Similitude exceptional theta correspondences

We construct and develop a similitude version of exceptional theta correspondences and show that the Howe duality theorem follows from that for the"isometry"case. We also extend basic tools such as the seesaw identity associated to seesaw dual pairs to the similitude setting.


Introduction
Let F be a local field and suppose (G, H) is a reductive dual pair in simple linear algebraic group E over F .Hence, G and H are subgroups of E which are mutual centralizers of each other.We thus have a homomorphism of algebraic groups If Ω is a minimal representation of E(F ), we may pull back Ω to the product group G(F ) × H(F ) and study the resulting branching problem: this is the usual set up of theta correspondence.
However, the map i is frequently not injective.Typically, one has (multiplicative type) central subgroups Z ֒→ G and Z ֒→ H which are identified under i.More precisely, one has Ker(i) = Z ∇ = {(z, z −1 ) : z ∈ Z}, so that one has an injecion (G × H)/Z ∇ ֒→ E. This already occurs in the setting of the classical theta correspondence for the symplecticorthogonal dual pair i : O(V ) × Sp(W ) −→ Sp(V ⊗ W ) whose kernel is the diagonally embedded Z = µ 2 .
In this short note, we explain how the non-injectivity of i can be exploited to extend the theta correspondence for G(F ) × H(F ) to the setting of similitude groups associated to G and H. Indeed, as we shall see, this non-injectivity is the reason for the existence of such a theory.
Here is a brief summary of the content of the paper.In §2, we associate to a dual pair G × H −→ E as above, and some additional data, a similitude dual pair G × H.After introducing the associated similitude version of theta correspondence in §3 and demonstrating some simple properties of this similitude theta correspondence in §4, we show in §5 that the Howe duality theorem holds for G × H if and only if the analogous theorem holds for G × H; this is Proposition 5.1 and should be considered the main result of this paper.The rather 2000 Mathematics Subject Classification.11S90, 17A75, 17C40.
long §6 is devoted to highlighting a few families of examples in the context of exceptional groups.In §7, we discuss how the useful notion of seesaw duality can be extended to the similitude setting, using the families of examples discussed in §6 as illustration.Finally, §8 discusses the similitude theta correspondences in the global setting and §9 collects together some basic results in Clifford theory that are used in the paper.

Similitude Groups
Let us first introduce the notion and construction of the relevant similitude groups.We will continue to work in the context of the introduction.
2.1.Initial data.The initial data needed for the construction of similitude groups is described in the following hypotheses: (a) the commutative group Z can be embedded into an induced torus: where K is an étale F -algebra of finite rank.

(b) One has
S := T /j(Z) ֒→ Res E/F (G m ), for some E/F .We shall fix these data in what follows.

Examples. Let us give some examples:
(1) when Z = r i=1 µ n i , one takes embeddings µ n i ֒→ G m , so that one has where the map T → T is z → z n i on the i-th coordinate.
(2) when Note that in this last example, the sequence above is not exact on the right, as the image of the last arrow is a codimension 1 torus.Consider however the special case when |n − [K : F ]| = 1.In this case, it turns out that the composite is an isomorphism of algebraic tori.Indeed, if x ∈ Res K/F (G m ) lies in the kernel of the above composite map, so that then on taking norms on both sides, we conclude that and hence In the examples we shall consider later, the rather peculiar condition |n − [K : F ]| = 1 will turn out to be satisfied.
2.3.Similitude groups.Now we make the following definition: Definition 2.1.The similitude group G associated to G and the data in (a) and (b) above is: The associated similitude homomorphism is the map given by the second projection and (b).
Then observe that one has: associated to H and the data in (a) and (b), as well as analogs of the two exact sequences above.
2.4.Surjectivity of similitude factor.Recalling that S = T /j(Z), we have a commutative diagram of short exact sequences of algebraic groups: On taking F -points, we obtain the following commutative diagram with exact rows: From this, we see that the similitude map sim G : G(F ) −→ S(F ) is surjective if and only if the map S(F ) −→ H 1 (F, G) is 0. Since the latter map factors as: Proposition 2.2.The map sim G : G(F ) −→ S(F ) is surjective if one of the following conditions holds: Observe that the first condition in the proposition depends only on the initial data (a) and (b) in §2.1, whereas the second condition depends only on the pair Z ⊂ G.

2.5.
Note that one has exact sequences: One denotes the image of the last arrow in each sequence above by H(F ) + and G(F ) + respectively.Then one has the following containments with finite index: In particular, G(F ) Definition 2.3.We shall call the pair of groups ( G(F ) + , H(F ) + ) constructed above a similitude dual pair.

Similitude Theta Correspondence
We now consider the theory of theta correspondence.Suppose that Ω is a minimal representation of E(F ) [GS].If π ∈ Irr(G(F )), then the big theta lift of π to H(F ) is by definition which is a smooth representation of H(F ).Likewise, one has a smooth G(F )-representation Θ(σ) for any σ ∈ Irr(H(F )).
Definition 3.1.We say that the dual pair (G, H) satisfies the Howe duality property, or the Howe duality theorem holds for (G, H), if Θ(π) is of finite length with a unique irreducible quotient (if nonzero) for any π ∈ Irr(G(F )) and likewise for Θ(σ) for any σ ∈ Irr(H(F )).
We would now like to extend the theta correspondence for (G(F ), H(F )) to the setting of the similitude dual pair ( G(F ) + , H(F ) + ).For this, we observe that one sees that the natural inclusion gives an isomorphism In particular, one has a natural homomorphism ι : Jsim ։ J sim ֒→ E.
On taking F -points and noting that T is an induced torus, so that H 1 (F, T ) is trivial, we see that Now one has the following short exact sequences: Given an irreducible representation π of G(F ) + , we may regard π∨ as a representation of Jsim (F ) via the natural surjection Jsim (F ) → G(F ) + given in the second short exact sequence above.Noting that G(F ) is a normal subgroup of Jsim (F ) by the first short exact sequence above, we set Θ(π) := (Ω ⊗ π∨ ) G(F ) , so that Θ(π) is a smooth representation of H(F ) + .Likewise for σ ∈ Irr( H(F ) + ), one has which is a smooth representation of G(F ) + .
One can rephrase the above definitions slightly by introducing the similitude minimal representation.Set Ω = ind Then Ω is a representation of the similitude dual pair G(F ) + × H(F ) + and we can defined Θ(π) and Θ(σ) in the usual way.We leave it to the reader to verify that the two definitions are the same.

Central Characters
Let us record some simple properties of the similitude theta correspondence.Proof.This is because T ∇ (F ) ⊂ Jsim (F ) acts trivially on Ω and so T (F ) ⊂ H(F ) + acts on in the same way as T (F ) ⊂ G(F ) + acts on π.
One can also decompose the J(F ) + -module Ω according to characters of the central subgroup T (F ) × T (F )/T ∇ (F ).Each character of T (F ) × T (F )/T ∇ (F ) is of the form χ ⊗ χ, where χ is a character of T (F ).We can consider the χ-isotypic quotient ΩT,χ of Ω. which can be described as follows.
One has an intermediate group We may extend the Jsim (F )-module Ω to the intermediate group Jsim (F ) • (T (F ) × T (F )) by letting T (F ) × T (F ) acts by χ ⊗ χ.This extension is well-defined because χ ⊗ χ is trivial on the intersection Jsim (F ) We denote this extended representation by Ω χ Then one has ΩT,χ ≃ ind It will sometimes be convenient to restrict to a fixed T (F )-central character χ, in which case one will be working with ΩT,χ .

The Howe Duality Property
In this section, we shall show: Proposition 5.1.The Howe duality property holds for the dual pair (G(F ), H(F )) if and only if the Howe duality property holds for the similitude dual pair ( G(F ) + , H(F ) + ).
5.1.Some lemmas.We shall show this in a series of lemmas, beginning with the following observation: Lemma 5.2.Let V be a not-necessarily-smooth representation of G(F ) + on which the central subgroup T (F ) acts by a character χ.Then: where the V i 's are irreducible smooth G(F )-modules which are pairwise inequivalent.Moreover, G(F ) + permutes the isomorphism classes of the V i 's transitively.Next we have: for some π i ∈ Irr(G(F )).Then we have: In particular, Θ(π) is nonzero if and only if Θ(π i ) = 0 for some (equivalently for all) i.
Proof.We compute: Here, we have used Lemma 5.2(i) which ensures that the contragredient of π as a representation of G(F ) is the same as its contragredient as a representation of G(F ), so that This completes the proof of the lemma.
By Lemma 5.3 and Lemma 5.2, we deduce: Corollary 5.5.Θ(π) has finite length as an H(F ) + -module if and only if Θ(π i ) has finite length as an H(F )-module for all i.
Assuming this finiteness condition holds, one has: as H(F )-modules.
5.2.Proof of Proposition 5.1.We can now prove Proposition 5.1.Suppose first that the Howe duality property holds for the pair (G, H).By symmetry, for any π ∈ Irr( G(F ) + ), we need to show that if Θ(π) is nonzero, then it has finite length and θ(π) is irreducible.

Suppose that π|
Then by Corollary 5.5, By hypothesis, Θ(π i ) has finite length for each i, and hence so does Θ(π) as a H(F )-module by Corollary 5.5.Moreover, by Lemma 5.3, Θ(π i ) is nonzero for each i and σ i := θ(π i ) is irreducible by hypothesis.
Let σ ⊂ θ(π) be an irreducible summand.Since (by (5.6)) If m = 1, then θ(π) = σ and we would have been done already.For general m, we argue as follows.Switching the roles of G and H in the above argument, it follows by Corollary 5.5 that By hypothesis, each θ(σ i ) is irreducible and hence θ(σ i ) = π i .But since π ⊂ θ(σ), we deduce that m ≤ n.Hence, we must have m = n, so that θ(π) = σ is irreducible.
This completes the proof of Proposition 5.1.

Examples
We give some interesting examples of similitude dual pairs and similitude theta correspondences.
6.1.Classical dual pairs.We begin by revisiting the case of classical dual pairs.
• (Symplectic-orthogonal) For a quadratic space V and a symplectic vector space W , have ι : Then Ker(ι) = µ ∇ 2 .Hence, we have Z = µ 2 and may take T = G m , so that H = GO(V ) and G = GSp(W ) are the usual similitude groups.Then one has Recall however that the Weil representation Ω is not a representation of Sp(V ⊗W ) but rather of its metaplectic cover Mp(V ⊗ W ). In the classical theta correspondence, one needs to first construct liftings of ι to Mp(V ⊗ W ): before one can restrict the Weil representation to O(V ) × Sp(W ).Such splittings exist if dim V is even and have been systematically constructed by Kudla.Likewise, to obtain a similitude theta correspondence, we would need to extend ι to: Such extensions have been constructed by B. Roberts [R].This accounts for the main complexity in the theory of similitude theta correspondences for symplecticorthogonal dual pairs.
• (Unitary) If V is a Hermtitian space and W a skew-Hermitian space relative to a quadratic extension E/F , one has where we have written E 1 for Res 1 E/F (G m ).In this case, we take T = Res E/F G m to get the usual similitude groups G = GU(V ) and H = GU(W ).
As in the symplectic-orthogonal case, one needs to construct a lifting before one can consider the similitude theta correspondence.However, the unitary case is somewhat better than the symplectic-orthogonal one.
More precisely, note that has image contained in U(V ⊗ E W ).However, the metaplectic covering is split over U(V ⊗ E W )! By fixing such a splitting, we thus obtain Thus, there is no need to do extra work beyond that needed for splitting the isometry dual pairs.6.2.Exceptional dual pairs.Given the somewhat sporadic nature of the geometry of exceptional groups, it is not surprising that it is harder to formulate a uniform theory of dual pairs in exceptional groups.Nonetheless, we shall attempt to do so for a few families of such dual pairs, by realizing the dual pairs as for two algebraic structures A and B.
6.3.Composition algebras and cubic norm structures.We refer to reader to [KMRT,Chap. VIII,§33] for the notion of composition algebras and [KMRT,Chap. 9,§37 and §38] for the notion of cubic norm structures (also known as Freuthendal-Jordan algebras).Let C be a composition algebra over F , so that dim C = 1, 2, 4 or 8. Let J be a nontrivial cubic norm structure over F , so that J has dimension 3, 6, 9, 15 or 27.Then one has an isometry dual pair [MS, Ru] Aut(C) × Aut(J) ֒→ E where E is a certain ambient adjoint group.We enumerate the most interesting (split) cases, where C or J have the maximal dimensions.
-For dim C = 8, one has Aut(C) = G 2 and as J varies, one has the dual pairs with Aut(J) and E given in the following table.
dim J 3 6 9 15 27 Aut(J) -For dim J = 27, one has Aut(J) = F 4 and as C varies, one has the dual pairs with Aut(C) and E given in the following table.
Observe that the map from Aut(C) × Aut(J) to E is injective.Thus, in such cases, we do not have a theory of similitude theta correspondence.
6.4.Twisted composition algebras.Let E be an étale cubic F -algebra.Then one has the notion of twisted composition algebras with respect to E/F ; the reader can consult [KMRT,Chap. 8,§36] for this.One way such a twisted composition algebra arises is to start with a composition F -algebra C and consider C ♭ = C ⊗ F E. Then a construction in [KMRT, §36C, Pg 499] equips C ♭ with the structure of a twisted composition algebra, built out of the composition algebra structure on C. In fact, if C ♭ is any twisted composition algebra, then dim E (C ♭ ) = 1, 2, 4 or 8 [KMRT,Cor. 36.4,Pg 492].
Let C ♭ 1 and C ♭ 2 be two twisted composition algebras relative to E/F .Then one has a dual pair [GS2] i where E is some ambient adjoint group.We enumerate the most interesting case with dim E C ♭ 1 maximal.Then one has Aut E (C ♭ 1 ) = Spin E 8 , a simply-connected quasi-split group of type D 4 determined by E. As C ♭ varies, one has the dual pair ) and E given in the following table (assuming E = F 3 so that the groups are split).
Observe that the group Aut E (C ♭ ) has center isomorphic to where the last isomorphism is given by Moreover, the map i : ).Thus, in this case, we have a theory of similitude theta correspondence which we shall now explicate.
We first pick the data (a) and (b) in §2.1 as in Example (3) of §2.2.Namely, we take T = Res E/F (G m ) and consider the natural embedding As in §2.2, we then have the embedding T /j(Z) whose cokernel has dimension 1.In this special case, since 3 − 2 = 1, it turns out that ), Hence, the similitude maps sim G and sim H on the group of F -rational points both take value in T (F ) = E × .
We would like to know what the groups G(F ) + and H(F ) + are.If F is nonarchimedean, then H 1 (F, Spin E 8 ) = 0 and by Proposition 2.2, we see that the map sim G : G(F ) −→ E × is surjective.Hence H(F ) + = H(F ) when F is nonarchimedean.The same is true for G(F ) + if dim E C ♭ = 8.We shall now consider the question of whether sim H : H(F ) −→ E × is surjective when dim E C ♭ = 2 or 4, ignoring the somewhat degenerate case when dim E C ♭ = 1.For this, it is convenient to give an alternative description of H for which it is easier to describe H(F ).
Then the group H(F ) is given by and the similitude character is From this, it is clear that sim H is surjective, so that G(F ) + = G(F ).
• When dim E C ♭ = 2 and recalling that T = Res E/F (G m ), we have an isomorphism: With this alternative description of H0 , all tori involved are induced tori, so that H0 (F ) where F × × E × is embedded into (E × ) 3 as in (ii) above.In this incarnation, the similitude character is given by sim In particular, sim H is surjective onto E × , so that G(F ) + = G(F ) in this case as well.
In particular, let us consider the case where the central character is trivial.Then we obtain for example the similitude dual pairs of adjoint type: 6.5.Jordan pairs.We refer the reader to [L,Introduction and Chap. 1] for the notion of Jordan pairs.Since this notion is perhaps less familiar to the reader than the notion of Jordan algebras, let us give some motivation and a brief introduction.
A Freuthendal-Jordan algebra or a cubic norm structure J comes equipped with a norm form, which is a cubic form N J : J −→ F .The similitude group of this cubic form sim(J, is called the structure group of J, and we will call the isometry group iso(J, N J ) of N J the reduced or special structure group (this consists of those pairs (g, t) with t = 1).If J is the 27-dimensional Jordan algebra, for example, sim(J, N J ) is the group GE 6 and iso(J, N J ) is the simply-connected E 6 .Now the group sim(J, N J ) acts irreducibly on J and its linear dual J * , but these two representations are not isomorphic (as their central characters are different) and there is no reason to favour one of these representation over the other (a more familiar example is: a group isomorphic to GL(V ) has two standard representations).Indeed, the trace bilinear form on J allows us to identify J with J * , and this defines an outer automorphism of sim(J, N J ) which is the inverse map on the center of sim(J, N J ) and which interchanges the two representations.In addition, in the context of E 6 , when one considers a quasi-split E 6 associated to a quadratic field extension K/F , the two 27-dimensional representations are fused together to give a single rational representation over F .Hence, such a quasi-split E 6 cannot be realized as the isometry group of a cubic form over F , unlike the split form.
The theory of Jordan pairs, introduced by Loos [L], treats both these representations J and J * on equal footing and realizes sim(J, N J ) as the automorphism group of an algebraic structure on the pair {J, J * }.More formally, a Jordan pair over F consists of the data: • a pair (J + , J − ) of F -vector spaces; • a pair of quadratic maps defined over F : satisfying the following axioms [L,Pg. 1,Def. 1.2] for ǫ = ± and any F -algebra K: for any x ∈ J ǫ (K) and y, z ∈ J −ǫ (K); (JP2) {Q ǫ (x)(y), y, z} ǫ = {x, Q −ǫ (y)(x), z} ǫ for any x, z ∈ J ǫ (K) and y ∈ J −ǫ (K); (JP3) where we have set for the linearization of the map (x, y) → Q ǫ (x)(y).These axioms may look a bit unwieldy for the uninitiated (including ourselves), but we will not seriously make use of them in this paper.
A homomorphism from one Jordan pair (J + , J − ) to another (V + , V − ) is a pair of linear maps φ ǫ : ) for x ∈ J ǫ and y ∈ J −ǫ .Hence, one has a notion of the automorphism group of a Jordan pair (J + , J − ), which is a subgroup of GL(J + ) × GL(J − ).Via the projection onto each of the factors, Aut(J + , J − ) has two natural representations.6.6.Examples.We give two pertinent examples here: and define: This defines a Jordan pair whose automorphism group is the subgroup Thus, this gives a description of the general linear group without favouring one of its two standard representations.
• For a cubic norm structure J, the pair inherits the structure of a Jordan pair from its Jordan algebra structure and the automorphism group of (J + , J − ) is precisely the structure group of J.More precisely, recall that in addition to the cubic norm form N J and an identity element 1 J , a cubic norm structure J comes equipped with -a nondegenerate symmetric bilinear trace form T : J × J → F ; -a quadratic map x → x # from J to itself, with linearization Given these, one sets and observes that U is quadratic in x and linear in y.The reader familiar with the notion of quadratic Jordan algebras will recognize that this is the U -operator in that theory.One then has a Jordan pair defined by setting: Since T gives an identification of J with J * , we may also describe this Jordan pair as (J, J * ).
6.7.Dual pairs.After this brief sidetrack, we can now introduce a family of dual pairs in exceptional group of the form [MS, Ru] i where J is a Freudenthal Jordan algebra.For split groups, this dual pair is obtained by removing from the extended Dynkin diagram of E (of type F 4 or E n ) the simple vertex joined to the unique vertex attached to the extra vertex.One sees that the extended Dynkin diagram breaks into two pieces, with one of them of type A 2 .
By the notion of Jordan pairs introduced above, we recognize that this dual pair is of the form Aut(V + , V − ) der × Aut(J + , J − ) der = SL(V ) × iso(J, N J ), where dim V = 3 and the superscript der signifies the derived group.
The following summarizes the algebras and groups which occur, where J(C) the space of 3 by 3 hermitian symmetric matrices with coefficients in a composition algebra C.
Observe that the centers of SL(V ) and iso(J, det) are isomorphic to µ 3 in all cases and is the space of 3 by 3 hermitian symmetric matrices with coefficients in a composition algebra C, and N J = det, then the similitude character sim : sim(J, N J ))(F ) → F × is surjective.
Proof.Let t ∈ F × .Let g : J → J be the linear transformation such that for any x ∈ J, y = g(x) is obtained from x by rescaling the entries in the following way.The entries in the upper-left 2 × 2 block are multiplied by t, the remaining diagonal entry is divided by t and the other entries are left unchanged.Then det(y) = t det(x).
Thus in this case G(F ) = G(F ) + and H(F ) = H(F ) + and the resulting similitude dual pair is Aut(V + , V − ) × Aut(J + , J − ) = GL(V ) × sim(J, N J ).Moreover, the group sim(J, N J )(F ) is given as follows: 6.8.Twisted Jordan pairs.It is in fact better to slightly repackage the definition of Jordan pairs introduced above, by setting ) where σ is the nontrivial automorphism of the F -algebra F × F given by switching the two components.The axioms (JP1-3) for a Jordan pair can be accordingly reformulated in terms of (J , Q ).A homomorphism φ : The advantage of such a repackaging is that it allows one to introduce twisted version (or F -rational forms) of Jordan pairs, where one replaces F × F by a separable quadratic field extension K/F .Moreover, in the definition of an automorphism of (J , Q ), we may consider those φ's which are (F × F, σ)-linear instead of (F × F )-linear.This gives rise to a larger Fautomorphism group Aut F (J , Q ) containing the subgroup Aut F ×F (J , Q ) as a normal subgroup of index 2; these extra F -automorphisms are outer automorphisms of Aut F ×F (J , Q ) and their action on J exchanges the two factors J + and J − .Now let K be a separable étale F -algebra with nontrivial automorphism σ ∈ Aut(K/F ).We shall define a notion of Jordan pair (J, Q) with respect to K/F , following a paper of de Medts [dM] where it was introduced under the guise of Hermitian cubic norm structure [dM, §4].Such a Jordan pair consists of: satisfying the reformulated analog of (JP1-3) (we will not elaborate further here).Let us give two examples: • Suppose (V, h) is a K-vector space equipped with a Hermitian form h : V × V σ → K.
Then defining Q by gives a Jordan pair relative to K/F .The automorphism group of (V, Q) is precisely the unitary group U(V, h).
• In [dM,Thm. 4.6], it was explained how a cubic norm structure J over K and a "σ-linear self adjoint autotopy" of J give rise to a Hermitian cubic norm structure on J, from which one can deduce a Jordan pair (J, Q) over K/F with Q given by the U -operator for the Hermitian cubic norm structure.If J is the exceptional Jordan algebra, then Aut K (J, Q) der is the quasi-split simply connected E K 6 associated to K/F and Aut K (J, Q)/Aut K (J, Q) der ≃ Res 1 K/F (G m ) is the anisotropic 1-dimensional torus associated to K.
Associated to these two examples and as a twisted version of the dual SL 3 × iso(J, N J ) introduced in the previous subsection, one has the quasi-split but non-split dual pair where we assume for simplicity that E is split.The centers of these groups are isomorphic to Z = Res 1 K/F (µ 3 ).Following Example (3) in §2, we choose T = Res K/F (G m ) and note that there is a short exact sequence The associated similitude dual pair is: where We note here that the similitude character on GU(V, h) is not the usual similitude character sim : GU(V, h) −→ G m .Rather, by construction, it is the homomorphism given by g → det(g)/sim(g).This is surjective onto K × , as one can see by restricting it to a maximal torus in a Borel subgroup (or by observing that SU(V, h) is simply-connected when F is nonarchimedean).Hence GAut K (J, Q)(F ) + = GAut K (J, Q)(F ).We will leave it to the reader to work out what GU(V, h)(F ) + is in the various cases.

Seesaw Duality
In the theory of theta correspondence, seesaw dual pairs and the associated seesaw dualities serve as useful tools.In this section, we examine how this seesaw duality is impacted by the extension to similitude dual pairs.Hence, suppose one has a seesaw diagram of dual pairs in E: For π ∈ Irr(G(F )) and σ ∈ Irr(H ′ (F )), the associated seesaw identity reads: Now let us make the following hypotheses: • The pair (G, H) gives rise to a corresponding similitude dual pair ( G, H), as we explained earlier in this paper; • for the other dual pair (G ′ , H ′ ), G ′ × H ′ maps injectively into E (so that there is no associated similitude pair).Observe also that Then we have: Proposition 7.1.Assume the above hypotheses.Then for π ∈ Irr( G(F )) and σ ∈ Irr(H ′ (F )), one has the similitude seesaw identity: Proof.For π ∈ Irr( G(F )) and σ ∈ Irr(H ′ (F )), we consider Hom G×H ′ (Ω, π ⊗ σ).
On the one hand, this is isomorphic to On the other hand, by Lemma 5.2(i), we also get Hence, one deduces that as desired.
In other words, there is no significant problem in extending the seesaw identity to the similitude setting, at least under the hypotheses in the proposition.
Let us give some examples of the above situation, using the exceptional dual pairs we discussed above.A first example is the seesaw diagram: where • J is a cubic norm structure over F with norm form N ; • V is a 3-dimensional vector space over F , giving rise to a Jordan pair (V , Q ); • O is the octonion F -algebra constructed as O = F 2 × V (as a structurable algebra [dM, §4], a notion we did not introduce).We have explained that the dual pair Aut(J) × Aut(O) does not have a similitude version, whereas the other pair SL(V ) × iso(J, N ) does.
As another example, we have:

where
• O is the octonion F -algebra giving rise to the twisted octonion algebra 1 is a twisted composition algebra relative to E/F giving rise to a a cubic norm structure (or Freudenthal-Jordan algebra) J = E ⊕C ♭ 1 via the Springer decomposition [KMRT,§38A,Pg. 522].
As above, the dual pair Aut(O) × Aut(J) does not have a similitude version whereas the pair Aut(O ♭ ) × Aut E (C ♭ ) does.
Observe that in forming both these seesaw diagrams, the initial data consists of giving two algebraic structures on the bottom row of the diagram, which then induces the algebraic structures in the top row.Moreover, observe that the two seesaw diagrams can be combined into a single one involving 3 dual pairs.

Global Theta Correspondence
In this section, we consider the global theta correspondence.Hence, let k be a number field with adele ring A. Suppose that we have a dual pair i : G × H −→ E as in the introduction, with Ker(i) = Z ∇ .
Let Ω be the global minimal representation of E and suppose one has an automorphic realization θ : Ω −→ A(E).
For an irreducible cuspidal representation π ⊂ A cusp (G) of G, one has the usual notion of global theta lifting.More precisely, for φ ∈ Ω and f ∈ π, one sets We would like to define its similitude global theta lifting to the space A( H+ ) of automorphic forms on H(A) + .For f ∈ π, φ ∈ Ω and y ∈ Jsim (A), we set θ(φ, f )(y) = Finally we combine the two lemmas to obtain the general case.
Proposition 9.4.Let U be an irreducible N -module, and G U its stabilizer in G. Let A U = G U /N .Let ÃU be the central extension of A U arising from the projective action of G U on U .Then where the sum is over all irreducible genuine representations E of ÃU .This is a decomposition of Ind G N U into isotypic summands.We remark that a study of the type of problems considered in this appendix can be found in the paper [T] of M. Tadić, where it was shown that the restriction of an irreducible representation of GL n (F ) to SL n (F ) is multiplicity-free.

Proof.
All these follows readily from the fact that T (F ) • G(F ) is an open subgroup of finite index in G(F ) + and the Clifford theory recounted in Appendix A below.More precisely, • (i) follows from the fact that an open compact subgroup of G(F ) • T (F ) is an open compact subgroup of G(F ) + ; • (ii) follows from Proposition 9.1(1); • (iii) follows from Proposition 9.1(2) and Proposition 9.4; • (iv) follows from Proposition 9.1(3).
[G]    θ(φ)(g • ι(y)) • f (g) dg.Next, we have U ⊗ Ind Ã C × (ǫ) ≃ Ind G N (U ) where U is viewed as G-module and ǫ : C → C is the inverse character.This isomorphism is realized byu ⊗ f → (g → f (g)π(g)(u)).Thus in order to decompose Ind G N (U ) it suffices to decompose Ind Ã C × (ǫ).Claim: The extension Ã is defined by a co-cycle with values in µ m .To see this, define a section s : A → Ã such that s(a), for every a ∈ A, acts on Ind Ã C × (ǫ) as a linear transformation of determinant 1.Then the extension Ã is determined by the co-cycle c(a, b) defined bys(a)s(b) = c(a, b)s(ab).After taking determinant of both sides, we get c(a, b) m = 1, as claimed.Using well known facts from representations of finite groups, we can now writeInd Ã C × (ǫ) ≃ ⊕ E dim(E)• E where the sum runs over all irreducible representations E of Ã such that C × ⊂ Ã acts by ǫ.Call such representations genuine.Lemma 9.3.Let U be an irreducible representation of N such that G U = G.With notation as above Ind G N U = E dim(E) • U ⊗ Ewhere the sum is over all irreducible genuine representations E of Ã. Representations U ⊗ E are irreducible and mutually non-isomorphic.Proof.It remains to prove the last sentence.Let E 1 and E 2 be any two genuine Ã-modules.Since U is irreducible N -module,Hom N (U ⊗ E 1 , U ⊗ E 2 ) ≃ Hom C (E 1 , E 2 ).Thus Hom G (U ⊗ E 1 , U ⊗ E 2 ) ≃ Hom Ã(E 1 , E 2 ).The lemma follows.We now briefly discuss the lemma when A is abelian.Then Ã is a two-step nilpotent group.The commutator of elements in Ã defines a skew-linear form onA × A a, b = [s(a), s(b)]Since the order of A is m, the skew form, and therefore the commutator, takes values in µ m .The dimension of any genuine irreducible representation E is equal to | Ā| where Ā is the quotient of A by the kernel of the skew form a, b .Furthermore, any two irreducible representations of G containing U are A-character twists one of another.Moreover, two character twists are isomorphic if and only if the characters coincide on the kernel of the skew form.
we have the similitude group and similitude character sim H : H The group Jsim .Going back to dual pairs, let us set Jsim