Decomposition of splitting invariants in split real groups

Transcription

1 Decomposition of splitting invariants in split real groups Tasho Kaletha Abstract For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic, Langlands and Shelstad construct in [LS] a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group. In applications of harmonic analysis and representation theory of reductive groups over local fields to questions in number theory, a central role is played by the theory of endoscopy. This theory associates to a given connected reductive group G over a local field F a collection of connected reductive groups over F, often denoted by H, which have smaller dimension (except when H = G, but are usually not subgroups of G. The geometric side of the theory is then concerned with transferring functions on G(F to functions on H(F in such a way that suitable linear combinations of their orbital integrals are comparable, while the spectral side is concerned with transferring packets of representations on H(F to packets of representations on G(F in such a way that suitable linear combinations of their characters are comparable. In both cases, the comparison involves certain normalizing factors, called geometric or spectral transfer factors. Over the real numbers, the theory of endoscopy was developed by Diana Shelstad in a series of profound papers including [S], [S2], [S3], [S4] (but see also [S6], [S7], [S8] for a more modern point of view and additional results, in which she defines geometric and spectral transfer factors and proves that these factors indeed give a comparison of orbital integrals and character formulas between G and H. A very subtle and complicated feature of the transfer factors was the need to assign a ±-sign to each maximal torus in G in a coherent manner, and Shelstad was able to prove that this is possible. A uniform and explicit definition of geometric transfer factors for all local fields was given in [LS]. An explicit construction of spectral transfer factors over the real numbers was given in [S7], while over the p-adic numbers their existence is still conjectural (see however [Kal] for a proof of the spectral transfer in a special case. The structure of transfer factors is quite complex both the geometric and the real spectral ones are a product of multiple terms of group-theoretic or Galois-cohomological nature. There are numerous choices involved in the construction of each individual term, but the product is independent of most choices. One term that is common to both the geometric and the real spectral transfer factors is called I. It is regarded as the most subtle and is the one that makes explicit the choice of coherent collection of signs in Shelstad s earlier work. At its heart is a Galois-cohomological object, called the splitting invariant. The splitting invariant is an element of H (F, T associated to any maximal torus T of a quasi-split semi-simple simply-connected group G, whose construction occupies the first half of [LS, Sec.2]. It depends on the choice of a splitting (T, B, {X α } α of G as well as a-data {a β } β R(T,G. AMS subject classification: Primary F7, 22E47; Secondary S37, F72, 7B22

2 This paper addresses the following question. If one has two maximal tori in a given real group which originate from the same endoscopic group, how can one compare their splitting invariants? While there will in general be no direct relation between H (F, T and H (F, T 2 for two maximal tori T and T 2 of G, if both those tori originate from H then there are certain natural quotients of their cohomology groups which are comparable, and it is the image of the splitting invariant in those quotients that is relevant to the construction of I. An example of a situation where this problem arises is the stabilization of the topological trace formula of Goresky-MacPherson. One is led to consider characters of virtual representations which occur as sums indexed over tori in G that originate from the same endoscopic group H, and each summand carries a I -factor associated to the corresponding torus. To describe the results of this paper, we take G to be a split simply-connected real group and (T, B, {X α } α be a fixed splitting. For a set A consisting of roots of T in G which are pairwise strongly orthogonal, let S A denote the element of the Weyl group of T given by the product of the reflections associated to the elements of A (the order in which the product is taken is irrelevant. We show that associated to A there is a canonical maximal torus T A of G and a set of isomorphisms of real tori T S A T A, where T S A is the twist by S A of T. Any maximal torus in G is G(R-conjugate to one of the T A, so it is enough to study the tori T A. We give an expression in purely roottheoretic terms for a certain -cocycle in Z (R, T S A. This cocycle has the property that its image in Z (R, T A under any of the isomorphisms T S A T A above is the same, and the class in H (R, T A of that image is the splitting invariant of T A (associated to a specific choice of a-data. Moreover, we prove a reduction theorem which shows that this cocycle is a product over α A of the cocycles associated to the canonical tori T {α}, thereby reducing the study of the splitting invariant of T A to those of the various T {α}. This product decomposition takes place inside the group Z (R, T S A that is, we show that the elements of Z (R, T s α associated to the various T {α} with α A also lie in Z (R, T S A and that their product is the element associated to T A. Finally we show that if A A and the tori T A and T A originate from the same endoscopic group, then the endoscopic characters on the cohomology groups H (R, T A and H (R, T A factor through certain explicitly given quotients of these groups, and the quotient of H (R, T A is canonically embedded into that of H (R, T A. This, together with the reduction theorem, allows for a direct comparison of the values that the endoscopic characters associate to the splitting invariants for T A and T A. Our techniques rely heavily on the study of sets of strongly orthogonal roots in root systems and the fact that each element of order 2 in the Weyl group of a root system is of the form S A for some set A consisting of strongly orthogonal roots. In a split real group the Galois action on any maximal torus is realized by such an element. This is the reason why we restrict our attention to such groups. It may be possible to use our techniques also in the case of non-split quasi-split groups which possess an anisotropic maximal torus, for then the Galois-action on any maximal torus is of the form S A, but we have not pursued this line of thought here. The paper is organized as follows: Section contains a few basic facts and serves mainly to fix notation for the rest of the paper. Section 2 contains proofs of general facts about subsets of strongly orthogonal roots in reduced root systems, which are needed as a preparation for the reduction theorem mentioned above. The study of the splitting invariants takes place in Section 3, where first the splitting invariant for the tori T {α} is computed, and after that the results of Section 2 are used to reduce the case of T A to that of T {α}. While the statement of the reduction theorem appears natural and clear, the proof contains some subtle points. First, one has to choose the Borel B in the splitting of G with care according to the strongly orthogonal set A. As remarked in 2

3 Section 3, this choice does not affect the splitting invariant, but it significantly affects its computation. Moreover, the root system G 2 exhibits a singular behavior among all reduced root systems as far as pairs of strongly-orthogonal roots are concerned. Section 4 contains explicit computations of the splitting invariants of the tori T {α} for all split almost-simple classical groups. In Section 5 we construct the aforementioned quotients of the cohomology groups and the embedding between them. Moreover we show that the endoscopic character factors through these quotients and is compatible with the constructed embedding. We would like to thank the referee for a careful reading of the manuscript and for helpful and insightful comments, in particular about the origin and motivation of the splitting invariant, which we have included in Section 3. Notation and preliminaries Throughout this paper G will stand for a split semi-simple simply-connected group over R and (B, T, {X α } will be a splitting of G. We write R = R(T, G for the set of roots of T in G, set α > if α R(T, B, denote by the set of simple roots in R(T, B and by Ω the Weyl-group of R, which is identified with N(T /T. Moreover we put Γ = Gal(C/R and denote by σ both the non-trivial element in that group, as well as its action on T. The notation g G will be shorthand for g G(C, and Int(gh = ghg.. sl 2 -triples For any α R(T, B we have the coroot α : G m T and its differential dα : G a Lie(T. We put H α := dα ( Lie(T. Given X α Lie(G α non-zero, there exists a unique X α Lie(G α so that [H α, X α, X α ] is an sl 2 -triple. The map ( H α, ( X α, ( X α gives a homomorphism sl 2 Lie(G which integrates to a homomorphism SL 2 G and one has sl 2 Lie(G exp SL 2 exp G The image of ( a c b d Notice that ( t t X α = α (t. SL2 under this homomorphism will be called ( a b d c Fact.. Let α, β R be s.t. α + β R and α β R. For any non-zero elements X α Lie(G α and X β Lie(G β, the homomorphisms ϕ Xα, ϕ Xβ : SL 2 G given by X α and X β commute. X α. 3

4 Proof: Since for any field k, SL 2 (k is generated by its two subgroups {( } {( } u u k u k u it is enough to show that, for any u, v C, each of exp(ux α and exp(ux α commutes with each of exp(vx β and exp(vx β. This follows from [Spr,..4] and our assumption on α, β..2 Chevalley bases For α let n α = exp(x α exp( X α exp(x α = ( lift n(µ N(T given by n(µ = n α n αq X α. Given µ Ω we have the where s α s αq = µ is any reduced expression (by [Spr,.2.9] this lift is independent of the choice of reduced expression. Notice n(µ N(T (R since T is split. Put X µ α := Int(n(µ X α Then X µ α Lie(G µα is a non-zero element. Lemma.2. If α, α and µ, µ Ω are s.t. µα = µ α then we have in Lie(G µα the equality X µ α = ( β,µα X µ α β> (µ β< µ β> Proof: By [Spr,.2.] the relation (µ µα = α implies X α = Int [ n ( ( µ µ ] X α The claim now follows from [LS, 2..A] and the following computation X µ α = Int ( n ( µ X α = Int [ n ( µ n ( ( µ µ ] X α = Int [ t ( µ, (µ µ n(µ ] X α = Int [ t ( µ, (µ µ ] X µ α = (µα ( t(µ, (µ µ X µ α Remark: We see that while the absolute value of X µ α only depends on the root µ α, its sign does depend on both µ and α. Definition.3. For γ R, µ, µ Ω put ɛ(µ, γ, µ := β> (µ β< µ β> ( β,γ 4

5 Remark: With this definition we can reformulate the above lemma as follows Corollary.4. If γ R and µ, µ Ω are s.t. µ γ, (µ γ then X µ (µ γ = ɛ(µ, γ, µ X µ µ γ Remark: If for each γ R we choose µ γ Ω so that µ γ γ, then {X µγ µ γ γ} γ R is a Chevalley system in the sense of [SGA3, exp XXIII 6]..3 Cayley-transforms Let α R(T, B and choose X α Lie(G α (R {}. Put ( iπ g α := exp 4 (X α + X α Then σ(g α = exp ( iπ 4 (X α + X α = g α We have g α = 2 2 ( i i ( i, g 2 α = i X α and σ(g α g α = g 2 α = exp ( iπ 2 (X α + X α. X α, g 4 α = ( X α = α ( Fact.5. The images of T under Int(g α and Int(g α are the same. They are a torus T defined over R and the transports of the Γ-action on T to T via Int(g α and Int(g α both equal s α σ. Proof: Int(g α T = Int(g α Int(g 2 αt = Int(g α s α T = Int(g α T σ(int(g α T = Int(σ(g α T = Int(g α T Int(σ(g α g α = Int(g 2 α = s α = Int(g 2 α = Int(σ(g α g α Note: Different choices of X α will lead to different (yet conjugate tori T. However, since we have fixed a splitting there is up to a sign a canonical X α. Changing the sign of X α changes g α to g α, hence T does not change. Thus we conclude: The choice of a splitting gives for each α R(T, B the following canonical data:. a pair {X, X } Lie(G α (R {} with X = X 2. a torus T α on which Γ acts via s α σ, 3. a pair ϕ, ϕ of isomorphisms T s α T α s.t. ϕ = ϕ s α, given by the Cayleytransforms with respect to X, X. Corollary.6. For α R(T, B let T α be the canonically given torus as above. For µ, µ Ω s.t. µ α, (µ α let ϕ, ϕ : T s α T α be the isomorphisms given by Int(g Xµ µ α and Int(g Xµ (µ α. Then ϕ ϕ, ɛ(µ, α, µ = = ϕ s α, ɛ(µ, α, µ = Proof: Clear. Notation: From now on we will write g µ,α instead of g Xµ µ α. This notation will only be employed in the case that α R(T, B and µ α. 5

6 2 Strongly orthogonal subsets of root systems In this section, a few technical facts about strongly orthogonal subsets of root systems are proved. Definition 2... α, β R are called strongly orthogonal if α + β R and α β R. 2. A R is called a strongly orthogonal subset (SOS if it consists of pairwise strongly orthogonal roots. 3. A R is called a maximal strongly orthogonal subset (MSOS if it is a SOS and is not properly contained in a SOS. A classification of the Weyl group orbits of MSOS in irreducible root systems was given in [AK]. In some cases, there exists more than one orbit. To handle these cases, we will use the following definition and lemma. Definition 2.2. Let A,A 2 be SOS in R. A 2 will be called adapted to A if span(a 2 span(a and for all distinct α, β A 2 {a A : (a, α } {a A : (a, β } = where ( is any Ω-invariant scalar product on the real vector space spanned by R. Note that any A is adapted to itself. Lemma 2.3. There exist representatives A,..., A k of the Weyl group orbits of MSOS s.t. A has maximal length and A 2,..., A k are adapted to A. Proof: This follows from the explicit classification in [AK]. Notation: If A is a SOS then all reflections with respect to elements in A commute. Their product will be denoted by S A. Definition 2.4. For a root system R, a choice of positive roots >, and a subset A of R let #(R, >, A be the following statement α, α 2 A β > α α 2 s α (β < = s α2 (β > and let ##(R, >, A be the following statement A, A 2 A β > A A 2 = S A (β < = S A2 (β > S A S A2 (β < Remark: We will soon show that these statements are equivalent. Moreover we will show that for any SOS A R we can choose > so that the triple (R, >, A verifies these statements. For this it is more convenient to work with #. For the applications however, we need ##. Lemma 2.5. Let R be a reduced root system and A R a SOS. There exists a choice of positive roots > s.t. #(R, >, A holds for any A adapted to A. 6

7 Proof: Let V denote the real vector space spanned by R, and (, be an Ω-invariant scalar product on V. The elements of A are orthogonal wrt (,. Extend A to an orthogonal basis (a,..., a n of V. Define the following notion of positivity on R α > (α, a i > for i = min{i : (α, a i } It is clear from the construction that with this notion #(R, >, A is satisfied for any A adapted to A. We just need to check that > defines a choice of positive roots, which we will now do. It is clear that for each α R precisely one of α > or α > is true. We will construct p V s.t. for all α R Let α > (α, p > m = min{ (α, a i : α R, i n, (α, a i } M = max{ (α, a i : α R, i n} Construct recursively real numbers p,..., p n s.t. p n = p i > M m and put p = p i a i. If α R is s.t. α > and i is the smallest i s.t. (α, a i then Thus (α, p = n i=i k>i p k p i (α, a i > mp i M p k > k>i α > = (α, p > The converse implication follows formally: (α > α > ( α, p > ((α, p > Remark: The truth value of the statement #(R, >, A and the notion of being adapted to A are unchanged if one replaces elements of A by their negatives. Thus we can always assume that the elements of A are positive. Remark: It is necessary to choose the set of positive roots based on A in order for #(R, >, A to be true. An example that #(R, >, A may be false is provided by V = R 3, R = D 3 with positive roots and, A =,,,,, β =, Fact 2.6. Let R = G 2 and > any choice of positive roots. All MSOS A of R lie in the same Weyl-orbit and moreover automatically satisfy #(R, >, A. Some of these A contain simple roots. 7

8 Proof: This is an immediate observation. Proposition 2.7. Let A R be a SOS and > be a choice of positive roots. Then the statements #(R, >, A and ##(R, >, A are equivalent. Proof: First, we show that # implies the following statement, to be called # : α, α 2 A β > α α 2 s α (β < = s α2 (β > s α s α2 (β < Let α, α 2 A and β > be s.t. s α (β <. Put β = s α (β. Then β > and s α (β = β <. Then # implies that s α2 s α (β = s α2 (β <. Next we show that # implies the following statement, to be called # 2 : α A A 2 A β > α A 2 s α (β < = S A2 (β > s α S A2 (β < We do this by induction of the cardinality of A 2, the case of A 2 singleton being precisely #. Now let α A, A 2 A \ {α }, and β > be s.t. s α (β <. Choose α 2 A 2 and put β := s α2 (β. Then by # we have β > and s α (β <. Applying the inductive hypothesis we obtain S A2 (β = S A2 \{α 2 }(β > and s α S A2 (β = s α S A2 \{α 2 }(β <. Now we show that # 2 implies the following statement, to be called : If A 2 A and β > are s.t. S A2 (β < then there exists α 2 A 2 s.t. s α2 (β <. To see this, let A 3 A 2 be a subset of minimal size s.t. S A3 (β <. Take α 3 A 3 and put β = S A3 \{α 3 }(β. By minimality of A 3 we have β >, and moreover s α3 (β = S A3 (β <. Then # 2 implies that s α3 (β = s α3 S A3 \{α 3 }(β <. Finally we show that # 2 implies the statement ##. Take A, A 2 A s.t. A A 2 = and β > s.t. S A (β <. By there exists α A s.t. s α (β <. Since α A 2 we get from # 2 that S A2 (β > and S A S A2 (β = s α S A2 S A \{α }(β <. This shows that # imples ##. The converse implication is trivial. Proposition 2.8. For an SOS A R, and a choice > of positive roots, let R + A = {β R : β > S Aβ < } Assume that > is chosen so that ##(R, >, A is true. Then if A, A A are disjoint, so are R + A and R + A, and R + A A = R + A R + A. Moreover, the action of S A on R preserves R + A. Proof: This follows immediately. Corollary 2.9. If A is a SOS and > is chosen so that #(R, >, A is true then R + A = R + α Proof: Clear. 8

9 Lemma 2.. Let R be a root system, V the real vector space spanned by it, Q V the root lattice, and (, a Weyl-invariant scalar product on V. If v Q is s.t. v min{ α : α R} where is the Euclidian norm arising from (, then v R and the above inequality is an equality. Proof: Choose a presentation v = n α α, α R n α Z s.t. α n α is minimal. First we claim that if α, β R contribute to this sum, then (α, β. If that were not the case, then by [Bou, Ch.VI,,no.3,Thm.] we have that γ := α + β R {} and we can replace the contribution α + β in the sum by γ, contradicting its minimality. Now, if γ R is any root contributing to the sum, we get v 2 = n α n β (α, β n 2 γ(γ, γ (γ, γ = γ 2 α,β R with equality precisely when v = γ. Lemma 2.. Let R be a root system and α, β R two strongly orthogonal roots. If α + β 2Q, then α, β belong to the same copy of G 2. Proof: Let V denote the real vector space spanned by R. Choose a Weyl-invariant scalar product (, and use it to identify V with its dual and regard R as a root system in V. Assume now that α + β 2Q. Note that α and β are orthogonal (but may not be strongly orthogonal elements of R. First we show that then α, β belong to the same irreducible piece of R. To that end, assume that R decomposes as R = R R 2 and V decomposes accordingly as V V 2. If α R and β R 2, then α V and β V 2. Then 2 (α + β Q implies 2 α Q, 2 β Q 2 (project orthogonally onto V resp V 2. This however contradicts the above lemma, because 2 α has length strictly less then the shortest elements in R. Knowing that α, β lie in the same irreducible piece we can now assume wlog that R is irreducible. Normalize (, so that the short roots in R have length. We have the following cases All elements of R have length. Then all elements of R have length 2. The length of 2 (α + β is 2, which by the above lemma is not a length of an element in Q. R contains elements of lengths and 2. Then R contains elements of lengths 2 and 2. If both α, β have length 2, then 2 (α + β has length, so is not in Q. If α has length 2 and β has length 2, then 2 (α + β has length 6 2, so again is not in Q. 9

10 If both α, β have length 2, then 2 (α + β has length 2 and thus could potentially be in Q. If it is, then by the above lemma it is also in R, so 2 (α + β must be an element of R. One immediately computes that [ 2 (α + β ] = α + β, but the latter is not an element of R because α, β are strongly orthogonal. R has elements of lengths and 3. Then R is G 2 and R is also G 2. As one sees immediately, up to the action of its Weyl-group, G 2 has a unique pair of orthogonal roots, which are then automatically strongly orthogonal and half their sum is also a root. 3 Splitting invariants Recall that we have fixed a split semi-simple and simply-connected group G over R and a splitting (T, B, {X α } of it. Given a maximal torus T, an element h G s.t. Int(hT = T, and a-data {a β } for R(T, G, Langlands and Shelstad construct in [LS, 2.3] a certain element of Z (Γ, T, whose image in H (Γ, T they call λ(t the splitting invariant of T. They show that this image is independent of the choice of h. The introduction of this invariant is motivated by a (general calculation in Langlands paper [L], which is the basis for [LS, 5.4]. It is also used by Shelstad to give an explicit formula for regular unipotent germs of p-adic groups [S5]. In this section we want to study this splitting invariant in such a way that enables us to see how it varies when the torus varies. It turns out that a certain type of a-data is very well suited for this. This a-data is determined by a Borel subgroup B T as follows: i, β > σ T (β < i, β < σ T (β > α β =, β > σ T (β >, β < σ T (β < where σ T denotes the Galois-action on X (T and β > means β R(T, B. We will call this a-data B-a-data. We would like to alert the reader of a similar yet inequivalent terminology that of based a-data which has been introduced by Shelstad and is also specified by a choice of a Borel subgroup. For based a-data the positive imaginary roots are assigned i while all other positive roots are assigned ; for B-a-data, any positive root whose Galois-conjugate is negative is assigned i. Therefore, a splitting invariant computed using based a-data will in general be different from one computed using B- a-data. The precise difference is given by [LS, 2.3.2]. It is however more important to note that according to [LS, Lemma 3.2.C] this difference disappears once the splitting invariant has been paired with an endoscopic character. Thus, as far applications to transfer factors are concerned, based a-data and B-a-data give the same result. In view of the reduction theorem which we will prove in section 3.2, it will be helpful to consider not just the cohomology class, but the actual cocycle constructed in [LS, 2.3]. We will denote this cocycle by λ(t, B, h to record its dependence on the B-adata and the element h, while the splitting (T, B, {X α } is not present in the notation because it is assumed fixed. Since we are working over R, we will identify a -cocycle and its value at σ Gal(C/R, and hence we will view λ(t, B, h as an element of T. Given h, there is an obvious choice for B, namely Int(hB. We will write λ(t, h for

11 λ(t, Int(hB, h. Note that in this notation, T is clearly redundant, because it equals Int(hT. However, we keep it so that the notation is close to that in [LS]. We would like to alert the reader of one potential confusion while the cohomology class of λ(t, B, h is independent of the choice of h, that of λ(t, h is not, because in the latter h influences not only the identification of T with T but also the choice of B-a-data for T. 3. The splitting invariant for T α Recall from section.3 that for each α R(T, B there is a canonical maximal torus T α and a pair of isomorphisms T s α T α. To fix one of the two, fix µ Ω s.t. µ α. Then Int(g µ,α is one of the two isomorphisms T s α T α. The goal of this section is to compute λ(t α, B, g µ,α for a given Borel B T α, and in particular λ(t α, g µ,α. We will give a formula for the latter in purely root-theoretic terms. Lemma 3.. With g := g µ,α we have where λ(t α, B, g = Int(g ( α (i a α Int(g s α (σ(δδ δ = β> µ β< and σ denotes complex conjugation on T. β (a β Int(g Proof: Put u = n(µ. We will first compute the cocycle λ(t α, B, gu. The notation will be as in [LS, 2.3]. The pullback of the Γ-action on T α to T via gu differs from σ by ω Tα (σ := Int ( (gu σ(gu = Int ( n(µ g σ(gn(µ = µ s α µ = s µ α Using that µ α is simple, we compute the three factors of Int(gu λ(t α, B, gu: x(σ = β (a β Int(gu Thus n(ω Tα (σ = β> ω Tα (σβ< = (µ α (a µ α Int(u g = Int(u (α (a α Int(g ( ( = Int(u X µ α ( i σ(gu (gu = Int(u g 2 = Int(u i X µ µ α λ(t α, B, gu = Int(guInt(u ( α (a α Int(g α (i From the proofs of [LS, 2.3.A] and [LS, 2.3.B] one sees that λ(t α, B, gu = Int(g ( δσ Tα (δ λ(t α, B, g X µ µ α where σ Tα is the transport of the action of complex conjugation on T α to T via g. This action is s α σ. Notice that the term λ σ T (λ appearing in the proof of [LS, 2.3.A] is trivial since for us u = n(µ and hence λ =. The claim now follows.

12 Before we turn to the computation of s α (σ(δδ we will need to take a closer look at the following set. Definition 3.2. For α > put R + α = { β R β > s α (β < } Lemma 3.3. Let α > and µ Ω be s.t. µ α. Then the sets { β R β > sα (β < µ β < } and { β R β > sα (β < µ β > β α } are disjoint and their union is R + α {α}. The map β s α (β is an involution on R + α {α} which interchanges the above two sets. Proof: Every β in the first set satisfies β α because µ α is positive. Hence the first set lies in R + α {α} and clearly the second also does. The fact that the two are disjoint and cover R + α {α} is obvious. Now to the bijection. Let β be an element in the first set, and consider β = s α (β. We have β α β α s α (β < β > β > s α β = β < µ β < µ β = µ s α ( β = s µ α( µ β > where the last inequality holds because µ α is simple and β > β α µ β µ α Remark: A similar observation appears in [LS2, 4.3]. Lemma 3.4. We have s α (σ(δδ = [ β ( a β Int(g sα β ( a ] sα β Int(g µ β< Proof: According to the proof of part (a of [LS, 2.3.B] the contributions to δs α (σ(δ are as follows: ( {β β > µ β < s α β < } : β (a β Int(g (2 {β β < µ β < µ s α β < s α β > } : β (a β Int(g (3 {β β > µ β < s α β > µ s α β > } : β (a β Int(g (4 {β µ β > s α β > µ s α β < } : β (a β Int(g We will use µ s α (β = s µ α(µ β and the fact that µ α is simple to show that the last two sets are empty. In set (3, the conditions µ β < and µ s α β > imply 2

13 µ β = µ α, i.e. β = α, which contradicts β >. In set (4, the conditions µ β > and µ s α β < imply β = α. Since α > this contradicts s α β >. Next we claim (2 = s α ((. We have from which we get µ β < µ s α β < µ β < µ β µ α (2 = { β β > s α β < µ β > β α} Now (2 = s α (( follows from Lemma 3.3. From these considerations it follows that δs α (σ(δ = β (a β Int(g β (a β Int(g β ( β (2 [ = β ( a β Int(g s α β ( a sα β Int(g ] β ( Let us recall our notation: α R is any positive root, µ Ω is s.t. µ α, and g = g µ,α is the Cayley-transform corresponding to X µ µ α. From Lemmas 3. and 3.4 we immediately get Corollary 3.5. λ(t α, B, g = Int(g α (ia α Int(g µ β< [ β ( a β Int(g sα β ( a ] sα β Int(g In the case B = Int(gB this formula becomes simpler. Corollary 3.6. λ(t α, g µ,α = Int(g µ,α α ( (β s α β (i µ β< Definition 3.7. Put ρ(µ, α := α ( (β s α β (i T µ β< By Corollary 3.6 and the work of [LS, 2.3] we know that ρ(µ, α Z (Γ, T s α and Int(g µ,α ρ(µ, α = λ(t α, g µ,α. Proposition ρ(µ, α = β (in(s α g 2 µ,α. 2. s α ρ(µ, α = ρ(µ, α, σ(ρ(µ, α = ρ(µ, α. 3

14 3. The image of ρ(µ, α under the two canonical isomorphisms T s α T α is the same. 4. If µ Ω is another Weyl-element s.t. (µ α, then In particular, ρ(µ, α = α ( ɛ(µ, α, µ ρ(µ, α. λ(t α, g µ,α = λ(t α, g µ,α Int(g µ,α [ α (ɛ(µ, α, µ ]. Proof: The first point follows from Corollary 3.6, because the right hand side is by construction Int(g µ,α λ(t α, g µ,α. The second point is evident from the structure of ρ. The third point is now clear because as remarked in section.3 the two canonical isomorphisms differ by precomposition with s α. For the last point, ρ(µ, α = β (in(s α g 2 µ,α If ɛ(µ, α, µ = + then g µ,α = g µ,α and the statement is clear. Assume now that ɛ(µ, α, µ =. Then g µ,α = g µ,α. We see ρ(µ, α = β (in(s α g 2 µ,αg 4 µ,α But g 4 µ,α = α (, hence the claim. 3.2 The splitting invariant for T A Fact 3.9. Let A be a SOS in R. Consider the set of automorphisms of G given by Int(g g = g µα,α, µ α Ω, µ α α The image of T under any element of that set is the same. Call it T A. Then any element of that set induces an isomorphism of real tori T S A T A Proof: Let Int(g, Int(g 2 be elements of the above set, with g i = g µ i α,α and let A A be the subset of those α s.t. g µ α,α g µ 2 α,α. For those α we have then g µ α,α = g, hence Int(g µ g 2 α,α 2 T = g2 µ T α,α = S A which normalizes T. This shows that the images of T under these two automorphisms are the same. Moreover, the transport of the Γ-action on T A to T via Int(g differs from σ by Int(σ(g g T = Int(g 2 T = S A. Definition 3.. For a SOS A R, we will call the set Int(g T g = g µα,α, µ α Ω, µ α α 4

15 the canonical set of isomorphisms T S A T A. More generally, if A A, we will call the set Int(g T A g = g µα,α, µ α Ω, µ α α \A the canonical set of isomorphisms T S A\A A T A. Fact 3.. Any maximal torus T G is G(R-conjugate to one of the T A. Proof: Choose g G s.t. Int(gT = T. The transport of the Γ-action on T to T via Int(g differs from σ by an element of Z (Γ, Ω = Hom(Γ, Ω and this element sends complex conjugation to an element of Ω of order 2. By [Bou, Ch.VI.Ex (5] there exists a SOS A s.t. this element equals S A. If Int(g A is one of the canonical isomorphisms T S A T A, then Int(g A g : T T A is an isomorphism of real tori. By [S, Thm. 2.] there exists g G(R s.t. Int(g T = T A. If we conjugate A by Ω to an A, then the tori T A and T A are also conjugate by G(R. Thus we may fix representatives A,..., A k for the Ω-orbits of MSOS in R and study the tori T A for A inside one of the A i. We assume that the fixed splitting (T, B, {X α } is compatible with the choice of representatives in the following sense ##(R, >, A i holds for all A i. If α, β A i lie in the same G 2 -factor then one of them is simple This can always be arranged, as Lemmas 2.3, 2.5, Fact 2.6 and Proposition 2.7 show. Notice that this condition does not reduce generality it is only a condition on B, but all Borels containing T are conjugate under N T (R and thus by [LS, 2.3.] the splitting invariants are independent of the choice of B. Lemma 3.2. If A, A are disjoint subsets of some A i then In particular, n(s A and n(s A commute. n(s A n(s A = n(s A A Proof: This follows immediately from [LS, Lemma 2..A], because by ## the set {β R : β > S A (β < S A S A (β > } is empty. Proposition 3.3. Let α, γ be distinct elements of one of the A i, and µ Ω be s.t. µ α. Then ρ(µ, α is fixed by s γ. Proof: We first show Claim: Proof: We have s γ ρ(µ, α = ρ(µ, αα (ɛ(s γ µ, α, µ s γ (ρ(µ, α = s γ α ( µ β< β (is α β (i 5

16 Now s γ preserves α, commutes with s α, and by Proposition 2.8 also preserves the set R + α, hence the last expression equals α ( s γ β (is α s γ β (i = α ( β (is α β (i µ β< = α ( s γ µ s γ β< β (is α β (i µ s γ β< = ρ(s γ µ, α = ρ(µ, α α ( ɛ(s γ µ, α, µ the last equality coming from Proposition 3.8. (claim We want to show α ( ɛ(s γ µ, α, µ =. Choose ν Ω s.t. ν γ. We will derive and compare two expressions for β (in(s α s γ g 2 µ,αg 2 ν,γ ( β R + {α,γ} By Corollary 2.9 we have β (i = β (i β (i β R + {α,γ} β R + γ By Proposition 2.8 s α is a permutation of the set R + γ, hence n(s α β (in(s α = β (i β R + γ β R + γ By Lemma 3.2, the elements n(s α and n(s γ of N(T commute. Moreover, by Fact. the elements g 2 µ,α and g 2 ν,γ commute. Thus we get on the one hand ( = β (in(s γ β (in(s α g 2 µ,αg 2 ν,γ β R + γ = ρ(ν, γint(g 2 ν,γ [ ρ(µ, α ] = ρ(ν, γρ(µ, αα (ɛ(s γ µ, α, µ where the last equality follows from above claim. Analogously, we obtain on the other hand ( = β (in(s α β (in(s γ g 2 ν,γg 2 µ,α β R + γ = ρ(µ, αint(g 2 µ,α [ ρ(ν, γ ] = ρ(µ, αρ(ν, γγ (ɛ(s α ν, γ, ν We conclude that α (ɛ(s γ µ, α, µ = γ (ɛ(s α ν, γ, ν We claim that both sides of this equality are trivial. Assume by way of contradiction that this is not the case. Then we have α ( = γ ( = ( (α γ = ( α +γ C X (T = C Q α + γ 2Q 6

17 where Q is the coroot-lattice of T, which coincides with X (T since G is simplyconnected. By Lemma 2. α, γ must lie in the same G 2 -factor of R. In this case {α, γ} is a MSOS for that G 2 -factor and by our assumption from the beginning of this section one of α, γ must be simple. Say wlog α is simple. By Proposition 3.8 ρ(µ, α = α (ɛ(µ, α, ρ(, α = α ( ɛ(µ, α, which is clearly fixed by s γ, thus we see = s γ (ρ(µ, αρ(µ, α = α (ɛ(s γ µ, α, µ = γ (ɛ(s α ν, γ, ν Corollary 3.4. Let A be a subset of some A i, α A and µ Ω s.t. µ α. Then ρ(µ, α Z (Γ, T S A and its image in T A under any canonical isomorphism T S A T A is the same. Proof: By Propositions 3.8 and 3.3 ρ = ρ(µ, α is fixed by s γ for any γ A. The first statement now follows from ρs A σ(ρ = ρσ(ρ = showing ρ Z (Γ, T S A. The second holds because any two canonical isomorphisms T S A T A differ by precomposition with S A for some A A Corollary 3.5. Let A be a subset of some A i. Choose, for each α A, µ α Ω s.t. µ α α. Put ρ({µ α }, A = ρ(µ α, α Then. ρ({µ α }, A is fixed by s γ for all γ A (even all γ A i 2. The image of ρ({µ α }, A under any of the canonical isomorphisms T S A T A is the same. Proof: Clear by the preceding Corollary. Proposition 3.6. Let A be a subset of some A i. For each α A choose µ α Ω s.t. µ α α and put g A = g µα,α. Then λ(t A, g A is the common image of ρ({µ α }, A under the canonical isomorphisms T S A T A. In particular λ(t A, g A = Int(g A {α} λ(t α, g µα,α is a decomposition of λ(t A, g A as a product of elements of Z (Γ, T A. Proof: The factors of the cocycle Int(g A λ(t A, g A Z (Γ, T S A associated to these choices are as follows: x(σ T = β (i = β (i A A where the second equality is due to Corollary 2.9, n(ω T (σ = n(s A = n(s α 7

18 where the second equality is due to Lemma 3.2, and σ(g A g A = σ(g α g α = Their product, which equals Int(g A λ(t A, g A, is thus x(σ T n(ω T (σσ(g A g A = β (i n(s α Just as in the proof of Proposition 3.3 we can rewrite this product as β (in(s α g 2 α Now we induct on the size of A, with A = being clear. Choose α A. Then β (in(s α g 2 α = β (in(s α β \{α (in(s α g 2 α g 2 α } \{α } = β (in(s α ρ(µ α, α g 2 α = = \{α } \{α } \{α } β (in(s α \{α } ρ(µ α, α ρ(µ α, α g 2 α \{α } k g 2 α \{α } g 2 α s α (ρ(µ α, α where the last equality follows from Proposition 3.3 and the inductive hypothesis. This shows that Int(g A ρ({µ α }, A = λ(t A, g A and the result follows. 4 Explicit computations In this section we are going to use the classification of MSOS given in [AK] to explicitly compute λ(t A, g A for the split simply-connected semi-simple groups associated to the classical irreducible root systems. By Propositions 3.8 and 3.6 it is enough to compute the cocycles ρ(µ, α for each α A i, and some µ Ω with µ α, where A,..., A k is a set of representatives for the Ω-classes of MSOS. We will use the notation from [AK], which is also the notation used in the Plates of [Bou, Ch.VI]. There is only one cosmetic difference in [Bou] the standard basis of R k is denoted by (ɛ i, in [AK] by (λ i, and we are using (e i. The dual basis will be denoted by (e i. One checks easily in each case that the choices of positive roots given in the Plates of [Bou, Ch.VI] and the MSOS given in [AK] satisfy condition # of section Case A n There is only one Ω-equivalence class of MSOS, and the representative given in [AK] is A = {e 2i e 2i i [(n + /2]} 8

19 All elements of this MSOS are simple roots and for each of them we can choose µ =. Then for any α A we have ρ(, α = α ( 4.2 Case B n If n = 2k + then there is a unique equivalence class of MSOS, represented by A = {e 2i ± e 2i i k} {e n } If n = 2k then there are two equivalence classes of MSOS, represented by A = {e 2i ± e 2i i k } {e n } A 2 = {e 2i ± e 2i i k} If α = e 2i e 2i or α = e n then α is simple, we can choose µ = and have ρ(, α = α ( If α = e 2i + e 2i then we take µ = s e2i and have µ α = e 2i e 2i. To compute ρ(µ, α we first observe {β R β > µ β < s α β < } = {β R β > µ β < } = {e 2i } {e 2i ± e j 2i < j} hence (β + s α β µ β< n = 2e 2i + s α(2e 2i + (e 2i e j + s α(e 2i e j + e 2i + e j + s α(e 2i + e j j=2i+ = 2(e 2i e 2i 2(e 2i e 2i (n 2i = 2(n + 2i(e 2i e 2i = 2(n + 2iα We get ρ(µ, α = α (( n 4.3 Case C n The root system family C n is the only family for which the number of equivalence classes of MSOS grows when n grows. Representatives for the equivalence classes of MSOS are given by A s = {e 2i e 2i i s} {2e i 2s + i n} s [n/2] If α = e 2i e 2i then α is simple and ρ(, α = α ( 9

20 If α = 2e i and we take µ = s ei e n we have µ α = 2e n. Again we first observe {β R β > µ β < s α β < } = {β R β > µ β < } = {e i e j i < j} hence We get (β + s α β = µ β< n n (e i e j + ( e i e j = 2 j=i+ ρ(µ, α = n e i ( j=i j=i+ e j 4.4 Case D n There is a unique equivalence class of MSOS represented by A = {e 2i ± e 2i i [n/2]} If α = e 2i e 2i or α = e n + e n then α is simple and ρ(, α = α ( If α = e 2i + e 2i with 2i n then we take µ = s e2i e n s e2i e n and have µ α = e n e n. Then hence {β R β > µ β < s α β < } = {β R β > µ β < } (β + s α β = 2 µ β< n j=2i+ Notice that 2e n Q, while e n Q. We get = {e 2i e j 2i < j} {e 2i e j 2i < j} 2e j 4Q, 2 n 4Q + 4e n, 2 n + ρ(µ, α = α ( 2e n(( n 5 Comparison of splitting invariants Now we would like to employ the results from the previous sections to compare splitting invariants of different tori. To describe more precisely our goal, let us recall briefly a few notions from the theory of endoscopy. An endoscopic triple (H, s, η for G consists of a quasi-split real group H, an embedding η : Ĥ Ĝ of the complex dual group of H to that of G, and an element s Z(Ĥ Γ. On this triple, one imposes the conditions that η identifies Ĥ with the connected centralizer of η(s in Ĝ and that the Ĝ-conjugacy class of η is fixed by Γ. This is part of the definition given in [S6, 5], and for our purposes we will not need the finer structure brought by the object H discussed there. The conditions on η imply that it induces (by duality an isomorphism η of complex tori 2

21 from the most split maximal torus in H to the most split maximal torus in G (the latter is in our case completely split. An isomorphism from some maximal torus in H to some maximal torus in G is then called admissible if it is of the form Ad(g η Ad(h for some g G, h H. If T H T is such an isomorphism defined over R, then we say that T originates from H, and using its dual isomorphism and the canonical embedding Z(Ĥ T H we obtain from s an element of T Γ, which by Tate-Nakayama-duality defines a character on H (Γ, T. Now we can state our goal. Given two maximal tori T, T 2 of G that originate from H, we would like to compare the results of pairing the endoscopic datum s against the splitting invariants for T and T 2. In order for this to make sense, we need to compare the groups in which these invariants live. To that end, we will show that the endoscopic characters on H (Γ, T and H (Γ, T 2 arising from s factor through certain quotients of these groups and that those quotients can be related. For a maximal torus T in G put X (T = {λ X (T σ T (λ = λ} IX (T = {λ σ T (λ λ X (T} where σ T is the action of σ on T. Recall the Tate-Nakayama isomorphism X (T IX (T = H T (Γ, X (T H (Γ, T given by taking cup-product with the canonical class in H 2 (Γ, C. Via this isomorphism, the canonical pairing T X (T C induces a pairing T H (Γ, T C The splitting invariant enters into the construction of the Langlands-Shelstad transfer factors via this pairing. Lemma 5.. Let A A be SOS in R. Then each element in the canonical set of isomorphisms T S A\A A T A (Definition 3. induces the same embedding i A,A : X (T A X (T A Proof: For an element ω Ω put X (T ω= = {λ X (T ω(λ = λ} For any SOS B X (T S B = = span Q (B X (T and any canonical isomorphism T S B T B identifies X (T S B = with X (T B (this identification will of course depend on the chosen isomorphism. Fix one canonical isomorphism T S A\A A T A. It is the composition of canonical isomorphisms T S A\A ϕ A T S ψ A T A and hence induces an inclusion as claimed, because X (T S A = X (T S A =. Moreover, any other canonical isomorphism T S A\A A T A is given by T S A\A A ϕ T S A S A T S A ψ T A 2

22 for A A \ A and clearly S A acts trivially on X (T S A =. The embedding i A,A induces an embedding ī A,A : X (T A IX (T A + i A,A (IX (T A X (T A i A,A(IX (T A + IX (T A and via the Tate-Nakayama isomorphism these quotients correspond to quotients of H (Γ, T A and H (Γ, T A respectively. We will argue that if the tori T A and T A originate in an endoscopic group H, then the endoscopic character factors through these quotients. This provides a means of comparing the values of the endoscopic character on the cohomology of both tori. Proposition 5.2. Let (H, s, η be an endoscopic triple for G and assume that T A and T A (A A originate from H, that is, there exist tori T, T 2 H and admissible isomorphisms T T A and T 2 T A. Write s TA T A and s TA T A for the images of s under the duals of these isomorphisms. Assume that there exists a canonical isomorphism j : T S A\A A T A s.t. ĵ(s TA = s TA (this can always be arranged. Then the characters s TA and s TA on H (Γ, T A resp. H (Γ, T A factor through the above quotients, and the pull-back of the character s TA via ī A,A equals s TA. Proof: We identify H (Γ, with HT (Γ, X ( via the Tate-Nakayama isomorphism. Because the element s TA X (T A C is Γ-invariant, its action on X (T A annihilates the submodule IX (T A. Thus, the action of j (s TA X (T A C on X (T A annihilates the submodule j (IX (T A. But we have arranged things so that j (s TA = s TA and we see that the action of s TA on X (T A via the standard pairing annihilates the submodule IX (T A + j (IX (T A. By the same argument, the action of s TA on X (T A annihilates the submodule IX (T A + j (IX (T A. Finally notice that by Lemma 5. the restriction of j to X (T A coincides with i A,A. References [AK] Y. Agaoka, E. Kaneda - Strongly orthogonal subsets in root systems. Hokkaido Math. J. 3 (22, n., [Bou] N. Bourbaki - Lie groups and Lie algebras. Springer, 22, ISBN [Kal] T. Kaletha, Endoscopic character identities for depth-zero supercuspidal L- packets, arxiv:99.533v [L] R.P. Langlands - Orbital integrals on forms of SL(3. I. Amer. J. Math. 5 (983, no. 2, [LS] [LS2] R.P. Langlands, D. Shelstad - On the definition of transfer factors. Math. Ann., vol. 278 (987, R.P. Langlands, D. Shelstad - Descent for transfer factors. The Grothendieck Festschrift, Vol. II, , Progr. Math., 87, Birkhäuser, Boston, MA, 99. [SGA3] M. Demazure, A. Grothendieck - SGA III. LNM 53, Springer, 97 [S] D. Shelstad - Characters and inner forms of a quasi-split group over R. Compositio Math. 39 (979, no.,

23 [S2] D. Shelstad - Orbital integrals and a family of groups attached to a real reductive group. Ann. Sci. École Norm. Sup. (4 2 (979, no., 3. [S3] D. Shelstad - Embeddings of L-groups. Canad. J. Math. 33 (98, no. 3, [S4] D. Shelstad - L-indistinguishability for real groups. Math. Ann. 259 (982, no. 3, [S5] [S6] [S7] [S8] D. Shelstad - A formula for regular unipotent germs. Orbites unipotentes et reprsentations, II. Astérisque No (989, D. Shelstad - Tempered endoscopy for real groups. I. Geometric transfer with canonical factors. Representation theory of real reductive Lie groups, , Contemp. Math., 472, Amer. Math. Soc., Providence, RI, 28. D. Shelstad - Tempered endoscopy for real groups. II. Spectral transfer factors. Automorphic forms and the Langlands program, , Adv. Lect. Math. (ALM, 9, Int. Press, Somerville, MA, 2. D. Shelstad - Tempered endoscopy for real groups. III. Inversion of transfer and L-packet structure. Represent. Theory 2 (28, [Spr] T. A. Springer - Linear algebraic groups. Prog. Math. 9, 98. Tasho Kaletha tkaletha@math.uchicago.edu University of Chicago 5734 S. University Avenue Chicago, Illinois