A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS: CUBIC VARIATION OF A THEME OF SNITZ

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1 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS: CUBIC VARIATION OF A THEME OF SNITZ WEE TECK GAN 1. Introduction This paper is largely inspired by the 2003 Ph.D. thesis [S] of Kobi Snitz, written under the direction of S. Kudla at the University of Maryland. Let us recall his results briefly. Let F be a number field with adele ring A. Let D be a quaternion division algebra defined over F and ramified precisely at a finite set S D of places of F. Let V D be the 3-dimensional quadratic space (D 0, N D ) where D 0 is the space of trace zero elements in D and N D is the reduced norm of D. Then the special orthogonal group SO(V D ) is naturally isomorphic to PD = D /F and the reduced norm induces a natural map N D : PD (A) A /A 2. Each étale quadratic algebra K of F gives rise to a quadratic idele class character χ K which we may regard by composition with N D as a character of PD (A). Recall that PD SL 2 forms a dual pair, with its associated Weil representation realized on the space S(V D (A)) of Schwarz functions on V D (A). Hence one may consider the theta lift of χ K to SL 2. Namely, if f S(V D (A)), set θ D (χ K, f)(g) = θ(f)(gh) χ K (h)dh PD (F)\PD (A) and let V D (χ K ) be the space of automorphic forms on SL 2 obtained for all f S(V D (A)). If χ K = 1 is the trivial character (so that K = F F), the Siegel-Weil formula associates to each f an explicit Eisenstein series E D (f) such that θ D (1, f) = E D (f). This should be thought of as providing an alternative construction of θ D (1, f). In [S], Snitz considered the analogous problem when χ K is a non-trivial character. In this case, the space V D (χ K ) is non-zero if and only if K is non-split at all places in S D. The function θ D (χ K, f) is contained in the space of cusp forms and one would like an alternative construction of this cusp form. To do so, one considers the rank 1 quadratic space V K = (K 0, N K ), where K 0 is the space of trace zero elements in K and N K is the norm map of K/F. The orthogonal group O(V K ) is naturally isomorphic to the finite group scheme S K with S K (F v ) = Gal(K v /F v ) = Z/2Z. Now let χ D = v χ D,v be the character of S K (A) such that χ D,v is non-trivial precisely when v S D. Observe that since #S D is even, χ D is a character of S K (F)\S K (A). Now S K SL 2 forms a dual pair with its associated Weil representation realized on S(V K (A)). Thus for each Schwarz function 1

2 2 WEE TECK GAN f S(V K (A)), one has the analogous theta lift θ K (χ D, f )(g) = and the space V K (χ D ) that these lifts generate. Now the main result of [S] is: S K(F)\S K(A) θ(f )(gh) χ D (h)dh Theorem 1.1. Given f S(V D (A)), there is a unique f S(V K (A))[χ D ] (the χ D -isotypic component of S(V K (A)) such that θ D (χ K, f) = θ K (χ D, f ). In fact, results of this type were first obtained some twenty years ago by Schulze-Pillot [SP1,2,3] in a more classical language and applied to the arithmetic of quadratic forms. Snitz s theorem provides a refinement of the results of Schulze-Pillot and exposes the representation theoretic underpinnings. The map f f can be explicitly described at the local level and can be expressed as a weighted orbital integral. Namely, we fix an F-algebra embedding ι : K D, which gives rise to a map of quadratic spaces V K V D ; this exists because of our assumption that K v is a field for all v S D. Fix also a non-zero element x 0 in the 1-dimensional F-vector space V K. Then, for each place v, a Schwarz function f v S(V Dv ) is mapped to the function f v S(V Kv ) given by: f v(r) = r v χ Kv ( r ) f v (h 1 v ι(r) h v ) χ Kv (N Dv (h v ))dh v, if r 0. x 0 x 0 ι(k v )\D v Since any two such embeddings as ι are conjugate under PD v by the Skolem-Noether theorem, the above definition is independent of the choice of ι. In any case, the above formula determines f v uniquely. Note that though this local map depends on the choice of x 0 V K, the associated global map (which is the tensor product of the local ones) does not. The purpose of this paper is to prove a cubic analog of the above theorem. As we explain below, there are a number of subtleties which are absent in the quadratic case and which makes the formulation of the result in the cubic case somewhat interesting. More precisely, assume henceforth that D is a division algebra of degree 3 over F, ramified precisely at the finite set S D of places of F. Unlike the case of quaternions, D is not uniquely specified by the set S D. For each place v S D, one needs to specify a local invariant inv(d v ) = ±1 Z/3Z with inv(d v ) = 0(mod3). v S D The definition of the local invariant inv(d v ) and the condition of global coherence they satisfy requires the specification of the global Artin isomorphism art F : π 0 (F \A ) W ab F. There are after all two natural choices of this (which are inverses of each other), depending on whether local uniformizers at each finite place are sent to the arithmetic or geometric Frobenius elements. In any case, we fix art F once and for all. If D is associated to the data (S D, {ǫ v }), then the opposite algebra D op of D has data (S D, { ǫ v }). The automorphism group of D is naturally isomorphic to PD. One subtlety in the cubic case is that since D = (D op ) via x x 1,

3 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 3 there is a natural isomorphism i D : PD = (PD op ). Thus the correspondence D PD is twoto-one. To resolve this, we should think of D as giving the group PD as well as a distinguished homomorphism N D : PD (A) A /A 3 induced by the reduced norm map. We shall write G D for the group PD equipped with the map N D. Now let E be a cyclic cubic field extension of F. Unlike the case of quadratic extensions, where K and χ K determine each other, E/F determines a pair of cubic characters {χ E, χ 1 E } of Gal(E/F). This is another subtlety in the cubic case. Using the isomorphism art F, we obtain by composition an idele class character χ E art F. Since art F is fixed, we shall suppress it from the notations and denote this idele class character by χ E as well. Then we may regard χ E = v χ E,v as a character of G D (A) by composition with the distinguished map N D. Observe that χ E N D op i D = χ 1 E N D as characters on PD. Thus i D identifies the pair (D, χ 1 E ) with (Dop, χ E ). Now let G 2 denote the split exceptional group of type G 2. Then G D G 2 is a dual pair in an inner form H D of type E 6. One may consider the theta lift of the character χ E of G D to G 2, using the minimal representation Π D of H D. This theta lift is non-zero iff E v is a field for all v S D, in which case the lift is cuspidal. Assuming this, for each f Π D, one thus has a cusp form θ D (χ E, f) on G 2. Our main result gives an alternative construction of θ D (χ E, f). We now describe the alternative construction. As in the quadratic case, the field E gives rise to a finite group scheme S E which is a twisted form of S 3 ; S E is simply the automorphism group scheme of the F-algebra E. We have: { S E (F v ) = Gal(E v /F v ) A 3, if E v is a field; =. S 3, if E v is split. Now we want to define an automorphic character χ D E = v χ Dv E,v of S E (A). Recall that if E v is a field, then χ E,v is a character of S E (F v ) by definition. We define the character χ D E by χ Dv E,v = χinv(dv) E,v where we recall that inv(d v ) is the local invariant of D v and for v / S D, inv(d v ) = 0. Observe that the character χ D E is trivial when restricted to S E(F) by the condition of global coherence and that Now we have the dual pair χ D E = (χdop E ) 1. S E G 2 H E S E := Spin E 8 S E. Given an element f Π E (the minimal representation of H E ), one thus has an automorphic form θ E (χ D E, f ). Our first result is: Theorem 1.2. Assume that E v is unramified for all v S D. Given f Π D, there is a unique element f Π E [χ D E ] (the χd E -isotypic part of Π E) such that θ D (χ E, f) = θ E (χ D E, f ).

4 4 WEE TECK GAN Observe that if we replace the Artin isomorphism art F by art F inv (where inv is the inverse map on WF ab), then the characters χ E and χ D E will be replaced by their inverses. Thus the theorem does not depend on the choice of art F. We hope that our exposition above makes the parallel between the quadratic and cubic cases transparent. Our second result is the honest Siegel-Weil formula which identifies the theta lift of the trivial character as an explicit Eisenstein series. In fact, the Siegel-Weil formula was needed in our proof of Theorem 1.2. In [G1], we have obtained an instance of such a formula for the dual pair F cpt 4 G 2 in the quaternionic form of E 8, with F cpt 4 an anisotropic group of type F 4. This is analogous to the situation in Weil s paper [W], in the sense that the theta integral and the Eisenstein series are both absolutely convergent. The case we treat here is analogous to that in the paper [KR] of Kudla-Rallis, where the Siegel-Weil formula was extended beyond Weil s range of convergence for anisotropic orthogonal groups. As in [KR], the relevant Eisenstein series may a priori have a pole (of order 2 here) at the point of interest. We show however that it is holomorphic there for the standard sections we are interested in. The main difficulty in the proof of the formula here is to show that the theta lift is orthogonal to the space of cusp forms. In [G1], this was easy because the theta lift of the trivial representation turns out to be non-unitary. This is no longer the case here. Indeed, there are cuspidal representations of G 2 which are nearly equivalent to the theta lift of the trivial representation. These cuspidal representations belong to the so-called cubic unipotent Arthur packets introduced in [GGJ]. We overcome this difficulty by using the results of [G2] which gives a complete classification of the near equivalence class of such representations. Acknowledgments: This work was done when the author visited the University of Maryland during the Fall semester of The author would like to thank Jeffrey Adams, Tom Haines, Steve Kudla and Niranjan Ramachandran for many stimulating discussions beyond the subject matter of this paper. He would especially like to thank Steve Kudla for explaining the results of Snitz s thesis. This research is partially supported by NSF grant DMS Groups and Dual Pairs We begin by describing the relevant groups and representations which play a role in this paper The group H J. Let J = D or E, and let N J (resp. T J ) denote the norm (resp. trace) map on J. The trace map T J induces a symmetric bilinear form on J given by T J (x, y) = T J (xy). As mentioned in the introduction, we have a linear algebraic group H J which is adjoint of type E 6 (respectively simply-connected of type D 4 ) when J = D (respectively J = E). The group H J has F-rank 2 and its relative root system Φ is of type G 2. Let T J be a maximal split torus of H J and for each root γ Φ, let U J,γ be the associated root subgroup. Then { U J,γ F, if γ is long; = J, if γ is short. Indeed, when J = E so that H E is quasi-split, one has a compatible system of such isomorphisms by means of a Chevalley-Steinberg system of épinglage. We choose a system of simple roots {α, β} for Φ, with α short and β long.

5 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 5 One can describe the Lie algebra Lie(H J ) uniformly. Namely, there is a Z/3Z-grading Lie(H J ) = h 1 h 0 h 1 such that h 0 = sl 3 l J h 1 = F 3 J h 1 = (F 3 J). Here, { D 0 D 0, if J = D; l J = E 0, if J = E. Using the symmetric bilinear form on J and the standard pairing on F 3, we shall identity h 1 with F 3 J. For the definition of the Lie brackets, we refer the reader to [Ru1] Parabolic subgroups. Since H J has F-rank 2, it has two conjugacy classes of F-rational maximal parabolic subgroups. One of them is the Heisenberg parabolic P J = M J N J. The unique simple root of its Levi subgroup M J is the short root α. The unipotent radical N J is a Heisenberg group with 1-dimensional center Z J. If NJ denotes the opposite unipotent radical, then V J := N J / Z J = U J, β U J, α β U J, 2α β U J, 3α β = F J J F. We denote an element of Ω J by (a, x, y, d) with a, d F and x, y J. The representation of M J on V J is irreducible and we let Ω J denote the minimal non-trivial M J - orbit. It is the orbit of the highest weight vector and is a cone in V J. The Zariski closure of Ω J is Ω J = Ω J {0}. The variety Ω J can be described as those (a, x, y, d) V J satisfying: (xy + yx)/2 = ad x # = a y y # = d x Here x x # is the quadratic map on J determined by the condition that x # x = N J (x). Note that the product makes J into a Jordan algebra. x y = xy + yx 2 Since J is a division algebra, N J (x) is non-zero if x is non-zero. Thus, if (a, x, y, d) Ω, then a and d are both non-zero, and (a, x, y, d) = a (1, x, x #, N J (x)). Thus the variety Ω is birational to F J, via (a, x, y, d) (a, x/a). The other maximal parabolic subgroup is denoted by Q J = L J U J. The unique simple root of its Levi subgroup L J is the long root β. When J = D, we have L D = GL 2 µ3 (SL 1 (D) µ3 SL 1 (D)). Thus there is a natural projection map of algebraic groups GL 2 (SL 1 (D) µ3 SL 1 (D)) L D.

6 6 WEE TECK GAN Since the unipotent radical U J is 3-step nilpotent, we shall often refer to Q J as the 3-step parabolic. Note that if χ Ω J, then the stabilizer in M J of the line spanned by χ is the parabolic subgroup M J Q J. Hence the stabilizer of χ itself is a codimension one subgroup of M J Q J Dual pairs. Now we can describe the dual pair G J G 2 H J. If e denotes the identity element in J, then sl 3 (F 3 e) (F 3 e) sl 3 l J (F 3 J) (F 3 J) is a Lie subalgebra isomorphic to Lie(G 2 ). Its centralizer is trivial when J = E, but when J = D, the centralizer is Lie(G D ) = D 0 which is embedded into l J = D 0 D 0 diagonally. This describes the dual pair G D G 2 H D. When J = E, the outer automorphism group Out(H E ) of H E is isomorphic to S E. We fix such an isomorphism as follows. As we have noted before, using a Chevalley-Steinberg system of épinglage for the quasi-split group H E, one has a natural isomorphism The isomorphism U α = ResE/F G a. Out(H E ) = S E is such that S E (F) = Gal(E/F) acts in the natural way on U α (F) = E. Moreover, with the Chevalley- Steinberg system of épinglage, one obtains a splitting S E = Out(HE ) Aut(H E ). As a result, one has a dual pair S E G 2 H E S E. We may consider the intersection of G 2 with the two parabolic subgroups introduced above. By working on the level of Lie algebras, one sees that G 2 P J = P where P = M N is a Heisenberg parabolic of G 2. Here, M = GL 2 whereas N is a 5-dimensional Heisenberg group whose center is that of N J. If we set V = N/ Z, then the M-orbits on V are parametrized naturally by cubic F-algebras, with the generic orbits corresponding to étale cubic algebras. The embedding V V J is given by F F F F F J J F. There is a natural M-equivariant projection V J V by (a, x, y, d) (a, T J (x), T J (y), d). Given an element χ = (a, b, c, d) V, let Ω J (χ) be the fiber of this projection map over χ. Lemma 2.1. (i) One has Ω J (χ) = {(a(1, x, x #, N J (x)) : x has characteristic polynomial x 3 (b/a)x 2 + (c/a)x (d/a)}. Thus, if the M-orbit of χ corresponds to an étale cubic algebra A, then Ω J (χ) is in bijection with the set of algebra embeddings of A into D. In particular, when J = E, Ω E (χ) is non-empty iff A = E, whereas when J = D, Ω D (χ) is non-empty iff A v is a field for all v S D.

7 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 7 (ii) When Ω J (χ) is non-empty, G J acts transitively on Ω J (χ). If J = E, this action is simply transitive, whereas if J = D, the stabilizer of of a point in Ω D (χ) is the maximal torus PA PD associated to the corresponding embedding A D. Likewise, the intersection of G 2 with the 3-step parabolic Q J is equal to Q = L U, which is the other maximal parabolic subgroup of G 2. The Levi subgroup L is isomorphic to GL 2 and its unipotent radical U is a 3-step nilpotent group. 3. Representations Now let v be a place of F and fix a non-trivial additive character ψ v of F. The group H J (F v ) has a distinguished representation Π J,v known as the minimal representation. For a systematic introduction to minimal representations, we refer the reader to [GS2]. We shall content ourselves with stating the properties of Π J,v that we need Minimal representations. Assume first that v is a finite place of F; this assumption will hold up to and including 3.4. Unlike the Weil representation, we do not have a completely explicit smooth model for Π J,v, but we do have a rather neat description of it which suffices for some purposes. One may identify V J with the set of unitary characters of N J (F v ), using the Killing form and the additive character ψ v. Recall that Ω J is the minimal non-trivial M J -orbit on V J. It was shown in [MS] that there is a P J (F v )-equivariant embedding i v : (Π J,v ) ZJ C (Ω J ), where P J (F v ) acts on C (Ω J ) by: { (m f)(χ) = δ PJ (m) sj f(m 1 χ) (n f)(χ) = χ(n) f(χ) where s J = { 2/11, if J = D; 1/5, if J = E. The image of i contains C c (Ω J ), and we have an exact sequence of P J (F v )-modules: 0 C c (Ω J ) (Π J,v ) ZJ (Π J,v ) NJ 0. Moreover, one can describe (Π J,v ) NJ as a representation of M J (F v ) as follows. When J = D, we have: { (Π D,v ) ND δ 2/11 = P D δ PD Π(M D ), if D v is unramified;, if D v is ramified, δ 2/11 P D where Π(M D ) is the minimal representation of M D (F v ) with center acting trivially. When J = E and E v is a field, then (Π E,v ) NE = χe,v δ 1/5 P E χ 1 E,v δ1/5 P E. When E v is split, the space of N Ev -coinvariants is somewhat complicated; we refer the reader to [GGJ, Prop. 4.4(ii)].

8 8 WEE TECK GAN From this, one sees that there is an injection j v : Π J,v Ind HJ P J χ J δ sj P J (unnormalized induction) where χ J is trivial unless J v = E v is a cubic field, in which case χ J is one of the two cubic characters associated to E v. Further, for each non-trivial character χ of N J (F v ), one has: { 1, if χ Ω J (F v ); dimhom NJ(F v)(π J,v, C χ ) = 0, otherwise. A non-zero element of this Hom space for χ Ω J (F v ) is given by L 0 χ,v(f) = i v (f)(χ). Note that the stabilizer of χ in M J (F v ) acts on the above Hom space, and recall from the end of 2.2 that this stabilizer is a codimension one subgroup of (M J Q J )(F v ). Let R J,χ denote the derived group of this stabilizer (which is also the derived group of M J Q J ). Then one knows that R J,χ (F v ) acts trivially on Hom NJ(F v)(π J,v, C χ ) (cf. [GS2, 11]) A P J -model. The above gives a rather complete description of (Π J,v ) ZJ as a P J (F v )-module. On the other hand, if χ is a non-trivial character of Z H (F v ), then as a representation of the derived group P J,der of P J, (Π J,v ) ZH,χ = ω χ where ω χ is the Weil representation Mp(N J /Z J ) N J (with central character χ) pulled back to P H,der (F v ). Thus, we have a short exact sequence of P J (F v )-modules: In fact, one has 0 ind PJ P J,der ω χ Π J,v (Π J,v ) ZJ 0. ind PJ P J,der ω χ Π J,v Ind PJ P J,der ω χ. The latter space can be realized as the space of P J (F v )-smooth functions on F v taking values in ω χ, while the former is the subspace of compactly supported such functions A unitary model. We do not know how to characterize the subspace Π J,v in the above space of P J,v -smooth functions. However, we do know that the functions in Π J,v are square-integrable for the natural inner product on Ind PJ P J,der ω χ and its unitary completion is Π J,v = L 2 (F v ; ω χ ). On this unitary model, one can write down the action of H J (F v ) by giving the action of the Weyl group element w β (which together with P J (F v ) generates H J (F v )). Let us realize the unitary Weil representation ω χ on L 2 (F v J v ); this is the Schrodinger model. Then Π J,v can be realized on L 2 (F v F v J v ) and the action of P J (F v ) on this unitary model can be found in [Ru1, Prop. 43]. The action of w β is now given by [Ru1, Prop. 47]: ( ) NJ (x) (w β f)(t, a, x) = χ f( a, a, x). a t

9 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS A Q J -model. One can analogously give a model for Π J,v in which the action of Q J (F v ) can be neatly described. This is easier to describe when J v is a division algebra and so we shall restrict ourselves to this situation here. Recall that the center C J of U J is two-dimensional and contains Z J. Indeed, C J = Z J U 3α+β. Moreover, the Levi subgroup L J (F v ) acts transitively on the non-trivial characters of C J (F v ). Let χ be the character of C J (F v ) defined by { χ ZJ = trivial; χ U3α+β = ψ v. The stabilizer in Q J of the line spanned by χ is precisely the minimal parabolic F-subgroup B J = P J Q J. Now by a version of Mackey theory, we have the following short exact sequence of Q J (F v )-modules: 0 ind QJ B J (C c (F v ) (Π J,v ) CJ,χ)) Π J,v (Π J,v ) CJ 0. To make this completely explicit, it remains to describe the B J -action on C c (F v ) (Π J,v ) CJ,χ and the structure of (Π J,v ) CJ. For this, note that both (Π J,v ) CJ,χ and (Π J,v ) CJ are quotients of (Π J,v ) ZJ. From our description of (Π J,v ) ZJ given in (3.1) and Lemma 2.1(i), it is now easy to see that (Π J,v ) CJ,χ = C c (Ω J,v ) and (Π J,v) CJ = (Π J,v ) UJ = (Π J,v ) NJ where Ω J,v = {(N J(x), x #, x, 1) : x J v } = J v. Thus Cc (F v ) (Π J,v) CJ,χ) = Cc (Ω J,v) = Cc (F v J v) and the action of B J (F v ) P J (F v ) is simply given by the restriction of the action of P J (F v ) on Cc (Ω J,v) described in (3.1) Archimedean case. Now assume that v is an archimedean place of F, so that J v and H J,v are split. In this case, instead of working with (admissible) (g, K)-modules, we shall largely work with their Casselman-Wallach globalizations which are smooth Frechet representations of moderate growth. Similarly, when we work with automorphic forms later on, we shall work with smooth automorphic forms of uniform moderate growth [W], rather than their K-finite versions. For a reference for this, see [W] or [C]. Let us recall the main facts we need concerning the minimal representation of H J (F v ). As in the non-archimedean case, the minimal representation Π J,v admits a unique (up to scaling) embedding j v : Π J,v Ind HJ P J χ J δ sj P J (unnormalized induction), where χ J is the trivial character here. Moreover, for each non-trivial character χ of N J (F v ), one has Hom NJ(F v)(π J,v, C χ ) 0 χ Ω J (F v ), where the Hom space considered here refers to the space of continuous linear functionals. When χ Ω J (F v ), one may extend χ to a character of R J,χ (F v ) N J (F v ) by insisting that R J,χ (F v ) acts trivially. Then using Bruhat theory (cf. [GRS, Thm. 6.2] for a similar proof), one can show the multiplicity-one result: dimhom RJ,χ N J(F v)(π J,v, C χ ) = 1. We shall denote a non-zero element of this Hom space for χ Ω J (F v ) by L 0 χ,v.

10 10 WEE TECK GAN 3.6. Automorphic realization of Π J. Now we consider the global setting. Let Π J = v Π J,v be the global minimal representation of H J (A). From the local results above, there is a H J (A)-equivariant embedding j : Π J I J (s J ) := Ind HJ P J χ J δ sj P J. Because Π J is self-contragredient, we have j : I J (1 s J ) = Ind HJ P J χ 1 J δ1 sj P J Π J. One can consider the Eisenstein series E(f, s, g) associated to the degenerate principal series I J (s). In this regard, we should mention that one can work with not-necessarily-k-finite Eisenstein series here, in view of a recent result of Lapid [La] which extends some standard results in the theory of Eisenstein series (such as meromorphicity and functional equation) from the K-finite setting to the smooth setting. Proposition 3.1. For f I J (1 s J ), the Eisenstein series E(f, s, g) has a pole of order 1 at s = 1 s J. The map f Res s=1 sj E(f, s, ) factors through j and induces a non-zero embedding of Π J into the space A 2 (H J ) of square-integrable automorphic forms on H J (A). This proposition was shown in an unpublished manuscript [Ru2] of Rummelhart when J = D. When J = E, it was shown in [GGJ, Prop. 5.1(ii)]. Using the functional equation satisfied by the Eisenstein series, we obtain: Corollary 3.2. If f Π J, then E(j(f), s, g) is holomorphic at s = s J and the map f E(j(f), s J, ) gives a non-zero H J -equivariant embedding θ J : Π J A 2 (H J ). Proof. If Φ s I J (s) is a flat section, then the functional equation for Eisenstein series gives: where E(M(s)Φ, 1 s, ) = E(Φ, s, ) M(s) : I J (s) I J (1 s) is a standard intertwining operator. At s = 1 s J, both E(Φ, s, ) and M(s) have a pole of order 1 (for some Φ). Moreover, the residue M (1 s J ) of M(s) at s = 1 s J is (up to scalars) the map j j, so that M (1 s J )(Φ) = j(j (Φ)) I J (s J ). Taking residues at s = 1 s J on both sides of the functional equation and using the proposition, we deduce the corollary Fourier coefficients of Π J. Let us fix a non-trivial unitary character ψ of F \A. Then the set of unitary characters of N J (A) which is trivial on N J (F) can be identified with V J (F) using ψ, the Killing form and the exponential map. In particular, it makes sense to say whether χ belongs to the minimal orbit Ω J (F). Suppose that χ is a character of N J (A) which is trivial on N J (F). Then for each f Π J, we may consider the Fourier coefficient θ J (f) NJ,χ of θ J (f) with respect to χ. It is a function on H J (A) given by θ J (f) NJ,χ(h) = χ(n) θ J (f)(nh)dn. The continuous linear functional on Π J given by N J(F)\N J(A) L χ : f θ J (f) NJ,χ(1)

11 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 11 is clearly an element of the space Hom NJ(A)(Π J, C χ ). Thus, in view of the local results of 3.1, the linear functional L χ is nonzero iff χ Ω J (F) or χ is trivial. Suppose then that χ Ω J (F). In this case, by the discussion at the end of 3.1, L χ is necessarily fixed by the finite-adelic group R J,χ (A fin ). Since L χ is clearly fixed by R J,χ (F), it follows by the weak approximation theorem that L χ is fixed by the adelic group R J,χ (A). In view of the local uniqueness results at the end of 3.1 and 3.5, we deduce that L χ is Eulerian, i.e. for a decomposable vector f = v f v Π J, one has L χ (f) = L 0 χ,v (f v) v where the L 0 χ,v s are the corresponding local functionals. In particular, we have: θ J (f) NJ,χ(h) = v L 0 χ,v(h v f v ) Representations of G 2. We introduce some notations for representations of G 2. If χ 1 and χ 2 are two characters of F v, we shall let π(χ 1, χ 2 ) denote the principal series of GL 2 unitarily induced from the character χ 1 χ 2 of the diagonal torus. We also let St denote the Steinberg representation. Recall that the Levi subgroups M and L of the two parabolic subgroups are both isomorphic to GL 2. We fix these isomorphisms so that δ P = det 3 and δ Q = det 5. Now if τ is a representation of GL 2 and R = P or Q, then we set I R (τ, s) = Ind G2 R δ1/2 R (τ det s )). If I R (τ, s) has a unique Langlands quotient, we denote this quotient by J R (τ, s). If τ is the trivial representation 1, we shall simply write I R (s) for I R (1, s). We shall also denote by R P or R Q the normalized Jacquet functor with respect to P and Q respectively. Thus, for example, R P (π) = δ 1/2 P π N. Similarly, if B = P Q is the Borel subgroup of G 2 and χ is a character of T = M L, we write I B (χ) for the normalized induction of χ. We shall be especially interested in the degenerate princiapl series I P (1/2). The following lemma describes the structure of this representation (cf. [M] for non-archimedean cases and [G2, Lemma 4.8 and 4.10] for archimedean cases). Lemma 3.3. (i) If v is p-adic, then I P,v (1/2) has a filtration such that 0 I 0,v I 1,v I 2,v = I P,v (1/2) I 0,v = Σv I 1,v /I 0,v = JQ,v (St, 1/2) I 2,v /I 1,v = JQ,v (π(1, 1), 1). Moreover, Σ v is the unique irreducible submodule and is a discrete series representation with 1- dimensional space of Iwahori-fixed vectors. The representation J Q,v (π(1, 1), 1) is the unique irreducible quotient. (ii) If v is real, then there is a non-split exact sequence 0 π κ I P,v (1/2) J Q,v (π(1, 1), 1) 0.

12 12 WEE TECK GAN (iii) If v is complex, I P,v ( 1 2 ) = J Q,v(π(1, 1), 1) is irreducible. (iv) In each case, the unique irreducible quotient J Q,v (π(1, 1), 1) is spherical Distinguished representations. Using the fixed additive character ψ v and the Killing form, we may identify V with the set of unitary characters of N(F v ). As we noted above, the M(F v )-orbits on V are indexed by cubic F v -algebras. If B v is an étale cubic algebra, let ψ Bv be a character of N(F v ) in the orbit indexed by B v. If π v is a representation of G 2, we say that π v is B v -distinguished if { 0, if B v Hom N (π v, C ψb ) = B v; v non-zero, if B v = B v. For example, it is known that the representation J Q,v (π(1, 1), 1) is Fv 3 -distinguished (cf. [GGJ]) The representation Σ v. We conclude this preliminary section with a closer look at the representation Σ v. The representation Σ v is a discrete series representation, but the key fact we need is that Σ v has a 1-dimensional space of Iwahori-fixed vectors. Let K v = G 2 (O Fv ) be a hyperspecial maximal compact subgroup of G 2 (F v ), and let K 1 be its first principal congruence subgroup so that K v /K 1 = G2 (F q ). Because Σ v has a 1-dimensional space of Iwahori-fixed vectors, Σ K1 v is an irreducible representation of G 2 (F q ) with a 1-dimensional space of Borel-fixed vectors. In particular, Σ K1 v occurs with multiplicity one in Ind G2(Fq) B(F q) (1) = I B,v(χ) K1 for any unramified character χ of the Borel subgroup B(F v ). We shall need the following proposition later when we study the effect of the standard intertwining operators on Σ v. Proposition 3.4. (i) Consider the natural Q(F q )-equivariant map Σ K1 v Ind G2(Fq) B(F q) 1 IndL(Fq) (B L)(F (1) q) = 1 St L, where the second map is restriction of functions and St L is the Steinberg module of L(F q ). The image of this map is equal to the Steinberg module St L. (ii) Similarly, the image of the P(F q )-equivariant map Σ K1 v is contained in the trivial submodule 1. Ind G2(Fq) B(F q) 1 IndM(Fq) (B M)(F (1) q) = 1 St M, Proof. (i) Clearly, this map is non-zero. If the image includes the trivial representation, then we would have a non-zero Q(F q )-equivariant projection Σ v 1. This would imply that Σ v has a non-zero Q(F q )-fixed vector f. This vector f must be the unique (up to scaling) B(F q )-fixed vector. Now because Σ K1 v I P,v (1/2) K1 = Ind G2(Fq) P(F q) (1), we see that the B(F q )-fixed vector in Σ K1 v is in fact fixed by P(F q ). Since P(F q ) and Q(F q ) generate G 2 (F q ), the vector f would be fixed by G 2 (F q ), contradicting the fact that Σ K1 v is irreducible and non-trivial. (ii) This follows immediately from the fact that Σ K1 v Ind G2(Fq) P(F q) 1,

13 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS Local Theta Lifts In this section, we recall various results concerning local theta correspondence for the two dual pairs of interest in this paper. The key result to take note here is Proposition Lifting from S E to G 2. Consider first the dual pair S E G 2. For each place v of F, we may write: Π E,v = η v η v θ Ev (η v ) for some representations θ Ev (η v ) of G 2 (F v ), as η v ranges over the irreducible characters of S E (F v ). By results of Huang-Magaard-Savin [HMS, Prop. 6.1] and Vogan [V, Thm ], we have: Proposition 4.1. (i) Each θ Ev (η v ) is non-zero and irreducible. (ii) If E v is split, then θ Ev (1 v ) = J Q,v (π(1, 1), 1). (iii) If E v is a field (so that v is finite) and {χ E,v, χ 1 E,v } the two cubic characters of S E(F v ) = Gal(E v /F v ), then θ Ev (1 v ) = J Q,v (π(χ E,v, χ 1 E,v ), 1). Further, θ Ev (χ E,v ) and θ Ev (χ 1 E,v ) are supercuspidal and contragredient to each other. When E v is unramified, θ Ev (χ E,v ) and θ Ev (χ 1 E,v ) are obtained by compact induction: θ Ev (χ E,v ) = ind G2(Fv) G 2(O Fv ) π(χ E,v) and θ Ev (χ 1 E,v ) = ind G2(Fv) G 2(O Fv ) π(χ 1 E,v ) where π(χ E,v ) and π(χ 1 E,v ) are two unipotent supercuspidal representations of G 2(F q ) and are the minimal K v -types of θ Ev (χ E,v ) and θ Ev (χ 1 E v ) respectively. Remarks: The results of [HMS] were established under the assumption that the residual characteristic of F v is not equal to 2. This is because the authors made use of a result of Moeglin-Waldspurger relating the leading term of the local character expansion of a representation to the twisted Jacquet modules supported by the representation. In fact, one does not need to appeal to the local character expansion to obtain control of the possible twisted Jacquet modules supported by the minimal representation, as explained in [GS2]. With this knowledge of the twisted Jacquet modules of the minimal representation, the results of [HMS] can be shown regardless of residual characteristic. In view of this proposition, we have the global decomposition Π E = η η θ E (η) as η = v η v runs over the irreducible characters of S E (A), and θ E (η) = v θ E,v (η v ). If we realize Π E as a submodule in A 2 (H E ) using θ E, we may consider the restriction of f Π E to the subgroup G 2. The following proposition was shown in [GGJ, Thm. 8.2(i)]:

14 14 WEE TECK GAN Proposition 4.2. The restriction of functions from H E (A) to G 2 (A) gives a G 2 (A)-equivariant map Moreover, the image is isomorphic to Π E A 2 (G 2 ). η SE(F) θ E (η). η In particular, the χ D E -isotypic part Π E[χ D E ] of Π E is an irreducible G 2 -module and admits an embedding into A 2 (G 2 ) Lifting from PGL 3 to G 2. Suppose that D v is split, so that we are looking at the dual pair PGL 3 G 2 H Dv. If v is non-archimedean, the theta correspondence for this dual pair was essentially completely determined in [GS1]. For archimedean v s, the results are somewhat fragmentary. Hence, we shall focus on the non-archimedean case here. For each irreducible representation π v of PGL 3, the maximal π v -isotypic quotient of Π Dv has the form π v Θ Dv (π v ) for some smooth representation Θ Dv (π v ) of G 2 (F v ). We let θ Dv (π v ) be the maximal semisimple quotient of Θ Dv (π v ). We are interested in the cases when π v is trivial or is a cubic character χ Ev. The following proposition was shown in [GS1]: Proposition 4.3. Let v be a finite place of F. (i) If π v = 1 v, then and thus is the unique irreducible quotient of Θ Dv (1 v ). (ii) If π v = χ E,v is a cubic character, then Θ Dv (1 v ) = I P,v (1/2) θ Dv (1 v ) = J Q,v (π(1, 1), 1) θ Dv (χ E,v ) = J Q,v (π(χ E,v, χ 1 E,v ), 1) Remarks: Indeed, if we realize Π Dv as a submodule of I D (2/11), then by restriction of functions, we have a natural map Res v : Π Dv I D (2/11) I P,v (1/2). This map is clearly PD v -invariant and G 2 -equivariant. Moreover, it is surjective, since the spherical vector of I D (2/11) restricts to that of I P,v (1/2) and the latter is generated by its spherical vector. In view of Prop. 4.3(i), this natural map is precisely the projection of Π Dv onto its maximal 1 v -isotypic quotient. When v is archimedean, we still have this natural map. However, we don t know if I P,v (1/2) is the maximal 1 v -isotypic quotient or not Lifting from PD v to G 2. Now assume that D v is a division algebra, so that v is finite. In this case, since PD v is compact, we may write: Π Dv = π v π v θ Dv (π v ) as π v runs over the irreducible representations of PD v. The theta correspondence here has been investigated by Savin [Sa], who showed:

15 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 15 Proposition 4.4. Assume that D v is a division algebra. Then θ Dv (1 v ) = Σ v is the unique irreducible submodule of I P,v (1/2). Further, if χ E,v is a cubic character associated to the cyclic cubic field E v, then θ Dv (χ E,v ) and θ Dv (χ 1 E,v ) are irreducible supercuspidal representations which are contragredient to each other. Moreover, when E v is unramified, we have: θ Dv (χ E,v ) = ind G2(Fv) G 2(O Fv ) π(χdv E,v ) and θ D v (χ 1 E,v ) = ind G2(Fv) G 2(O Fv ) π((χdv E,v ) 1 ) where π(χ E,v ) and π(χ 1 E,v ) are the two unipotent supercuspidal representations of G 2(F q ) encountered in Prop. 4.1(ii) Remarks: As above, the projection Π Dv Σ v can be realized by Res v : Π Dv I PD,v I P,v (1/2). This time, however, the map is not surjective: its image is the unique irreducible submodule Σ v Identity of local theta lifts. In view of Propositions 4.1, 4.3 and 4.4, we have the following crucial proposition: Proposition 4.5. Assume that (i) v is a finite place; (ii) if D v is ramified, then E v is the unramified cubic field extension of F v. Then we have: as representations of G 2 (F v ). θ Dv (χ E,v ) = θ Ev (χ inv(dv) E,v ) This proposition is the local analog of the global Theorem 1.2. Ideally, we would like to remove the two conditions in the proposition, since it will shorten the proof of Theorem 1.2 considerably. Unfortunately, the local theta correspondence at archimedean places is not very well understood. However, we shall be able to finesse this local difficulty by using some interplay of local and global considerations. As for removing the second condition, we only have the following result to offer: Proposition 4.6. Assume that D v is a division algebra (so that v is finite). Then we have as representations of Q(F v ). θ Dv (χ E,v ) = θ Ev (χ inv(dv) E,v ) 4.7. An explicit isomorphism. We shall prove Prop. 4.6 by writing down an explicit isomorphism using the unitary model of Π J,v described in (3.3). Recall that we have Π D,v = L 2 (F v F v D v ) and Π E,v = L 2 (F v F v E v ). Let us focus on the χ E,v -isotypic space in Π D,v. The action of PD v on L 2 (F v F v D v ) is via its adjoint action on D v. Now a Schwarz function on D v which is a χ E,v -eigenform for PD v is completely determined by its restrictions to E v D v. Thus under restriction of functions, we have Now we can define a map S(D v )[χ E,v ] = S(E v )[χ Dv E v ]. Ξ : Π D,v [χ E,v ] Π E,v [χ Dv E v ]

16 16 WEE TECK GAN by Ξ(f)(t, a, x) = N E (x 0 ) v f(t, a, x). Here is the trace zero part of x E v. Then we have x 0 = x 1 3 T E(x) Proposition 4.7. Assume that D v is a division algebra. The map Ξ is bijective and Q(F v )-equivariant. Thus it gives an explicit isormorphism of Q(F v )-modules. θ Dv (χ E,v ) = θ Ev (χ inv(dv) E,v ) Proof. As we explained in (3.3), [Ru1, Props. 43 and 47] give the formulas for the action of a set of generators of G 2 (F v ) on both minimal representations. Using these formulas, it is straightforward to check the Q(F v )-equivariance. We omit the details since the result is not used in the rest of the paper Remarks: (i) By composing Ξ with the projection map from Π D,v to its χ E,v -isotypic subspace, we have a map : Π D,v Π D,v [χ E,v ] This is given by: (f)(t, a, x) = N E (x 0 ) E v \D v Ξ Π E,v [χ Dv E v ]. f(t, a, h 1 xh) χ E,v (N Dv (h))dh. Compare this with the map in Snitz s thesis [S] described after Theorem 1.1. (ii) What prevents one from showing that Ξ is G 2 -equivariant? The only remaining thing to check is that Ξ is equivariant for the action of the Weyl group element w α. According to [Ru1, Prop. 43], the action of w α on Π J,v is essentially given by the Fourier transform over F v J v. Thus the identity reduces to an equality of the type: Ξ(w α f) = w α Ξ(f) (Fourier transform of f along D v ) Ev = Fourier transform of f v Ev along E v. In Snitz s thesis [S], one has the quadratic analog of this identity, with D v a quaternion algebra [S, Prop. 46]. The verification of this identity in [S] is one of the most non-trivial computation there (see [S, Pg ]). Unfortunately, because of our unfamiliarity with cubic division algebras, we are not able to verify the desired identity in the cubic case by a direct computation.

17 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 17 In this section, we consider the dual pair 5. Siegel-Weil Formula PD G 2 H D and proves the honest Siegel-Weil formula which identifies the theta lift of the trivial representation of PD. Recall that Corollary 3.2 gives us an embedding We may thus consider the theta integral I(f)(g) = θ D : Π D A 2 (H D ). G D(F)\G D(A) θ D (f)(gh)dh which converges absolutely and defines an element of A(G 2 ). Indeed, I defines a G D (A)-invariant and G 2 (A)-equivariant map from Π D to A(G 2 ). Here, the Haar measure dh on G D (A) is chosen so that dh = 1. G D(F)\G D(A) On the other hand, we have another such map from Π D to A(G 2 ) given as follows. There is a natural map Res : Π D I D (2/11) I P (1/2) given by restriction of functions. The map Res : Π D I P (1/2) is non-zero, G D (A)-invariant and G 2 (A)-equivariant. The main theorem of this section is: Theorem 5.1. For any f Π D, the Eisenstein series E(Res(f), s, g) is holomorphic at s = 1 2 and E(Res(f), 1/2, ) is nonzero if and only if Res(f) is nonzero. Moreover, I(f)(g) = E(Res(f), 1/2, g). The rest of the section is devoted to the proof of this theorem Constant terms of Π D. We begin with the left hand side of the Siegel-Weil formula, i.e. the theta integral. For this, we need to recall some properties of the automorphic realization θ D of the minimal representation. In particular, we need to know the constant terms of θ(π D ) along the unipotent radicals of the two maximal parabolic subgroups of H D. The following proposition is proved in the course of constructing the automorphic realization θ D of Π D : Proposition 5.2. For f Π D I D (2/11), θ D (f) ND (g) = f(g). When pulled back to a function on GL 2 (SL 1 (D) µ3 SL 1 (D)), θ(f) UD is the product of an Eisenstein series of GL 2 and a constant function on SL 1 (D) µ3 SL 1 (D).

18 18 WEE TECK GAN 5.2. The representation Res(Π D ). Now we come to the Eisenstein series side of the Siegel-Weil formula. The following lemma describes the image of the restriction map Res and is a consequence of the remarks in (4.3) and (4.5). Lemma 5.3. Let S D be the set of places v where D v is non-split. Then the image of Res is: ( ) Res(Π D ) = Σ v I P,v ( 1 2 ). v S D v/ S D We note that #S D The Eisenstein series E(Res(f), s, g). Let us consider the Eisenstein series E(Φ, s, g) associated to a standard section Φ s I P (s). It was shown by Ginzburg-Jiang [GJ] that E(Φ, s, g) has at most a double pole at s = 1/2 and this double pole is attained, for example, by the spherical section. However, in view of Lemma 5.3, we are only interested in those standard sections which pass through Res(Π). For these sections, the leading term of the Laurent expansion vanishes and so we are led to study the higher order terms in the Laurent expansion E(Φ, s, g) = A 2(Φ) (s 1 + A 1(Φ)(g) 2 )2 s Proposition 5.4. (i) The maps A 2 and A 1 are both zero on Res(Π D ), but A 0 is non-zero on Res(Π D ). (ii) For f Π D, the constant term of A 0 (Res(f)) along N is given by: A 0 (Res(f)) N (g) = Res(f)(g). (iii) The map A 0 : Res(Π D ) A(G 2 ) is G 2 -equivariant. Proof. The proof of (i) and (ii) is a fairly standard exercise in the theory of Eisenstein series, albeit a non-trivial and somewhat intricate one. To simplify notation, let us write S for S D. We shall work with a distinguished standard section Φ s = v Φ v,s defined as follows. For v S, let Φ v,s be the standard section associated to a non-zero vector in the 1-dimensional space of Iwahori-fixed vectors. For v / S, we let Φ v,s be the spherical section. By Lemma 5.3, the representation Res(Π D ) is generated by the distinguished vector Φ 1/2. Thus it suffices to show E(Φ, s, g) is holomorphic at s = 1/2 with constant term Φ 1/2 (g). The constant term of E(Φ, s, g) along the unipotent radical N 0 of the Borel subgroup B is given by [MW, Pg , Prop. II 1.7]: E N0 (Φ, s, g) = Φ s (g) + M(w β, s)(φ)(g) + M(w α w β, s)(φ)(g)+ +M(w β w α w β, s)(φ)(g) + M(w α w β w α w β, s)(φ)(g) + M(w β w α w β w α w β, s)(φ)(g). Here, w α and w β are representatives in G 2 (Z) of the simple reflections in the Weyl group of G 2 associated to the simple roots α and β. Moreover, M(w, s) is the standard intertwining operator associated to an element w in the Weyl group and we are interested in its analytic behaviour at s = 1/2. Indeed, since I P (1/2) I B (χ 0 )

19 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 19 where χ 0 = α + β, M(w, 1/2) is the restriction to I P (1/2) of the standard intertwining operator M(w, χ 0 ) : I B (χ 0 ) I B (w χ 0 ). These intertwining operators factor as: M(w, s)(φ)(g) = M S (w, s)(φ S )(g S ) M S (w, s)(φ S )(g S ) where M S (w, s)(φ S )(g S ) = M v (w, s)(φ v )(g v ) v S and M S (w, s)(φ S )(g S ) = v/ S M v (w, s)(φ v )(g v ). As discovered by Langlands in his classic monograph Euler Products [L], the value of M S (w, s)(φ S )(1) can be calculated by the Gindikin-Karpelevich formula, yielding a product of local zeta functions. These unramified calculations were carried out by Ginzburg-Jiang in [GJ]. We tabulate the results below, describing the behaviour of M S (w, s)(φ S )(1) at s = 1/2. w M S (w, 1 2 )(ΦS )(1) Behaviour at s = holomorphic and non-zero ζ w S (1) β ζ S (2) pole of order 1 ζ w α w S (1) β ζ S (2) ζs (2) ζ S (3) pole of order 1 w β w α w β ζ S (1) ζ S (2) ζs (2) ζ S (3) ζs (1) ζ S (2) pole of order 2 w α w β w α w β ζ S (1) ζ S (2) ζs (2) ζ S (3) ζs (1) ζ S (2) ζs (1) ζ S (2) pole of order 3 w β w α w β w α w β ζ S (1) ζ S (2) ζs (2) ζ S (3) ζs (1) ζ S (2) ζs (1) ζ S (2) ζs (0) ζ S (1) zero of order #S 3 Observe that if S is empty and Φ is the spherical section, the last 2 terms in the table have poles of order 3. However, because the residues of the complete zeta function at s = 0 and s = 1 sum to zero, the Eisenstein series only has a pole of order at most 2 at s = 1/2. At the places in S, the local intertwining operators can be analyzed by decomposing them as a composition of operators associated to the simple reflections w α and w β, and then appealing to results about intertwining operators for SL 2. We tabulate the results below before explaining how they are proved. w Behaviour of M S (w, 1 2 )(Φ S) at s = holomorphic and non-zero w β zero of order #S w α w β zero of order of order #S w β w α w β zero of order 2 #S w α w β w α w β zero of order 2 #S w β w α w β w α w β zero of order #S

20 20 WEE TECK GAN By the above two tables, and using the fact that #S 2, one sees easily that if w 1, M(w, s)(φ)(g) vanishes at s = 1/2. This proves (i) and (ii) of the proposition. To deduce (iii), one considers the difference F(h) = A 0 (Res(g f))(h) A 0 (Res(f))(hg) as a function of h (with g fixed). It is an automorphic form whose cuspidal support is the Borel subgroup. However, by (ii), its constant term along N 0 is (g f)(h) f(hg) = 0. Thus F is zero and (iii) is proved. Hence, the proposition is proved as long as we can justify the second table above. This justification is the main difficulty of the proof. The key inputs are Prop. 3.4 and the following simple observation. Observation: Let w γ be a reflection in a simple root γ and let G γ be the associated SL 2. Suppose that f I B (χ) and consider M(w γ, χ) : I B (χ) I B (w γ χ). Let M (w γ, χ) be the leading term in the Laurent expansion of M(w γ, χ) at χ, so that M (w γ, χ) is a G 2 -equivariant map. Then M v (w γ, χ)(f) = 0 if and only if for all f in the K v -span of f, the restriction of f to G γ is in the kernel of the standard intertwining operator on G γ = SL2. (Here, we recall from 3.10 that K v = G 2 (O v ) is a hyperspecial maximal compact subgroup of G 2 (F v )). Let us fix v S. We shall exploit the above observation by taking f = Φ v to be the Iwahori-fixed vector in Σ v, so that the K v -span of Φ v is Σ K1 v. We consider each Weyl group element in turn. w = w β. Consider the simple reflection w β. The restriction of Φ v to G β lies in the principal series of SL 2 which has the trivial representation as the quotient and the Steinberg representation as the submodule. Thus M v (w β, s)(φ v ) is holomorphic at s = 1/2, and is either non-zero or has a zero of order 1 there. We need to show that it has a zero of order 1. By the observation, we need to show that the restriction of Σ K1 v to G β lies in the Steinberg submodule. But this is precisely what Prop. 3.4 tells us. We have thus proved the desired result when w = w β. Consider the derivative M v(w β, 1/2) of M w (w β, s) at s = 1/2. It gives a map M v (w β, 1/2) : I B (χ 0 ) I B (w β χ 0 ). This map is the second term in the Laurent expansion of M v (w β, s) at s = 1/2 and so it is not G 2 (F v )-intertwining. However, it is still K v -intertwining and is non-zero when restricted to Σ v. Thus we have a K v -intertwining map M v(w β, 1/2) : I B (χ 0 ) K1 I B (w β χ 0 ) K1. For any unramified character χ of B, we have a canonical identification Thus we get a G 2 (F q )-equivariant map A key point here is that since Σ K1 v a non-zero scalar on Σ v. I B (χ) K1 = Ind G2(Fq) B(F q) 1. M v (w β, 1/2) : Ind G2(Fq) B(F q) 1 IndG2(Fq) B(F 1 q) occurs with multiplicity one in I B (χ) K1, this map acts by

21 A SIEGEL-WEIL FORMULA FOR AUTOMORPHIC CHARACTERS 21 w = w α w β. Now consider M v (w α w β, s)(φ v ). We may factor M v (w α w β, χ 0 ) as M v (w α, w β χ 0 ) M v (w β, χ 0 ). Because M (w β 1/2) acts as a non-zero scalar on Σ K1 v, we now need to know the effect of M v (w α, w β χ 0 ) on Σ K1 v I B (w β χ 0 ) K1. When one restricts the elements of I B (w β (χ 0 )) to G α, one gets an irreducible principal series at which the standard intertwining operator of SL 2 is holomorphic. Thus M v (w α, w β χ 0 ) is holomorphic and non-zero. Hence M v (w α w β, s)(φ v ) also has a zero of order 1 at s = 1/2. w = w β w α w β. Next we consider the operator M v (w β, w α w β χ 0 ). The restriction of the elements of I B (w α w β χ 0 ) to G β again lies in the reducible principal series with the trivial representation as a quotient. By the observation above and Prop. 3.4 again, we see that this operator has a zero of order 1 when restricted to the space Σ K1 v I B (w α w β χ 0 ). Thus M v (w β w α w β, s)(φ v ) has a zero of order 2 at s = 1/2. w = w α w β w α w β. Consider the operator M v (w α, w β w α w β χ 0 ). The restriction of the elements of I B (w β w α w β χ 0 ) to G α lies in the reducible principal series with the trivial representation as a quotient. On the other hand, by Prop. 3.4, the restrictions of the elements in Σ K1 v are K α -spherical. Thus this operator is holomorphic and non-zero when restricted to the space Σ K1 v and M v (w α w β w α w β, s)(φ v ) has a zero of order 2 at s = 1/2. w = w β w α w β w α w β. Finally, we consider M v (w β, w α w β w α w β χ 0 ). The restriction of the elements of I B (w α w β w α w β χ 0 ) to G β lies in the unitary principal series (induced from the trivial character of the torus). The corresponding SL 2 -standard intertwining operator has a pole of order 1. Thus, M v (w β w α w β w α w β, s)(φ v ) has a zero of order 1. This completes our justification of the second table above and proves the proposition Proof of Theorem 5.1. We can now give the proof of Theorem 5.1. Writing I(f) as the sum of its Eisenstein and cuspidal components: I(f) = I(f) Eis + I(f) cusp, we first show that I(f) Eis = A 0 (Res(f)). For this, it suffices to show that { I(f) N = A 0 (Res(f)) N I(f) U = A 0 (Res(f)) U. and It is not difficult to see that I(f) N (g) = Now the equality I(f) U (g) = G D(F)\G D(A) G D(F)\G D(A) θ(f) ND (gh)dh θ(f) UD (gh)dh. I(f) N (g) = A 0 (Res(f)) N (g) is an immediate consequence of Propositions 5.4 and 5.2, since both sides are equal to f(g). This implies that I(f) U A 0 (Res(f)) U is a cusp form on L. But Prop. 5.2 shows that I(f) U is orthogonal to cusp forms and clearly so is A 0 (Res(f)) U. Thus, we deduce that I(f) U A 0 (Res(f)) U = 0 and I(f) Eis = A 0 (Res(f)).