ W a . Also, we see that |I ˜ w I / I| = q ℓ ( ˜ w ) for ˜ w ∈ ˜ W . Hence we haveΨ( m ( ̟ i ) w j ) vol( I ′ m ( ̟ i ) w j I ′ ) = ( − ℓ ( m ( ̟ i ) w j ) · vol( I ′ ) = 1 q + 1 × ( j = 0, − j = 1,so that Z = 1 q + 1 · ∞ X i = −∞ Φ( m ( ̟ i )) − ∞ X i = −∞ Φ( m ( ̟ i ) w ) ! . 04 ATSUSHI ICHINO AND KARTIK PRASANNA Combining this with Lemma 6.17, we obtain Z = 1 q + 1 · ∞ X i =0 q − i + ∞ X i =1 q − i − ∞ X i =0 q − i − − ∞ X i =1 q − i +1 ! = 1 q + 1 · q − − q − − q − − q − = q − q + 1) if B and B are split, and Z = 1 q + 1 · ∞ X i =0 q − i + ∞ X i =1 q − i + ∞ X i =0 q − i − + ∞ X i =1 q − i +1 ! = 1 q + 1 · q − + q − + q − − q − = 1 q − B and B are ramified. (cid:3) The case (1d). In this case, π B = χ ◦ ν , where χ is unitary unramified. We have L ( s, π, ad) = ζ ( s + 1). Lemma 6.24. We have Z = q ( q − q + 1) · L (1 , π, ad) ζ (2) . Proof. We retain the notation of §§ Z = Z B dg = 1 . (cid:3) The case (ds). In this case, π B = DS k and Φ = Φ l , where l = ( k if B and B are split, k − B and B are ramified. Lemma 6.25. (i) If B and B are split, then we have Z = 4 πk . (ii) If B and B are ramified, then we have Z = 4 πk − . Proof. We retain the notation of §§ C × with a subgroup of B × .Then we have Z = 4 π · Z C Z ∞ Z C Φ( κ m ( e t ) κ )Ψ( κ m ( e t ) κ ) sinh(2 t ) dκ dt dκ = 4 π · Z ∞ Φ( m ( e t ))Ψ( m ( e t )) sinh(2 t ) dt, where • dt is the Lebesgue measure; • dκ and dκ are the Haar measures on C such that vol( C ) = 1.It is known that Ψ( m ( e t )) = cosh( t ) − k . Combining these with Lemma 6.19, we obtain Z = 4 π · Z ∞ cosh( t ) − k − l − sinh(2 t ) dt = 8 π · Z ∞ cosh( t ) − k − l − sinh( t ) dt = 8 π · Z ∞ t − k − l − dt = 8 πk + l . (cid:3) The case (fd). In this case, π B = Sym k − and Φ = Φ k − . Lemma 6.26. We have Z = 1 k − . Proof. We retain the notation of §§ Z = Z B | φ k − ( g ) | dg = 1 k − . (cid:3) The main conjecture on the arithmetic of theta lifts On the choices of u , J and J . We suppose now that we are given a totally real number field F and two quaternion algebras B and B over F . Let us define for convenience: d B r B = Y q | d B , q ∤ d B q , d B r B = Y q | d B , q ∤ d B q , d B ∪ B = Y q | d B d B q , d B ∩ B = Y q | ( d B , d B ) q and Σ B r B = Σ B r Σ B , Σ B r B = Σ B r Σ B , Σ B ∪ B = Σ B ∪ Σ B , Σ B ∩ B = Σ B ∩ Σ B . For the constructions so far (especially the constructions of splittings), the only condition neededis:(7.1) At every place v of F , at least one of u , J , J , J is a square.However, to formulate the main conjecture we will need to make a more careful choice. In this section,we show that we can make such a choice that satisfies a number of useful auxiliary conditions. 06 ATSUSHI ICHINO AND KARTIK PRASANNA Proposition 7.1. Suppose that ℓ is a rational prime that is coprime to d B ∪ B and { f , · · · , f n } is acollection of primes of O F (possibly empty) that are coprime to ℓ d B ∪ B . Then we can find elements u, J , J ∈ F such that the following hold: (i) u, J , J lie in O F . (ii) At every place v of F , at least one of u , J , J , J is a square. (iii) u ≪ , so that E := F + F i , i = u is a CM field. (iv) u is a unit at any prime q that is unramified in E . (v) If q is a prime of F dividing , then E q is the unique unramified quadratic extension of F q if q | d B ∪ B and E q /F q is split otherwise. (vi) • B ≃ E + E j , with j = J and ij = − j i . • B ≃ E + E j , with j = J and ij = − j i . • B ≃ E + E j with j = J = J J and ij = − ji . (vii) • If q | d B r B , then J is a uniformizer at q and J is the square of a unit. • If q | d B r B , then J is a uniformizer at q and J is the square of a unit. • If q | d B ∩ B , then J and J are both uniformizers at q such that J /J is the square of aunit. (viii) u , J , J and J are squares of units at the primes in { f , . . . , f n } and at all primes l of F above ℓ . Let K denote the quadratic extension of F given by(7.2) K = F + F j . Note that the condition (viii) above implies that both E and K are split at the primes in { l | ℓ } ∪{ f , . . . , f n } .Prop. 7.1 will suffice for the current paper. The following enhancement of it will be useful in [31],[32]. Proposition 7.2. Let ℓ and f , . . . , f n be as in the previous proposition. Suppose that the prime ℓ satisfies the following conditions: • ℓ is unramified in F . • ℓ > and for any q | d B r B · d B r B , we have N q , ± ℓ ) . Then we can choose u , J , J such that in addition to (i) through (viii) above, we have: (ix) If E or K is ramified at a prime p , then N p , ± ℓ ) . The following Lemmas 7.3 and 7.4 will be useful in the proofs of Prop. 7.1 and Prop. 7.2 respectively. Lemma 7.3. Let F be a number field, Ξ f a finite subset of Σ fin and Ξ ∞ ⊆ Σ ∞ a set of real infiniteplaces. Let I be an ideal in O F prime to the primes in Ξ f . Then there exists a prime ideal q ⊂ O F such that I · q = ( α ) is principal with α satisfying: (a) α is a square of a unit at the primes in Ξ f . (b) σ v ( α ) < for v in Ξ ∞ and σ v ( α ) > for any real place v of F not in Ξ ∞ .Further, q can be picked to avoid any finite set of primes. Lemma 7.4. Let F be a number field and ℓ > a rational prime unramified in F . Suppose that Ξ f isa finite subset of Σ fin all whose elements are prime to ℓ and Ξ ∞ ⊆ Σ ∞ is a set of real infinite places. Let I be an ideal in O F prime to ℓ and the primes in Ξ f . Then there exists a prime ideal q ⊂ O F suchthat I · q = ( α ) is principal with α satisfying: (a) α is a square of a unit at the primes in Ξ f and at all primes l above ℓ . (b) σ v ( α ) < for v in Ξ ∞ and σ v ( α ) > for v any real place of F not in Ξ ∞ . (c) N q , ± ℓ ) .Further, q can be picked to avoid any finite set of primes. We first prove Lemma 7.3 and then explain the modifications needed to prove Lemma 7.4. Proof (of Lemma 7.3). Let m be the product of all the real places of F and the primes in Ξ f , eachraised to a sufficiently large power so that the local units congruent to 1 (mod m ) are squares. For α ∈ F × , let ι ( α ) denote the principal fractional ideal generated by α . Also let F m , denote the set ofelements in F × that are congruent to 1 (mod × m ). If U F denotes the units in F and U F, m the unitscongruent to 1 (mod × m ), then there is an exact sequence:1 → U F U F, m → F × F m , → ι ( F × ) ι ( F m , ) → . Let H be the Hilbert class field of F and H m the ray class field of F of conductor m . Then F ⊂ H ⊂ H m and there is a canonical isomorphism Gal( H m /H ) ≃ ι ( F × ) ι ( F m , ) . Pick an element β ∈ F × such that β ≡ m ) and such that β is negative at the real places inΞ ∞ and positive at the real places not in Ξ ∞ . Let σ ( β ) ∈ Gal( H m /H ) be the element corresponding to[ ι ( β )] via the isomorphism above. Let σ I denote the image of I in Gal( H m /F ) under the Artin map.By Tchebotcharev, there exists a prime ideal q in O F that is prime to m and such that σ q = σ − I · σ ( β ) in Gal( H m /F ) . In particular this implies that σ q = σ − I in Gal( H/F ), so there exists α ∈ F × such that q · I = ( α ).Then σ ( α ) = σ ( β ) , which is the same as saying that[ ι ( α )] = [ ι ( β )] in ι ( F × ) ι ( F m , ) . The exact sequence above implies then that there is a unit u ∈ U F such that[ u · α ] = [ β ] in F × F m , . Replacing α by u · α , we see that it has the required properties. (cid:3) Proof (of Lemma 7.4). We modify the proof of Lemma 7.3.Let { l , . . . , l r } be the primes of F lying over ℓ . Let m be the product of all the real places of F , the primes in Ξ f (each raised to a sufficiently large power so that the local units congruent to 1(mod m ) are squares) and the primes l , . . . , l r . Fix for the moment an element w ∈ ( o F / l ) × . By theapproximation theorem, we may pick β ∈ F × such that • β is negative at the places in Ξ ∞ and positive at the real places not in Ξ ∞ . • β ≡ m ). • β ≡ w (mod l ). 08 ATSUSHI ICHINO AND KARTIK PRASANNA Let σ I denote the image of I in Gal( H ml /F ) under the Artin map. By Tchebotcharev, there existsa prime ideal q in O F that is prime to m · l and such that σ q = σ − I · σ ( β ) in Gal( H ml /F ) . As before then, there exists α ∈ F × such that q · I = ( α ) and a unit u ∈ U F such that[ u · α ] = [ β ] in F × F ml , . Replacing α by u · α , we see that α satisfies the requirements (a), (b) of the lemma. It remains to showthat w can be chosen so that q satisfies (c). Clearly q is prime to ℓ . ButN q · N I = ± N( α ) = ± N( β ) ≡ ± N F l / F ℓ ( w ) (mod ℓ ) . Since N I is fixed and we only need N q 6≡ ± ℓ ), it suffices to show that the subgroup n N F l / F ℓ ( w ) : w ∈ F × l o ⊂ F × ℓ contains at least 3 elements. But this subgroup is just ( F × ℓ ) (since l is unramified over ℓ ) and hascardinality ℓ − > ℓ > (cid:3) Now we prove Prop. 7.1 and then explain the modifications needed to prove Prop. 7.2. Proof (of Prop. 7.1). Let f = f · · · f n and S = 2 ℓ d B ∪ B f . We begin by using Lemma 7.3 above topick: • A prime ideal q B r B (prime to S ) such that d B r B · q B r B = ( α B r B ), with α B r B satisfyingthe following conditions: · α B r B is a square of a unit at the primes dividing ℓ d B f and the primes above 2 not dividing d B r B . · For v ∈ Σ ∞ , σ v ( α B r B ) < , if v ∈ Σ B r B ; σ v ( α B r B ) > , if v Σ B r B . • A prime ideal q B r B (prime to S ) such that d B r B · q B r B = ( α B r B ), with α B r B satisfyingthe following conditions: · α B r B is a square of a unit at the primes dividing ℓ d B f and the primes above 2 not dividing d B r B . · For v ∈ Σ ∞ , σ v ( α B r B ) < , if v ∈ Σ B r B ; σ v ( α B r B ) > , if v Σ B r B . • A prime ideal q B ∩ B (prime to S ) such that d B ∩ B · q B ∩ B = ( α B ∩ B ), with α B ∩ B satisfyingthe following conditions: · α B ∩ B is a square of a unit at the primes dividing ℓ d B r B d B r B f and the primes above 2not dividing d B ∩ B . · For v ∈ Σ ∞ , σ v ( α B ∩ B ) < , if v ∈ Σ B ∩ B ; σ v ( α B ∩ B ) > , if v Σ B ∩ B . Let R denote the ideal R := q B r B · q B r B · q B ∩ B . Next, we use the approximation theorem to pick α ∈ F × satisfying the following properties:(I) α ≫ − α is a square of a unit at the primes dividing ℓ Rf . (III) If q is a prime dividing d B ∪ B , then − α is a unit at q but not a square. If further q divides 2,then we also require that √− α generate the unique unramified extension of F q .(IV) If q is a prime dividing 2 but not dividing d B ∪ B , then − α is a square of a unit at q .Let m := 2 a ℓ · d B ∪ B f · R · Y v ∈ Σ ∞ v, with the power 2 a being chosen large enough so that locally at any prime above 2, the units congruentto 1 modulo 2 a are squares. By Tchebotcharev, there exists a prime ideal Q ⊂ O F (prime to m ) suchthat σ ( α ) · σ Q = 1 in Gal( H m /F ) . This implies that ( α ) · Q = ( β ) , for some β ≡ × m ). Now, take u := − α − β, J := α B r B · α B ∩ B , J := α B r B · α B ∩ B . Since ( u ) = Q , ( J ) = d B · q B r B · q B ∩ B , ( J ) = d B · q B r B · q B ∩ B , we see that u, J , J lie in O F , which shows that (i) is satisfied. Let E/F be the quadratic extension E = F + F i with i = u . Since α ≫ β ≫ 0, we have u ≪ 0, which shows (iii), whence E is aCM quadratic extension of F . The conditions (III) and (IV) above imply that if q is a prime above 2,then E q is the unique unramified quadratic extension of F q if q divides d B ∪ B and otherwise is split,which shows (v). Since ( u ) = Q , it follows that E is ramified exactly at the prime Q , and in particularsatisfies (iv). Now we check that B ≃ E + E j , j = J , ij = − j i . To show this, it suffices to check that the Hilbert symbol ( u, J ) v equals − v atwhich B is ramified. At the archimedean places this is clear since u ≪ J is negative exactly atthe places at which B is ramified. As for the finite places, we only need to check this for v dividing2 uJ , since outside of these primes B is split and ( u, J ) = 1 since both u and J are units at suchplaces. At the primes dividing q B r B · q B ∩ B , the algebra B is split and u is a square of a unit, sothis is clear. For q | d B , the algebra B is ramified, J is a uniformizer and by (III) above, we have( u, J ) q = ( − α, J ) q = − 1. Next we consider the primes q above 2. If q | d B , this is done already. If q | d B r B , then J is a square at q , so ( u, J ) q = 1 as required. This leaves the primes q above 2 whichdo not divide d B ∪ B . At such primes, u is a square of a unit, so ( u, J ) q = 1. The only prime left is Q at which the required equality follows from the product formula! The isomorphism B ≃ E + E j follows similarly, and then the isomorphism B ≃ E + E j follows from the equality B = B · B inthe Brauer group. This completes the proof of (vi). The conditions (vii) and (viii) are easily verified,which leaves (ii).Finally, we check that (ii) is satisfied, namely that at every place v of F , at least one of u , J , J or J is a square. At the archimedean places, this is obvious. At the primes dividing d B ∪ B , this followsfrom (vii). Let q be a finite prime not dividing d B ∪ B . If such a q divides 2, then all of u, J , J , J aresquares at q . So let q be prime to 2 d B ∪ B . If E is split at q , then u is a square at q . If E is inert at q , then J , J lie in N E q /F q ( E × q ) since B and B are split at q . If J and J are both not squares atsuch q , it must be the case that J = J J is a square. Finally, we deal with q = Q , the only ramifiedprime in E . At this prime, J and J are both units. Again, if both of them are non-squares, theirproduct must be a square. This completes the proof. (cid:3) We will now prove Prop 7.2. 10 ATSUSHI ICHINO AND KARTIK PRASANNA Proof (of Prop. 7.2). We will show that we can pick u, J , J such that (ix) is satisfied in addition to(i)-(viii). The proof is almost the same as that of Prop. 7.1 with some minor modifications whichwe now describe. First, we pick as before the prime ideals q B r B , q B r B , q B ∩ B and the elements α B r B , α B r B , α B ∩ B . Using Lemma 7.4, we can ensure that for q = q B r B and q B r B , we haveN q , ± ℓ ) . Fix a prime l of F above ℓ . Let R be as before and then using the approximation theorem, pick α ∈ F × satisfying the properties (I) through (IV) of the proof of Prop. 7.1 and the following additionalconditions:(V) − α ≡ w (mod l ), where w ∈ F × l is an element such that N F l / F ℓ ( w ) = ± − α ≡ l ) for all primes l = l dividing ℓ .Note that this uses the assumption that ℓ is unramified in F and ℓ > 5. Next, as before, we pick m , β and Q and set: u := − α − β, J := α B r B · α B ∩ B , J := α B r B · α B ∩ B . It is easy to see (using arguments similar to those of Prop. 7.1) that (i)-(viii) hold. We verify (ix) now.The field E is only ramified at Q . Also,N Q = ± N( β ) · N( α − ) ≡ ± N( α − ) ≡ ± N F l / F ℓ ( w − ) , ± ℓ ) , which proves what we need for the field E . Now let us consider the field K . Since J = J J , we have( J ) = d B · q B r B · q B ∩ B · d B · q B r B · q B ∩ B = d B r B · d B r B · q B r B · q B r B · d B ∩ B · q B ∩ B . We claim that K is ramified exactly at the primes dividing d K := d B r B · d B r B · q B r B · q B r B . This will follow if we show that K/F is unramified at the primes q over 2 that do not divide d B r B · d B r B . We claim that J is a square (and hence K is split) at such primes q . Indeed, if a prime q above 2 does not divide d B ∪ B , then α B r B , α B r B and α B ∩ B are all squares of units at q , hence J , J and J are squares at q . On the other hand, if a prime q above 2 divides d B ∩ B , then α B r B , α B r B are squares of units at q and thus J = α B r B · α B r B · α B ∩ B is a square at q .Thus it suffices to consider the primes q dividing d K . For q dividing either d B r B or d B r B , itfollows from the assumptions in the statement of the proposition that N q , ± ℓ ). For q equalto either q B r B or q B r B , the same follows from the choice of these prime ideals. (cid:3) The main conjecture. Finally, in this section we come to the main conjecture. Our startingdata will be the totally real field F , the automorphic representation Π (from the introduction) ofconductor N = N s · N ps and the two quaternion algebras B and B . We assume that Π admits aJacquet–Langlands transfer to both B and B . We also assume the following condition holds: • N is prime to 2 D F/ Q , where D F/ Q denotes the different of F/ Q .The conjecture will in addition depend on several auxiliary choices which we now make completelyexplicit in the following series of steps.(i) Let ℓ be a rational prime such that ( ℓ, N (Π)) = 1. (ii) Pick u , J and J satisfying all the conditions of Prop. 7.1, taking { f , . . . , f n } to be the set ofprimes of F dividing 2 D F/ Q .(iii) Set E = F + F i where i = u . An explicit model for B i , i = 1 , , is B i = E + E j i where j i = J i and α j i = j i α ρ for α ∈ E . Likewise an explicit model for B = B · B is B = E + E j where j = J := J J .(iv) Let B ′ denote any one of the quaternion algebras B , B or B . Let d B ′ denote the discriminantof B ′ and define N B ′ by N = d B ′ · N B ′ . Thus d B ′ divides N s , and N ps divides N B ′ .(v) Given the choices of u , J and J , in Sec. 5.5 we have picked local maximal orders and oriented Eichler orders of level N B ′ . (Here the orientation is only picked at places dividing N ps .) Let O B ′ (resp. O B ′ ( N B ′ )) denote the unique maximal order (resp. Eichler order of level N B ′ )corresponding to these choices. Also denote by o B ′ the corresponding orientation on O B ′ ( N B ′ ).This defines open compact subgroups K B ′ = Q v K B ′ v and ˜ K B ′ = Q v ˜ K B ′ v of B ′× ( A f ) with K B ′ v = ker (cid:2) o B ′ ,v : ( O B ′ ( N B ′ ) ⊗ O F O F,v ) × → ( O F,v / N ps O F,v ) × (cid:3) and ˜ K B ′ v = ( O B ′ ( N B ′ ) ⊗ O F O F,v ) × . For future use, we record that with our choices of u , J and J and local orders, we have O E ⊗ Z ( ℓ ) ⊆ O B ′ ⊗ Z ( ℓ ) . (vi) As in § L and isomorphisms φ B , φ B and φ B satisfying(1.17). We recall that φ B ′ gives an isomorphism: O B ′ ⊗ O L, ( ℓ ) ≃ Y σ M ( O L, ( ℓ ) )for B ′ = B , B and B . As explained in § K B ( G B , X B ), Sh K B ( G B , X B ) and Sh K B ( G B , X B ) as well as sections s B , s B and s B of these bundles that are ℓ -normalized. For B ′ = B, B or B let us denote the correspondingbundle by V B ′ k B ′ ,r to indicate that is a bundle on X B ′ .(vii) So far, we have not had to pick a base point of X B ′ but we will now need to do so. Fixisomorphisms E ⊗ F,σ R ≃ C for all infinite places σ of F as in § B ′ = B, B , B we define h B ′ as follows: take the composite maps C → Y σ ∈ Σ ∞ C ≃ E ⊗ R → B ′ ⊗ R where the first map sends z ( z σ ) σ with z σ = z if σ is split in B ′ and z σ = 1 is σ is ramified in B ′ .(viii) For B ′ = B, B or B , let F B ′ = Lift h B ′ ( s B ′ ).(ix) For each σ , we pick a vector v B ′ σ,k B ′ ,σ ∈ V σ,k B ′ ,σ ,r satisfying (1.12) and that is integrally normalizedwith respect to the ℓ -integral structure given bySym k B ′ ,σ O L, ( ℓ ) ⊗ det( O L, ( ℓ ) ) r − kB ′ ,σ . (x) Let v B ′ k B ′ = ⊗ σ v B ′ σ,k B ′ ,σ . Now we can define φ F B ′ = ( F B ′ ( g ) , v B ′ k B ′ ). Let f B ′ denote the correspond-ing element of π B ′ : f B ′ ( g ) = φ F B ′ ( g ) · ν B ′ ( g ) − r/ . Then f B ′ is a new-vector as defined in § ℓ . 12 ATSUSHI ICHINO AND KARTIK PRASANNA From Defn. 1.11, Prop. 1.13 and Prop. 1.17, we find that hh s B ′ , s B ′ ii ˜ K B ′ = 2 d B ′ h F [ K B ′ : ˜ K B ′ ] · h F B ′ , F B ′ i h B ′ = 2 d B ′ h F [ K B ′ : ˜ K B ′ ] · rank V B ′ k B ′ ,r h v B ′ k B ′ , v B ′ k B ′ i h B ′ · h f B ′ , f B ′ i , where d B ′ is the number of infinite places of F where B ′ is split, and K B ′ is the maximal open compactsubgroup of B ′× ( A f ) defined by K B ′ = Q v K B ′ ,v , with K B ′ ,v = ( O B ′ ⊗ O F,v ) × .In order to state the main conjecture, we will need to renormalize the measure and the Schwartzfunction in the definition of the theta lift. First we renormalize the Schwartz function. Let ϕ v be theSchwartz function on S ( X v ) defined in § ϕ = ⊗ v ϕ v where ϕ v = p C ϕv · ϕ v with C ϕv = v is unramified principal series ,q v − q n v +1 v if Π v is ramified principal series with conductor q n v v ,q − v if Π v is special ,k v !2 k v π k v if v ∈ Σ ∞ r (Σ B, ∞ ∪ Σ B , ∞ ∪ Σ B , ∞ ) , ( k v − k v − π k v − if v ∈ Σ B , ∞ ∩ Σ B , ∞ ( k v − k v − π k v − if v ∈ Σ B, ∞ . As for the measure used in the theta lift (4.1) we renormalize the measure on B (1) ( A ) to[ K B : ˜ K B ] · rank V Bk B ,r · standard measure on B (1) ( A ) . With this choice of measure, let us define α ( B , B ) by(7.3) θ ϕ ( f B ) = α ( B , B ) · ( f B × f B ) . Theorem 7.5. Suppose B = M ( F ) . Then (7.4) | α ( B , B ) | · hh s B , s B ii ˜ K B · hh s B , s B ii ˜ K B = Λ(1 , Π , ad) · hh s B , s B ii ˜ K B in C × /R × ( ℓ ) . Proof. Recall that Λ( s, Π , ad) = Q v ∈ Σ ∞ L ( s, Π v , ad) · L ( s, Π , ad) with L ( s, Π v , ad) = Γ R ( s + 1)Γ C ( s + k v − , where Γ R ( s ) = π − s Γ( s ) and Γ C ( s ) = 2(2 π ) − s Γ( s ). Since L (1 , Π v , ad) = ( k v − k v − π k v +1 for v ∈ Σ ∞ , it follows from Proposition 6.6 that h f, f i = 2 | D F | · Y v C Λ v · Λ(1 , Π , ad) , where the constant C Λ v is defined by the table below. We also define a constant C B ′ v for B ′ = B , B , B by the table, so that the following hold: hh s B ′ , s B ′ ii ˜ K B ′ = 2 d B ′ h F · Y v C B ′ v · h f B ′ , f B ′ ih v B ′ k B ′ , v B ′ k B ′ i h B ′ , | α ( B , B ) | = Y v ( C Bv ) C ϕv · | α ( B , B ) | . Π v B ,v B ,v B v C B v C B v C Bv C Λ v C ϕv C v ur spl spl spl 1 1 1 1 1 1rps spl spl spl q n v − v ( q v + 1) q n v − v ( q v + 1) q n v − v ( q v + 1) q v q v +1 q v − q nv +1 v q nv − v ( q v − q v +1) st spl spl spl q v + 1 q v + 1 q v + 1 q v q v +1 q − v q v ( q v +1) st ram ram spl 1 1 q v + 1 q v q v +1 q − v q v st ram spl ram 1 q v + 1 1 q v q v +1 q − v q v st spl ram ram q v + 1 1 1 q v q v +1 q − v q v ds spl spl spl 1 1 1 kv +2 k v !2 kv π kv kv +2 π kv k v ! ds ram ram spl k v − k v − kv +2 ( k v − kv − π kv − kv π kv − ( k v − · ( k v − ds ram spl ram k v − k v − kv +2 ( k v − kv − π kv − kv − π kv − ( k v − · ( k v − ds spl ram ram 1 k v − k v − kv +2 ( k v − kv − π kv − kv − π kv − ( k v − · ( k v − By Theorem 6.7, we have | α ( B , B ) | · h f B , f B i · h f B , f B i = | D F | · Y v C v · h f B , f B i · h f, f i with the constant C v above, and hence | α ( B , B ) | Q v ( C Bv ) C ϕv · h v B k B , v B k B i h B · hh s B , s B ii ˜ K B d B h F · Q v C B v · h v B k B , v B k B i h B · hh s B , s B ii ˜ K B d B h F · Q v C B v = | D F | · Y v C v · h v Bk B , v Bk B i h B · hh s B , s B ii ˜ K B d B h F · Q v C Bv · | D F | · Y v C Λ v · Λ(1 , Π , ad) . Now the theorem follows from this and the fact that C B v C B v C Bv C Λ v C ϕv C v = 1for all v . (cid:3) We now motivate the main conjecture of this paper. Let us setΛ(Π) := Λ(1 , Π , ad) . Thus from (7.4), we see that for B = B , we have(7.5) | α ( B , B ) | · q B (Π , ℓ ) · q B (Π , ℓ ) = Λ(Π) · q B (Π , ℓ ) in C × /R × ( ℓ ) , and consequently,(7.6) | α ( B , B ) | · q B (Π) · q B (Π) = Λ(Π) · q B (Π) in C × /R × ( ℓ ) . 14 ATSUSHI ICHINO AND KARTIK PRASANNA If we combine this with Conjecture A(ii) of the introduction, we are lead to the following conjecturalexpression for | α ( B , B ) | : | α ( B , B ) | = Λ(Π) · Λ(Π) Q v ∈ Σ B c v (Π)Λ(Π) Q v ∈ Σ B c v (Π) · Λ(Π) Q v ∈ Σ B c v (Π) = Y v ∈ Σ B ∩ Σ B c v (Π) in C × /R × ( ℓ ) . Combining this last expression with Conjecture A(i) suggests Conjecture D of the introduction onthe arithmetic nature of the constants α ( B , B ). We restate it below for the convenience of the reader. Conjecture . Suppose that B = B and Σ B ∩ Σ B ∩ Σ ∞ = ∅ , that is B and B have no infiniteplaces of ramification in common. Then(i) α ( B , B ) lies in Q × .(ii) Moreover, α ( B , B ) belongs to R ( ℓ ) .(iii) If in addition B and B have no finite places of ramification in common, then α ( B , B ) lies in R × ( ℓ ) .Note that this conjecture makes absolutely no reference to the constants c v (Π). However, we shallshow now that the truth of this conjecture (for all ℓ prime to N (Π)) implies the truth of Conj. A. Theorem 7.7. Suppose that Conj. 7.6 is true for all ℓ prime to N (Π) . Then Conj. A is true.Remark . The proof below will show that the validity of Conj. 7.6 for a single ℓ implies a versionof Conj. A with R × replaced by R × ( ℓ ) . Proof. Recall from Remark 3 of the introduction that(7.7) q M ( F ) = Λ(Π) in C × /R × . Note that if | Σ Π | = 0 or 1, then Π does not transfer to any non-split quaternion algebra, so theconjecture follows from (7.7).If | Σ Π | = 2, say Σ Π = { v, w } , then there is a unique non-split quaternion algebra B with Σ B ⊆ Σ Π ,given by Σ B = Σ Π . In this case, we need to pick two elements c v (Π) and c w (Π) in C × /R × such thatthe relation q B (Π) = Λ(Π) c v (Π) · c w (Π)is satisfied (in addition to (7.7)), and there are obviously many ways to do this. Since at most one ofthe places in Σ Π (say v ) is a finite place, we can also make this choice so that c v (Π) lies in R . Notethat in this case, the invariants c v (Π) and c w (Π) are not uniquely determined by the single relationabove, so in order to get canonical invariants one would need to rigidify the choices by imposing otherconstraints on them. We do not pursue this here.Thus we may assume that | Σ Π | ≥ 3. We need to first define the constants c v (Π) in this case. First,for any subset Σ ⊆ Σ Π of even cardinality let us define c Σ (Π) ∈ C × /R × by c Σ (Π) := Λ(Π) q B Σ (Π) , where B Σ denotes the unique quaternion algebra ramified exactly at Σ. Note that from (7.7), we have(7.8) c ∅ (Π) = 1 in C × /R × . Now let v be any element in Σ Π . We will define c v (Π) as follows. Pick any two other elements u, w ∈ Σ Π and define c v (Π) to be the unique element in C × /R × such that(7.9) c v (Π) = c { v,u } (Π) · c { v,w } (Π) c { u,w } (Π) . We will show that the truth of Conj. 7.6 for a single ℓ implies that c v (Π) is well defined in C × /R × ( ℓ ) ,that it lies in R ( ℓ ) if v is a finite place and that the relation(7.10) q B (Π) = Λ(Π) Q v ∈ Σ B c v (Π) in C × /R × ( ℓ ) is satisfied. It follows from this that the truth of Conj. 7.6 for all ℓ prime to N (Π) implies that the c v (Π) is well defined in C × /R × , that it lies in R if v is a finite place and that the relation q B (Π) = Λ(Π) Q v ∈ Σ B c v (Π) in C × /R × is satisfied, which would complete the proof of the theorem.Thus let ℓ be any prime not dividing N (Π) and let us assume the truth of Conj. 7.6 for this fixed ℓ .If Σ and Σ are two distinct subsets of Σ Π of even cardinality and if B and B are the correspondingquaternion algebras, the relation (7.6) gives | α ( B , B ) | · c Σ (Π) = c Σ (Π) · c Σ (Π) in C × /R × ( ℓ ) . If moreover Σ and Σ are disjoint, then Conj. 7.6 implies that α ( B , B ) lies in R × ( ℓ ) . Thus we getthe key multiplicative relation:(7.11) c Σ (Π) · c Σ (Π) = c Σ (Π) in C × /R × ( ℓ ) , if Σ ∩ Σ = ∅ , including the case Σ = Σ = ∅ on account of (7.8). We can use this to check that c v (Π) defined via(7.9) is independent of the choice of u and w , viewed as an element in C × /R × ( ℓ ) . Since the definitionis symmetric in u and w , it suffices to show that it remains invariant under changing u to some other u ′ distinct from u and w . However, this follows from the equality c { v,u } (Π) · c { u ′ ,w } (Π) = c { v,u,u ′ ,w } (Π) = c { v,u ′ } (Π) · c { u,w } (Π) in C × /R × ( ℓ ) , which is implied by (7.11).Next we check that if v is a finite place, then c v (Π) lies in R ( ℓ ) . If B , B and B are the quaternionalgebras with Σ B = { v, u } , Σ B = { v, w } and Σ B = { u, w } , then B = B · B and c v (Π) = Λ(Π) q B (Π) · Λ(Π) q B (Π)Λ(Π) q B (Π) = Λ(Π) · q B (Π) q B (Π) · q B (Π) = | α ( B , B ) | in C × /R × ( ℓ ) . Since B and B have no infinite places of ramification in common, it follows from (i) and (ii) of Conj.7.6 that c v (Π) lies in R ( ℓ ) .Finally, let us check that c u (Π) · c v (Π) = c { u,v } (Π) in C × /R × ( ℓ ) if u, v are distinct elements in Σ B .Indeed, picking any w distinct from u and v , we have c u (Π) · c v (Π) = c { u,v } (Π) · c { u,w } (Π) c { v,w } (Π) · c { v,u } (Π) · c { v,w } (Π) c { u,w } (Π) = c { u,v } (Π) in C × /R × ( ℓ ) , 16 ATSUSHI ICHINO AND KARTIK PRASANNA as claimed. From this, (7.8) and (7.11) it follows immediately that for any subset Σ ⊆ Σ Π of evencardinality, we have c Σ (Π) = Y v ∈ Σ c v (Π) in C × /R × ( ℓ ) , from which (7.10) follows immediately. (cid:3) Appendix A. Polarized Hodge structures, abelian varieties and complex conjugation In this section we discuss polarizations and the action of complex conjugation on Hodge structuresattached to abelian varieties. This material is completely standard, so the purpose of this section issimply to carefully fix our conventions and motivate some of our constructions in Chapter 1.If A is a complex abelian variety, there is a natural Hodge structure on Λ = H ( A, Z ). If V = Λ ⊗ Q ,we have V C = H ( A, C ) = V − , ⊕ V , − where V − , = Lie( A ) and V , − = F ( V ) = V − , is identified with H ( A, O A ) ∨ . In fact, the exactsequence 0 → V , − → V C → V − , → → H ( A, Ω A ) → H ( A, C ) → H ( A, O A ) → H ( A, C ). As a complex torus, A is recovered as A = V , − \ V C / Λ ≃ V − , / Λ . Let h : S → GL( V R ) be the homomorphism of the Deligne torus into GL( V R ) corresponding to theHodge structure on H ( A, Z ). Let C = h ( i ). Recall that according to our conventions, the operator C ⊗ V R ⊗ R C acts on V − , as i and on V , − as − i . We write F for F V = V , − so that¯ F = V − , = Lie( A ). Then the composite maps(A.1) Λ ⊗ R → Λ ⊗ C → F, Λ ⊗ R → Λ ⊗ C → ¯ F are R -linear isomorphisms.Let Ψ be a skew-symmetric form Ψ : Λ × Λ → Z (1)whose R -linear extension Ψ R : V R × V R → R (1) satisfiesΨ R ( Cv, Cw ) = Ψ R ( v, w ) . Define B : Λ × Λ → Z , B ( v, w ) = 12 πi Ψ( v, w ) . Remark A.1 . Note that the discussion up to this point was in fact independent of a choice of i .However, in the definition of B above and in the sequel, we need to fix such a choice. For any element x + yi ∈ C let us also set Im( x + yi ) = yi, im( x + yi ) = y. Let B R and B C denote the R -linear and C -linear extensions of B to V R and V C respectively. Let B C denote the hermitian form on V C given by B C ( v, w ) := B C ( v, C ¯ w ) . Finally, we let B F and B ¯ F denote the bilinear forms on F and ¯ F obtained from B R via the isomorphisms(A.1) above. Proposition A.2. The forms B C and B C have the following properties: (i) The subspaces F and ¯ F of V C are isotropic for B C . (ii) The form B C pairs F × ¯ F to zero. (iii) 2 · im( B C ) | F = B F and · im( B C ) | ¯ F = − B ¯ F . 18 ATSUSHI ICHINO AND KARTIK PRASANNA Proof. For v, w ∈ F , we have B C ( v, w ) = B C ( h ( i ) v, h ( i ) w ) = B C ( − iv, − iw ) = − B C ( v, w ) , so F is isotropic for B C . The argument for ¯ F is similar. Part (ii) follows immediately from part (i).For part (iii), suppose v, w ∈ F . Then2 Im( B C )( v, w ) = B C ( v, w ) − B C ( v, w ) = B C ( v, C ¯ w ) − B C (¯ v, Cw )= B C ( v, i ¯ w ) − B C (¯ v, − iw ) = i ( B C ( v, ¯ w ) + B C (¯ v, w )) . On the other hand, under the isomorphism V R ≃ F , the element v ∈ F corresponds to v + ¯ v ∈ V R .Thus B F ( v, w ) = B R ( v + ¯ v, w + ¯ w ) = B C ( v + ¯ v, w + ¯ w ) = B C ( v, ¯ w ) + B C (¯ v, w )from part (i). This shows that 2 · im( B C ) | F = B F . The proof for ¯ F is similar. (cid:3) Proposition A.3. The following are equivalent: (i) The bilinear form ( v, w ) B R ( v, Cw ) on V R is positive definite. (ii) The hermitian form B C on V C is positive definite and induces by restriction positive definitehermitian forms on both F and ¯ F .Proof. Let v, w ∈ V C . Suppose v = v + v and w = w + w with v , w ∈ F and v , w ∈ ¯ F . Then B C ( v, w ) = B C ( v + v , w + w ) = B C ( v , w ) + B C ( v , w )= B C ( v , C ¯ w ) + B C ( v , C ¯ w )so in particular, B C ( v, v ) = B C ( v , C ¯ v ) + B C ( v , C ¯ v ) . On the other hand, B R ( v + ¯ v , C ( v + ¯ v )) = B C ( v , C ¯ v ) + B C (¯ v , Cv )= B C ( v , C ¯ v ) − B C ( C ¯ v , Cv )= B C ( v , C ¯ v ) − B C ( C ¯ v , v )= 2 B C ( v , C ¯ v ) . Likewise, B R ( v + ¯ v , C ( v + ¯ v )) = 2 B C ( v , C ¯ v ) . The implication (i) ⇐⇒ (ii) is clear from this. (cid:3) Definition A.4. We will say that Ψ or B is a polarization if either of the equivalent conditions of theproposition above are satisfied. Remark A.5 . In the classical theory of complex abelian varieties, one considers hermitian forms H on F or ¯ F whose imaginary part im H equals a given skew-symmetric form. A polarization correspondsto the choice of a skew-symmetric form such that H is either positive or negative definite. This canlead to some confusion: note for example that the skew-symmetric form B F is the imaginary part ofthe positive definite form 2 · B C | F , while the skew-symmetric form B ¯ F is the imaginary part of thenegative definite form − · B C | ¯ F . We will always use the form B C which is positive definite on both F and ¯ F . Appendix B. Metaplectic covers of symplectic similitude groups B.1. Setup. Let F be a local field of characteristic zero. Fix a nontrivial additive character ψ of F .Let V be a 2 n -dimensional symplectic space over F . Let GSp( V ) and Sp( V ) := ker ν be the similitudegroup and the symplectic group of V respectively, where ν : GSp( V ) → F × is the similitude character.Fix a complete polarization V = X ⊕ Y . Choose a basis e , . . . , e n , e ∗ , . . . , e ∗ n of V such that X = F e + · · · + F e n , Y = F e ∗ + · · · + F e ∗ n , hh e i , e ∗ j ii = δ ij . Using this basis, we may writeGSp( V ) = (cid:26) g ∈ GL n ( F ) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) n − n (cid:19) t g = ν ( g ) · (cid:18) n − n (cid:19) (cid:27) . For ν ∈ F × , we define d ( ν ) = d Y ( ν ) ∈ GSp( V ) by d ( ν ) := (cid:18) n ν · n (cid:19) . Let P = P Y be the maximal parabolic subgroup of Sp( V ) stabilizing Y : P = (cid:26) (cid:18) a ∗ t a − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) a ∈ GL n ( F ) (cid:27) . We have a Bruhat decomposition Sp( V ) = n a j =0 P τ j P, where τ j := n − j − j n − j j . For h ∈ Sp( V ), put j ( h ) := j if h ∈ P τ j P . We define a map x : Sp( V ) −→ F × / ( F × ) by x ( p τ j p ) := det( a a ) mod ( F × ) , where p i = (cid:18) a i ∗ t a − i (cid:19) ∈ P. In particular, we have x ( p hp ) = x ( p ) x ( h ) x ( p ) for p , p ∈ P and h ∈ Sp( V ).Let z Y = z Sp Y be the 2-cocycle given by z Y ( h , h ) := γ F ( 12 ψ ◦ q ( Y , Y h − , Y h ))for h , h ∈ Sp( V ). Lemma B.1. We have • z Y ( h, h − ) = 1 for h ∈ Sp( V ) , • z Y ( p h p, p − h p ) = z Y ( h , h ) for p, p i ∈ P and h i ∈ Sp( V ) , • z Y ( τ i , τ j ) = 1 , • z Y ( τ n , n ( β ) τ n ) = γ F ( ψ ◦ q β ) for n ( β ) = (cid:0) n β n (cid:1) with β ∈ Hom( X , Y ) if q β is non-degenerate,where q β is a symmetric bilinear form on X defined by q β ( x, y ) = hh x, yβ ii . 20 ATSUSHI ICHINO AND KARTIK PRASANNA Proof. See [61, Theorem 4.1, Corollary 4.2]. (cid:3) Suppose that V = V ⊕ V , where each V i is a non-degenerate symplectic subspace over F . If V i = X i ⊕ Y i is a complete polarization and X = X ⊕ X , Y = Y ⊕ Y , then we have z Y ( h , h ′ ) · z Y ( h , h ′ ) = z Y ( h h , h ′ h ′ )for h i , h ′ i ∈ Sp( V i ) (see Theorem 4.1 of [61]).B.2. Action of outer automorphisms on the -cocycle. For ν ∈ F × , let α ν = α Y ,ν be the outerautomorphism of Sp( V ) given by α ν ( h ) = d ( ν ) · h · d ( ν ) − for h ∈ Sp( V ). This induces an action of F × on Sp( V ) and thus we have an isomorphismSp( V ) ⋊ F × −→ GSp( V ) . ( h, ν ) ( h, ν ) Y := h · d ( ν )Note that ( h, ν ) Y · ( h ′ , ν ′ ) Y = ( h · α ν ( h ′ ) , ν · ν ′ ) Y . There exists a unique automorphism ˜ α ν of Mp( V ) such that ˜ α ν | C = id C and the diagramMp( V ) ˜ α ν / / (cid:15) (cid:15) Mp( V ) (cid:15) (cid:15) Sp( V ) α ν / / Sp( V )commutes. This implies that there exists a unique function v Y : Sp( V ) × F × −→ C such that ˜ α ν ( h, z ) = ( α ν ( h ) , z · v Y ( h, ν ))for ( h, z ) ∈ Mp( V ) Y . Since ˜ α ν is an automorphism, we have z Y ( α ν ( h ) , α ν ( h ′ )) = z Y ( h, h ′ ) · v Y ( hh ′ , ν ) · v Y ( h, ν ) − · v Y ( h ′ , ν ) − for h, h ′ ∈ Sp( V ) and ν ∈ F × . Lemma B.2. We have v Y ( h, ν ) = ( x ( h ) , ν ) F · γ F ( ν, ψ ) − j ( h ) for h ∈ Sp( V ) and ν ∈ F × .Proof. See [2, Proposition 1.2.A]. For convenience, we recall the proof of [2, Proposition 1.2.A]. Warn-ing: our convention differs from that in [2] . Note that z Y ( p, h ) = z Y ( h, p ) = 1 for p ∈ P and h ∈ Sp( V ). This implies that v Y ( php ′ , ν ) = v Y ( p, ν ) · v Y ( h, ν ) · v Y ( p ′ , ν )for p, p ′ ∈ P and h ∈ Sp( V ). Moreover, there exist a character ξ ν of F × and an element γ ν ∈ C suchthat v Y ( p, ν ) = ξ ν ( x ( p )) , v Y ( τ j , ν ) = γ jν . To determine ξ ν and γ ν , we may assume that dim V = 2 as explained in the proof of [2, Proposition1.2.A]. Put n ( x ) := (cid:18) x (cid:19) . If x = 0, then we have n ( x ) = (cid:18) x − (cid:19) (cid:18) − 11 0 (cid:19) (cid:18) x x − (cid:19) , so that v Y ( n ( x ) , ν ) = ξ ν ( x ) · γ ν . Let x, y ∈ F such that x = 0, y = 0, x + y = 0. Since α ν ( n ( x )) = n ( νx ), we have z Y ( n ( νx ) , n ( νy )) z Y ( n ( x ) , n ( y )) = v Y ( n ( x + y ) , ν ) v Y ( n ( x ) , ν ) · v Y ( n ( y ) , ν ) = ξ ν (cid:18) x + yxy (cid:19) · γ − ν . By [61, Corollary 4.3], we have z Y ( n ( x ) , n ( y )) = γ F ( 12 xy ( x + y ) · ψ )and hence z Y ( n ( νx ) , n ( νy )) z Y ( n ( x ) , n ( y )) = γ F ( ν xy ( x + y ) · ψ ) γ F ( xy ( x + y ) · ψ )= γ F ( ν xy ( x + y ) , ψ ) γ F ( xy ( x + y ) , ψ )= γ F ( ν , ψ ) · ( xy ( x + y ) , ν ) F = γ F ( ν, ψ ) · ( xy ( x + y ) , ν ) F . Thus we obtain γ F ( ν, ψ ) · (cid:18) x + yxy , ν (cid:19) F = ξ ν (cid:18) x + yxy (cid:19) · γ − ν . Taking x = y = 2, we have γ ν = γ F ( ν, ψ ) − and hence ξ ν ( a ) = ( a, ν ) F for all a ∈ F × . (cid:3) B.3. Metaplectic groups. For each ν ∈ F × , we have an automorphism ˜ α ν of Mp( V ). This inducesan action of F × on Mp( V ) and thus we have a topological groupMp( V ) ⋊ F × . We define a bijection Mp( V ) Y ⋊ F × −→ GMp( V ) Y := GSp( V ) × C (( h, z ) , ν ) (( h, ν ) Y , z ) 22 ATSUSHI ICHINO AND KARTIK PRASANNA as sets. Via this bijection, we regard GMp( V ) Y as a group. Note that the diagramMp( V ) Y ⋊ F × / / (cid:15) (cid:15) GMp( V ) Y (cid:15) (cid:15) Sp( V ) ⋊ F × / / GSp( V )commutes. Let z GSp Y be the 2-cocycle associated to GMp( V ) Y . By definition, one can see that z GSp Y ( g, g ′ ) = z Sp Y ( h, α ν ( h ′ )) · v Y ( h ′ , ν )for g = ( h, ν ) Y , g ′ = ( h ′ , ν ′ ) Y ∈ GSp( V ). In particular, the restriction of z GSp Y to Sp( V ) × Sp( V ) isequal to z Sp Y . Thus we omit the superscripts GSp and Sp from the notation.We shall see that the isomorphism class of GMp( V ) Y does not depend on the choice of the completepolarization. If there is no confusion, we write GMp( V ) = GMp( V ) Y .B.4. Change of polarizations. Let V = X ′ + Y ′ be another complete polarization. Fix an element h ∈ Sp( V ) such that X ′ = X h and Y ′ = Y h . Let α be the inner automorphism of GSp( V ) given by α ( g ) = h · g · h − for g ∈ GSp( V ). Note that α | Sp( V ) is an inner automorphism of Sp( V ). We have d Y ′ ( ν ) = h − · d Y ( ν ) · h , α Y ′ ,ν = α − ◦ α Y ,ν ◦ α . By [39, Lemma 4.2], we have z Y ′ ( h, h ′ ) = z Y ( α ( h ) , α ( h ′ ))for h, h ′ ∈ Sp( V ), and an isomorphism Mp( V ) Y −→ Mp( V ) Y ′ , ( h, z ) ( h, z · µ ( h ))where µ ( h ) = z Y ( h , hh − ) · z Y ( h, h − )for h ∈ Sp( V ). Lemma B.3. We have v Y ′ ( h, ν ) = v Y ( α ( h ) , ν ) for h ∈ Sp( V ) and ν ∈ F × .Proof. We have z Y ′ ( α Y ′ ,ν ( h ) , α Y ′ ,ν ( h ′ )) = z Y ′ (( α − ◦ α Y ,ν ◦ α )( h ) , ( α − ◦ α Y ,ν ◦ α )( h ′ ))= z Y (( α Y ,ν ◦ α )( h ) , ( α Y ,ν ◦ α )( h ′ ))= z Y ( α ( h ) , α ( h ′ )) · v Y ( α ( h ) · α ( h ′ ) , ν ) · v Y ( α ( h ) , ν ) − · v Y ( α ( h ′ ) , ν ) − = z Y ′ ( h, h ′ ) · v Y ( α ( hh ′ ) , ν ) · v Y ( α ( h ) , ν ) − · v Y ( α ( h ′ ) , ν ) − . Thus the assertion follows from the characterization of v Y ′ . (cid:3) Lemma B.4. We have z Y ′ ( g, g ′ ) = z Y ( α ( g ) , α ( g ′ )) for g, g ′ ∈ GSp( V ) . Proof. Let g = ( h, ν ) Y ′ , g ′ = ( h ′ , ν ′ ) Y ′ ∈ GSp( V ). Then we have z Y ′ ( g, g ′ ) = z Y ′ ( h, α Y ′ ,ν ( h ′ )) · v Y ′ ( h ′ , ν )= z Y ( α ( h ) , ( α ◦ α Y ′ ,ν )( h ′ )) · v Y ( α ( h ′ ) , ν )= z Y ( α ( h ) , ( α Y ,ν ◦ α )( h ′ )) · v Y ( α ( h ′ ) , ν )= z Y (( α ( h ) , ν ) Y , ( α ( h ′ ) , ν ′ ) Y ) . Since α ( g ) = h · h · d Y ′ ( ν ) · h − = h · h · h − · d Y ( ν ) = ( α ( h ) , ν ) Y , the assertion follows. (cid:3) Put µ ( g ) = z Y ( g, h − ) · z Y ( h , gh − ) = z Y ′ ( h − gh , h − ) · z Y ′ ( h − , g ) − for g ∈ GSp( V ). Note that µ depends on the choice of h . By a direct calculation, one can see that z Y ′ ( g, g ′ ) = z Y ( g, g ′ ) · µ ( gg ′ ) · µ ( g ) − · µ ( g ′ ) − for g, g ′ ∈ GSp( V ). Thus we obtain an isomorphismGMp( V ) Y −→ GMp( V ) Y ′ . ( g, z ) ( g, z · µ ( g )) Appendix C. Splittings: the case dim B V = 2 and dim B W = 1C.1. Setup. Let F be a number field. Recall that E = F + F i , B = E + E j , B = E + E j , B = E + E j ,u := i , J := j , J := j , J := j , where J = J J . Recall that V = B ⊗ E B and W = B are a right skew-hermitian B -space and a left hermitian B -space respectively, and V = V ⊗ B W is an F -space with a symplectic form hh· , ·ii = 12 tr B/F ( h· , ·i ⊗ h· , ·i ∗ ) . Recall that V = X + Y is a complete polarization, where X = F e + F e + F e + F e , Y = F e ∗ + F e ∗ + F e ∗ + F e ∗ . The actions of B , B , B on V are given as follows: 24 ATSUSHI ICHINO AND KARTIK PRASANNA • B -action e i = u e ∗ e i = − uJ e ∗ e i = − uJ e ∗ e i = uJ e ∗ e ∗ i = e e ∗ i = − J e e ∗ i = − J e e ∗ i = 1 J e e j = e e j = J e e j = J e e j = J e e ∗ j = − J e ∗ e ∗ j = − J e ∗ e ∗ j = − J e ∗ e ∗ j = − e ∗ e ij = − uJ e ∗ e ij = uJ e ∗ e ij = uJ e ∗ e ij = − uJ e ∗ e ∗ ij = e e ∗ ij = − e e ∗ ij = − e e ∗ ij = e • B -action ie = u e ∗ ie = uJ e ∗ ie = − uJ e ∗ ie = − uJ e ∗ ie ∗ = e ie ∗ = 1 J e ie ∗ = − J e ie ∗ = − J e j e = e j e = J e j e = e j e = J e j e ∗ = − J e ∗ j e ∗ = − e ∗ j e ∗ = − J e ∗ j e ∗ = − e ∗ ij e = uJ e ∗ ij e = uJ e ∗ ij e = − uJ e ∗ ij e = − uJ e ∗ ij e ∗ = − e ij e ∗ = − e ij e ∗ = 1 J e ij e ∗ = 1 J e • B -action ie = u e ∗ ie = − uJ e ∗ ie = uJ e ∗ ie = − uJ e ∗ ie ∗ = e ie ∗ = − J e ie ∗ = 1 J e ie ∗ = − J e j e = e j e = e j e = J e j e = J e j e ∗ = − J e ∗ j e ∗ = − J e ∗ j e ∗ = − e ∗ j e ∗ = − e ∗ ij e = uJ e ∗ ij e = − uJ e ∗ ij e = uJ e ∗ ij e = − uJ e ∗ ij e ∗ = − e ij e ∗ = 1 J e ij e ∗ = − e ij e ∗ = 1 J e Let α i ∈ B × i and α ∈ B × . We write α = a + b i + c j + d ij , α = a + b i + c j + d ij , α = a + b i + c j + d ij , where a i , a, b i , b, c i , c, d i , d ∈ F . Then we have α e α e α e α e α e ∗ α e ∗ α e ∗ α e ∗ = g e e e e e ∗ e ∗ e ∗ e ∗ , g = a c b u d uJ c J a d uJ b uJ a c − b uJ − d uJc J a − d uJ − b uJb − d a − c J − d b J − c a − b J d J a − c J d J − b J − c a , α e α e α e α e α e ∗ α e ∗ α e ∗ α e ∗ = g e e e e e ∗ e ∗ e ∗ e ∗ , g = a c b u d uJ a c − b uJ − d uJc J a d uJ b uJ c J a − d uJ − b uJb − d a − c J − b J d J a − c J − d b J − c a d J − b J − c a , e α e α e α e α e ∗ α e ∗ α e ∗ α e ∗ α = g e e e e e ∗ e ∗ e ∗ e ∗ , g = a c bu − duJa cJ − buJ duJcJ a duJ − buJ cJ a − duJ buJb d a − cJ − bJ − d a − cJ − d − bJ − cJ ad bJ − c a . For a ∈ GL ( F ) and b ∈ Sym ( F ), put m ( a ) := (cid:18) a t a − (cid:19) , n ( b ) := (cid:18) b1 (cid:19) . Fix a place v of F . In §§ C.2, C.3, we shall suppress the subscript v from the notation. Thus F = F v will be a local field of characteristic zero.C.2. The case u ∈ ( F × v ) or J ∈ ( F × v ) . First we explicate Morita theory. Fix an isomorphism i : B −→ M ( F )of F -algebras such that i ( α ∗ ) = i ( α ) ∗ for α ∈ B . Put e := i − (cid:18) (cid:19) , e ′ := i − (cid:18) (cid:19) , e ′′ := i − (cid:18) (cid:19) . Then we have e ∗ = i − (cid:18) (cid:19) 26 ATSUSHI ICHINO AND KARTIK PRASANNA and e = e, ee ′ = e ′ , ee ′′ = 0 , ee ∗ = 0 ,e ′ e = 0 , ( e ′ ) = 0 , e ′ e ′′ = e, e ′ e ∗ = e ′ ,e ′′ e = e ′′ , e ′′ e ′ = e ∗ , ( e ′′ ) = 0 , e ′′ e ∗ = 0 ,e ∗ e = 0 , e ∗ e ′ = 0 , e ∗ e ′′ = e ′′ , ( e ∗ ) = e ∗ . Thus we obtain B = F e + F e ′ + F e ′′ + F e ∗ , eB = F e + F e ′ , Be = F e + F e ′′ , eBe = F e and (cid:20) e · α e ′ · α (cid:21) = i ( α ) · (cid:20) ee ′ (cid:21) for α ∈ B .Now we consider an F -space W † := eW . Since eBe ∗ = F e ′ and ( e ′ ) ∗ = − e ′ , we have h x, y i ∈ F e ′ , h y, x i = −h x, y i for x, y ∈ W † . Hence we can define a symplectic form h· , ·i † : W † × W † −→ F by h x, y i ∗ = h x, y i † · e ′ for x, y ∈ W † . Moreover, we see that h· , ·i † is non-degenerate.We have W † = F e + F e ′ and h e, e i † = h e ′ , e ′ i † = 0 , h e, e ′ i † = 1 . Thus we may identify W † with the space of row vectors F so that h x, y i † = x y − x y for x = ( x , x ), y = ( y , y ) ∈ W † . Lemma C.1. The restriction to W † induces an isomorphism GU( W ) ∼ = −→ GSp( W † ) . Proof. One can see that the restriction to W † induces a homomorphism GU( W ) → GSp( W † ). Since B · W † = B · eW = BeB · W = B · W = W, this homomorphism is injective. Let h ∈ GSp( W † ). Since W = W † ⊕ e ′′ W , we can define a map˜ h : W → W by ˜ h ( x ) = ( h ( x ) if x ∈ W † , e ′′ h ( e ′ x ) if x ∈ e ′′ W .Then one can check that ˜ h ∈ GU( W ). This yields the lemma. (cid:3) Thus we may identify GU( W ) with GSp( W † ). Similarly, we consider an F -space V † := V e with anon-degenerate symmetric bilinear form h· , ·i † : V † × V † −→ F defined by 12 · h x, y i = h x, y i † · e ′′ for x, y ∈ V † . As in Lemma C.1, the restriction to V † induces an isomorphismGU( V ) ∼ = −→ GO( V † ) . Thus we may identify GU( V ) with GO( V † ).One can see that the natural map V † ⊗ F W † −→ V ⊗ B W is an isomorphism. Thus we may identify V with V † ⊗ F W † . Lemma C.2. We have hh· , ·ii = h· , ·i † ⊗ h· , ·i † . Proof. Let a = h x, x ′ i † and b = h y, y ′ i † for x, x ′ ∈ V † and y, y ′ ∈ W † . Then we have hh x ⊗ y, x ′ ⊗ y ′ ii = 12 · tr B/F ( h x, x ′ i · h y, y ′ i ∗ )= tr B/F ( ae ′′ · be ′ )= ab · tr B/F ( e ∗ )= ab. (cid:3) Thus we obtain a commutative diagramGU( V ) × GU( W ) / / (cid:15) (cid:15) GSp( V )GO( V † ) × GSp( W † ) / / GSp( V ) . Let W † = X + Y be a complete polarization given by X = F e, Y = F e ′ . Put X ′ = V † ⊗ F X, Y ′ = V † ⊗ F Y. Then we have a complete polarization V = X ′ + Y ′ . Put s ′ ( h ) := γ j ( h ) for h ∈ GSp( W † ), where γ = ( B and B are split, − B and B are ramified,and j ( h ) = ( i ( h ) = ( ∗ ∗ ∗ ),1 otherwise. Lemma C.3. We have z Y ′ ( h, h ′ ) = s ′ ( hh ′ ) · s ′ ( h ) − · s ′ ( h ′ ) − for h, h ′ ∈ GSp( W † ) . 28 ATSUSHI ICHINO AND KARTIK PRASANNA Proof. The lemma follows from [39, Theorem 3.1, case 1 + ] and [64, Proposition 2.1]. We shall give aproof for the sake of completeness.Recall that dim F V † = 4 and det V † = 1. By [39, Theorem 3.1, case 1 + ], we have(C.1) z Y ′ ( h, h ′ ) = s ′ ( hh ′ ) · s ′ ( h ) − · s ′ ( h ′ ) − for h, h ′ ∈ Sp( W † ).Let g, g ′ ∈ GSp( W † ). For ν ∈ F × , put d ( ν ) = (cid:18) ν (cid:19) ∈ GSp( W † ) . We write g = h · d ( ν ) , g ′ = h ′ · d ( ν ′ )with h, h ′ ∈ Sp( W † ) and ν, ν ′ ∈ F × . Then we have z Y ′ ( g, g ′ ) = z Y ′ ( h, h ′′ ) · v Y ′ ( h ′ , ν ) , where h ′′ = d ( ν ) · h ′ · d ( ν ) − . By (C.1), we have z Y ′ ( h, h ′′ ) = s ′ ( hh ′′ ) · s ′ ( h ) − · s ′ ( h ′′ ) − . We have s ′ ( h ) = s ′ ( g ), and since j ( h ′′ ) = j ( h ′ ), we have s ′ ( h ′′ ) = s ′ ( h ′ ) = s ′ ( g ′ ). Moreover, since gg ′ = hh ′′ · d ( νν ′ ), we have s ′ ( hh ′′ ) = s ′ ( gg ′ ). Thus we obtain z Y ′ ( h, h ′′ ) = s ′ ( gg ′ ) · s ′ ( g ) − · s ′ ( g ′ ) − . By Lemma B.2, we have v Y ′ ( h ′ , ν ) = ( x Y ′ ( h ′ ) , ν ) F · γ F ( ν, ψ ) − j Y ′ ( h ′ ) , where x Y ′ and j Y ′ are as in § B.1 with respect to the complete polarization V = X ′ + Y ′ . Sincedim F V † = 4 and det V † = 1, one can see that x Y ′ ( h ′ ) ≡ F × ) and j Y ′ ( h ′ ) = 4 · j ( h ′ ). Hencewe have v Y ′ ( h ′ , ν ) = 1 . This completes the proof. (cid:3) Lemma C.4. We have z Y ′ ( g, g ′ ) = 1 for g, g ′ ∈ GO( V † ) .Proof. For g, g ′ ∈ GO( V † ) , we have z Y ′ ( g, g ′ ) = z Y ′ ( h, h ′′ ) · v Y ′ ( h ′ , ν ) , where h = g · d Y ′ ( ν ) − , h ′ = g ′ · d Y ′ ( ν ′ ) − , h ′′ = d Y ′ ( ν ) · h ′ · d Y ′ ( ν ) − ,ν = ν ( g ) , ν ′ = ν ( g ′ ) . We have h, h ′ ∈ P Y ′ and z Y ′ ( h, h ′′ ) = 1. Since g ′ ∈ GO( V † ) , we have x Y ′ ( h ′ ) ≡ det g ′ ≡ F × ) , so that v Y ′ ( h ′ , ν ) = 1 by Lemma B.2. This completes the proof. (cid:3) Lemma C.5. We have z Y ′ ( g, h ) = z Y ′ ( h, g ) = 1 for g ∈ GO( V † ) and h ∈ GSp( W † ) .Proof. See [2, Proposition 2.2.A]. We shall give a proof for the sake of completeness.For g ∈ GO( V † ) and h ∈ GSp( W † ), we have z Y ′ ( g, h ) = z Y ′ ( g ′ , h ′′ ) · v Y ′ ( h ′ , ν ) , z Y ′ ( h, g ) = z Y ′ ( h ′ , g ′′ ) · v Y ′ ( g ′ , ν ′ ) , where g ′ = g · d Y ′ ( ν ) − , g ′′ = d Y ′ ( ν ′ ) · g ′ · d Y ′ ( ν ′ ) − , ν = ν ( g ) ,h ′ = h · d ( ν ′ ) − , h ′′ = d ( ν ) · h ′ · d ( ν ) − , ν ′ = ν ( h ) . Since g ′ , g ′′ ∈ P Y ′ , we have z Y ′ ( g ′ , h ′′ ) = z Y ′ ( h ′ , g ′′ ) = 1. As in the proof of Lemma C.3, we have v Y ′ ( h ′ , ν ) = 1. As in the proof of Lemma C.4, we have v Y ′ ( g ′ , ν ′ ) = 1. This completes the proof. (cid:3) We define a map s ′ : GO( V † ) × GSp( W † ) → C by s ′ ( g ) = γ j ( h ) for g = ( g, h ) ∈ GO( V † ) × GSp( W † ). By Lemmas C.3, C.4, C.5, we see that z Y ′ ( g , g ′ ) = s ′ ( gg ′ ) · s ′ ( g ) − · s ′ ( g ′ ) − for g , g ′ ∈ GO( V † ) × GSp( W † ).Recall that we have two complete polarizations V = X + Y = X ′ + Y ′ , where X = F e + F e + F e + F e , Y = F e ∗ + F e ∗ + F e ∗ + F e ∗ , X ′ = F e e + F e e ′′ + F e e + F e e ′′ , Y ′ = F e e ′ + F e e ∗ + F e e ′ + F e e ∗ . Fix h ∈ Sp( V ) such that X ′ = X h and Y ′ = Y h , and put s ( g ) := s ′ ( g ) · µ ( g ) , where µ ( g ) := z Y ( h gh − , h ) · z Y ( h , g ) − for g ∈ GU( V ) × GU( W ). Then we have z Y ( g , g ′ ) = s ( gg ′ ) · s ( g ) − · s ( g ′ ) − for g , g ′ ∈ GU( V ) × GU( W ).C.2.1. The case u ∈ ( F × v ) . Choose t ∈ F × such that u = t . We take an isomorphism i : B → M ( F )determined by i (1) = (cid:18) (cid:19) , i ( i ) = (cid:18) t − t (cid:19) , i ( j ) = (cid:18) J (cid:19) , i ( ij ) = (cid:18) t − tJ (cid:19) . Then we have e = 12 + 12 t i , e ′ = 12 j + 12 t ij , e ′′ = 12 J j − tJ ij , e ∗ = 12 − t i . 30 ATSUSHI ICHINO AND KARTIK PRASANNA Put h = − t − tJ − tJ − − tJ − − t tJ tJ − tJ ∈ Sp( V ) . Then we have − t e e tJ e e t e e ′′ − t e e ′′ e e ∗ e e ∗ J e e ′ e e ′ = h e e e e e ∗ e ∗ e ∗ e ∗ , and hence X ′ = X h and Y ′ = Y h . Lemma C.6. Let g i := α − i ∈ GU( V ) with α i = a i + b i i + c i j i + d i ij i ∈ B × i . Then we have µ ( g i ) = if b i = d i = 0 , γ F ( J j , ψ ) · (( a i b i + c i d i J i ) ν i J i , J j ) F if ( b i , d i ) = (0 , and b i − d i J i = 0 , ( − ( b i − d i J i ) ν i J i , J j ) F if ( b i , d i ) = (0 , and b i − d i J i = 0 ,where ν i = ν ( α i ) and { i, j } = { , } .Proof. We only consider the case i = 1; the other case is similar. Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h g h − , h ) = z Y ( h g h − · d − , d · h · d − ) · v Y ( h , ν ) . Since Y ′ g = Y ′ , we have h g h − · d − ∈ P Y and hence z Y ( h g h − · d − , d · h · d − ) = 1 . We have h = n ( b ) · τ · n ( b ), where b = 12 tJ · − J J J − , b = t · − J J − J , so that x Y ( h ) ≡ F × ) and j Y ( h ) = 4. Hence we have v Y ( h , ν ) = 1. Thus we obtain z Y ( h g h − , h ) = 1 . Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . Next assume that ( b , d ) = (0 , 0) and b − d J = 0. Then we have b = 0, d = 0, and ν = a − c J = 0. Since( a d − b c ) · ( a b + c d J ) = a b d + a c d J − a b c − b c d J = a b d − b c d J = ν b d = 0 , we have a d − b c = 0 and a b + c d J = 0. We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a = b d J b d J , b = a d − b c b d · J − J . Hence we have z Y ( h , g · d − ) = z Y ( τ · n ( b ) , m ( a ) · n ( b ) · τ )= z Y ( τ · m ( a ) , m ( a ) − · n ( b ) · m ( a ) · n ( b ) · τ )= z Y ( τ , n ( b + b ) · τ ) , where b = a − · b · t a − = tb · − d J J − d Jd J − d J . Since τ − · n ( b ) · τ ∈ P Y , we have z Y ( h , g · d − ) = z Y ( τ , n ( b ) · τ ) = γ F ( 12 ψ ◦ q ) , where q is a non-degenerate symmetric bilinear form associated to a d − b c b d · (cid:18) J − J (cid:19) . Since det q ≡ − J mod ( F × ) and h F ( q ) = ( a d − b c b d · J , J ) F , we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( a d − b c b d · J , J ) F = γ F ( J , ψ ) − · ( a d − b c b d · J , J ) F = γ F ( J , ψ ) − · ( ν a b + c d J · J , J ) F = γ F ( J , ψ ) − · (( a b + c d J ) ν J , J ) F . Finally assume that ( b , d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b = 1 b − d J · a b + c d J ( a d + b c ) J ( a d + b c ) J ( a b + c d J ) J − ( a b + c d J ) J − ( a d + b c ) J − ( a d + b c ) J − ( a b + c d J ) J . Hence we have z Y ( h , g · d − ) = z Y ( τ · n ( b ) , n ( b ) · τ ) = z Y ( τ , n ( b ) · n ( b ) · τ ) = γ F ( 12 ψ ◦ q ) , 32 ATSUSHI ICHINO AND KARTIK PRASANNA where q is a non-degenerate symmetric bilinear form associated to b + b . We have b + b = (cid:18) b ′ − J · b ′ (cid:19) , where b ′ = t · (cid:18) − J (cid:19) + 1 b − d J · (cid:18) a b + c d J ( a d + b c ) J ( a d + b c ) J ( a b + c d J ) J (cid:19) . Since det b ′ = ν J b − d J = 0 , we have det q ≡ F × ) and h F ( q ) = (det b ′ , J ) F · ( − , − J ) F = ( − ν J b − d J , J ) F · ( − , − F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · ( − ν J b − d J , J ) F · ( − , − F = ( − ( b − d J ) ν J , J ) F . This completes the proof. (cid:3) Lemma C.7. Let g := α ∈ GU( W ) with α = a + b i + c j + d ij ∈ B × . Then we have µ ( g ) = ( ν, J ) F if b = d = 0 , γ F ( J , ψ ) · ( ab − cdJ, J ) F if ( b, d ) = (0 , and b − d J = 0 , ( − ( b − d J ) J, J ) F if ( b, d ) = (0 , and b − d J = 0 ,where ν = ν ( α ) .Proof. Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h gh − , h ) = z Y ( h gh − · d − , d · h · d − ) · v Y ( h , ν ) . As in the proof of Lemma C.6, we have v Y ( h , ν ) = 1. We have h gh − = a + bt − c + dt t a + bt c + dt t a + bt c + dt t a + bt − c + dt t − c − dt ) tJ a − bt c − dt ) tJ a − bt c − dt ) tJ a − bt − c − dt ) tJ a − bt . If c − dt = 0, then we have h gh − · d − ∈ P Y and hence z Y ( h gh − · d − , d · h · d − ) = 1. If c − dt = 0,then we have h gh − · d − ∈ P Y · τ · n ( b ), where b = a − bt νtJ ( c − dt ) · − − . We have d · h · d − ∈ n ( ν − · b ) · τ · P Y , where b is as in the proof of Lemma C.6. Hence we have z Y ( h gh − · d − , d · h · d − ) = z Y ( τ · n ( b ) , n ( ν − · b ) · τ ) = z Y ( τ , n ( b ) · τ ) , where b = ν − · b + b . Put r = a − btc − dt . We have b = 12 νtJ · − J − rJ rr J − r − = a · b · t a , where a = rJ rJ , b = 12 νtJ · − J r J − J J − r J , and hence z Y ( τ , n ( b ) · τ ) = z Y ( τ , m ( a ) · n ( b ) · m ( a − ) · τ ) = z Y ( τ , n ( b ) · τ ) . If J − r = 0, then we have z Y ( τ , n ( b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degenerate symmetricbilinear form associated to 12 νtJ · (cid:18) − J J (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( − νt , νtJ ) F = ( − νt, J ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( − νt, J ) F = γ F ( J , ψ ) · (2 νt, J ) F . Note that γ F ( J , ψ ) = γ F ( J , ψ ) and (2 νt, J ) F = (2 νt, J ) F since J = r ∈ ( F × ) . If J − r = 0,then we have z Y ( τ , n ( b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degenerate symmetric bilinear formassociated to b . We have det q ≡ F × ) and h F ( q ) = (det q , νtJ ) F · ( − J, r J − F · ( J , J − r J ) F · ( − J ( r J − , J ( J − r J )) F = ( − J, J − r ) F · ( − J, − J ) F · ( J , J − r ) F · ( J , J ) F · ( J − r , J − r ) F = ( − J, J − r ) F · ( − J, − F · ( J , J − r ) F · ( J , − F · ( J − r , − F = ( JJ , J − r ) F · ( − JJ , − F = ( J , J − r ) F · ( J , − F · ( − , − F = ( J , r − J ) F · ( − , − F . Note that ( J , r − J ) F = ( J , r − J ) F since ( J, r − J ) F = 1. Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · ( J , r − J ) F · ( − , − F = ( J , r − J ) F . Thus we obtain z Y ( h gh − , h ) = c − dt = 0, γ F ( J , ψ ) · (2 νt, J ) F if c − dt = 0, ( a − bt ) − ( c − dt ) J = 0,(( a − bt ) − ( c − dt ) J, J ) F if c − dt = 0, ( a − bt ) − ( c − dt ) J = 0.Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . 34 ATSUSHI ICHINO AND KARTIK PRASANNA First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . Next assume that ( b, d ) = (0 , 0) and b − d J = 0. Then we have b = 0 and d = 0. We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a = b b − dJ − dJ , b = ad + bcbd · − J J . Hence we have z Y ( h , g · d − ) = z Y ( τ · n ( b ) , m ( a ) · n ( b ) · τ )= z Y ( τ · m ( a ) , m ( a ) − · n ( b ) · m ( a ) · n ( b ) · τ )= z Y ( τ , n ( b + b ) · τ ) , where b is as in the proof of Lemma C.6 and b = a − · b · t a − = tb · − − dJJ dJdJ b J − dJ − b J . We write b + b = b + b , where b = tb · − − dJJ dJdJ − dJ , b = r ′ · J − J , r ′ = 2 t − ad + bcbd . Since τ − · n ( b ) · τ ∈ P Y , we have z Y ( h , g · d − ) = z Y ( τ , n ( b ) · τ ) . If r ′ = 0, then we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , τ ) = 1. If r ′ = 0, then we have z Y ( τ , n ( b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degenerate symmetric bilinear form associated to r ′ · (cid:18) J − J (cid:19) . Since det q ≡ − J mod ( F × ) and h F ( q ) = ( r ′ · J , J ) F , we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( r ′ · J , J ) F = γ F ( J , ψ ) − · ( r ′ · J , J ) F . Finally assume that ( b, d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b = 1 b − d J · ab − cdJ − ( ad − bc ) J − ( ab − cdJ ) J ( ad − bc ) J ( ad − bc ) J − ( ab − cdJ ) J − ( ad − bc ) J ( ab − cdJ ) J . Hence we have z Y ( h , g · d − ) = z Y ( τ · n ( b ) , n ( b ) · τ ) = z Y ( τ , n ( b + b ) · τ ) . Put l = ab − cdJ − ( b − d J ) t, l ′ = ( ad − bc ) J, r ′′ = l J − l ′ ( b − d J ) = (( a − bt ) − ( c − dt ) J ) Jb − d J . We have b + b = 1 b − d J · l − l ′ − lJ l ′ l ′ − lJ − l ′ lJ , and if l = 0, then we have b + b = a · b · t a , where a = 1 l · l l − J − l ′ J − l ′ , b = 1 b − d J · l ( l J − l ′ ) l − ( l J − l ′ ) lJ − lJ . If l = l ′ = 0, then we have z Y ( τ , n ( b + b ) · τ ) = z Y ( τ , τ ) = 1. If ( l, l ′ ) = (0 , 0) and r ′′ = 0, thenwe have l = 0 and l ′ = 0, so that z Y ( τ , n ( b + b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degeneratesymmetric bilinear form associated to lb − d J · (cid:18) − J (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( lb − d J , J ) F = ( ab − cdJb − d J − t, J ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( ab − cdJb − d J − t, J ) F = γ F ( J , ψ ) − · ( ab − cdJb − d J − t, J ) F . Note that γ F ( J , ψ ) = γ F ( J , ψ ) and ( ab − cdJb − d J − t, J ) F = ( ab − cdJb − d J − t, J ) F since r ′′ = 0 and hence J ∈ ( F × ) . If r ′′ = 0, then we have z Y ( τ , n ( b + b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degeneratesymmetric bilinear form associated to b + b . We have det q ≡ F × ) . Also, we have h F ( q ) = ( lb − d J , ( l J − l ′ ) lb − d J ) F · ( − lJ b − d J , − ( l J − l ′ ) lJ b − d J ) F · ( ( l J − l ′ ) l ( b − d J ) , ( l J − l ′ ) l J ( b − d J ) ) F = ( lb − d J , − ( l J − l ′ )) F · ( − lJ b − d J , − ( l J − l ′ )) F · ( l J − l ′ , l J − l ′ ) F = ( − J , − ( l J − l ′ )) F · ( − , l J − l ′ ) F = ( J , − ( l J − l ′ )) F · ( − , − F = ( J , − r ′′ ) · ( − , − F if l = 0, and h F ( q ) = ( − , − F = ( J , − r ′′ ) · ( − , − F if l = 0. Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · ( J , − r ′′ ) · ( − , − F = ( J , − r ′′ ) . 36 ATSUSHI ICHINO AND KARTIK PRASANNA Note that ( J , − r ′′ ) = ( J , − r ′′ ) since ( J, − r ′′ ) = ( J, l ′ − l J ) = 1. Thus we obtain z Y ( h , g ) = b = d = 0,1 if ( b, d ) = (0 , b − d J = 0, ad + bc − bdt = 0, γ F ( J , ψ ) − · ((2 t − ad + bcbd ) · J , J ) F if ( b, d ) = (0 , b − d J = 0, ad + bc − bdt = 0,1 if ( b, d ) = (0 , b − d J = 0, ab − cdJ − ( b − d J ) t = ad − bc = 0, γ F ( J , ψ ) − · ( ab − cdJb − d J − t, J ) F . if ( b, d ) = (0 , b − d J = 0,( ab − cdJ − ( b − d J ) t, ad − bc ) = (0 , a − bt ) − ( c − dt ) J = 0,( − (( a − bt ) − ( c − dt ) J ) Jb − d J , J ) F if ( b, d ) = (0 , b − d J = 0 , ( a − bt ) − ( c − dt ) J = 0.Now we compute µ ( g ) = z Y ( h gh − , h ) · z Y ( h , g ) − . Recall that u = t and ν = a − b u − c J + d uJ = 0. We have( a − bt ) − ( c − dt ) J = a + b u − c J − d uJ − t ( ab − cdJ ) . First assume that b = d = 0. Then we have c − dt = c and( a − bt ) − ( c − dt ) J = a − c J = ν = 0 . Hence we have µ ( g ) = 1 · ν, J ) F if c = 0, and µ ( g ) = (( a − bt ) − ( c − dt ) J, J ) F · ν, J ) F if c = 0. Next assume that ( b, d ) = (0 , 0) and b − d J = 0. Then we have b = 0, d = 0, ν = a − c J = 0,and J ∈ ( F × ) . Since ( a − bt ) − ( c − dt ) J = a − c J − t ( ab − cdJ ) and(( a − bt ) − ( c − dt ) J ) · bd − ( ad + bc − bdt ) · ( ab − cdJ )= ( a − c J − abt + 2 cdtJ ) · bd − ( a bd + ab c − ab dt − acd J − bc dJ + 2 bcd tJ )= − ab c + acd J = 0 , we have ( a − bt ) − ( c − dt ) J = 0 ⇐⇒ ad + bc − bdt = 0 . If c − dt = 0, then we have ν = a − b u = ( a + bt )( a − bt ) = 0, so that ( a − bt ) − ( c − dt ) J = 0.Hence we have µ ( g ) = 1 · γ F ( J , ψ ) · ((2 t − ad + bcbd ) · J , J ) F = γ F ( J , ψ ) · (( 2 cd − ad + bcbd ) · J , J ) F = γ F ( J , ψ ) · ( ad − bcbd , J ) F = γ F ( J , ψ ) · ( abd − b cd , J ) F = γ F ( J , ψ ) · ( ab − cdJ, J ) F . If c − dt = 0 and ( a − bt ) − ( c − dt ) J = 0, then we have ν − t ( ab − cdJ ) = ( a − bt ) − ( c − dt ) J = 0and hence µ ( g ) = γ F ( J , ψ ) · (2 νt, J ) F · γ F ( J , ψ ) · ( ab − cdJ, J ) F . If c − dt = 0 and ( a − bt ) − ( c − dt ) J = 0, then we have( a − bt ) − ( c − dt ) J = ( ad + bdbd − t ) · ( ab − cdJ )and hence µ ( g ) = (( a − bt ) − ( c − dt ) J, J ) F · γ F ( J , ψ ) · ((2 t − ad + bcbd ) · J , J ) F = γ F ( J , ψ ) · ( − ( ab − cdJ ) · J , J ) F = γ F ( J , ψ ) · ( ab − cdJ, J ) F . Finally assume that ( b, d ) = (0 , 0) and b − d J = 0. Recall that( ab − cdJ − ( b − d J ) t ) − ( ad − bc ) J = (( a − bt ) − ( c − dt ) J ) · ( b − d J ) . If c − dt = 0, then we have ν = a − b u = ( a + bt )( a − bt ) = 0 and( a − bt ) − ( c − dt ) J = ( a − bt ) = 0 . Hence we have µ ( g ) = 1 · ( − (( a − bt ) − ( c − dt ) J ) Jb − d J , J ) F = ( − ( a − bt ) Jb − d J , J ) F = ( − ( b − d J ) J, J ) F . If c − dt = 0 and ( a − bt ) − ( c − dt ) J = 0, then we have ν + 2( b − d J ) u = a + b u − c J − d uJ = 2( ab − cdJ ) t, so that ab − cdJ − ( b − d J ) t = 0 . Hence we have µ ( g ) = γ F ( J , ψ ) · (2 νt, J ) F · γ F ( J , ψ ) · ( ab − cdJb − d J − t, J ) F = ( − , J ) F · ( 2( ab − cdJ ) tb − d J · ν − νu, J ) F = ( − , J ) F · ( ν + 2( b − d J ) ub − d J · ν − νu, J ) F = ( − , J ) F · ( ν b − d J , J ) F = ( − ( b − d J ) , J ) F = ( − ( b − d J ) J, J ) F . 38 ATSUSHI ICHINO AND KARTIK PRASANNA If c − dt = 0 and ( a − bt ) − ( c − dt ) J = 0, then we have µ ( g ) = (( a − bt ) − ( c − dt ) J, J ) F · ( − (( a − bt ) − ( c − dt ) J ) Jb − d J , J ) F = ( − Jb − d J , J ) F = ( − ( b − d J ) J, J ) F . This completes the proof. (cid:3) C.2.2. The case J ∈ ( F × v ) . Choose t ∈ F × such that J = t . We take an isomorphism i : B → M ( F )determined by i (1) = (cid:18) (cid:19) , i ( i ) = (cid:18) u (cid:19) , i ( j ) = (cid:18) t − t (cid:19) , i ( ij ) = (cid:18) − ttu (cid:19) . Then we have e = 12 + 12 t j , e ′ = 12 i − t ij , e ′′ = 12 u i + 12 tu ij , e ∗ = 12 − t j . Put h = t J t J − t − t J t tJ − tJ − tJ ∈ Sp( V ) . Then we have e e e e − e e ′′ J e e ′′ u e e ′ − uJ e e ′ e e ∗ e e ∗ = h e e e e e ∗ e ∗ e ∗ e ∗ , and hence X ′ = X h and Y ′ = Y h . Lemma C.8. Let g i := α − i ∈ GU( V ) with α i = a i + b i i + c i j i + d i ij i ∈ B × i . Then we have µ ( g i ) = if b i = d i = 0 , γ F ( J j , ψ ) · (( a i b i + c i d i J i ) ν i J i , J j ) F if ( b i , d i ) = (0 , and b i − d i J i = 0 , ( − ( b i − d i J i ) ν i J i , J j ) F if ( b i , d i ) = (0 , and b i − d i J i = 0 ,where ν i = ν ( α i ) and { i, j } = { , } .Proof. We only consider the case i = 1; the other case is similar. Note that J ≡ J mod ( F × ) since J ∈ ( F × ) . Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h g h − , h ) = z Y ( h g h − · d − , d · h · d − ) · v Y ( h , ν ) . Since Y ′ g = Y ′ , we have h g h − · d − ∈ P Y and hence z Y ( h g h − · d − , d · h · d − ) = 1 . We have h = m ( a ) · n ( b ) · τ · m ( a ), where a = − t − tJ , b = 12 t · J J , a = − tJ − t , so that x Y ( h ) ≡ − J mod ( F × ) and j Y ( h ) = 2. Hence we have v Y ( h , ν ) = ( − J , ν ) F · γ F ( ν , ψ ) − = ( J , ν ) F . Thus we obtain z Y ( h g h − , h ) = ( J , ν ) F = ( J , ν ) F . Moreover, if b = d = 0, then we have( J , ν ) F = ( J , a − c J ) F = 1 . Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . Next assume that ( b , d ) = (0 , 0) and b − d J = 0. Then we have b = 0 and d = 0. As in theproof of Lemma C.6, we have a b + c d J = 0. We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a and b are as in the proof of Lemma C.6. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , m ( a ) · n ( b ) · τ ) = z Y ( τ , m ( a ) · m ( a ) · n ( b ) · τ ) . We have m ( a ) · m ( a ) · n ( b ) · τ ∈ m ( a ) · n ( b ) · τ ′ · P Y , where a = b d J b d J , b = ( a d − b c ) J b · d tb tb , and τ ′ = − 11 0 − 111 0 11 0 . Hence we have z Y ( τ , m ( a ) · m ( a ) · n ( b ) · τ ) = z Y ( τ , m ( a ) · n ( b ) · τ ′ ) = z Y ( τ · m ( a ) · n ( b ) , τ ′ ) . Since τ · m ( a ) · n ( b ) · τ − ∈ P Y , we have z Y ( τ · m ( a ) · n ( b ) , τ ′ ) = z Y ( τ , τ ′ ) = 1 . On the other hand, since J ∈ ( F × ) and J ∈ ( F × ) , we have γ F ( J , ψ ) = 1 and(( a b + c d J ) J , J ) F = 1 . 40 ATSUSHI ICHINO AND KARTIK PRASANNA Finally assume that ( b , d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b is as in the proof of Lemma C.6. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , n ( b ) · τ )= z Y ( τ · m ( a ) , n ( b ) · m ( a ) − · τ )= z Y ( τ · m ( a ) · n ( b ) · m ( a ) − , τ ) . Since τ · m ( a ) · n ( b ) · m ( a ) − · τ − ∈ P Y , we have z Y ( h , g · d − ) = z Y ( τ , τ ) = 1 . On the other hand, we have( − ( b − d J ) J , J ) F = ( d J − b J , J ) F = 1 . This competes the proof. (cid:3) Lemma C.9. Let g := α ∈ GU( W ) with α = a + b i + c j + d ij ∈ B × . Then we have µ ( g ) = ( ν, J ) F if b = d = 0 , γ F ( J , ψ ) · ( ab − cdJ, J ) F if ( b, d ) = (0 , and b − d J = 0 , ( − ( b − d J ) J, J ) F if ( b, d ) = (0 , and b − d J = 0 , × ( if b + dt = 0 , ( u, J ) F if b + dt = 0 ,where ν = ν ( α ) .Proof. Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h gh − , h ) = z Y ( h gh − · d − , d · h · d − ) · v Y ( h , ν ) . As in the proof of Lemma C.8, we have v Y ( h , ν ) = ( ν, J ) F . We have h gh − = a + ct ( b − dt ) u a + ct − ( b − dt ) uJ a + ct − b − dt a + ct b − dt J b + dt ) a − ct − b + dt ) J a − ct − b + dt ) u a − ct b + dt ) uJ a − ct . If b + dt = 0, then we have h gh − · d − ∈ P Y and hence z Y ( h gh − · d − , d · h · d − ) = 1. If b + dt = 0,then we have h gh − · d − ∈ P Y · τ · n ( b ), where b = a − ct ν ( b + dt ) · − J − u uJ . We have d · h · d − ∈ m ( a ) · n ( ν − · b ) · τ · P Y , where a and b are as in the proof of LemmaC.8. Hence we have z Y ( h gh − · d − , d · h · d − ) = z Y ( τ · n ( b ) , m ( a ) · n ( ν − · b ) · τ )= z Y ( τ · m ( a ) , m ( a ) − · n ( b ) · m ( a ) · n ( ν − · b ) · τ )= z Y ( τ , n ( b ) · τ ) , where b = ν − · b + a − · b · t a − . Put r = a − ctb + dt . We have b = 12 νt · rt − rtJ J J rJ tu − rtu . We write b = b + b , where b = 12 νt · rt − rtJ J J , b = r νuJ · J − . Since τ − · n ( b ) · τ ∈ P Y , we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ) . If r = 0, then we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , τ ) = 1. If r = 0, then we have z Y ( τ , n ( b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degenerate symmetric bilinear form associated to r νuJ · (cid:18) J − (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( rJ νuJ , − r νuJ ) F = ( J , − νru ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( J , − νru ) F = γ F ( J , ψ ) · ( J , νru ) F . Thus we obtain z Y ( h gh − , h ) = ( ν, J ) F if b + dt = 0,( ν, J ) F if b + dt = 0, a − ct = 0, γ F ( J , ψ ) · (2 u · a − ctb + dt , J ) F if b + dt = 0, a − ct = 0.Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . 42 ATSUSHI ICHINO AND KARTIK PRASANNA Next assume that ( b, d ) = (0 , 0) and b − d J = 0. Then we have b = 0, d = 0, and ν = a − c J = 0.Since ( ad + bc ) · ( ab − cdJ ) = a bd − acd J + ab c − bc dJ = a bd − bc dJ = νbd = 0 , we have ad + bc = 0 and ab − cdJ = 0. We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a and b areas in the proof of Lemma C.7. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , m ( a ) · n ( b ) · τ ) = z Y ( τ · m ( a ) , n ( b ) · τ ) , where a is as in the proof of Lemma C.8 and a = a · a = b b − ( b + dt ) tJ − ( b + dt ) t . If b + dt = 0, then we have τ · m ( a ) · τ − ∈ P Y and hence z Y ( τ · m ( a ) , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ) = γ F ( 12 ψ ◦ q ) , where q is a non-degenerate symmetric bilinear form associated to ad + bcbd · (cid:18) − J J (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( − ad + bcbd · J , ad + bcbd · J ) F = ( J , ad + bcbd ) F = ( J , νab − cdJ ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( J , νab − cdJ ) F = γ F ( J , ψ ) − · ( J , ( ab − cdJ ) ν ) F . If b + dt = 0 ,then we have m ( a ) · n ( b ) · τ ∈ m ( a ) · n ( b ) · τ ′′ · P Y , where a = − − ( b + dt ) tbJ ( b + dt ) tb , b = ( ad + bc ) b ( b + dt ) d · − J 01 0 , and τ ′′ = − . Since τ · m ( a ) · n ( b ) · τ − ∈ P Y , we have z Y ( τ · m ( a ) , n ( b ) · τ ) = z Y ( τ , m ( a ) · n ( b ) · τ ′′ ) = z Y ( τ , τ ′′ ) = 1 . Finally assume that ( b, d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b isas in the proof of Lemma C.7. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ) , where b = a · b · t a = 1 b − d J · ab − cdJ − ( a − ct )( b + dt ) t − ( ab − cdJ ) J ( a − ct )( b + dt ) t ( a − ct )( b + dt ) t − a − ct )( b + dt ) J − ( a − ct )( b + dt ) t a − ct )( b + dt ) J . We write b = b + b , where b = 1 b − d J · ab − cdJ − ( a − ct )( b + dt ) t − ( ab − cdJ ) J ( a − ct )( b + dt ) t ( a − ct )( b + dt ) t − ( a − ct )( b + dt ) t , b = 2( a − ct ) b − dt · − J J . Since τ · b · τ − ∈ P Y , we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ). If a − ct = 0, then we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , τ ) = 1. If a − ct = 0, then we have z Y ( τ , n ( b ) · τ ) = γ F ( ψ ◦ q ), where q is a non-degenerate symmetric bilinear form associated to2( a − ct ) b − dt · (cid:18) − J J (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( − a − ct ) b − dt · J , a − ct ) b − dt · J ) F = ( J , a − ct ) b − dt ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( J , a − ct ) b − dt ) F = γ F ( J , ψ ) − · ( J , a − ct ) b − dt ) F . Thus we obtain z Y ( h , g ) = b = d = 0, γ F ( J , ψ ) − · ( J , ( ab − cdJ ) ν ) F if ( b, d ) = (0 , b − d J = 0, b + dt = 0,1 if ( b, d ) = (0 , b − d J = 0, b + dt = 0,1 if ( b, d ) = (0 , b − d J = 0, a − ct = 0, γ F ( J , ψ ) − · ( J , a − ct ) b − dt ) F if ( b, d ) = (0 , b − d J = 0, a + ct = 0.Now we compute µ ( g ) = z Y ( h gh − , h ) · z Y ( h , g ) − . Recall that J = t and ν = a − b u − c J + d uJ = 0. First assume that b = d = 0. Then we have µ ( g ) = ( ν, J ) F · ν, J ) F . Next assume that ( b, d ) = (0 , 0) and b − d J = 0. Then we have ν = a − c J = ( a + ct )( a − ct ) = 0.Since b − d J = ( b + dt )( b − dt ), we have b + dt = 0 ⇐⇒ b − dt = 0 . 44 ATSUSHI ICHINO AND KARTIK PRASANNA If b + dt = 0, then we have µ ( g ) = ( ν, J ) F · γ F ( J , ψ ) · ( J , ( ab − cdJ ) ν ) F = γ F ( J , ψ ) · ( J , ab − cdJ ) F . If b + dt = 0, then we have( a − ct )( b + dt ) = 2( a − ct ) dt = 2( adt − cdJ ) = 2( ab − cdJ ) . Hence we have µ ( g ) = γ F ( J , ψ ) · (2 u · a − ctb + dt , J ) F · γ F ( J , ψ ) · (2( a − ct )( b + dt ) , J ) F · ( u, J ) F = γ F ( J , ψ ) · ( ab − cdJ, J ) F · ( u, J ) F . Finally assume that ( b, d ) = (0 , 0) and b − d J = 0. Then we have b + dt = 0. If a − ct = 0, then wehave ν = − b u + d uJ and hence µ ( g ) = ( ν, J ) F · − b + d J, J ) F · ( u, J ) F = ( − ( b − d J ) J, J ) F · ( u, J ) F . If a − ct = 0, then we have µ ( g ) = γ F ( J , ψ ) · (2 u · a − ctb + dt , J ) F · γ F ( J , ψ ) · ( J , a − ct ) b − dt ) F = γ F ( J , ψ ) · ( u ( b + dt )( b − dt ) , J ) F = ( − , J ) F · ( b − d J, J ) F · ( u, J ) F = ( − ( b − d J ) J, J ) F · ( u, J ) F . This completes the proof. (cid:3) C.3. The case J i ∈ ( F × v ) . We only consider the case i = 1; the other case is similar. Choose t ∈ F × such that J = t . We take an isomorphism i : B −→ M ( F )of F -algebras determined by i (1) = (cid:18) (cid:19) , i ( i ) = (cid:18) u (cid:19) , i ( j ) = (cid:18) t − t (cid:19) , i ( ij ) = (cid:18) − t tu (cid:19) . Note that i ( α ∗ ) = i ( α ) ∗ for α ∈ B . Let v := 12 e + 12 t e , v ∗ := e ∗ + t e ∗ = 1 u e i − tu e i . Then we have V = v B + v ∗ B and h v , v i = h v ∗ , v ∗ i = 0 , h v , v ∗ i = 1 . Moreover, we see that (cid:2) α · v α · v ∗ (cid:3) = (cid:2) v v ∗ (cid:3) · i ( α )for α ∈ B , and (cid:2) α · v α · v ∗ (cid:3) = (cid:2) v v ∗ (cid:3) · ( α + βt j )for α = α + β j ∈ B with α, β ∈ E . We regard V ′ := V as a left B -space by putting α · x ′ := ( x · α ∗ ) ′ for α ∈ B and x ′ ∈ V ′ . Here we let x ′ denote the element in V ′ corresponding to x ∈ V . We letGL( V ′ ) act on V ′ on the right. We define a skew-hermitian form h· , ·i ′ : V ′ × V ′ −→ B by h x ′ , y ′ i ′ := h x, y i . Note that h α x ′ , β y ′ i ′ = α h x ′ , y ′ i ′ β ∗ for α , β ∈ B . For x ′ ∈ V ′ and g ∈ GL( V ), put x ′ · g := ( g − · x ) ′ . Then we have an isomorphism GL( V ) −→ GL( V ′ ) ,g [ x ′ x ′ · g ]so that we may identify GU( V ) with GU( V ′ ) via this isomorphism. Let V ′ = X ′ + Y ′ be a completepolarization given by X ′ = B · v ′ , Y ′ = B · ( v ∗ ) ′ . Note that (cid:20) v ′ · α ( v ∗ ) ′ · α (cid:21) = t i ( α ) − · (cid:20) v ′ ( v ∗ ) ′ (cid:21) for α ∈ B . We may identify V ′ with the space of row vectors B so that h x ′ , y ′ i ′ = x y ∗ − x y ∗ for x ′ = ( x , x ), y ′ = ( y , y ) ∈ V ′ . Then we may writeGU( V ′ ) = (cid:26) g ∈ GL ( B ) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) − (cid:19) t g ∗ = ν ( g ) · (cid:18) − (cid:19) (cid:27) . Similarly, we have a right B -space W ′ := W with a hermitian form h· , ·i ′ : W ′ × W ′ −→ B. We let GL( W ′ ) act on W ′ on the left. Now we consider an F -space V ′ := W ′ ⊗ B V ′ with a symplectic form hh· , ·ii ′ := 12 tr B/F ( h· , ·i ′ ⊗ h· , ·i ′∗ ) . We let GL( V ′ ) act on V ′ on the right. For x = x ⊗ y ∈ V and g ∈ GL( V ), put x ′ := y ′ ⊗ x ′ ∈ V ′ and x ′ · g := ( x · g ) ′ . 46 ATSUSHI ICHINO AND KARTIK PRASANNA Lemma C.10. We have an isomorphism GSp( V ) −→ GSp( V ′ ) . g [ x ′ x ′ · g ] Moreover, this isomorphism induces a commutative diagram GU( V ) × GU( W ) / / (cid:15) (cid:15) GSp( V ) (cid:15) (cid:15) GU( W ′ ) × GU( V ′ ) / / GSp( V ′ ) . Proof. For x , x ∈ V and y , y ∈ W , we have hh y ′ ⊗ x ′ , y ′ ⊗ x ′ ii ′ = 12 tr B/F ( h y ′ , y ′ i ′ · h x ′ , x ′ i ′∗ )= 12 tr B/F ( h y , y i · h x , x i ∗ )= 12 tr B/F ( h x , x i · h y , y i ∗ )= hh x ⊗ y , x ⊗ y ii . Also, for g = ( g, h ) ∈ GL( V ) × GL( W ) and x = x ⊗ y ∈ V , we have x ′ · g = (( x ⊗ y ) · ( g, h )) ′ = ( g − x ⊗ yh ) ′ = ( yh ) ′ ⊗ ( g − x ) ′ = h − y ′ ⊗ x ′ g. This completes the proof. (cid:3) Thus we may identify GSp( V ) with GSp( V ′ ) and GU( V ) × GU( W ) with GU( W ′ ) × GU( V ′ ) respec-tively.Let V ′ = X ′ + Y ′ be a complete polarization given by X ′ = W ′ ⊗ B X ′ , Y ′ = W ′ ⊗ B Y ′ . Put s ′ ( g ) := γ j ( g ) for g ∈ GU( V ′ ) , where γ = ( B and B are split, − B and B are ramified,and j ( g ) = ( g = ( ∗ ∗ ∗ ),1 othersiwe. Lemma C.11. We have z Y ′ ( g, g ′ ) = s ′ ( gg ′ ) · s ′ ( g ) − · s ′ ( g ′ ) − for g, g ′ ∈ GU( V ) .Proof. The proof is similar to that of Lemma C.3. If B is ramified, then we have(C.2) z Y ′ ( g, g ′ ) = s ′ ( gg ′ ) · s ′ ( g ) − · s ′ ( g ′ ) − for g, g ′ ∈ U( V ) by [39, Theorem 3.1, case 2 + ]. If B is split, then we see that (C.2) also holds byusing Morita theory as in § C.2 and [39, Theorem 3.1, case 1 − ]. Let g, g ′ ∈ GU( V ) . For ν ∈ F × , put d ( ν ) = (cid:18) ν (cid:19) ∈ GU( V ) . We write g = h · d ( ν ) , g ′ = h ′ · d ( ν ′ )with h, h ′ ∈ U( V ) and ν, ν ′ ∈ F × . Then we have z Y ′ ( g, g ′ ) = z Y ′ ( h, h ′′ ) · v Y ′ ( h ′ , ν ) , where h ′′ = d ( ν ) · h ′ · d ( ν ) − . By (C.2), we have z Y ′ ( h, h ′′ ) = s ′ ( hh ′′ ) · s ′ ( h ) − · s ′ ( h ′′ ) − . We have s ′ ( h ) = s ′ ( g ), and since j ( h ′′ ) = j ( h ′ ), we have s ′ ( h ′′ ) = s ′ ( h ′ ) = s ′ ( g ′ ). Moreover, since gg ′ = hh ′′ · d ( νν ′ ), we have s ′ ( hh ′′ ) = s ′ ( gg ′ ). Thus we obtain z Y ′ ( h, h ′′ ) = s ′ ( gg ′ ) · s ′ ( g ) − · s ′ ( g ′ ) − . By Lemma B.2, we have v Y ′ ( h ′ , ν ) = ( x Y ′ ( h ′ ) , ν ) F · γ F ( ν, ψ ) − j Y ′ ( h ′ ) , where x Y ′ and j Y ′ are as in § B.1 with respect to the complete polarization V ′ = X ′ + Y ′ . Since thedeterminant over F of the automorphism x x · α of B is ν ( α ) for α ∈ B × , we have x Y ′ ( h ′ ) ≡ F × ) . Noting that either c = 0 or c ∈ B × for h ′ = (cid:0) a bc d (cid:1) , one can see that j Y ′ ( h ′ ) = 4 · j ( h ′ ).Hence we have v Y ′ ( h ′ , ν ) = 1 . This completes the proof. (cid:3) Lemma C.12. We have z Y ′ ( h, h ′ ) = 1 for h, h ′ ∈ GU( W ) .Proof. The proof is similar to that of Lemma C.4.For g, g ′ ∈ GU( W ), we have z Y ′ ( g, g ′ ) = z Y ′ ( h, h ′′ ) · v Y ′ ( h ′ , ν ) , where h = g · d Y ′ ( ν ) − , h ′ = g ′ · d Y ′ ( ν ′ ) − , h ′′ = d Y ′ ( ν ) · h ′ · d Y ′ ( ν ) − ,ν = ν ( g ) , ν ′ = ν ( g ′ ) . We have h, h ′ ∈ P Y ′ and z Y ′ ( h, h ′′ ) = 1. Since the determinant over F of the automorphism x α · x of B is ν ( α ) for α ∈ B × , we have x Y ′ ( h ′ ) ≡ F × ) , so that v Y ′ ( h ′ , ν ) = 1 by Lemma B.2. Thiscompletes the proof. (cid:3) Lemma C.13. We have z Y ′ ( g, h ) = z Y ′ ( h, g ) = 1 for g ∈ GU( V ) and h ∈ GU( W ) . 48 ATSUSHI ICHINO AND KARTIK PRASANNA Proof. The proof is similar to that of Lemma C.5.For g ∈ GU( V ) and h ∈ GU( W ), we have z Y ′ ( g, h ) = z Y ′ ( g ′ , h ′′ ) · v Y ′ ( h ′ , ν ) , z Y ′ ( h, g ) = z Y ′ ( h ′ , g ′′ ) · v Y ′ ( g ′ , ν ′ ) , where g ′ = g · d ( ν ) − , g ′′ = d ( ν ′ ) · g ′ · d ( ν ′ ) − , ν = ν ( g ) ,h ′ = h · d Y ′ ( ν ′ ) − , h ′′ = d Y ′ ( ν ) · h ′ · d Y ′ ( ν ) − , ν ′ = ν ( h ) . Since h ′ , h ′′ ∈ P Y ′ , we have z Y ′ ( g ′ , h ′′ ) = z Y ′ ( h ′ , g ′′ ) = 1. As in the proof of Lemma C.12, we have v Y ′ ( h ′ , ν ) = 1. As in the proof of Lemma C.11, we have v Y ′ ( g ′ , ν ′ ) = 1. This completes the proof. (cid:3) We define a map s ′ : GU( V ) × GU( W ) → C by s ′ ( g ) = γ j ( g ) for g = ( g, h ) ∈ GU( V ) × GU( W ). By Lemmas C.11, C.12, C.13, we see that z Y ′ ( g , g ′ ) = s ′ ( gg ′ ) · s ′ ( g ) − · s ′ ( g ′ ) − for g , g ′ ∈ GU( V ) × GU( W ).Recall that we may identify V with V ′ , and we have two complete polarizations V = X + Y = X ′ + Y ′ ,where X = F e + F e + F e + F e , Y = F e ∗ + F e ∗ + F e ∗ + F e ∗ , X ′ = v · B, Y ′ = v ∗ · B. Put h = 12 12 t 12 12 t − t − t t t − t − t ∈ Sp( V ) . Then we have v t vj − u vi − tuJ vijv ∗ − tJ v ∗ jv ∗ i − t v ∗ ij = h e e e e e ∗ e ∗ e ∗ e ∗ , and hence X ′ = X h and Y ′ = Y h . Put s ( g ) := s ′ ( g ) · µ ( g ) , where µ ( g ) := z Y ( h gh − , h ) · z Y ( h , g ) − for g ∈ GU( V ) × GU( W ). Then we have z Y ( g , g ′ ) = s ( gg ′ ) · s ( g ) − · s ( g ′ ) − for g , g ′ ∈ GU( V ) × GU( W ). Lemma C.14. Let g := α − ∈ GU( V ) with α = a + b i + c j + d ij ∈ B × . Then we have µ ( g ) = if b = d = 0 , γ F ( J , ψ ) · (( a b + c d J ) ν J , J ) F if ( b , d ) = (0 , and b − d J = 0 , ( − ( b − d J ) ν J , J ) F if ( b , d ) = (0 , and b − d J = 0 , × ( if b − d t = 0 , ( u, J ) F if b − d t = 0 ,where ν = ν ( α ) .Proof. Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h g h − , h ) = z Y ( h g h − · d − , d · h · d − ) · v Y ( h , ν ) . We have h = τ · m ( a ), where a = 12 12 t 12 12 t − t − t , so that x Y ( h ) ≡ − F × ) and j Y ( h ) = 2. Hence we have v Y ( h , ν ) = ( − , ν ) F · γ F ( ν , ψ ) − = 1 . We have h g h − = a + c t ( b + d t ) u a + c t − ( b + d t ) uJ a + c t − b + d t a + c t ( b + d t )2 J b − d t ) a − c t − b − d t ) J a − c t − b − d t ) u a − c t b − d t ) uJ a − c t . If b − d t = 0, then we have h g h − · d − ∈ P Y and hence z Y ( h g h − · d − , d · h · d − ) = 1. If b − d t = 0, then we have h g h − · d − ∈ P Y · τ · n ( b ), where b = a − c t ν ( b − d t ) · − J − u uJ . Since d · h · d − ∈ τ · P Y , we have z Y ( h g h − · d − , d · h · d − ) = z Y ( τ · n ( b ) , τ ) . If a − c t = 0, then we have z Y ( τ · n ( b ) , τ ) = z Y ( τ , τ ) = 1. If a − c t = 0, then we have z Y ( τ · n ( b ) , τ ) = γ F ( ψ ◦ q ), where q is a non-degenerate symmetric bilinear form associated to b = a − c t ν ( b − d t ) · (cid:18) − u uJ (cid:19) . 50 ATSUSHI ICHINO AND KARTIK PRASANNA We have det q ≡ − J mod ( F × ) and h F ( q ) = ( − a − c t ν u ( b − d t ) , a − c t ν uJ ( b − d t ) ) F = ( − a − c t ν u ( b − d t ) , J ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( − a − c t ν u ( b − d t ) , J ) F = γ F ( J , ψ ) · ( a − c t ν u ( b − d t ) , J ) F . Thus we obtain z Y ( h g h − , h ) = b − d t = 0,1 if b − d t = 0, a − c t = 0, γ F ( J , ψ ) · ( a − c t ν u ( b − d t ) , J ) F if b − d t = 0, a − c t = 0.Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . Next assume that ( b , d ) = (0 , 0) and b − d J = 0. Then we have b = 0 and d = 0. As in theproof of Lemma C.6, we have( a d − b c ) · ( a b + c d J ) = ν b d = 0 . We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a and b are as in the proof of Lemma C.6. Hencewe have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , m ( a ) · n ( b ) · τ ) = z Y ( τ · m ( a ) , n ( b ) · τ ) , where a = a · a = b + d t tb + d t t b − d t − t b − d t − t . If b − d t = 0, then we have τ · m ( a ) · τ − ∈ P Y and hence z Y ( τ · m ( a ) , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ) = γ F ( 12 ψ ◦ q ) , where q is a non-degenerate symmetric bilinear form associated to a d − b c b d · (cid:18) J − J (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( a d − b c b d · J , − a d − b c b d · J ) F = ( a d − b c b d , J ) F = ( ν a b + c d J , J ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( ν a b + c d J , J ) F = γ F ( J , ψ ) − · (( a b + c d J ) ν , J ) F . If b − d t = 0, then we have m ( a ) · n ( b ) · τ ∈ n ( b ) · τ ′′ · P Y , where b = b b − d t ) · − J and τ ′′ is as in the proof of Lemma C.9. Since τ · n ( b ) · τ − ∈ P Y , we have z Y ( τ · m ( a ) , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ′′ ) = z Y ( τ , τ ′′ ) = 1 . Finally assume that ( b , d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b is as in the proof of Lemma C.6. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ) , where b = a · b · t a = 1 b − d J · ( a + c t )( b + d t )2 − ( a + c t )( b + d t ) J a − c t )( b − d t ) − a − c t )( b − d t ) J . We write b = b + b , where b = a + c t b − d t ) · − J , b = 2( a − c t ) b + d t · − J . Since τ · b · τ − ∈ P Y , we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , n ( b ) · τ ). If a − c t = 0, then we have z Y ( τ , n ( b ) · τ ) = z Y ( τ , τ ) = 1. If a − c t = 0, then we have z Y ( τ , n ( b ) · τ ) = γ F ( ψ ◦ q ),where q is a non-degenerate symmetric bilinear form associated to2( a − c t ) b + d t · (cid:18) − J (cid:19) . We have det q ≡ − J mod ( F × ) and h F ( q ) = ( 2( a − c t ) b + d t , − a − c t ) b + d t · J ) F = ( 2( a − c t ) b + d t , J ) F . Hence we have γ F ( 12 ψ ◦ q ) = γ F ( 12 ψ ) · γ F ( − J , ψ ) · ( 2( a − c t ) b + d t , J ) F = γ F ( J , ψ ) − · ( 2( a − c t ) b + d t , J ) F . Thus we obtain z Y ( h , g ) = b = d = 0, γ F ( J , ψ ) − · (( a b + c d J ) ν , J ) F if ( b , d ) = (0 , b − d J = 0, b − d t = 0,1 if ( b , d ) = (0 , b − d J = 0, b − d t = 0,1 if ( b , d ) = (0 , b − d J = 0, a − c t = 0, γ F ( J , ψ ) − · ( a − c t ) b + d t , J ) F if ( b , d ) = (0 , b − d J = 0, a − c t = 0. 52 ATSUSHI ICHINO AND KARTIK PRASANNA Now we compute µ ( g ) = z Y ( h g h − , h ) · z Y ( h , g ) − . Recall that J = t and ν = a − b u − c J + d uJ = 0. First assume that b = d = 0. Then we have µ ( g ) = 1 · . Next assume that ( b , d ) = (0 , 0) and b − d J = 0. Then we have ν = a − c J = ( a + c t )( a − c t ) = 0. Since b − d J = ( b + d t )( b − d t ), we have b − d t = 0 ⇐⇒ b + d t = 0 . If b − d t = 0, then we have µ ( g ) = 1 · γ F ( J , ψ ) · (( a b + c d J ) ν , J ) F = γ F ( J , ψ ) · (( a b + c d J ) ν J , J ) F . If b − d t = 0, then we have( a − c t )( b − d t ) = − a − c t ) d t = 2( − a d t + c d J ) = 2( a b + c d J ) . Hence we have µ ( g ) = γ F ( J , ψ ) · ( a − c t ν u ( b − d t ) , J ) F · γ F ( J , ψ ) · (2 ν ( a − c t )( b − d t ) , J ) F · ( u, J ) F = γ F ( J , ψ ) · (( a b + c d J ) ν , J ) F · ( u, J ) F = γ F ( J , ψ ) · (( a b + c d J ) ν J , J ) F · ( u, J ) F . Finally assume that ( b , d ) = (0 , 0) and b − d J = 0. Then we have b − d t = 0. If a − c t = 0,then we have µ ( g ) = 1 · . On the other hand, since ν = − b u + d uJ , we have( − ( b − d J ) ν J , J ) F · ( u, J ) F = ( ν u · ν J , J ) F · ( u, J ) F = ( ν J , J ) F = 1 . If a − c t = 0, then we have µ ( g ) = γ F ( J , ψ ) · ( a − c t ν u ( b − d t ) , J ) F · γ F ( J , ψ ) · ( 2( a − c t ) b + d t , J ) F = γ F ( J , ψ ) · ( ν u ( b + d t )( b − d t ) , J ) F = ( − , J ) · (( b − d J ) ν , J ) F · ( u, J ) F = ( − ( b − d J ) ν , J ) F · ( u, J ) F = ( − ( b − d J ) ν J , J ) F · ( u, J ) F . This completes the proof. (cid:3) Lemma C.15. Let g := α − ∈ GU( V ) with α = a + b i + c j + d ij ∈ B × . Then we have µ ( g ) = if b = d = 0 , γ F ( J , ψ ) · (( a b + c d J ) ν J , J ) F if ( b , d ) = (0 , and b − d J = 0 , ( − ( b − d J ) ν J , J ) F if ( b , d ) = (0 , and b − d J = 0 ,where ν = ν ( α ) . Proof. Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h g h − , h ) = z Y ( h g h − · d − , d · h · d − ) · v Y ( h , ν ) . Since Y ′ g = Y ′ , we have h g h − · d − ∈ P Y and hence z Y ( h g h − · d − , d · h · d − ) = 1 . As in the proof of Lemma C.14, we have v Y ( h , ν ) = 1. Thus we obtain z Y ( h g h − , h ) = 1 . Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . Next assume that ( b , d ) = (0 , 0) and b − d J = 0. Then we have b = 0 and d = 0. As in theproof of Lemma C.6, we have a b + c d J = 0. We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a = d d b b , b = a d − b c b d · J − J . Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , m ( a ) · n ( b ) · τ ) = z Y ( τ , m ( a ) · m ( a ) · n ( b ) · τ ) , where a is as in the proof of Lemma C.14. We have m ( a ) · m ( a ) · n ( b ) · τ ∈ m ( a ) · n ( b ) · τ ′ · P Y , where a = d b d b , b = ( a d − b c ) J b d · , and τ ′ is as in the proof of Lemma C.8. Hence we have z Y ( τ , m ( a ) · m ( a ) · n ( b ) · τ ) = z Y ( τ , m ( a ) · n ( b ) · τ ′ ) = z Y ( τ · m ( a ) · n ( b ) , τ ′ ) . Since τ · m ( a ) · n ( b ) · τ − ∈ P Y , we have z Y ( τ · m ( a ) · n ( b ) , τ ′ ) = z Y ( τ , τ ′ ) = 1 . On the other hand, since J ∈ ( F × ) , we have γ F ( J , ψ ) = 1 and(( a b + c d J ) ν J , J ) F = 1 . Finally assume that ( b , d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b = 1 b − d J · a b + c d J ( a d + b c ) J − ( a b + c d J ) J − ( a d + b c ) J ( a d + b c ) J ( a b + c d J ) J − ( a d + b c ) J − ( a b + c d J ) J . 54 ATSUSHI ICHINO AND KARTIK PRASANNA Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , n ( b ) · τ )= z Y ( τ · m ( a ) , n ( b ) · m ( a ) − · τ )= z Y ( τ · m ( a ) · n ( b ) · m ( a ) − , τ ) . Since τ · m ( a ) · n ( b ) · m ( a ) − · τ − ∈ P Y , we have z Y ( h , g · d − ) = z Y ( τ , τ ) = 1 . On the other hand, since J ∈ ( F × ) , we have( − ( b − d J ) ν J , J ) F = 1 . This completes the proof. (cid:3) Lemma C.16. Let g := α ∈ GU( W ) with α = a + b i + c j + d ij ∈ B × . Then, for any i ∈ { , } , wehave µ ( g ) = ( ν, J i ) F if b = d = 0 , γ F ( J i , ψ ) · ( ab − cdJ, J i ) F if ( b, d ) = (0 , and b − d J = 0 , ( − ( b − d J ) J, J i ) F if ( b, d ) = (0 , and b − d J = 0 ,where ν = ν ( α ) .Proof. Put d := d Y ( ν ) ∈ GSp( V ). We have z Y ( h gh − , h ) = z Y ( h gh − · d − , d · h · d − ) · v Y ( h , ν ) . Since Y ′ g = Y ′ , we have h gh − · d − ∈ P Y and hence z Y ( h gh − · d − , d · h · d − ) = 1 . As in the proof of Lemma C.14, we have v Y ( h , ν ) = 1. Thus we obtain z Y ( h gh − , h ) = 1 . Now we compute z Y ( h , g ). We have z Y ( h , g ) = z Y ( h , g · d − ) . First assume that b = d = 0. Then we have g · d − ∈ P Y and hence z Y ( h , g · d − ) = 1 . On the other hand, since J ∈ ( F × ) , we have ( ν, J ) F = 1 and( ν, J ) F = ( ν, J ) F = ( a − c J, J ) F = 1 . Next assume that ( b, d ) = (0 , 0) and b − d J = 0. Then we have b = 0 and d = 0. As in the proof ofLemma C.9, we have ab − cdJ = 0. We have g · d − ∈ m ( a ) · n ( b ) · τ · P Y , where a and b are asin the proof of Lemma C.7. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , m ( a ) · n ( b ) · τ ) = z Y ( τ , m ( a ) · m ( a ) · n ( b ) · τ ) , where a is as in the proof of Lemma C.14. We have m ( a ) · m ( a ) · n ( b ) · τ ∈ m ( a ) · n ( b ) · τ ′ · P Y , where a = bt − dJ btdJ , b = − ( ad + bc ) J bd · , and τ ′ is as in the proof of Lemma C.8. Hence we have z Y ( τ , m ( a ) · m ( a ) · n ( b ) · τ ) = z Y ( τ , m ( a ) · n ( b ) · τ ′ ) = z Y ( τ · m ( a ) · n ( b ) , τ ′ ) . Since τ · m ( a ) · n ( b ) · τ − ∈ P Y , we have z Y ( τ · m ( a ) · n ( b ) , τ ′ ) = z Y ( τ , τ ′ ) = 1 . On the other hand, since J ∈ ( F × ) and J ∈ ( F × ) , we have γ F ( J , ψ ) = γ F ( J , ψ ) = 1 and( ab − cdJ, J ) F = ( ab − cdJ, J ) F = 1 . Finally assume that ( b, d ) = (0 , 0) and b − d J = 0. We have g · d − ∈ n ( b ) · τ · P Y , where b isas in the proof of Lemma C.7. Hence we have z Y ( h , g · d − ) = z Y ( τ · m ( a ) , n ( b ) · τ )= z Y ( τ · m ( a ) , n ( b ) · m ( a ) − · τ )= z Y ( τ · m ( a ) · n ( b ) · m ( a ) − , τ ) . Since τ · m ( a ) · n ( b ) · m ( a ) − · τ − ∈ P Y , we have z Y ( h , g · d − ) = z Y ( τ , τ ) = 1 . On the other hand, since J ∈ ( F × ) , we have ( − ( b − d J ) J, J ) F = 1 and( − ( b − d J ) J, J ) F = ( − ( b − d J ) J, J ) F = ( b − d J, J ) F = 1 . This completes the proof. (cid:3) C.4. The product formula. Suppose that F is a number field. First we fix quaternion algebras B and B over F . Next we fix a quadratic extension E of F such that E embeds into B and B . Let B be the quaternion algebra over F which is the product of B and B in the Brauer group. Then E also embeds into B .Fix a finite set Σ of places of F containingΣ ∞ ∪ Σ ∪ Σ E ∪ Σ B ∪ Σ B ∪ Σ B . Here Σ ∞ is the set of archimedean places of F , Σ is the set of places of F lying above 2, and Σ • isthe set of places v of F such that • v is ramified over F v for • = E , B , B , B .We write B i = E + E j i . Put J i = j i and J = J J . We may write B = E + E j such that j = J .Then, for each place v of F , we have • J ∈ N E v /F v ( E × v ) if v / ∈ Σ B , • J ∈ N E v /F v ( E × v ) if v / ∈ Σ B , • J ∈ N E v /F v ( E × v ) if v / ∈ Σ B .By using the weak approximation theorem and replacing j i by α i j i with some α i ∈ E × if necessary,we may assume that J ∈ ( F × v ) or J ∈ ( F × v ) or J ∈ ( F × v ) for all v ∈ Σ. Lemma C.17. We have u ∈ ( F × v ) or J ∈ ( F × v ) or J ∈ ( F × v ) or J ∈ ( F × v ) for all v / ∈ Σ . 56 ATSUSHI ICHINO AND KARTIK PRASANNA Proof. Let v / ∈ Σ. We may assume that v is inert in E . Assume that J i / ∈ ( F × v ) for i = 1 , 2. Since J i ∈ N E v /F v ( E × v ), we have J i ∈ ε · ( F × v ) for i = 1 , 2, where ε ∈ o × F v but ε / ∈ ( F × v ) . Hence we have J = J J ∈ ( F × v ) . This yields the lemma. (cid:3) Thus, for each place v of F , we can define a map s v : GU( V v ) × GU( W v ) −→ C by s v := s ′ v · µ v , where s ′ v and µ v are as in §§ C.2, C.3. Here, for • = u , J , J , J with • ∈ ( F × v ) , wehave chosen t ∈ F × v such that • = t . Recall that z Y v ( g , g ′ ) = s v ( gg ′ ) · s v ( g ) − · s v ( g ′ ) − for g , g ′ ∈ GU( V v ) × GU( W v ). Proposition C.18. (i) Let g i := α − i ∈ GU( V v ) with α i = a i + b i i + c i j i + d i ij i ∈ B × i,v . Then wehave s v ( g i ) = if b i = d i = 0 , γ F v ( J j , ψ v ) · (( a i b i + c i d i J i ) ν i J i , J j ) F v if ( b i , d i ) = (0 , and b i − d i J i = 0 , ( − ( b i − d i J i ) ν i J i , J j ) F v if ( b i , d i ) = (0 , and b i − d i J i = 0 ,where ν i = ν ( α i ) and { i, j } = { , } . (ii) Let g := α ∈ GU( W v ) with α = a + b i + c j + d ij ∈ B × v . Then we have s v ( g ) = ( ν, J ) F v if b = d = 0 , γ F v ( J , ψ v ) · ( ab − cdJ, J ) F v if ( b, d ) = (0 , and b − d J = 0 , ( − ( b − d J ) J, J ) F v if ( b, d ) = (0 , and b − d J = 0 ,where ν = ν ( α ) .Proof. If u ∈ ( F × v ) , then B i,v is split and the assertion follows from Lemmas C.6 and C.7.Assume that J ∈ ( F × v ) . Let i : B v → M ( F v ) be the isomorphism as in § C.2. Since i ( α ) = (cid:18) a + ct b − dtu ( b + dt ) a − ct (cid:19) , we have j ( g ) = ( b + dt = 0,1 if b + dt = 0.Since ( u, J ) F v = ( B ,v is split, − B ,v is ramified,the assertion follows from Lemmas C.8 and C.9.Assume that J i ∈ ( F × v ) . We only consider the case i = 1; the other case is similar. Let i : B ,v → M ( F v ) be the isomorphism as in § C.3. Since t i ( α ) = (cid:18) a + c t u ( b + d t )2( b − d t ) a − c t (cid:19) , we have j ( g ) = ( b − d t = 0,1 if b − d t = 0. Also, we have j ( g ) = 0. Since ( u, J ) F v = ( B v is split, − B v is ramified,the assertion follows from Lemmas C.14, C.15, and C.16. (cid:3) Recall that, for almost all v , we have a maximal compact subgroup K v of Sp( V v ) and a map s Y v : K v → C such that z Y v ( k, k ′ ) = s Y v ( kk ′ ) · s Y v ( k ) − · s Y v ( k ′ ) − for k, k ′ ∈ K v . Put K v := G(U( V v ) × U( W v )) ∩ K v . Then K v is a maximal compact subgroup of G(U( V v ) × U( W v )) for almost all v . Lemma C.19. We have s v | K v = s Y v | K v for almost all v .Proof. Recall that s v ( g ) = s ′ v ( g ) · µ v ( g ) for g ∈ GU( V v ) × GU( W v ), where s ′ v : GU( V v ) × GU( W v ) −→ C is the map as in §§ C.2, C.3 and µ v ( g ) = z Y v ( h gh − , h ) · z Y v ( h , g ) − for g ∈ GSp( V v ) with some h ∈ Sp( V v ) such that X ′ v = X v h and Y ′ v = Y v h . By the uniqueness ofthe splitting, we have s Y v = s Y ′ v · µ v | K v for almost all v . On the other hand, by definition, one can see that s ′ v | K v = s Y ′ v | K v for almost all v . This yields the lemma. (cid:3) Proposition C.20. Let γ ∈ GU( V ) ( F ) × GU( W )( F ) . Then we have s v ( γ ) = 1 for almost all v and Y v s v ( γ ) = 1 . Proof. Let γ , γ ∈ GU( V ) ( F ) × GU( W )( F ). Suppose that s v ( γ i ) = 1 for almost all v and Q v s v ( γ i ) =1 for i = 1 , 2. Since s v ( γ γ ) = s v ( γ ) · s v ( γ ) · z Y v ( γ , γ ), the product formulas for the quadraticHilbert symbol and the Weil index imply that s v ( γ γ ) = 1 for almost all v and Q v s v ( γ γ ) = 1.Hence the assertion follows from Proposition C.18. (cid:3) Appendix D. Splittings for the doubling method: quaternionic unitary groups D.1. Setup. Let F be a number field and B a quaternion algebra over F . Recall that • V is a 2-dimensional right skew-hermitian B -space with det V = 1, • W is a 1-dimensional left hermitian B -space, • V := V ⊗ B W is an 8-dimensional symplectic F -space, • V = X ⊕ Y is a complete polarization over F . 58 ATSUSHI ICHINO AND KARTIK PRASANNA We consider a 2-dimensional left B -space W (cid:3) := W ⊕ W equipped with a hermitian form h ( x, x ′ ) , ( y, y ′ ) i := h x, y i − h x ′ , y ′ i for x, x ′ , y, y ′ ∈ W . Put W + := W ⊕ { } and W − := { } ⊕ W . We regard GU( W ± ) as a subgroupof GL( W ) and identify it with GU( W ) via the identity map. Note that the identity map GU( W − ) → GU( W ) is an anti-isometry. We have a natural map ι : G(U( W ) × U( W )) −→ GU( W (cid:3) )and seesaw dual pairs GU( W (cid:3) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ G(U( V ) × U( V )) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ G(U( W ) × U( W )) GU( V ) . Put W △ := { ( x, x ) ∈ W (cid:3) | x ∈ W } , W ▽ := { ( x, − x ) ∈ W (cid:3) | x ∈ W } . Then W (cid:3) = W ▽ ⊕ W △ is a complete polarization over B . Choosing a basis w , w ∗ of W (cid:3) such that W ▽ = B w , W △ = B w ∗ , h w , w ∗ i = 1 , we may write GU( W (cid:3) ) = (cid:26) g ∈ GL ( B ) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) (cid:19) t g ∗ = ν ( g ) · (cid:18) (cid:19) (cid:27) . For ν ∈ F × , put d ( ν ) = d W △ ( ν ) := (cid:18) ν (cid:19) ∈ GU( W (cid:3) ) . Similarly, we consider a 16-dimensional F -space V (cid:3) := V ⊗ B W (cid:3) = V ⊕ V equipped with a symplecticform(D.1) hh ( x, x ′ ) , ( y, y ′ ) ii := hh x, y ii − hh x ′ , y ′ ii for x, x ′ , y, y ′ ∈ V . Put V + := V ⊕ { } and V − := { } ⊕ V . We regard Sp( V ± ) as a subgroup of GL( V )and identify it with Sp( V ) via the identity map. Note that the identity map Sp( V − ) → Sp( V ) is ananti-isometry. We have a natural map ι : Sp( V ) × Sp( V ) −→ Sp( V (cid:3) ) . Put V △ := V ⊗ B W △ = { ( x, x ) ∈ V (cid:3) | x ∈ V } , X (cid:3) := X ⊕ X , V ▽ := V ⊗ B W ▽ = { ( x, − x ) ∈ V (cid:3) | x ∈ V } , Y (cid:3) := Y ⊕ Y . Then V (cid:3) = V ▽ ⊕ V △ = X (cid:3) ⊕ Y (cid:3) are complete polarizations over F .For the rest of this section, we fix a place v of F and suppress the subscript v from the notation. Thus F = F v will be a local field of characteristic zero. We may lift the natural map ι : Sp( V ) × Sp( V ) → Sp( V (cid:3) ) to a unique homomorphism˜ ι : Mp( V ) × Mp( V ) −→ Mp( V (cid:3) )such that ˜ ι ( z , z ) = z z − for z , z ∈ C . D.2. Splitting z V △ . First assume that B is split. Fix an isomorphism i : B → M ( F ). Put e = i − ( ) and e ′ = i − ( ). Then W (cid:3) † := eW (cid:3) is a 4-dimensional symplectic F -space and therestriction GU( W (cid:3) ) → GSp( W (cid:3) † ) is an isomorphism. Using a basis e w , e ′ w , e ′ w ∗ , − e w ∗ of W (cid:3) † , wewrite GSp( W (cid:3) † ) = (cid:26) h ∈ GL ( F ) (cid:12)(cid:12)(cid:12)(cid:12) h (cid:18) − (cid:19) t h = ν ( h ) · (cid:18) − (cid:19) (cid:27) . Then the restriction GU( W (cid:3) ) → GSp( W (cid:3) † ) is given by (cid:18) a bc d (cid:19) (cid:18) τ − (cid:19) · (cid:18) i ( a ) i ( b ) i ( c ) i ( d ) (cid:19) · (cid:18) τ (cid:19) , where τ = τ = (cid:0) − (cid:1) . Note that x ∗ = τ · t x · τ − for x ∈ M ( F ). Also, W † := eW is a 2-dimensionalsymplectic F -space and V † := V e is a 4-dimensional quadratic F -space. We define a mapˆ s : G(U( V ) × U( W (cid:3) )) −→ C by ˆ s ( g ) = γ ˆ ( h ) for g = ( g, h ) ∈ G(U( V ) × U( W (cid:3) )), where γ = ( V † is isotropic, − V † is anisotropic,and ˆ ( h ) = c = 0,1 if c = 0 and det c = 0,2 if det c = 0, h = (cid:18) a bc d (cid:19) ∈ GSp( W (cid:3) † ) . Since dim F V † = 4 and det V † = 1, we have z V △ ( h, h ′ ) = ˆ s ( hh ′ ) · ˆ s ( h ) − · ˆ s ( h ′ ) − for h, h ′ ∈ U( W (cid:3) ) by [39, Theorem 3.1, case 1 + ].Next assume that B is ramified. We define a mapˆ s : G(U( V ) × U( W (cid:3) )) −→ C by ˆ s ( g ) = 1for g ∈ G(U( V ) × U( W (cid:3) )). Since dim B V = 2 and det V = 1, we have z V △ ( h, h ′ ) = ˆ s ( hh ′ ) · ˆ s ( h ) − · ˆ s ( h ′ ) − for h, h ′ ∈ U( W (cid:3) ) by [39, Theorem 3.1, case 2 − ]. Lemma D.1. We have z V △ ( g , g ′ ) = ˆ s ( gg ′ ) · ˆ s ( g ) − · ˆ s ( g ′ ) − for g , g ′ ∈ G(U( V ) × U( W (cid:3) )) .Proof. Let g i = ( g i , h i ) ∈ G(U( V ) × U( W (cid:3) )) and put h ′ i = h i · d ( ν ( h i )) − ∈ U( W (cid:3) ). Then we have h h = h ′ h ′′ · d ( ν ( h h )), where h ′′ = d ( ν ( h )) · h ′ · d ( ν ( h )) − . Since V △ · g = V △ , V △ · d ( ν ) = V △ for g ∈ GU( V ) and ν ∈ F × , we have V △ · g − = V △ · h − = V △ · h ′− , V △ · g − g − = V △ · h − h − = V △ · h ′′− h ′− . 60 ATSUSHI ICHINO AND KARTIK PRASANNA Hence we have q ( V △ , V △ · g − , V △ · g ) = q ( V △ · g − , V △ · g − g − , V △ )= q ( V △ · h ′− , V △ · h ′′− h ′− , V △ )= q ( V △ , V △ · h ′′− , V △ · h ′ ) , so that z V △ ( g , g ) = z V △ ( h ′ , h ′′ ) = ˆ s ( h ′ h ′′ ) · ˆ s ( h ′ ) − · ˆ s ( h ′′ ) − . By definition, we have ˆ s ( h ′ ) = ˆ s ( g ), ˆ s ( h ′′ ) = ˆ s ( h ′ ) = ˆ s ( g ), andˆ s ( h ′ h ′′ ) = ˆ s ( h h · d ( ν ( h h )) − ) = ˆ s ( g g ) . This completes the proof. (cid:3) D.3. Splitting z Y ′ (cid:3) . Let V = X ′ ⊕ Y ′ be the complete polarization given in § C.2, C.3. Put X ′ (cid:3) := X ′ ⊕ X ′ , Y ′ (cid:3) := Y ′ ⊕ Y ′ . Then V (cid:3) = X ′ (cid:3) ⊕ Y ′ (cid:3) is a complete polarization. Noting that the symplectic form on V (cid:3) = V ⊕ V isgiven by (D.1), we have z Y ′ (cid:3) ,ψ ( ι ( g , g ) , ι ( g ′ , g ′ )) = z Y ′ ,ψ ( g , g ′ ) · z Y ′ ,ψ − ( g , g ′ ) = z Y ′ ,ψ ( g , g ′ ) · z Y ′ ,ψ ( g , g ′ ) − for g i , g ′ i ∈ Sp( V ), where we write z Y ′ = z Y ′ ,ψ to indicate the dependence of the 2-cocycle on ψ . TheWeil representation ω (cid:3) ψ of Mp( V (cid:3) ) can be realized on the Schwartz space S ( X ′ (cid:3) ) = S ( X ′ ) ⊗ S ( X ′ ) . As representations of Mp( V ) Y ′ × Mp( V ) Y ′ , we have ω (cid:3) ψ ◦ ˜ ι = ω ψ ⊗ ( ω ψ ◦ ˜ j Y ′ ) , where ˜ j Y ′ is the automorphism of Mp( V ) Y ′ = Sp( V ) × C defined by˜ j Y ′ ( g, z ) = ( j Y ′ ( g ) , z − ) , j Y ′ ( g ) = d Y ′ ( − · g · d Y ′ ( − . Fix h ′ ∈ Sp( V (cid:3) ) such that X ′ (cid:3) = V ▽ · h ′ and Y ′ (cid:3) = V △ · h ′ . Put µ ′ ( g ) = z V △ ( g, h ′− ) · z V △ ( h ′− , h ′ g h ′− ) − for g ∈ Sp( V (cid:3) ). Then we have z Y ′ (cid:3) ( g, g ′ ) = z V △ ( g, g ′ ) · µ ′ ( gg ′ ) · µ ′ ( g ) − · µ ′ ( g ′ ) − for g, g ′ ∈ Sp( V (cid:3) ). Put G := { ( g, h , h ) ∈ GU( V ) × GU( W ) × GU( W ) | ν ( g ) = ν ( h ) = ν ( h ) } . We have natural maps G ֒ → G(U( V ) × U( W (cid:3) )) , G ֒ → G(U( V ) × U( W )) × G(U( V ) × U( W )) . Lemma D.2. We have ˆ s · µ ′ = s ′ ⊗ ( s ′ ◦ j Y ′ ) on G , where s ′ : GU( V ) × GU( W ) → C is the map defined in § C.2, C.3. The proof of this lemma will be given in the next two sections. D.3.1. The case u ∈ ( F × ) or J ∈ ( F × ) . Recall that X ′ = V † ⊗ F X and Y ′ = V † ⊗ F Y , where X = F e and Y = F e ′ , and W † = X + Y is a complete polarization over F . We have X ′ (cid:3) = V † ⊗ F X (cid:3) , Y ′ (cid:3) = V † ⊗ F Y (cid:3) , where X (cid:3) = X ⊕ X and Y (cid:3) = Y ⊕ Y . We have d Y ′ ( − 1) = id ⊗ d Y ( − 1) and j Y ′ = id ⊗ j Y , where d Y ( ν ) = (cid:18) ν (cid:19) ∈ GSp( W † )and j Y ( h ) = d Y ( − · h · d Y ( − 1) for h ∈ GSp( W † ). In particular, we have j Y ′ (G(U( V ) × U( W ))) = G(U( V ) × U( W )) . Let ι : G(Sp( W † ) × Sp( W † )) → GSp( W (cid:3) † ) be the natural map. We may take w = 12 (1 , − , w ∗ = (1 , . Since ( e, , e )( e ′ , , − e ′ ) = h · e w e ′ w e ′ w ∗ − e w ∗ , h = − − − − ∈ Sp( W (cid:3) † ) , we have ι ( h , h ) = h − · ι ♮ ( h , j Y ( h )) · h . Here, using a basis e, e ′ of W † , we identify GSp( W † ) with GL ( F ) and put ι ♮ ( h , h ) = a b a b c d c d , h i = (cid:18) a i b i c i d i (cid:19) . Since X (cid:3) = eW ▽ · h and Y (cid:3) = eW △ · h , we may take h ′ = id ⊗ h . Proof of Lemma D.2. Let g = ( g, h , h ) ∈ G and ν = ν ( g ). Put g i = ( g, h i ) ∈ G(U( V ) × U( W )). Bydefinition, we have s ′ ( g ) · s ′ ( j Y ′ ( g )) = γ j ( h ) · γ j ( j Y ( h )) , where j is as in § C.2.Put h = ι ( h , h ) ∈ GU( W (cid:3) ). We identify g with ( g, h ) ∈ G(U( V ) × U( W (cid:3) )). Since ˆ ( h ) =ˆ ( d ( ν ) − · h ), we have ˆ s ( g ) = ˆ s ( d ( ν ) − · h ) . Put g ′ = ( g, d ( ν )) ∈ G(U( V ) × U( W (cid:3) )). Then we have V △ · g ′ = V △ and g = g ′ · d ( ν ) − · h, h ′ gh ′− = h hh − · g = h hh − · d ( ν ) − · g ′ . Hence, by Lemma D.1, we have z V △ ( g , h ′− ) = z V △ ( d ( ν ) − · h, h − )= ˆ s ( d ( ν ) − · hh − ) · ˆ s ( d ( ν ) − · h ) − · ˆ s ( h − ) − ,z V △ ( h ′− , h ′ gh ′− ) = z V △ ( h − , h hh − · d ( ν ) − )= ˆ s ( hh − · d ( ν ) − ) · ˆ s ( h − ) − · ˆ s ( h hh − · d ( ν ) − ) − . Since ˆ ( d ( ν ) − · hh − ) = ˆ ( hh − · d ( ν ) − ), we haveˆ s ( d ( ν ) − · hh − ) = ˆ s ( hh − · d ( ν ) − ) . 62 ATSUSHI ICHINO AND KARTIK PRASANNA Since h hh − = ι ♮ ( h , j Y ( h )), we have ˆ ( h hh − · d ( ν ) − ) = j ( h ) + j ( j Y ( h )) andˆ s ( h hh − · d ( ν ) − ) = γ j ( h )+ j ( j Y ( h )) . Thus we obtain ˆ s ( g ) · µ ′ ( g ) = ˆ s ( d ( ν ) − · h ) · z V △ ( g , h ′− ) · z V △ ( h ′− , h ′ gh ′− ) − = ˆ s ( h hh − · d ( ν ) − )= γ j ( h )+ j ( j Y ( h )) . This completes the proof. (cid:3) D.3.2. The case J i ∈ ( F × ) . We only consider the case i = 1; the other case is similar. As in § C.3, weregard V and W as left and right B -spaces, respectively. Recall that X ′ = W ⊗ B X and Y ′ = W ⊗ B Y ,where X = B v and Y = B v ∗ , and V = X + Y is a complete polarization over B . As in § D.1, we definea 4-dimensional left skew-hermitian B -space V (cid:3) = V ⊕ V and a complete polarization V (cid:3) = V ▽ ⊕ V △ over B . Using a basis v := 12 ( v , − v ) , v := 12 ( v ∗ , − v ∗ ) , v ∗ := ( v ∗ , v ∗ ) , v ∗ := ( − v , − v )of V (cid:3) , we write GU( V (cid:3) ) = (cid:26) g ∈ GL ( B ) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) − (cid:19) t g ∗ = ν ( g ) · (cid:18) − (cid:19) (cid:27) . We may identify V (cid:3) with W ⊗ B V (cid:3) . Under this identification, we have V ▽ = W ⊗ B V ▽ , X ′ (cid:3) = W ⊗ B X (cid:3) , V △ = W ⊗ B V △ , Y ′ (cid:3) = W ⊗ B Y (cid:3) , where X (cid:3) = X ⊕ X and Y (cid:3) = Y ⊕ Y . We have d Y ′ ( − 1) = id ⊗ d Y ( − 1) and j Y ′ = id ⊗ j Y , where d Y ( ν ) = (cid:18) ν (cid:19) ∈ GU( V ) and j Y ( g ) = d Y ( − · g · d Y ( − 1) for g ∈ GU( V ). In particular, we have j Y ′ (G(U( V ) × U( W ))) = G(U( V ) × U( W )) . Let ι : G(U( V ) × U( V )) → GU( V (cid:3) ) be the natural map. Since ( v , , v )( v ∗ , , − v ∗ ) = g · v v v ∗ v ∗ , g = − − − − ∈ U( V (cid:3) ) , we have ι ( g , g ) = g − · ι ♮ ( g , j Y ( g )) · g . Here, regarding V as a left B -space and using a basis v , v ∗ of V , we identify GU( V ) with a subgroupof GL ( B ) and put ι ♮ ( g , g ) = a b a b c d c d , g i = (cid:18) a i b i c i d i (cid:19) . Since X (cid:3) = V ▽ · g and Y (cid:3) = V △ · g , we may take h ′ = id ⊗ g . When B is split, we define a map ˆ s ′ : U( V (cid:3) ) −→ C by ˆ s ′ ( g ) = 1for g ∈ U( V (cid:3) ). Then we have z V △ ( g, g ′ ) = ˆ s ′ ( gg ′ ) · ˆ s ′ ( g ) − · ˆ s ′ ( g ′ ) − for g, g ′ ∈ U( W (cid:3) ) by [39, Theorem 3.1, case 1 − ]. When B is ramified, we define a mapˆ s ′ : U( V (cid:3) ) −→ C by ˆ s ′ ( g ) = ( − ˆ ′ ( g ) for g ∈ U( V (cid:3) ), whereˆ ′ ( g ) = c = 0,1 if c = 0 and ν ( c ) = 0,2 if ν ( c ) = 0, g = (cid:18) a bc d (cid:19) ∈ U( V (cid:3) ) , and ν : M ( B ) → F is the reduced norm. Since dim B W = 1, we have z V △ ( g, g ′ ) = ˆ s ′ ( gg ′ ) · ˆ s ′ ( g ) − · ˆ s ′ ( g ′ ) − for g, g ′ ∈ U( W (cid:3) ) by [39, Theorem 3.1, case 2 + ]. Proof of Lemma D.2. Let g = ( g, h , h ) ∈ G and ν = ν ( g ). Put g i = ( g, h i ) ∈ G(U( V ) × U( W )). Bydefinition, we have j ( j Y ( g )) = j ( g ), where j is as in § C.3. Hence we have s ′ ( g ) · s ′ ( j Y ′ ( g )) = γ j ( g ) · γ j ( j Y ( g )) = 1 . Here γ = ( B is split, − B is ramified.Also, by definition, we have ˆ s ( g ) = 1 . Now we compute z V △ ( g , h ′− ). We identify g with ( g, ι ( h , h )) ∈ G(U( V ) × U( W (cid:3) )), where ι : G(U( W ) × U( W )) → GU( W (cid:3) ) is the natural map. Put g ′ = ( g, ι ( h , h )) ∈ G(U( V ) × U( W (cid:3) )) and h = h − h ∈ U( W ). Then we have g = g ′ · ι ( h − , V (cid:3) = V ⊗ B W (cid:3) = W ⊗ B V (cid:3) ,we identify g ′ with ( h , ι ( g, g )) ∈ G(U( W ) × U( V (cid:3) )). Since V △ · g ′ = V △ , we have z V △ ( g , h ′− ) = z V △ ( ι ( h − , , g − ) . Put τ := − ∈ U( V (cid:3) ) . Then we have g τ = − − − 12 12 − − − 64 ATSUSHI ICHINO AND KARTIK PRASANNA and V △ · τ − g − = V △ , so that z V △ ( ι ( h − , , g − ) = z V △ ( ι ( h − , , τ ) . Under the identification V (cid:3) = V ⊕ V = W ⊗ B V (cid:3) , we have x ⊗ ( y, ± y ) · ι ( h − , 1) = ( x ⊗ y, ± x ⊗ y ) · ι ( h − , hx ⊗ y, ± x ⊗ y )= 12 (( h ± x ⊗ y, ( h ± x ⊗ y ) + 12 (( h ∓ x ⊗ y, − ( h ∓ x ⊗ y )= 12 ( h ± x ⊗ ( y, y ) + 12 ( h ∓ x ⊗ ( y, − y )for x ∈ W and y ∈ V . Thus we obtain x ⊗ v · ι ( h − , 1) = 12 ( h + 1) x ⊗ v − 14 ( h − x ⊗ v ∗ ,x ⊗ v · ι ( h − , 1) = 12 ( h + 1) x ⊗ v + 14 ( h − x ⊗ v ∗ ,x ⊗ v ∗ · ι ( h − , 1) = 12 ( h + 1) x ⊗ v ∗ + ( h − x ⊗ v ,x ⊗ v ∗ · ι ( h − , 1) = 12 ( h + 1) x ⊗ v ∗ − ( h − x ⊗ v . First assume that B is split. Fix an isomorphism i : B → M ( F ). Put e = i − ( ), e ′ = i − ( ),and e ′′ = i − ( ). Then W † := W e is a 2-dimensional symplectic F -space and the restrictionGU( W ) → GSp( W † ) is an isomorphism. Also, V (cid:3) † := eV (cid:3) is an 8-dimensional quadratic F -space.We identify V (cid:3) with W † ⊗ F V (cid:3) † . Put f := 2 e ′′ and x := e v , x := e ′ v , x := e v , x := e ′ v , y := e ′ v ∗ , y := − e v ∗ , y := e ′ v ∗ , y := − e v ∗ . Using a basis e ⊗ x , f ⊗ x , . . . , e ⊗ x , f ⊗ x , f ⊗ y , − e ⊗ y , . . . , f ⊗ y , − e ⊗ y of V (cid:3) , we identify Sp( V (cid:3) ) with Sp ( F ). We define h ∈ GL ( F ) by (cid:20) hehf (cid:21) = h · (cid:20) ef (cid:21) . Then we have h · a · t h = a , where a = (cid:18) − (cid:19) ∈ GL ( F ) . Moreover, we have ι ( h − , 1) = d − · h ′ · d and τ = d − · − − · d = τ · − aa 1 − aa , where d = a a a a , h ′ = ˙ h ¨ h ˙ h − ¨ h ˙ h − ¨ h ˙ h ¨ h h ˙ h − h ˙ h − h ˙ h h ˙ h , ˙ h = ( h + ), ¨ h = ( h − ), and τ = − . If h = 1, then we have z V △ ( ι ( h − , , τ ) = z V △ (1 , τ ) = 1. Assume that h = 1 and det ¨ h = 0. Sincedet h = 1, we have tr ¨ h = 0. Hence we may take a ∈ SL ( F ) such that a · ¨ h · a − = x , where x = (cid:18) x (cid:19) with some x = 0. Put m = a a a a t a − t a − t a − t a − . Noting that a · ˙ h · a − = x + and a · t a − · a − = a , we have m · ι ( h − , · m − = d − · h ′′ · d ,where h ′′ = x + xx + − xx + − xx + x x x + − x x + − x x + x x + . We have z V △ ( ι ( h − , , τ ) = z V △ ( ι ( h − , , τ )= z V △ ( m · ι ( h − , , τ · τ − · m − · τ )= z V △ ( m · ι ( h − , · m − , τ )= z V △ ( d − · h ′′ · d , τ ) . 66 ATSUSHI ICHINO AND KARTIK PRASANNA Put e = (cid:18) (cid:19) , e ∗ = (cid:18) (cid:19) , a = (cid:18) − x − − x (cid:19) , and m = a − a − a a t a − − t a − − t a − t a − . Then we have a · x · a = − e , t a − · a − · x = e ∗ , and hence m · d − · h ′′ · d = h ′′′ , where h ′′′ = x ′ − e − x ′ − e − x ′ − e x ′ − ee ∗ x ′′ e ∗ − x ′′ e ∗ − x ′′ e ∗ x ′′ , x ′ = − ea − + a , and x ′′ = e ∗ a + t a − . We have z V △ ( d − · h ′′ · d , τ ) = z V △ ( m · d − · h ′′ · d , τ ) = z V △ ( h ′′′ , τ ) . Put b = (cid:18) x − (cid:19) and n = b1 − b1 − b1 b1 . We have h ′′′ · n = x ′ x ′ b − e − x ′ x ′ b − e − x ′ x ′ b − e x ′ x ′ b − ee ∗ ( e ∗ b + x ′′ ) e ∗ − ( e ∗ b + x ′′ ) e ∗ − ( e ∗ b + x ′′ ) e ∗ ( e ∗ b + x ′′ ) . Since e ∗ b + x ′′ = (cid:18) − x 01 0 (cid:19) , we have V △ · h ′′′ · n · τ − = V △ , where τ = e − e ∗ e − e ∗ e − e ∗ e − e ∗ e ∗ ee ∗ ee ∗ ee ∗ e . Hence we have z V △ ( h ′′′ , τ ) = z V △ ( h ′′′ , τ · τ − · n · τ ) = z V △ ( h ′′′ · n , τ ) = z V △ ( τ , τ ) = 1 . Assume that h = 1 and det ¨ h = 0. We have ι ( h − , t a t ¨ h − ˙ h − t a t ¨ h − ˙ h − t a t ¨ h − ˙ h t a t ¨ h − ˙ h a − ¨ h − a − ¨ h − a − ¨ h a − ¨ h · τ · n , where τ = (cid:18) − (cid:19) , n = ¨ h − ˙ ha1 − ¨ h − ˙ ha1 − ¨ h − ˙ ha1 ¨ h − ˙ ha1 . Hence we have z V △ ( ι ( h − , , τ ) = z V △ ( τ · n , τ ) = z V △ ( τ , n · τ ) . Since V △ · τ − n τ = V △ , we have z V △ ( τ , n · τ ) = z V △ ( τ , τ ) = 1 . Thus we obtain z V △ ( g , h ′− ) = 1 . Next assume that B is ramified. Choose a basis e , e , e , e of W over F . We may assume that tr B/F ( h e i , e j i ) = a i · δ ij with some a i ∈ F × . Put e ′ i := e i · a − i . Using a basis e ⊗ v , . . . , e ⊗ v , e ⊗ v , . . . , e ⊗ v , e ′ ⊗ v ∗ , . . . , e ′ ⊗ v ∗ , e ′ ⊗ v ∗ , . . . , e ′ ⊗ v ∗ 268 ATSUSHI ICHINO AND KARTIK PRASANNA of V (cid:3) , we identify Sp( V (cid:3) ) with Sp ( F ). We define h ∈ GL ( F ) by he he he he = h · e e e e . Then we have h · a · t h = a , where a := a a a a ∈ GL ( F ) . Moreover, we have ι ( h − , 1) = d − · h ′ · d and τ = d − · τ · d = τ · a − a , where d = a a , h ′ = ˙ h − ¨ h ˙ h ¨ h h ˙ h − h ˙ h , ˙ h = ( h + ), ¨ h = ( h − ), and τ = − . If h = 1, then we have z V △ ( ι ( h − , , τ ) = z V △ (1 , τ ) = 1. Assume that h = 1. Since B is ramified, h − x α · x of B with some α ∈ B × . In particular, we have ¨ h ∈ GL ( F ).We have ι ( h − , 1) = a t ¨ h − ˙ h − a t ¨ h − ˙ h a − ¨ h − a − ¨ h · τ · n , where τ = (cid:18) − (cid:19) , n = − ¨ h − ˙ ha1 ¨ h − ˙ ha1 . Hence we have z V △ ( ι ( h − , , τ ) = z V △ ( τ · n , τ ) = z V △ ( τ , n · τ ) . Since V △ · τ − n τ = V △ , we have z V △ ( τ , n · τ ) = z V △ ( τ , τ ) = 1 . Thus we obtain z V △ ( g , h ′− ) = 1 . Now we compute z V △ ( h ′− , h ′ gh ′− ). Put g ′′ = ( d Y ( ν ) , ι ( h , h )) ∈ G(U( V ) × U( W (cid:3) )) and g ′ = g · d Y ( ν ) − ∈ U( V ) . Then we have g = g ′ · g ′′ . Via the identification V (cid:3) = V ⊗ B W (cid:3) = W ⊗ B V (cid:3) ,we identify g ′ with ι ( g ′ , g ′ ) ∈ U( V (cid:3) ). Since V △ · h ′ = Y ′ (cid:3) = Y ⊗ B W (cid:3) , we have V △ · h ′ g ′′ h ′− = V △ and hence z V △ ( h ′− , h ′ gh ′− ) = z V △ ( g − , g · ι ( g ′ , g ′ ) · g − )= ˆ s ′ ( ι ( g ′ , g ′ ) · g − ) · ˆ s ′ ( g − ) − · ˆ s ′ ( g · ι ( g ′ , g ′ ) · g − ) − . Hence, if B is split, then we have z V △ ( h ′− , h ′ gh ′− ) = 1 . Assume that B is ramified. We write g ′ = (cid:0) a bc d (cid:1) . Since g − = − 12 12 12 − − − , g · ι ( g ′ , g ′ ) · g − = ι ♮ ( g ′ , j ′ Y ( g ′ )) = a ba − bc d − c d , and ι ( g ′ , g ′ ) · g − = a − a b b c − c d d c c d − d − a − a − b b , we have ˆ s ′ ( g − ) = − , ˆ s ′ ( g · ι ( g ′ , g ′ ) · g − ) = 1 , ˆ s ′ ( ι ( g ′ , g ′ ) · g − ) = − . Hence we have z V △ ( h ′− , h ′ gh ′− ) = 1 . Thus we obtain µ ′ ( g ) = z V △ ( g , h ′− ) · z V △ ( h ′− , h ′ gh ′− ) − = 1 . This completes the proof. (cid:3) D.4. Splitting z Y (cid:3) . Let V = X ⊕ Y be the complete polarization given in § C.1. Put X (cid:3) := X ⊕ X , Y (cid:3) := Y ⊕ Y . Then V (cid:3) = X (cid:3) ⊕ Y (cid:3) is a complete polarization. As in § D.3, we have z Y (cid:3) ( ι ( g , g ) , ι ( g ′ , g ′ )) = z Y ( g , g ′ ) · z Y ( g , g ′ ) − for g i , g ′ i ∈ Sp( V ). The Weil representation ω (cid:3) ψ of Mp( V (cid:3) ) can be realized on the Schwartz space S ( X (cid:3) ) = S ( X ) ⊗ S ( X ) . As representations of Mp( V ) Y × Mp( V ) Y , we have ω (cid:3) ψ ◦ ˜ ι = ω ψ ⊗ ( ω ψ ◦ ˜ j Y ) , where ˜ j Y is the automorphism of Mp( V ) Y = Sp( V ) × C defined by˜ j Y ( g, z ) = ( j Y ( g ) , z − ) , j Y ( g ) = d Y ( − · g · d Y ( − . Put J := (( j , j ) , j ). Here we view ( j , j ) ∈ GU( V ) and j ∈ GU( W ). Lemma D.3. We have j Y ( g ) = J · g · J − for g ∈ GU( V ) × GU( W ) . In particular, we have j Y (G(U( V ) × U( W ))) = G(U( V ) × U( W )) . 70 ATSUSHI ICHINO AND KARTIK PRASANNA Proof. Let g = (( α − , α − ) , α ) ∈ GU( V ) × GU( W ) with α i = a i + b i i + c i j i + d i ij i ∈ B × i and α = a + b i + c j + d ij ∈ B × . By § C.1, we see that j Y ( g ) = (( β − , β − ) , β ), where β i = a i − b i i + c i j i − d i ij i , β = a − b i + c j − d ij . On the other hand, since j i i = − ij i and ji = − ij , we have j i · α i · j − i = β i and j · α · j − = β . Thisyields the lemma. (cid:3) As in § C.2, C.3, fix h ∈ Sp( V ) such that X ′ = X h and Y ′ = Y h , and define a map s : GU( V ) × GU( W ) → C by s := s ′ · µ , where µ ( g ) = z Y ( h g h − , h ) · z Y ( h , g ) − . Put ˆ h := h ′ · ι ( h , h ) − ∈ Sp( V (cid:3) ). Then we have V ▽ · ˆ h = X ′ (cid:3) · ι ( h , h ) − = X (cid:3) , V △ · ˆ h = Y ′ (cid:3) · ι ( h , h ) − = Y (cid:3) . Put ˆ µ ( g ) = z V △ ( g, ˆ h − ) · z V △ (ˆ h − , ˆ h g ˆ h − ) − for g ∈ Sp( V (cid:3) ). Then we have z Y (cid:3) ( g, g ′ ) = z V △ ( g, g ′ ) · ˆ µ ( gg ′ ) · ˆ µ ( g ) − · ˆ µ ( g ′ ) − for g, g ′ ∈ Sp( V (cid:3) ). Lemma D.4. We have ˆ s · ˆ µ = s ⊗ ( s ◦ j Y ) on G .Proof. For g = ( g, h , h ) ∈ G , we identify g with ι ( g , g ) ∈ Sp( V (cid:3) ), where g i = ( g, h i ) ∈ G(U( V ) × U( W )) ⊂ Sp( V ). Then, by a direct calculation, one can see thatˆ µ ( g ) · µ ′ ( g ) − = z Y (cid:3) ( ι ( h , h ) · g · ι ( h , h ) − , ι ( h , h )) · z Y (cid:3) ( ι ( h , h ) , g ) − = z Y ( h g h − , h ) · z Y ( h g h − , h ) − · z Y ( h , g ) − · z Y ( h , g )= µ ( g ) · µ ( g ) − . Hence, by Lemma D.2, we haveˆ s · ˆ µ = ˆ s · µ ′ · ( µ ⊗ µ − ) = ( s ′ · µ ) ⊗ (( s ′ ◦ j Y ′ ) · µ − )on G . Since s ′ ◦ j Y ′ = s ′ = s ′− , we have ˆ s · ˆ µ = s ⊗ s − on G . By Proposition C.18 and Lemma D.3, we have s ( j Y ( g )) = s ( g ) − for g = α − i ∈ GU( V ) with α i ∈ B × i and g = α ∈ GU( W ) with α ∈ B × . Also, we have z Y ( j Y ( g ) , j Y ( g )) = z Y ( g , g ) − for g , g ∈ GSp( V ). Since s ( g g ) = s ( g ) · s ( g ) · z Y ( g , g ) for g , g ∈ GU( V ) × GU( W ), we have s ◦ j Y = s − on GU( V ) × GU( W ). This completes the proof. 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