On the stabilization of relative trace formulae: descent and the fundamental lemma
aa r X i v : . [ m a t h . N T ] J un ON THE STABILIZATION OF A RELATIVE TRACE FORMULA: THEFUNDAMENTAL LEMMA
SPENCER LESLIE
Abstract.
We introduce the notions of relative endoscopic data, transfer factors, and thefundamental lemma for certain symmetric spaces associated to unitary groups over a non-archimedean field of characteristic zero. The main result is a proof of this fundamental lemma.For this, we prove descent results delicate enough to reduce this statement to the infini-tesimal analogue, which we have previously established. Along the way, we show that p -adicsymmetric spaces enjoy a notion of topological Jordan decomposition, which may be of inde-pendent interest, and prove a relative version of a lemma of Kazhdan that played a crucial rolein the proof of the Langlands-Shelstad fundamental lemma. Contents
1. Introduction 12. Preliminaries 53. The symmetric space 74. Topological Jordan decompositions and descent 135. Orbital integrals and relative endoscopy 186. The infinitesimal theory and the very regular locus 227. Descent to the very regular case 29Appendix A. Endoscopy for unitary Lie algebras 34References 371.
Introduction
In this paper, we develop a relative theory of endoscopy for symmetric spaces of the formU(2 n ) / U( n ) × U( n ) and prove the endoscopic fundamental lemma for the unit element ofthe spherical Hecke algebra. This is the first such result in the literature, and relies on theinfinitesimal version (the “Lie algebra case”) previously established in [Les19a]. These resultswill play a central role in the stabilization of the elliptic part of the relative trace formulaassociated to the relevant automorphic period integrals.1.1. Global motivation.
Let
E/F be a quadratic extension of number fields, A E and A F theassociated rings of adeles. Let W and W be two n dimensional Hermitian spaces over E . Thedirect sum W ⊕ W is also a Hermitian space and we have the embedding of unitary groupsU( W ) × U( W ) ֒ → U( W ⊕ W ) . Let π be an irreducible cuspidal automorphic representation of U( W ⊕ W )( A F ). Then π issaid to be distinguished by the subgroup U( W ) × U( W ) if the period integral Z [U( W ) × U( W )] ϕ ( h ) dh (1) Date : June 24, 2020.2010
Mathematics Subject Classification.
Primary 11F70; Secondary 11F55, 11F85.
Key words and phrases.
Fundamental lemma, endoscopy, symmetric spaces, topological Jordandecompositions. is not equal to zero for some vector ϕ in the π -isotypic subspace of automorphic forms onU( W ⊕ W )( A F ). Here, [H] = H( F ) \ H( A F ) for any F -group H. We call these unitary Friedberg-Jacquet periods in homage to [FJ93].These periods have recently appeared in the literature in several ways (for example, [IP18],[PWZ19], [GS20], and indirectly in [LZ19]). We would therefore like to study automorphic formsdistinguished by these subgroups. Wei Zhang has conjectured a comparison of relative traceformulas (first suggested in a less precise form in [GW14]) which relates these periods to special L -values of certain L -functions (see [Les19a] for a discussion about some related conjectures),so we consider the relative trace formula associated to unitary Friedberg–Jacquet periods onU( W ⊕ W )( A F ).However, unlike other relative trace formulas that have studied in the literature, this relativetrace formula is not stable . Spectrally, this should be related to the non-factorizability of theperiod integrals for certain automorphic representations (see [AP06] for another example relatedto Galois periods with an non-stable relative trace formula). Geometrically, this instabilitymanifests in that when we consider the action of U( W ) × U( W ) on the symmetric space Q := U( W ⊕ W ) / U( W ) × U( W ) , invariant polynomials distinguish only geometric (or stable) orbits . Our goal is to stabilize thegeometric side of this relative trace formula by considering relative analogues of the theory ofendoscopy in order to establish Zhang’s conjecture (as well as its generalizations).In [Les19b], we introduced a potential theory of relative endoscopy for the infinitesimal sym-metric space (that is, the tangent space at the distinguished U ( W ) × U ( W )-fixed point of Q ( F )), and proved the existence of smooth transfer for many test functions. We then estab-lished the fundamental lemma for the unit function in this infinitesimal setting in [Les19a] viaa combination of local harmonic analysis and a global argument relying on a new comparisonof relative trace formulas. The motivation for considering the linearized case first was two-fold:(1) the infinitesimal setting is simpler from both invariant-theoretic and Galois-cohomologicalperspectives, and(2) we expect to be able to reduce the theory on the symmetric space Q to this infinitesimalsetting via descent.This latter expectation is largely motivated by the case of (twisted) endoscopy, where both theexistence of smooth transfer and the fundamental lemma for the unit function were ultimatelyreduced to the Lie algebra ([Wal95],[Wal97],[Wal08] [Hal95], [Wal06], [Ngˆo10]). Additionally,C. Zhang’s proof of smooth transfer in the context of the Guo-Jacquet comparison of relativetrace formulas ([Zha15]) similarly reduces to the infinitesimal setting.This paper accomplishes the first step in this reduction by extending the relative theoryof endoscopy to the symmetric space Q ( F ) and proving the fundamental lemma for the unitelement by descending to the main result of [Les19a]. To carry this out, we establish severaldescent results for a general class of symmetric spaces over p -adic fields. We also establish theexistence of smooth transfers for many test functions (see Proposition 5.7).In forthcoming work, we use these results to stabilize the elliptic part of the relative traceformula associated to unitary Friedberg–Jacquet periods, subject to the existence of smoothtransfer for all Schwartz functions. We wish to emphasize however that the main result here isthe core identity showing that our proposed theory of relative endoscopy gives rise to a globalstabilization of the relative trace formula.1.2. Main Result.
Let us now state the main result. For brevity, we refer the reader to Section5.2 for the relevant notations for orbital integrals and transfer factors.Assume
E/F is an unramified quadratic extension of p -adic fields with odd residue charac-teristic large enough depending only on F/ Q p (for example, if F = Q p the only constraint is p = 2). Let O F (resp. O E ) denote the ring of integers in F (resp. E ). We consider the Lemma 7.4 is the only source of restriction, and is standard. We remark that when F is a number field, thiscondition is satisfied at all but finitely many places which may be explicitly calculated. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 3 symmetric pair (G n , H n ) = (U( V n ⊕ V n ) , U( V n ) × U( V n )) , where V n is a split Hermitian space of dimension n . Concretely, we set V n = E n and equip itwith the Hermitian form represented by the identity matrix I n . This affords the self-dual latticeΛ n = O nE ⊂ V n , with respect to which we obtain hyperspecial subgroupsH n ( O F ) := U(Λ n ) × U (Λ n ) ⊂ H n ( F )and G n ( O F ) := U(Λ n ⊕ Λ n ) ⊂ G n ( F ) . Set Q n := G n / H n and let Q n ( O F ) to be the associated characteristic function.In Section 5.2, we introduce the notion of an unramified elliptic relative endoscopic datumΞ a,b = ( ξ a,b , α, β ), following the approach of [Les19b]; here ξ a,b is a certain unramified ellipticendoscopic datum of G n , α (resp. β ) is a Hermitian form on E a (resp. E b ). This gives rise to a(pure inner form of an) elliptic endoscopic groups G a,α × G b,β of G n and a symmetric subgroupH a,α × H b,β of G a,α × G b,β . Let Q a,α × Q b,β := G a,α / H a,α × G b,β / H b,β be the associated endoscopic symmetric space.For a regular semi-simple element x ∈ Q rssn ( F ) (see Section 2.1), the endoscopic datumdetermines a character κ , with respect to which we define the relative κ -orbital integralO κ ( x, f ) = X x ′ ∼ st x κ (inv( x, x ′ )) O( x ′ , f ) , where x ′ runs over rational H n ( F )-orbits that lie in the same stable orbit of x, and inv( x, x ′ ) isthe cohomological invariant assoicated to the rational orbit of x ′ (see Section 2.1). When κ = 1is the trivial character, set SO = O κ .We show in Section 5.2 that there is a good notion of the matching of regular semi-simpleelements x ∈ Q rssn ( F ) and ( x a , x b ) ∈ ( Q a,α ( F ) × Q b,β ( F )) rss , and transfer factors ∆ rel : ( Q a,α ( F ) × Q b,β ( F )) rss × Q rssn ( F ) −→ C in the sense that we can define the notion of smooth transfer of κ -orbital integrals on Q n ( F )and stable orbital integrals on Q a,α ( F ) × Q b,β ( F ) (Definition 5.6) and prove the existence ofsmooth transfers for many test functions (Proposition 5.7).We now state our main result. Theorem 1.1. If ( α, β ) = ( I a , I b ) , the functions Q n ( O F ) and Q a ( O F ) ⊗ Q b ( O F ) match. Oth-erwise, Q n ( O F ) matches .More precisely, for any regular semi-simple x ∈ Q n ( F ) and matching elements ( x a , x b ) ∈Q a,α ( F ) × Q b,β ( F ) , if κ is the character associated to the endoscopic datum, then ∆ rel (( x a , x b ) , x ) O κ ( x, Q n ( O F ) ) = ( SO(( x a , x b ) , Q a ( O F ) ⊗ Q b ( O F ) ) : ( α, β ) = ( I a , I b ) , α, β ) = ( I a , I b ) . (2)1.3. Sketch of the proof.
The proof of Theorem 1.1 uses a method of descent analogous tothe techniques used in the case of twisted endoscopy [Wal08]. As we work with a symmetricspace and not a group, several results must be extended to our relative setting. Once accom-plished, descent enables us to reduce Theorem 1.1 to an infinitesimal analogue. We recall thisinfinitesimal theory and the fundamental lemma for the Lie algebra of Q n , which is the mainresult of [Les19a], in Section 6.Setting W = V n ⊕ V n , we consider the Cayley transforms c ν : End( W ) GL( W ), where ν = ±
1. These exponential-like maps are well suited for the study of the symmetric space Q n .In particular, we introduce certain open subset Q ♥ ,νn ( F ) ⊂ Q n ( F ), which we refer to as the ν -very regular locus. We show in Section 6.5 that the Theorem 1.1 may be readily reduced to SPENCER LESLIE the Lie algebra result via the Cayley transform whenever x ∈ Q ♥ , n ( F ) ∪ Q ♥ , − n ( F ). Combinedwith certain elementary vanishing properties of orbital integrals (Lemma 5.10), it follows that(2) is known unless x ∈ Q rssn ( O F ) lies in a certain codimension 2 subvariety; see Remark 6.10.To handle these remaining cases, we develop two descent tools for p -adic symmetric spacesin Section 4. Firstly, we show in Section 4.2 that there is a good notion of a topological Jordandecomposition for certain points in Q n ( F ) . More precisely, for elements x ∈ Q n ( F ) which are strongly compact , there is a decomposition x = x as x tu with x as , x tu ∈ Q n ( F )where x as is the absolutely semi-simple part of x and x tu is the topologically unipotent part of x .This is a topological analogue of the fact that symmetric spaces are well-behaved with respectto the Jordan decomposition [Ric82, Lemma 6.2]. The proof relies heavily on the existence ofthe symmetrization map s : Q n −→ U( W )realizing the symmetric space as a closed subvariety of U( W ). In particular, this enables usto make sense of taking products of elements of Q n ( F ) , as well as the definitions of absolutelysemi-simplicity and topological unipotence. Our result holds for any connected symmetric spaceover a p -adic field. It is an interesting question whether a more general theory exists for p -adicspherical varieties.To make use of this decomposition, we establish in Proposition 4.9 a relative analogue ofa result of Kottwitz [Kot86, Proposition 7.1] (itself a generalization of a lemma of Kazhdan[Kaz84]) showing that stabilizer group schemes of absolutely semi-simple elements x as ∈ Q n ( O F )have smooth group O F -scheme models, implying that the stable orbits of such elements arestrongly constrained. This has several consequences for orbital integrals, the most direct beingProposition 7.2 which reduces the computation of the κ -orbital integrals at x occurring in (2)to those on a smaller-dimensional symmetric space Q x as , known as the descendant of Q n at x as . Remark . We establish Proposition 4.9 in greater generality (that of a nice, simply-connectedsymmetric pair; see Section 4.3) than is needed for this paper as it may be of independentinterest. For example, it applies to certain Galois pairs (Res
E/F
G, G ) with G a reductive groupover F . We will pursue the consequences of the results of this paper to Galois pairs in futurework.In Section 7, we apply these results in the remaining degenerate cases and establish thenecessary descent of the transfer factors. The point is that the topologically unipotent part x tu lies in the very regular locus of the descendant at x as , allowing us to pass to the Lie algebra.This concludes the final cases of the fundamental lemma.1.4. Outline.
In Section 2, we fix notation and our choice of Haar measures. Section 3 studiesthe basic geometry of the symmetric space Q , introducing the contraction map R used in definingour notion of endoscopic symmetric space, and computing all semi-simple descendants of thesymmetric space. Section 4 develops the needed tools for the descent arguments of Section7. More precisely, we prove the compatibility of the topological Jordan decomposition withthe symmetric space in Section 4.2. We then prove a relative version of Kazhdan’s lemma inProposition 4.9.In Section 5, we define elliptic relative endoscopic data and the relevant symmetric spaces.We then define the matching of stable orbits and transfer factors, following the infinitesimaltheory developed in [Les19b], and state the main theorem as Theorem 5.9. These notions relyon the theory of endoscopy for unitary Lie algebras, which we review in Appendix A for theconvenience of the reader. Section 6 recalls the infinitesimal theory and fundamental lemma(stated as Theorem 6.1). We then compare this infinitesimal theory to the symmetric spacecase using the Cayley transform in Section 6.4, deducing the fundamental lemma over the veryregular locus in Section 6.5. We complete the proof of Theorem 1.1 in Section 7 by applyingthe tools of Section 4 to study the remaining cases. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 5
Acknowledgements.
I would like to thank Jayce Getz for suggesting studying relativenotions of endoscopy and for many helpful conversations and suggestions. I also want to thankYiannis Sakellaridis for several insightful conversations and for his interest in this work. Thiswork was partially supported by an AMS-Simons Travel Award and by NSF grant DMS-1902865.2.
Preliminaries
Invariant theory.
For any field F and any non-singular affine algebraic variety Y over F with G a connected reductive algebraic group over F acting algebraically on Y, we set Y rss tobe the invariant-theoretic regular semi-simple locus. That is, x ∈ Y rss if and only if its G-orbitis of maximal possible dimension and Zariski-closed. We also recall the semi-simple locus Y ss ofpoints with Zariski-closed orbits. When F is a local field of characteristic zero, and we endow Y ( F ) the the Hausdorff topology, it is known [AG09, Theorem 2.3.8] that x ∈ Y ss ( F ) if andonly if G( F ) x ⊂ Y ( F ) is closed in the Hausdorff topology.For x, x ′ ∈ Y rss ( F ), we say that x ′ is in the rational G( F ) -orbit of x if there exists g ∈ G( F )such that g · x = x ′ . Fixing an algebraic closure F alg , two semi-simple points x, x ′ ∈ Y ss ( F ) are said to lie in thesame stable orbit if g · x = x ′ for some g ∈ G( F alg ) and such that the cocycleinv( x, x ′ ) := [ τ g − τ ( g )] ∈ Z ( F, G)lies in Z ( F, G x ), where G x ⊂ G x is the connected component of the identity of the stabilizerof x in G . For the symmetric spaces we consider, the semi-simple stabilizers are all connected(see Lemma 3.9), so that this cocycle constraint is automatic.A standard computation shows that the set O st ( x ) of rational orbits in the stable orbit of x are in natural bijection with D (G x /F ) := ker (cid:2) H ( F, G x ) → H ( F, G) (cid:3) . Here we ignore the dependence on G in the notation. If F is non-archimedean of characteristiczero, this is a finite abelian group and O st ( x ) is naturally a D (G x /F )-torsor.2.2. Local fields.
We fix a non-archimedean local field F of characteristic zero and assumethat the residue characteristic p is odd. A further assumption on p will arise in Section 7 forthe purposes of descent, but the results of the prior sections (aside from the main result, whichrelies on that section) are valid without this restriction.We set | · | F to be the normalized valuation so that if ̟ is a uniformizer, then | ̟ | − F = O F / p F ) =: q is the cardinality of the residue field k := O F / p F . Here p F denotes the unique maximal idealof O F . Let F alg denote a fixed algebraic closure of F and O F alg ⊂ F alg its ring of integers. For a ∈ O F alg , we let a ∈ k alg denote its image in the residue field, and use similar notation for k .For any quadratic ´etale algebra E/F of local fields, we set η E/F : F × → C × for the characterassociated to the extension by local class field theory. We also let Nm E/F : E × −→ F × denotethe norm map.Throughout the article, all tensor products are over C unless otherwise indicated.2.3. Groups and Hermitian spaces.
For a field F and for n ≥
1, we consider the algebraicgroup GL n of invertible n × n matrices. Suppose that E/F is a quadratic ´etale algebra andconsider the restriction of scalars Res
E/F (GL n ). For any F -algebra R and g ∈ Res
E/F (GL n )( R ),we set g g to be the Galois involution associated to the extension E/F ; we also denote this involution by σ . Set X n ( F ) = { x ∈ GL n ( E ) : t x = x } . SPENCER LESLIE
Note that GL n ( E ) acts on X n via g ∗ x = gx t g, x ∈ X n , g ∈ GL n ( E ) , where t g denotes the transpose. We let V n be a fixed set of orbit representatives. For any x ∈ X n , set h· , ·i x to be the Hermitian form on E n associated to x . Denote by V x the associatedHermitian space and U( V x ) the corresponding unitary group. Note that if g ∗ x = x ′ then V x t g −→ V x ′ is an isomorphism of Hermitian spaces. Thus, V n gives a set of representatives { V x : x ∈ V n } ofthe equivalence classes of Hermitian vector space of dimension n over E . We will abuse notationand identify this set with V n . If we are working with a fixed but arbitrary Hermitian space, weoften drop the subscript. For any Hermitian space, we set U ( V ) = U( V )( F ) . When
E/F is an unramified quadratic extension of p -adic fields, we fix V n = ( E n , I n ) as ourrepresentative of split Hermitian spaces.2.4. Measures and centralizers.
We will only consider integration with respect to unimod-ular groups G ( F ), so we fix a Haar measure dg throughout. Several definitions will depend onthe choice of such a measure, so we make a few conventions here.When E/F is unramified, V n = ( E n , I n ) our split Hermitian space, and Λ n = O nE ⊂ V n isthe standard self-dual lattice, we always fix the Haar measures giving the hyperspecial maximalsubgroups GL(Λ n ) ⊂ GL( V n ) and U (Λ n ) ⊂ U ( V n ) volume 1 . Outside of this setting, we mayfix an arbitrary Haar measure as the precise choices will not affect the results of this paper.We need also to consider the measures on regular semi-simple centralizers. Fix a Hermitanform x and consider U ( V ) = U ( V x ). We will be interested in the twisted Lie algebra H erm ( V ) = { δ ∈ End( V ) : h δv, u i = h v, δu i} . The group U ( V ) acts on this space by the adjoint action, and an element δ is regular semi-simpleif its centralizer is a maximal torus T δ ⊂ U ( V ). To construct T δ note that there is a naturaldecomposition F [ δ ] := F [ X ] / ( char δ ( X )) = m Y i =1 F i , where F i /F is a field extension and char δ ( X ) denotes the characteristic polynomial of δ . Setting E i = E ⊗ F F i , we have E [ δ ] = Y i E i = Y i ∈ S E i × Y i ∈ S F i ⊕ F i , where S = { i : F i + E } . Lemma 2.1.
Let δ ∈ H erm ( V ) be regular semi-simple, let T δ denote the centralizer of δ in U ( W ) . Then T δ ∼ = Z U( V ) ( F ) E [ δ ] × /F [ δ ] × , where Z U ( V ) ( F ) denotes the center of U ( V ) . Moreover, H ( F, T δ ) = Q S Z / Z and D ( T δ /F ) = ker (cid:2) H ( F, T δ ) → H ( F, U ( V )) (cid:3) = ker Y S Z / Z → Z / Z , where the map on cohomology is the summation of the factors.Proof. This is proved, for example, in [Rog90, 3.4]. (cid:3)
Set T S ∼ = Z U( V ) ( F ) Q i ∈ S E × i /F × i for the unique maximal compact subgroup of T δ . Wechoose the measure dt on T δ giving this subgroup volume 1. We will study orbital integralsover regular semi-simple orbits and always use the measures introduced here to define invariantmeasures on these orbits. By a slight abuse of notation, we will not acknowledge this in ournotation. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 7
Remark . This convention fixes measures on various rational orbits in a given stable orbitcompatibly in the sense of transfer of measures along an inner twisting (see [Rog90, Chapter3]). 3.
The symmetric space
Let W and W be two n -dimensional Hermitian spaces with respect to our fixed quadraticextension of p -adic E/F . Set W = W ⊕ W be the 2 n -dimensional Hermitian space. Let ǫ ∈ U( W ) be an element of order 2 inducing the eigenvalue decomposition W = W ⊕ W . Wethen have the involution θ ( g ) = ǫgǫ on both U( W ) and GL( W ), with corresponding fixed-pointsubgroups U( W ) × U( W ) ⊂ U( W ) and GL( W ) × GL( W ) ⊂ GL( W ) . Set G = U( W ) and H = U( W ) × U( W ). In this section, we study the associated symmetricspace Q ( F ) := U ( W ) /U ( W ) × U ( W ) . Set σ ( g ) = θ ( g ) − , and s ( g ) = gσ ( g ), known as the symmetrization map. Then s : G → Ggives a map of varieties inducing an isomorphism G / H ∼ −→ Q . In particular, we have a naturalembedding Q ⊂ G , and the H action on Q is realized as conjugation on the image of s .3.1. The linear symmetric space.
It is useful to first consider the base change of the variety Q to E , the E -points of which are isomorphic to S ( F ) = Res E/F ( Q E )( F ) = GL( W ) / GL( W ) × GL( W ) . Consider the symmetrization map s : GL( W ) −→ GL( W ). Since H ( F, GL( W ) × GL( W )) =0, we have a surjection on F -points and an identification s : GL( W ) / GL( W ) × GL( W ) ∼ −→ S ( F ) . Given an element x = s ( g ) ∈ S ( F ), write x = (cid:18) A BC D (cid:19) . Then the block matrices satisfy thepolynomial relations A = I n + BC, D = I n + CB, AB = BD, CA = DC.
These relations are not sufficient to cut out S . Set P ⊂
GL( V ⊕ V ) to be the subvariety ofelements satisfying σ ( g ) = g . Unwinding the definition, this is the variety of x ∈ GL( W ) suchthat ǫx is an involution. We have a decomposition of P into irreducible components P = n G i =1 P i where if for any x ∈ P ( F ), we have an eigenspace decomposition W = W x, ⊕ W x, − , for the involution ǫx , we have P i ( F ) = { x ∈ P ( F ) : dim( W x, − ) = i } . It is clear that S = P n . In general, a computation of the characteristic polynomial of ǫx distin-guishes these components.If we realize the embedding GL( W ) × GL( W ) ⊂ GL( W ) in a block-diagonal fashion, theaction of ( g, h ) ∈ GL( W ) × GL( W ) is given by( g, h ) · x = (cid:18) gAg − gBh − hCg − hDh − (cid:19) Consider the invariant map χ : S → A n given by sending x ∈ S ( F ) to the coefficients of themonic polynomial χ x ( t ) = det( tI − A ). SPENCER LESLIE
Lemma 3.1.
The pair ( A n , χ ) is a categorical quotient for (GL( V ) × GL( V ) , S ) .Proof. As the statement is geometric, we may assume that E = E alg . We make use of Igusa’scriterion [Zha14, Section 3]: let a reductive group H act on an irreducible affine variety X . Let Q be a normal irreducible variety, and let π : X → Q be a morphism that is constant on H orbits such that(1) Q − π ( X ) has codimension at least two,(2) there exists a nonempty open subset Q ′ ⊂ Q such that the fiber π − ( q ) of q ∈ Q ′ contains exactly one orbit.Then ( Q, π ) is a categorical quotient of (
H, X ). To show that this is the case, we make use of thefollowing set of representatives of the semi-simple GL( W ) × GL( W )-orbits on S due to Jacquetand Rallis [JR96, Lemma 4.3]: each semi-simple element x ∈ S is GL( W ) × GL( W )-conjugateto an element of the form x ( A, n , n ) := A A − I m I n − I n A + I m A I n
00 0 0 0 0 − I n , (3)with n = m + n + n and A ∈ g l m ( E ) semi-simple without eigenvalues ± x ( A, n , n ) is regular if and only if n = n = 0 and A is regular in g l n ( E ).With this set of semi-simple orbit representatives, it is clear that for a given tuple ( a , . . . , a n ),one may form the polynomial p ( b i ) ( t ) = t n + a t n − + · · · + a n = m Y i =1 ( t − α i ) × ( t − n ( t + 1) n , for certain α i ∈ E . Setting A = diag( α , . . . , α m ), we see that χ ( x ( A, n , n )) = ( b , · · · b n ) sothat χ is surjective. Moreover, if A ∈ g l n ( E ) is regular semi-simple, then the uniqueness state-ment of Jacquet and Rallis implies that there is a unique orbit in the fiber over the coefficientsof det( tI − A ). This implies the second criterion for the open set Q ′ = { ( b , . . . , b n ) : p ( b i ) ( t ) = det( tI − A ) for some A ∈ g l rssn ( E ) − D ∪ D − } , where for any a ∈ E , D a = { X ∈ g l n : det( aI n − X ) = 0 } . This completes the proof of thelemma. (cid:3) A similar argument gives the following lemma for the quotient by the GL( W )-factor. Lemma 3.2.
Let R : S → g l ( W ) denote the map (cid:18) A BC D (cid:19) A. Then ( g l ( W ) , R ) is a categorical quotient for the GL ( W ) -action on S given by h · (cid:18) A BC D (cid:19) = (cid:18) A Bh − hC hDh − (cid:19) . The map R is GL( W ) -equivariant with respect to the adjoint action on g l ( W ) .Proof. As in the proof of the previous lemma, the set of GL( W ) × GL( W )-orbit representativesgiven by (3) shows that the map π is surjective. The uniqueness of orbits over a non-emptysubset follows from the associated statement in Lemma 3.1. (cid:3) We end with the following elementary linear algebra computation.
N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 9
Lemma 3.3.
Suppose that x = (cid:18) A BC D (cid:19) ∈ S ( F ) . The characteristic polynomial of x is car x ( t ) = det( t − At + I n ) . In particular, the eigenvalues of x are given by Ω( x ) := { α ± p α − α is a root of χ x ( t ) } . (cid:3) Unitary symmetric space.
In the unitary case, we need to account for various pure innerforms as the symmetrization map is no longer surjective. Indeed,we have the exact sequence ofpointed sets1 → U ( W ) × U ( W ) → U ( W ) → Q ( F ) → H ( F, U( W ) × U( W )) ϕ −→ H ( F, U( W )) → . (4)Surjectivity of ϕ follows from the surjectivity of the map H ( k, U( W )) ∼ −→ H ( k, U( W )). Lemma 3.4.
Let
E/F be a quadratic extension of p -adic fields. There exist two isomorphismclasses of decomposition W ⊕ W = W = W ′ ⊕ W ′ . We have a bijection of F -points Q ( F ) = U ( W ) /U ( W ) × U ( W ) G U ( W ) /U ( W ′ ) × U ( W ′ ) (5) where the first quotient is identified with the image of s : U ( W ) → Q ( F ) . Proof.
This is a basic Galois cohomology calculation. We omit the details. (cid:3)
We pause to introduce some notation. The symmetrization map takes the form s ( g ) = qǫg ∗ ǫ ,where ∗ denotes the adjoint map such that U ( W ) = { g ∈ GL( W ) : gg ∗ = I W } . Writing this out, we have s ( g ) = (cid:18) AA ∗ − BB ∗ CA ∗ − DB ∗ BD ∗ − AC ∗ DD ∗ − CC ∗ (cid:19) for g = (cid:18) A BC D (cid:19) .Here we need to be precise about the overloaded notation. For A ∈ End( W ), A ∗ is the adjointoperator with respect to the Hermitian form Φ on W : h Av, w i = h v, A ∗ w i for all v, w ∈ W ;similarly with D ∈ End( V ). For B ∈ Hom E ( W , W ), the endomorphism B ∗ ∈ Hom E ( W , W )is defined by h B ( v ) , w i = h v, B ∗ ( w ) i , for all w ∈ W , v ∈ W ;the map C C ∗ is analogous. In particular, any element x ∈ Q ( F ) may be written x = (cid:18) A B − B ∗ D (cid:19) , where A ∈ H erm ( W ), D ∈ H erm ( W ), and B ∈ Hom E ( W , W ). As before, the blocks satisfythe polynomial relations A = I n − BB ∗ , D = I n − B ∗ B, AB = BD, B ∗ A = DB ∗ . As in the linear case, we define the morphism χ : Q → A n by sending x to the coefficients ofthe monic polynomial χ x ( t ) = det( tI − A ). Lemma 3.5.
The pair ( A n , χ ) is a categorical quotient for the U( W ) × U( W ) -action on Q .Proof. As the assertion is geometric, we may assume without loss that F = F alg . But over F alg , this setting is that of to the symmetric space in Lemma 3.1. (cid:3) We also have a unitary version of Lemma 3.2:
Lemma 3.6.
Define the contraction map R : Q → H erm ( W ) given by (cid:18) A B − B ∗ D (cid:19) A. The pair ( H erm ( W ) , R ) is a categorical quotient for the U( W ) -action on Q . A useful consequence of the orbit computations (3) is that if X ∈ Q ( F ) is regular semi-simple, then det( B ) = 0 and A is regular semi-simple in H erm ( V ). Let Q iso ⊂ Q denote theZariski-open subvariety cut out by this determinant condition. The superscript iso refers to thefact that x ∈ Q iso ( F ) if and only if I − R ( x ) ∈ Iso( W , W ) . Setting H erm ( W ) rr = { A ∈ H erm ( W ) : I − A is non-singular } , we obtain a map R : Q iso −→ H erm ( W ) rr . Here the superscript rr references the fact thatthat H erm ( W ) rr contains the image of the relatively regular locus of U ( W ). Lemma 3.7.
The restriction R : Q iso → H erm ( W ) rr is a U( W ) -torsor. Moreover, for x ∈ Q iso ( F ) , we have an isomorphism H x ∼ −→ U( W ) R ( x ) given by ( h , h ) h .Proof. This is analogous to Lemma 3.6 of [Les19b], and is proved in the same way. (cid:3)
Let W ′ denote the twist of the Hermitian space W realized as the same underlying vectorspace equipped with a Hermitian form of the other isomorphism class. If we set W ′ = W ⊕ W ′ ,then W = W ′ . Setting Q ′ = U( W ′ ) / U( W ) × U( W ′ ) , a simple Galois cohomology computation implies the decomposition H erm ( W ) rr = U ( W ) \Q iso ( F ) G U ( W ′ ) \ ( Q ′ ) iso ( F ) . (6)The following lemma explains how to detect the which orbit A ∈ H erm ( W ) rr lies in. Lemma 3.8.
Identify H ( F, U( W )) = F × / Nm E/F ( E × ) via the discriminant map ( W , Φ) d (Φ) ∈ F × / Nm E/F ( E × ) , where d (Φ) := ( − n ( n − / det(Φ) . Then X is in the image of R : Q iso ( F ) −→ H erm ( W ) rr if and only if d ( I − X ) ≡ d ( h· , · , i ) · d ( h· , · , i ) (mod Nm E/F ( E × )) Proof.
The claim follows from the definition of the map R in Lemma 3.6 and the relation I n − A = BB ∗ . (cid:3) Naturally, combining (5) and (6) we can express the quotient H erm ( W ) rr as a disjoint unionof four components of the form U ( W ) \ U ( W ⊕ W ) iso /U ( W ) × U ( W ) , where U( W ⊕ W ) iso denotes the preimage of Q iso under the symmetrization map. Here, W i for i = 2 , , n subject to the constraint that W ⊕ W = W ⊕ W is an equality of 2 n -dimensional Hermitian spaces. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 11
Descendants.
For this subsection only, we let
E/F denote a quadratic extension of fieldsof either odd or zero characteristic. Let x ∈ Q ss ( F ) denote a semi-simple element.We compute the possible pairs of centralizers (G x , H x ), known as the descendant of (G , H) at x . An important corollary of this computation is that all the stabilizers H x ⊂ H are connectedreductive groups (see Lemma 4.8) . We remark that our descent argument in Section 7 onlyencounters descendants of the form (1) and (2) below. Regardless, the general form will beuseful for later applications toward smooth transfer. Lemma 3.9.
For any x ∈ Q ss ( F ) , there is an orthogonal decomposition of WW = V ′ ⊕ V ⊕ V − , with V (resp. V − ) is the -eigenspace (resp. − -eigenspace) of x and V ′ is the orthogonalcompliment of V ⊕ V − in W . The involution θ ( g ) = ǫgǫ preserves this decomposition, and thesymmetric pair ( U ( W ) x , ( U ( W ) × U ( W )) x ) is a product of the following symmetric pairs:(1) ( U ( V ) , U ( V , ) × U ( V , − )) , where V is the -eigenspace of x , and V , ± = { v ∈ V : ǫv = ± v } ; (2) ( U ( V − ) , U ( V − , ) × U ( V − , − )) , where V − is the − -eigenspace of x , and V − , ± = { v ∈ V − : ǫv = ± v } ; (3) (GL( V ) , U ( V ′ )) , where V ′ is a non-degenerate Hermitian space over E ′ /F ′ . Here, F ′ isa finite extension of F and E ′ = EF ′ is the associated quadratic extension;(4) ( U ( V ′ ) × U ( V ′ ) , U ( V ′ )) , with U ( V ′ ) embedded diagonally;(5) (GL( V ′ ) × GL( V ′ ) , GL( V ′ )) , with GL( V ′ ) embedded diagonally.Proof. We begin by decomposition W = M i V i where each V i is a subspace upon which the minimal polynomial of x | V i is irreducible. For each i , let E i be the finite extension of E cut out by x ; we have then V i ∼ = E n i i for some n i . We set α i ∈ E × i for the eigenvalue of x on E i .Let P ( t ) = car x ( t ) denote the characteristic polynomial of x and let P i denote the minimalpolynomial of x | V i . Noting that x ∈ U ( W ) = ⇒ x ∗ = x − , we have P ( t ) = t dim( W ) P (0) P ( t − ), where P ( t ) denotes the action of the non-trivial Galois elementof Gal( E/F ) on the coefficients. This implies a product decomposition P ( t ) = Y i ∈ I P i ( t ) n i Y ( j,j ′ ) ∈ J (cid:0) P j ( t ) P j ′ ( t ) (cid:1) n j , where for each i ∈ I , P i ( t ) = t dim( V i ) P i (0) P i ( t − ) , and for each pair ( j, j ′ ) ∈ J P j ′ ( t ) = t dim( V j ) P j (0) P j ( t − ) . Thus, for each i ∈ I , we obtain a Galois element ( · ) : E i −→ E i induced by E i ∼ = E [ t ] / ( P i ( t )) −→ E [ t ] / ( P i ( t )) ∼ = E i α i t t α − i . This fact already follows over the algebraic closure from the orbit computation (3).
Setting F i to be the field fixed by this involution, we obtain a quadratic extension E i /F i and notethat α i = α − i . It is now easy to see that the Hermitian form on W restricts to a non-degenerateHermitian form on V i with respect to the quadratic extension E i /F i .For each ( j, j ′ ) ∈ J, a similar argument shows an isomorphism E j ∼ −→ E j ′ . Under the identi-fication, we find that α j ′ = α − j , and the restriction of the Hermitian form on W to V j ⊕ V j ′ isnon-degenerate, the direct summands of the decomposition being maximal isotypic subspaces.Thus, we obtain the product U ( W ) x = Y i ∈ I U ( V i ) × Y ( j,j ′ ) ∈ J GL( V j ) . We now compute the group ( U ( W ) × U ( W )) x . For simplicity, fix i and set E ′ = E i , V ′ = V i ,and let α = α i denote the associated eigenvalue. Since ǫxǫ = x − , we see that ǫ either fixeseach V ′ or ǫ ( V ′ ) is the α − -eigenspace. Set C ( α ) = { α, α − , α, α − } . Case 1: C ( α ) = { α } . In this case, α = α − , so that α = ± . Clearly, E ′ = E and ǫ ( V ′ ) = V ′ . This induces aneigenvalue decomposition V ′ = V ′ ⊕ V ′− . A simple exercise shows that the restriction of the Hermitian pairing is non-degenerate on eacheigenspace, so we obtain the symmetric pair( U ( V ′ ) , U ( V ′ ) × U ( V ′− )) . Case 2: C ( α ) = { α, α } . In this case, α = α − but α = α . Then ǫ ( V ′ ) is the α -eigenspace, and we find the symmetricpair ( U ( V ′ ) × U ( ǫV ′ ) , U ( V ′ )) , with respect to the embedding g ( g, σ ( g )) . Case 3: C ( α ) = { α, α − } . In this case, α = α , so that ǫ ( V ′ ) is the α − -eigenspace. This produces the pair(GL( V ′ ) , U ( V ′′ )) , where V ′′ = { ( w, ǫw ) : w ∈ V ′ } ֒ → V ′ ⊕ ǫ ( V ′ ) . The projection U ( V ′′ ) GL( V ′ ) × GL( ǫV ′ )GL( W ′ ) p produces an embedding U ( V ′′ ) ֒ → GL( V ′ ) where the resulting form on V ′ is given by h w, ǫv i ,for w, v ∈ V ′ . Case 4: C ( α ) = { α, α − , α, α − } . Finally, we have the case that all eigenvalues are distinct. Then ǫ ( V ′ ) is the the α − -eigenspace, and the spaces V ′ and ǫ ( V ′ ) belong to distinct pairs ( V j , V j ′ ) with ( j, j ′ ) ∈ J .Thus, we have the pair (GL( V ′ ) × GL( ǫV ′ ) , GL( V ′ ))with the embedding g ( g, σ ( g )) . This exhausts the cases and establishes the lemma. (cid:3)
N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 13 Topological Jordan decompositions and descent
Our proof of the fundamental lemma relies on refined descent argument, analogous to theapproach in the case of twisted endoscopy [Wal08], [Hal93]. For this purpose, we develop thenotion of a topological Jordan decomposition for certain elements in p -adic symmetric spacesand establish a relative version of a result of Kottwitz. This allows us to descend the orbitalintegrals to certain degenerations (or descendants) of Q ( F ), where the comparison may bereduced to the infinitesimal result of [Les19a]; see Section 6.4.1. Topological Jordan decomposition.
Let G be a connected reductive algebraic groupover F. We recall the notion of the topological Jordan decomposition as defined in [Hal93]; seealso [Spi08, Definition 2.15].For any profinite group K with a normal pro-p-subgroup L of finite index, the prime-to-ppart of the order of K/L is independent of the choice of L ; denote this integer by c K . Now forour reductive p-adic group G( F ), if we fix representatives of the finitely many conjugacy classesof maximal compact subgroups K , . . . , K d , we may set c G to be the least common multiple of c K i . Definition 4.1.
We call a semi-simple element γ ∈ G( F ) absolutely semi-simple if γ c G = 1.Note that such an element a fortiori satisfies the following equivalent criteria(1) γ lies in a compact subgroup of G( F ), and(2) the eigenvalues of ρ ( γ ) are units in F alg for some faithful finite-dimensional rationalrepresentation ρ : G( F ) → GL( V ) defined over F alg . Such elements are known as strongly compact elements of G( F ); an element is simply compact if its image in G ad ( F ) is strongly compact. Definition 4.2.
We say that an element γ ∈ G( F ) is topologically unipotent iflim n →∞ γ q n = 1 , where q = | k | is the size of the residue field of F. For each strongly compact element γ , there exists a unique decomposition γ = γ as γ tu = γ tu γ as , where γ as is absolutely semi-simple and γ tu is topologically unipotent; this is known as the topological Jordan decomposition . This may be constructed as follows [Hal93]: let l be a positiveinteger such that q l ≡ c ), and set γ as = lim m →∞ γ q lm and γ tu = γγ − as . Lemma 4.3.
The product γ = γ as γ tu gives the topological Jordan decomposition of γ . We refer the reader to [Hal93] and [Spi08] for more information on topological Jordan de-compositions.4.2.
The case of symmetric spaces.
Suppose now that (G , H) is a connected symmetricpair over F with the associated involution θ , and set σ ( g ) = θ ( g ) − . We have the embeddingof algebraic varieties Q := G / H −→ G g s ( g ) = gσ ( g ) . In this section, we study the relation between this quotient map and the topological Jordandecomposition of strongly compact elements of G( F ). The arguments combine the algebraicproperties of symmetric spaces with the structure of p -adic groups. Lemma 4.4.
Suppose that x ∈ G( F ) is topologically unipotent and that σ ( x ) = x and set V ( x ) ⊂ G for the Zariski closure of the cyclic subgroup of G generated by x . Then there exists y ∈ V ( x )( F alg ) such that σ ( y ) = y and x = y = s ( y ) . In particular, x ∈ Q ( F ) . Proof.
Let x = x s x u denote the Jordan decomposition of x . The lemma holds for x u = y u by[Ric82, Lemma 6.1]. Here, y u ∈ V ( x u )( F alg ) ⊂ Z (G x ) so that y u commutes with x s .Assume for the moment that the result holds for x s as well. Then there is a semi-simpleelement y s ∈ V ( x s )( F alg ) such that y s = x s . Since y s ∈ V ( x s )( F alg ) ⊂ Z (G x )( F alg ), we see y s y u = y u y s , so that ( y s y u ) = y s y u = x. Therefore, it suffices to assume that x = x s is semi-simple.In this case, we may pass to any finite extension and assume that there exist maximally θ -split maximal torus T ( F ) ⊂ G( F ) containing x [Ric82, Theorem 7.5]. Let A ⊂ T denote themaximal θ -split torus contained in T . Then x ∈ A ( F ) and it suffices to show that x is a squarein A ( F ). There is an isomorphism A ( F ) ∼ −→ ( F × ) n x ( x , . . . , x n )and θ = ( θ , . . . , θ n ) acts via inversion on each factor. Since x is topologically unipotent,1 A = lim m →∞ x q m = (cid:16) lim m →∞ x q m , . . . , lim m →∞ x q m n (cid:17) = (1 , . . . , . Therefore, we have x i ∈ p F for each i . This subgroup of O × F is a finitely generated Z p -module, hence is 2-divisible. Selecting y i ∈ p E such that y i = x i , we obtain an element y ∈ A ( F ) such that y = x . Since s ( y ) = yθ ( y ) − = y = x, we see that x ∈ Q ( F ) . (cid:3) Proposition 4.5.
For any strongly compact element x ∈ G( F ) , let x = x as x tu be the topologicalJordan decomposition. Then x ∈ Q ( F ) if and only if x as , x tu ∈ Q ( F ) . Proof.
To begin, note that if θ ( x ) = x − , then θ ( x as ) θ ( x tu ) = θ ( x ) = x − as x − tu . Uniqueness of the topological Jordan decomposition then forces θ ( x as ) = x − as and θ ( x tu ) = x − tu .A similar argument works for the converse, so that x ∈ P ( F ) if and only if x as , x tu ∈ P ( F ) . Suppose first that there exists v ∈ G( F alg ) such that s ( v ) = x . The previous lemma statesthat there is a y tu ∈ G( F alg ) such that s ( y tu ) = y tu = x tu . We claim that y tu commutes with x .Indeed, since G x is a Zariski-closed subgroup of G and x tu ∈ Z (G x ) (see [Spi08, Lemma 2.25]),we see that y tu ∈ V ( x tu )( F alg ) ⊂ Z (G x )( F alg ). Therefore, s ( y − tu v ) = y − tu s ( v ) y − tu = y − tu x = x as . Conversely, if s ( y as ) = x as and y tu is as in Lemma 4.4, then y tu x as = x as y tu as they both lie in Z (G x )( F alg ) [Spi08, Lemma 2.25]. This implies s ( y tu y as ) = y tu s ( y as ) y tu = x as y tu = x. (cid:3) A relative Kazhdan’s lemma.
Suppose that x = x as x tu ∈ Q rss ( F ) is strongly compact.Under certain additional assumptions on (G , H), the next proposition enables us to reduce thefundamental lemma for Q to an analogous statement on the descendant associated to x as .We assume that the symmetric pair (G , H) arises from a symmetric pair over the ring ofintegers O F in the sense that there is a smooth group scheme G over O F and an involutiveautomorphism θ : G −→ G such that θ : G −→ G arises as the generic fiber. Set H = G θ . This gives a smooth groupscheme over O F and H F ∼ = H . N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 15
Definition 4.6.
We define a symmetric pair (G , H) over a field k to be simply connected iffor every field extension K and every semi-simple point x ∈ (G / H)( K ), the centralizer (H K ) x is connected. We say a symmetric pair ( G , H ) over O F is simply connected if both the genericand special fibers are simply connected. Definition 4.7.
We say that a symmetric pair ( G , H ) over O F is nice if the ring of invariants O F ( Q ) H is a finitely generated O F -algebra such that for either x ∈ Spec( O F ) O F ( Q ) H ⊗ O F K x ∼ = K x ( Q x ) H x . If this holds, the scheme A := Spec( O F ( Q ) H ) has the property that for each x ∈ Spec O F , thefiber A x is the categorical quotient for the H x action on Q x .Our primary example is G = U( V n ⊕ V n ) and H = U( V n ) × U( V n ) , where V n is a split Hermitianspace of dimension n for an unramified extension E/F.
If we fix a self-dual lattice Λ n ⊂ V n andconsider the associated group O F -schemes, G = U (Λ n ⊕ Λ n ) and H = U (Λ n ) × U (Λ n ), theinvolution θ extends naturally to an automorphism of O F -schemes θ : G −→ G with G θ = H . Lemma 4.8.
The symmetric pair ( G , H ) = ( U (Λ n ⊕ Λ n ) , U (Λ n ) × U (Λ n )) is nice and simplyconnected.Proof. Lemma 3.9 shows that the base change of this pair to any field has connected semi-simplestabilizers, showing that the pair is simply connected. To see that the variety is nice, we appealto our explicit construction of the categorical quotient map χ : G / H −→ A nF x = (cid:18) A B − B ∗ D (cid:19) ( a ( x ) , . . . , a n ( x )) , where { a i ( x ) } are the coefficients of the characteristic polynomial of A . This map is clearly de-fined over O F , and O F [ a , . . . , a n ] provides the necessary integral model A = Spec( O F [ a , . . . , a n ]) . (cid:3) We suspect that smooth symmetric pairs over O F are always nice, but do not have a proof.It is relatively easy to check once a concrete model for the categorical quotient is constructed.Other examples of nice simply-connected pairs are Galois pairs associated to simply-connectedgroups such as the symmetric pair (Res E/F
SL( n ) , SL( n )) for an unramified quadratic extension E/F . Proposition 4.9.
Suppose that ( G , H ) is a nice simply-connected symmetric pair over O F andsuppose that γ ∈ Q ( O F ) is absolutely semi-simple. Let (G , H) denote the pair over F . Thecentralizer H γ is unramified and arises as the generic fiber of a smooth connected reductivegroup scheme H γ ⊂ H over O F .Moreover, if γ and γ ′ ∈ Q ( O F ) lie in the same stable H( F ) -orbit, they are conjugate by anelement in H ( O F ) . Proof.
Since γ ∈ Q ( O F ) is absolutely semi-simple as an element of G ( O F ), [Kot86, Proposition7.1] implies that G γ is a smooth group scheme over O F with reductive fibers. It is evidentlystable under θ , so we consider the automorphism θ : G γ −→ G γ . The fixed point scheme is given by G θγ ( R ) = { g ∈ G γ ( R ) : θ ( g ) = g } = H γ ( R ) , for any O F -algebra R . It follows from [Edi92, Proposition 3.4] that H γ is a smooth groupscheme over O F . By our assumption that ( G , H ) is simply connected, the smooth group scheme H γ has connected reductive fibers. In particular, H γ = H γ,F is unramified and H γ ( O F ) = H γ ( F ) ∩ H ( O F ) is a hyperspecial maximal subgroup.Now suppose that γ and γ ′ ∈ Q ( O F ) lie in the same stable orbit. Since the stabilizers areconnected, this implies that γ ′ = h · γ for some g ∈ H( F alg ). Viewed as elements of Q ( F ), itfollows that γ and γ ′ have the same invariant a ∈ A ( F ), where A := Spec( F ( Q ) H ) denotes thecategorical quotient. By the assumption of niceness, the quotient map Q −→
Spec( F ( Q ) H ) ∼ = A rk( Q ) has a natural O F -model, which we call A . This O F -scheme satisfies the property that for eachpoint x ∈ Spec( O F ), the fiber A x is the categorical quotient of Q x with respect to H x . We havethe commutative diagram Q ( O F ) Q ( F ) A ( O F ) A ( F ) , where the horizontal arrows are the natural inclusions. In particular, a ∈ A ( O F ) . Define now the O F -scheme given by Y ( R ) = { g ∈ G ( R ) : gγg − = γ ′ } for any O F -algebra R . By the proof of Proposition 7.1 of [Kot86], we know that Y is smoothas an O F -scheme and that Y ( O F ) = ∅ . It is simple to check that the involution θ preserves Y and so another application of [Edi92, Proposition 3.4] implies that Y θ is a smooth scheme over O F such that for any O F -algebra RY θ ( R ) = { g ∈ G ( R ) : gγg − = γ ′ , θ ( g ) = g } = { g ∈ H ( R ) : gγg − = γ ′ } . Thus, it suffices to show that Y θ ( O F ) = ∅ . Let γ and γ ′ denote the images of γ and γ ′ in Q ( k ) . These elements are semi-simple as γ and γ ′ are absolutely semi-simple as elements of G ( O F ). Since the quotient map A ( O F ) −→ A ( k ) a a is functorial, γ and γ ′ have the same invariant a ∈ A ( k ) . As there is a unique stable semi-simpleorbit in the fiber over a ∈ A ( k ) (see [AG09, Theorem 2.2.2]), it follows that γ and γ ′ lie in thesame stable orbit under the action of H ( k ) . By the assumption that ( G , H ) is a simply-connected symmetric pair, the stabilizer H γ,k is connected. Lang’s theorem [Lan56, Theorem 1] now implies that γ and γ ′ lie in the same H ( k )-orbit. That is, Y θ ( k ) = ∅ . The smoothness of Y θ over O F and Hensel’s lemma now gives that Y θ ( O F ) = ∅ . (cid:3) The main application of this result will come in Proposition 7.2. We derive a few moreconsequences here to be used in Section 7. Suppose (G , H) are as in the proposition; we continueto use calligraphic font of integral models and make use of the canonical inclusion G ( O F ) ⊂ G( F ) . Definition 4.10.
Let O ⊂ G( F ) be a closed H( F ) × H( F )-orbit. We say that O is a good orbit if σ ( O ) = O , where we remind the reader that σ ( g ) = θ ( g ) − . We say that a closed H( F )-orbit in Q ( F ) isgood if it is the image of a good H( F ) × H( F )-orbit under the symmetrization map.This terminology is inspired by the notion of a good symmetric pair from [AG09, Section 7]. Corollary 4.11.
Suppose that γ ∈ Q ( O F ) is absolutely semi-simple. The orbit O γ := H( F ) · γ is good. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 17
Proof.
First, let x ∈ Q ( F ) be any semi-simple element such that there exists g ∈ G( F ) suchthat s ( g ) = x . Lemma 7.1.4 of [AG09] implies that there exists g ′ in the stable H × H-orbit of g such that σ ( g ) = h g ′ h . Similarly, σ ( g ′ ) = h gh . This further implies that s ( g ′ ) = g ′ ( h gh ) = ( gh ) − ( gh g ′ h )( gh ) = h − ( g − s ( g ) g ) h . That is, s ( g ′ ) lies in the same H( F )-orbit as˜ s ( g ) := g − s ( g ) g = σ ( g ) g. In particular, s ( g ) lies in the same stable orbit as ˜ s ( g ) . Now let γ ∈ Q ( O F ) be absolutely semi-simple as in the statement. Lang’s theorem [Lan56,Theorem 2] implies that there exists g ∈ G ( O F ) with s ( g ) = γ and we must show that the H ( F ) × H ( F )-orbit of g is good. Note that˜ γ = ˜ s ( g ) = σ ( g ) g also lies in Q ( O F ) . Proposition 4.9 now implies that γ and ˜ γ lie in the same H ( O F )-orbit. Usingthe same notation as above, this implies that σ ( g ) = h gh for some h , h ∈ H ( F ) . In particular, g and g ′ lie in the same H ( F ) × H ( F )-orbit, so that theorbit of g is good. (cid:3) Corollary 4.12.
Suppose that γ ∈ Q ( O F ) is absolutely semi-simple. Then there exists g ∈G γ ( O F ) such that s ( g ) = γ .Proof. As before, Lang’s theorem implies that there exists g ∈ G ( O F ) such that s ( g ) = γ .By Corollary 4.11, the H ( F ) × H ( F )-orbit of g ∈ G ( O F ) is good. In particular, there exist h , h ∈ H ( F ) such that σ ( g ) = h g h . Set g ′ := g h and note that γ = s ( g ′ ) . We claim that g ′ σ ( g ′ ) = σ ( g ′ ) g ′ . Indeed, σ ( g ′ ) g ′ = h − σ ( g ) g h = h − σ ( g ) σ ( σ ( g )) h = h − ( h g h )( h − σ ( g ) h − ) h = g σ ( g ) = g h ( h − σ ( g )) = g ′ σ ( g ′ ) . This now implies that ( g ′ ) − γg ′ = ( g ′ ) − g ′ σ ( g ′ ) g ′ = σ ( g ′ ) g ′ = gσ ( g ′ ) = γ, so g ′ ∈ G γ ( F ). Inspecting the previous argument, we find the equation σ ( g ) g = h g ′ σ ( g ′ ) h − = h γh − . Noting that σ ( g ) g ∈ Q ( O F ), Proposition 4.9 now implies that h ∈ H ( O F ) H γ ( F ) . Write h = hh ′ for h ∈ H ( O F ) and h ′ ∈ H γ ( F ), and set g := g h ∈ G ( O F ). Then clearly s ( g ) = γ and g = g ′ ( h ′ ) − ∈ G γ ( F ) (cid:3) Orbital integrals and relative endoscopy
Let
E/F be a quadratic extension of p -adic local fields. We now return to the set up ofSection 3.2: let W and W denote two n -dimensional Hermitian vector spaces over E , andset W = W ⊕ W . Set G = U( W ) and denote by θ : G → G, the unitary involution withH = U( W ) × U( W ) = G θ . Let Q = G / H be the associated symmetric space, and let s : G −→ Q denote the symmetrization map.We also fix a second decomposition W = W ⊕ W , with dim( W ) = dim( W ) = n. The subgroup H = U( W ) × U( W ) ⊂ U( W ) is a pure innerform of H =: H . Fix an element x ∈ Q ( F ) such that the stabilizer in G under the twisted-conjugation action is H and set s : G −→ Q to be s ( g ) = gx σ ( g ) . Orbital integrals.
We define the local relative orbital integrals on the level of the group,as these are the orbital integrals that come most directly from the relative trace formula, thenreduce via the symmetrization map to orbital integrals on the symmetric space. First, we needto introduce the following terminology.
Definition 5.1.
We say that γ ∈ G( F ) is relatively (resp. regular) semi-simple if s ( γ ) ∈ Q ( F )is (resp. regular) semi-simple with respect to the H ( F )-action. Definition 5.2.
For f ∈ C ∞ c (G( F )), and γ ∈ G( F ) a relatively semi-simple element, we definethe relative orbital integral of f byRO( γ, f ) = x (H × H ) γ ( F ) \ H ( F ) × H ( F ) f ( h − γh ) d ˙ h d ˙ h , (7)where d ˙ h d ˙ h denotes the invariant measure determined by our choice of Haar measures onH i ( F ) and (H × H ) γ ( F ).We now explain the reduction to orbital integrals on the symmetric space for regular semi-simple orbits. Assume that γ is regular relatively semi-simple. If we write s ( γ ) = (cid:18) A B − B ∗ D (cid:19) , then (H × H ) γ ∼ = H s ( γ ) . Setting x = s ( γ ), Lemma 3.7 implies that H x ∼ −→ U( W ) A , where A = R ( x ) ∈ H erm ( W ). In particular, Lemma 2.1 implies that (H × H ) γ is a rank n torus.The pushforward map ( s ) ! : C ∞ c (G( F )) −→ C ∞ c ( Q ( F )) given by( s ) ! ( f )( s ( g )) = Z H ( F ) f ( gh ) dh is surjective onto the sub-module C ∞ c, ( Q ( F )) of functions whose support is contained in thethe image of s . Setting Φ := ( s ) ! ( f ) and x = s ( γ ), the isomorphism (H × H ) γ ∼ = H x andabsolute convergence of the relative orbital integral givesO( x, Φ) := Z H x ( F ) \ H( F ) Φ( h − xh ) d ˙ h = RO( γ, f ) , (8)where d ˙ h denotes the invariant measure on H x ( F ) \ H( F ) induced from our choice of Haarmeasures. More generally, for any f ∈ C ∞ c ( Q ( F )) and regular semi-simple x ∈ Q ( F ) , we setO( x, f ) = Z H x ( F ) \ H( F ) f ( h − xh ) d ˙ h to be the orbital integral of f at x. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 19
Application of the contraction map.
The inclusion Q rss ⊂ Q iso and Lemma 3.7 implythat the restriction of the contraction map R : Q −→ H erm ( W ) to the regular semi-simplelocus is a U( W )-torsor. The next lemma enables us to use R to study κ -orbital integrals atregular semi-simple elements of Q ( F ) in the next subsection. Lemma 5.3.
Let x ∈ Q rss ( F ) and set R ( x ) = y ∈ H erm ( W ) . The isomorphism φ : H x ∼ −→ U( W ) y induces an isomorphism D (H x /F ) ∼ −→ D (U( W ) y /F ) (9) where D (H x /F ) = ker (cid:2) H ( F, H x ) −→ H ( F, H) (cid:3) and D (U( W ) y /F ) = ker (cid:2) H ( F, U( W ) y ) −→ H ( F, U( W )) (cid:3) . Proof.
This is analogous to the Lie algebra version [Les19b, Lemma 3.9]; we include a slightlymore hands-on argument afforded by our restriction to the p -adic setting. Consider the com-mutative diagram H ( F, H x ) H ( F, U( W )) × H ( F, U( W )) H ( F, U( W ) y ) H ( F, U( W )) . ι x φ ∗ p ∗ ι y where p : U( W ) × U( W ) −→ U( W ) is the projection, φ : H x ∼ −→ U( W ) y is the inducedisomorphism, and φ ∗ and p ∗ are the maps induced on cohomology.If α ∈ D (H x /F ), then ι δ φ ( α ) = p ( ι x ( α )) = 1 . This allows us to extend the diagram to1 D (H x /F ) H ( F, H x ) H ( F, U( W )) × H ( F, U( W ))1 D (U( W ) y /F ) H ( F, U( W ) y ) H ( F, U( W )) , ι x φ ∗ p ∗ ι δ where the arrow D ( H x /F ) → D (U( W ) y /F ) is an injection. To show it is surjective, weshow that it induces a surjection on rational orbits in the given stable orbit. Suppose that y ′ ∈ H erm ( W ) is stably conjugate to y : this gives the elementinv( y, y ′ ) = [ τ h − τ ( h )] ∈ D (U( W ) y /F )where h ∈ U( W )( F alg ) such that y ′ = hyh − . Since R ( x ) = y and d ( I − ( y ′ ) ) ≡ d ( I − y ) (mod Nm E/F ( E × )) , Lemmas 3.7 and 3.8 combine to imply that there exists x ′ ∈ Q rss ( F ) such that R ( x ′ ) = y ′ . Then R ( x ′ ) = y ′ = hyh − = R (cid:18)(cid:18) h I (cid:19) · x (cid:19) . The U( W )-torsor statement of Lemma 3.7 now implies that there exists h ′ ∈ U( W )( F alg ) such that (cid:18) I ( h ′ ) − (cid:19) · x ′ = (cid:18) h I (cid:19) · x = ⇒ x ′ = (cid:18) h h ′ (cid:19) · x ;that is, x ′ is stably conjugate to x andinv( x, x ′ ) = [ τ ( h − τ ( h ) , ( h ′ ) − τ ( h ′ ))] ∈ D (H x /F )maps to inv( y, y ′ ) ∈ D (U( W ) y /F ) . (cid:3) We can only assert the existence of h ′ ∈ U( W )( F alg ) as the translate of x need not be rational. Relative endoscopy.
Recall that V n denotes a fixed set of representatives of the isometryclasses of Hermitian form over E on E n . Since E/F is an extension of p -adic fields, |V n | = 2.If E/F is unramified, we assume that the split Hermitian form is represented by I n and set V n := ( E n , I n ) for the split Hermitian space.Following [Les19b], we have the following definition. Definition 5.4.
We define an (elliptic) relative endoscopic datum of the symmetric space Q tobe a triple Ξ a,b = ( ξ a,b , α, β ), where ξ a,b = (U( V a ) × U( V b ) , s, η )is an elliptic endoscopic triple for U ( W ) and α ∈ V a (resp. β ∈ V b ). Setting Q a,α := U( V a ⊕ V α ) / U( V a ) × U( V α ) and Q b,β := U( V b ⊕ V β ) / U( V b ) × U( V β ) , we define the associated endoscopic symmetric space to be Q a,α × Q b,β .For a fixed endoscopic datum, the endoscopic symmetric space is equipped with a contractionmap as in Lemma 3.6 R α,β : Q a,α × Q b,β −→ H erm ( V a ) ⊕ H erm ( V b ) , (cid:18)(cid:18) A B − B ∗ D (cid:19) , (cid:18) A B − B ∗ D (cid:19)(cid:19) ( A , A ) . Let x ∈ Q rss ( F ) and ( x a , x b ) ∈ ( Q a,α × Q b,β ) rss ( F ). Definition 5.5.
We say that x matches ( x a , x b ) (or that x is an image of ( x a , x b )) if R ( x ) = y ∈ H erm ( W ) and R α,β ( x a , x b ) = ( y a , y b ) ∈ H erm ( V a ) ⊕ H erm ( V b )match in the sense of Definition A.1. When W ∼ = V a ⊕ V b , we say that ( x, ( x a , x b )) are a goodmatching pair if ( y, ( y a , y b )) are.As this notion of matching is defined via pushing forward regular elements to the HermitianLie algebra, it is well-defined only up to stable orbits. That is, if x ′ lies in the stable orbit of x and ( x ′ a , x ′ b ) lies in the stable orbit of ( x a , x b ) then x matches ( x a , x b ) ⇐⇒ x ′ matches ( x ′ a , x ′ b ) . Given matching elements ( x, ( x a , x b )), we define the transfer factor via pullback as∆ rel (( x a , x b ) , x ) := ∆(( y a , y b ) , y ) , (10)where the right-hand side is the Langland-Shelstad-Kottwitz transfer factor from Section A.2.2.5.2.1. Smooth transfer.
Fix x ∈ Q rss ( F ) and let Ξ a,b = ( ξ a,b , α, β ) be a relative endoscopicdatum. The endoscopic triple Ξ = (U( V a ) × U( V b ) , s, η ) of U ( W ) determines a character κ : D (U( W ) R ( x ) /F ) −→ C × via the construction of Lemma A.2. By Lemma 5.3, we may pull this character back along theisomorphism D (H x /F ) ∼ −→ D (U( W ) R ( x ) /F ) , to obtain a character which we also call κ : D (H x /F ) −→ C × . We now define the relative κ -orbital integral to be O κ ( x, f ) := X x ′ κ (inv( x, x ′ )) O( x ′ , f ) , where x ′ ∈ Q st ( x ) ranges over the representatives of the U ( W ) × U ( W )-orbits stably conjugateto x . By the proof of Lemma 5.3, inv( x, x ′ ) = inv( R ( x ) , R ( x ′ )). Definition 5.6.
Let f ∈ C ∞ c ( Q ( F )) and let f α,β ∈ C ∞ c ( Q a,α ( F ) × Q b,β ( F )). We say that f and f α,β are smooth transfers of each other (or match) if the following conditions are satisfied:(1) For any matching orbits x ∈ Q rss ( F ) and ( x a , x b ) ∈ Q a,α ( F ) × Q b,β ( F ), we have anidentify SO(( x a , x b ) , f α,β ) = ∆ rel (( x a , x b ) , x ) O κ ( x, f ) . (11) N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 21 (2) If there does not exist x ∈ Q rss ( F ) matching ( x a , x b ) ∈ Q a,α ( F ) × Q b,β ( F ), thenSO(( x a , x b ) , f α,β ) = 0 . (12)We conjecture that transfers always exist. For test functions supported in Q iso ( F ), we mayreadily deduce the existence of transfers. Proposition 5.7.
Let f ∈ C ∞ c ( Q ( F )) and assume supp( f ) ⊂ Q ( F ) iso . Then there exists f α,β ∈ C ∞ c ( Q a,α ( F ) × Q b,β ( F )) such that f and f α,β match.Proof. The argument relies on the properties of the contraction map on Q iso to reduce thestatement to the existence of smooth transfer on the Hermitian Lie algebra. It is very similarto the proof of Proposition 4.5 of [Les19b]; we omit the details. (cid:3) Remark . Using the discussion in 5.1, we may pull all these notions back to statements ofrelative κ -orbital integrals on U ( W ) and the (pure inner forms of) endoscopic groups U ( V a ⊕ V α )and U ( V b ⊕ V β ). As this is not the perspective used throughout this article, we leave these detailsto the interested reader.5.3. The fundamental lemma for the unit element.
Now assume that
E/F is an un-ramified quadratic extension of p -adic fields and assume that our Hermitian spaces satisfy V n = W = W . In this unramified setting, we will append our groups with a subscript n todifferentiate by rank of the associated symmetric space; for example, H n ( F ) = U ( V n ) × U ( V n ).Fixing a self-dual lattice Λ n ⊂ V n , let G n and H n denote the corresponding smooth groupschemes over O F . We obtain hyperspecial subgroupsH n ( O F ) := U(Λ n ) × U (Λ n ) ⊂ H n ( F )and G n ( O F ) := U(Λ n ⊕ Λ n ) ⊂ G n ( F ) . Set G n ( O F ) to be the associated the associated characteristic function.Now consider the symmetric space Q n := G n / H n . Lang’s theorem [Lan56, Theorem 2] tellsus that H (Spec( O F ) , H n ) = 0, so that we have a short exact sequence of pointed sets1 −→ H n ( O F ) −→ G n ( O F ) −→ Q n ( O F ) −→ . This is compatible with the sequence (4) on F -points and implies the equality Q n ( O F ) = s ! ( G n ( O F ) ) ∈ C ∞ c ( Q n ( F )) . (13)Now suppose that Ξ a,b = ( ξ a,b , α, β ) is an elliptic relative endoscopic datum. Our measuresconventions in Section 2.4 ensure that the given hyperspecial maximal subgroups of U ( V n ) × U ( V n ) and ( U ( V a ) × U ( V a )) × ( U ( V b ) × U ( V b ))each have volume 1. We now state the main result of this article. Theorem 5.9.
Assume that the characteristic of F satisfies the assumption of Lemma 7.4.If ( α, β ) = ( I a , I b ) , the functions Q n ( O F ) and Q a ( O F ) ⊗ Q b ( O F ) match. Otherwise, Q n ( O F ) matches . Combining (13) with (8), one obtains a matching of κ -orbital integrals between the testfunctions G n ( O F ) and G a ( O F ) ⊗ G b ( O F ) . Note that G a × G b is an unramified elliptic endoscopic group of G n . Proof of Theorem 5.9.
The proof of this theorem occupies Sections 6 and 7. For thereaders convenience, we summarize the components of the argument here.We begin with the following simple reduction.
Lemma 5.10.
Suppose that ( x a , x b ) ∈ Q a ( O F ) × Q b ( O F ) . Then there exists an image x ∈Q n ( O F ) . In particular, if x ∈ Q rssn ( F ) is not in the same stable orbit as an element x ′ ∈Q n ( O F ) , then x is not the image of an integral element.Proof. In accordance with our conventions on split Hermitian forms, we may fix a basis of V n , V a , and V b such that the forms are represented by the respective identity matrices. Writing( x a , x b ) = (cid:18)(cid:18) A B − B ∗ D (cid:19) , (cid:18) A B − B ∗ D (cid:19)(cid:19) , it is clear that x = A B A B − B ∗ D − B ∗ D ∈ Q n ( O F )is an image of ( x a , x b ) . Indeed, R n ( x ) = (cid:18) A A (cid:19) = (cid:18) R a ( x a ) R b ( x b ) (cid:19) , showing that x and ( x a , x b ) are a nice matching pair. (cid:3) Now suppose that Ξ a,b is a relative endoscopic datum and suppose that x ∈ Q rss ( F ) matches( x a , x b ) ∈ Q a,α ( F ) × Q b,β ( F ) . Lemma 5.10 shows that if the stable orbit of x fails to meet Q n ( O F ), which forces O κ ( x, Q n ( O F ) ) = 0 , then the same is true of ( x a , x b ). This is only meaningful if ( α, β ) = ( I a , I b ), in which case itforces SO(( x a , x b ) , Q a ( O F ) ⊗ Q b ( O F ) ) = 0 . This proves the claim in this case.We may now assume that x ∈ Q n ( O F ). For ν = ±
1, we define the ν -very regular locus by Q ♥ ,νn ( O F ) = { x ∈ Q rssn ( O F ) : x ∈ Q n ( k ) − D ν ( k ) } , where D ν ( k ) = { X ∈ End(Λ n /̟ Λ n ) : det( νI n − X ) = 0 } . If x ∈ Q ♥ ,νn ( O F ), the claim is shown in Proposition 6.9. If x / ∈ Q ♥ ,νn ( O F ) for either ν = ± , wemust apply the descent techniques developed in Section 4. The claim is shown in Proposition7.1, which follows from the descent formulas in Lemmas 7.6, 7.7, and 7.8. This completes theproof of Theorem 5.9.6. The infinitesimal theory and the very regular locus
In this and the follow section, we establish Theorem 5.9 by reducing to Theorem 6.1 below.We begin by recalling the infinitesimal result, then introducing the Cayley transform. Throughthis quasi-exponential map, we may reduce Theorem 5.9 to the Lie algebra case for certain opensubsets of Q n ( O F ), the so-called very regular locus. In Section 7, we apply the descent methodsof Section 4 to complete the proof. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 23
The Lie algebra of the symmetric space.
Consider the Lie algebra u ( W ) of U ( W ).The differential of the involution θ acts on u ( W ) by the same action and induces a Z / Z -grading u ( W ) = u ( W ) ⊕ u ( W ) , where u ( W ) i is the ( − i -eigenspace of the map θ . Then the pair( U ( W ) × U ( W ) , u ( W ) )is called the infinitesimal symmetric space associated to Q . This means that if x ∈ Q n ( O F ) ⊂Q n ( F ) denotes the distinguished U ( W ) × U ( W )-fixed point, then T x ( Q n )( F ) ∼ = u ( W ) , with U ( W ) × U ( W ) acting by restriction of the adjoint representation. We have a naturalisomorphism of H ( F )-representations u ( W ) ∼ = Hom E ( W , W ) (cid:18) X − X ∗ (cid:19) X, where the action on the right-hand side is given by pre- and post-composition; we frequentlyidentify u ( W ) and Hom E ( W , W ) via this map in the sequel.We also recall the (infinitesimal) contraction map r : u ( W ) −→ H erm ( W ) given by r ( X ) = − XX ∗ . This gives a categorical quotient of the U ( W )-action on u ( W ) . Relative endoscopy for the Lie algebra.
We briefly recall the notions of relative en-doscopy from [Les19b]. Fix a elliptic relative endoscopic datum Ξ a,b = ( ξ a,b , α, β ) . As before,we denote V α = ( E a , α ) and V β = ( E b , β ) and consider the Lie algebras u ( V a ⊕ V α ) and u ( V b ⊕ V β ) , and associated symmetric pairs( U ( V a ) × U ( V α ) , u ( V a ⊕ V α ) ) and ( U ( V b ) × U ( V β ) , u ( V b ⊕ V β ) ) . In [Les19b], we defined the direct sum of these symmetric pairs to be an infinitesimal endoscopicsymmetric pair associated to Ξ a,b . It is clear that it is the infinitesimal symmetric pair associatedto the endoscopic symmetric space associated to Ξ a,b defined in Section 6.2. In particular, thetheory we develop here is compatible with that of [Les19b].This space comes equipped with the contraction map (see [Les19b, Section 3] for details) r α,β : u ( V a ⊕ V α ) ⊕ u ( V b ⊕ V β ) −→ H erm ( V a ) ⊕ H erm ( V b )( δ a , δ b ) ( r ( δ a ) , r ( δ b ))We say that a regular semi-simple element δ ∈ u ( W ) rss matches the pair( δ a , δ b ) ∈ [ u ( V a ⊕ V α ) ⊕ u ( V b ⊕ V β ) ] rss if r ( δ ) ∈ H erm ( W ) and r α,β ( δ a , δ b ) ∈ H erm ( V a ) ⊕ H erm ( V b ) match in the sense of DefinitionA.1; we similarly define when ( δ, ( δ a , δ b ) is a good matching pair. For matching elements ( δ a , δ b )and δ , we define the transfer factor˜∆ rel (( δ a , δ b ) , δ ) := ∆( r α,β ( δ a , δ b ) , r ( δ )) , where the right-hand side is the Langlands-Shelstad-Kottwitz transfer factor the the twistedLie algebra. We defined and studied the notion of matching of orbital integrals in [Les19b]; thisagain is mirrored in Section 5 so we omit a full recap. The infinitesimal fundamental lemma.
We now and for the remainder of the paperassume that
E/F is an unramified extension of non-archimedean local fields of characteristiczero. Suppose that V n = W = W is split, and let Λ n ⊂ V n be a self-dual lattice. In this case, u ( W ) = Hom E ( V n , V n ) = End( V n )and the ring of endomorphisms End(Λ n ) ⊂ End( V n ) of the lattice Λ n is a compact open subset.The following was proved in [Les19a]. Theorem 6.1. If ( α, β ) = ( I a , I b ) , the functions End(Λ n ) and End(Λ a ) ⊗ End(Λ b ) match. Oth-erwise, End(Λ n ) matches . Our goal is to show that this result implies Theorem 5.9. We begin by considering certainequivariant birational maps u ( W ) −→ Q ( F ) which play the role of an exponential map for thesymmetric space.6.4. The Cayley transform.
We recall the Cayley transform. For any ξ ∈ E , we define D ξ = { X ∈ End( W ) : det( ξI − X ) = 0 } . Lemma 6.2.
The Cayley transform c ± : End( W ) − D −→ GL( W ) X (1 + X )(1 − X ) − induces a U ( V ) × U ( V ) -equivariant isomorphism between u ( W ) − D and Q ( F ) − D ± . Theimages of u ( W ) − D under c ± form a finite cover by open subsets of Q ( F ) − D ∩ D − . Inparticular, the images contain the regular semi-simple locus of Q ( F ) .Proof. It is well known [Zha14, Lemma 3.4] that for any ξ ∈ E the Cayley map c ξ ( X ) = ξ (1 + X )(1 − X ) − induces a GL( W )-equivariant isomorphism between g l ( W ) − D and GL( W ) − D ξ . Indeed, theinverse is of the same form: for g ∈ GL( W ) − D ξ , set β ξ ( g ) = − ( ξ − g )( ξ + g ) − . This gives the required inverse transformation. It is easy to check that the constraint that c ξ ( X ) ∈ U ( W ) whenever X ∈ u ( W ) forces ξξ = 1.We now show that for the transform to compatible with the involution θ , we need ξ = ± θ ( X ) = − X , then θ ( c ξ ( X )) = − ξ (1 − X )(1 + X ) − = ( − ξ − (1 + X )(1 − X ) − ) − = c ξ − ( X ) − . Thus, θ ( c ξ ( X )) = c ξ ( X ) − if and only if ξ = 1.For the final statement, we can check over F alg . Using (3), we see that any x ∈ Q rss ( F )lies in the same H( F alg )-orbit as an element of the form x ( A, ,
0) where A ∈ g l ( W ) is regularsemi-simple with eigenvalues avoiding ±
1. Considering Lemma 3.3, the same is true of roots of car x ( t ) , implying that Q rss ⊂ Q − D ∪ D − . (cid:3) The following lemma is a simply matrix computation.
Lemma 6.3.
Let δ = δ ( X ) ∈ u ( W ) − D , where X ∈ Hom E ( W , W ) such that δ ( X ) = (cid:18) X − X ∗ (cid:19) . Then for ν = ± , c ν ( δ ( X )) = ν (cid:18) ( I − XX ∗ )( I + XX ∗ ) − X ( I + X ∗ X ) − − X ∗ ( I + XX ∗ ) − ( I − X ∗ X )( I + X ∗ X ) − (cid:19) . N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 25
In particular, we have a commutative diagram u ( W ) QH erm ( W ) H erm ( W ) , c ν r R c ν where by abuse of notation we let c ν denote the Cayley transform on both u ( W ) and H erm ( W ) . (cid:3) We now consider the effect of the Cayley transform on the invariant polynomial maps.
Lemma 6.4.
Let δ ∈ End( W ) − D and set x = c ν ( δ ) . Then car x ( t ) = ( t + ν ) dim( W ) car δ (1) − (cid:18) car δ (cid:18) t − νt + ν (cid:19)(cid:19) . (14) Proof.
We may assume that F = F alg . It is a straightforward exercise that if λ ∈ F alg × is aneigenvalue of multiplicity m ( λ ) of δ , then λ ′ := ν (cid:18) λ − λ (cid:19) is an eigenvalue of x with the same multiplicity. The rational function car δ (cid:18) t − νt + ν (cid:19) then vanishes on the eigenvalues of x. In particular,( t + ν ) dim( W ) car δ (1) − (cid:18) car δ (cid:18) t − νt + ν (cid:19)(cid:19) is a monic polynomial with the correct roots, and so must be car x ( t ) . (cid:3) This allows us to compare invariant polynomials: for δ ∈ u ( W ) , the categorical quotientmap u ( W ) −→ A n is given by δ ( a ( δ ) , . . . , a n ( δ )) , where ( a , . . . , a n ) are the coefficients of car r ( δ ) ( t ) (see [Les19b, Section 3]); for x ∈ Q ( F ) , Lemma 3.1 tells us that the appropriate characteristic polynomial is car R ( x ) ( t ).Before we apply this to comparing transfer factors, we check that the Cayley transformpreserves our notions of matching orbits. Lemma 6.5.
Suppose that δ ∈ u ( W ) rss − D ± , and let x = c ν ( δ ) ∈ Q rss ( F ) . Fix an ellipticdatum Ξ a,b . Then ( δ a , δ b ) ∈ u ( V a ⊕ α ) ⊕ u ( V b ⊕ V β ) matches δ if and only if ( x a , x b ) = ( c ν ( δ a ) , c ν ( δ b )) ∈ Q a,α ( F ) × Q b,β ( F ) matches x .Proof. It is enough to prove the forward direction by the invertability of the Cayley transform.We denote r ( δ ) = γ, r α ( δ a ) = γ a , r β ( δ b ) = γ b and R ( x ) = y, R α ( x a ) = y b , R β ( x b ) = y b . We first assume that ( γ a , γ b ) and γ are a nice matching pair in the sense of Section A.1.Thus, we may assume that W ∼ = V a ⊕ V b and that there is an embedding ϕ a,b : H erm ( V a ) ⊕ H erm ( V b ) ֒ → H erm ( V ) such that ϕ a,b ( γ a , γ b ) = γ . The claim now follows if we can show thatthe diagram H erm ( V a ) ⊕ H erm ( V b ) H erm ( W ) H erm ( V a ) ⊕ H erm ( V b ) H erm ( W ) c ν ⊕ c ν c ν commutes. Up to conjugation, we are free to assume that embedding ϕ a,b has block diagonalimage ϕ a,b ( A, B ) = (cid:18)
A B (cid:19) . In this case, the statement is obvious as a simple matrix calculation gives (cid:18) c ν ( A ) c ν ( B ) (cid:19) = c ν (cid:18) A B (cid:19) . The claim follows whenever ( y a , y b ) and y are a good match.For the general case, we note that a similar argument proves compatibility between Jacquet-Langlands transfers and the Cayley transform, allowing us to reduce to the case of a goodmatching pair. (cid:3) The transfer factors on the symmetric space and its Lie algebra are closely related via theCayley transform. The key calculation is given by the following lemma.
Lemma 6.6.
With notation as in the previous lemma with x = c ν ( δ ) , consider the relativediscriminants ˜ D a,b ( δ ) = Y t a ,t b ( t a − t b ) and D a,b ( x ) = Y z a ,z b ( z a − z b ) , where t a (resp. z a ) runs over the roots of the invariant of δ (resp. x ) arising from δ a (resp. y a )and likewise with t b (resp. y b ), then ˜ D a,b ( δ ) = ( ν a · b Y z a ,z b z a + ν )( z b + ν ) D a,b ( x ) . Proof.
Lemma 6.4 implies that if ˜ χ δ ( t ) is the invariant of δ , then χ x ( t ) = ( t + ν ) n χ δ (1) − (cid:18) χ δ (cid:18) t − νt + ν (cid:19)(cid:19) is the invariant of x . It is clear that ± z is a root of χ x ( t ), then t = z − νz + ν is a root of ˜ χ δ ( t ) of the same multiplicity.Applying this to the discriminants, we have˜ D a,b ( δ ) = Y t a ,t b ( t a − t b )= Y z a ,z b (cid:18) z a − νz a + ν − z b − νz b + ν (cid:19) = Y z a ,z b ν z a − z b )( z a + ν )( z b + ν ) . Counting the number of factors gives the coefficient ( ν a · b . (cid:3) N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 27
The ν -very regular locus. Let ν = ±
1. If the pairs ( x, ( x a , x b )) and ( δ, ( δ a , δ b )) are asin Lemmas 6.5 and 6.6, we set C a,b,ν ( x, δ ) := ( ν a · b Y z a ,z b z a + ν )( z b + ν ) . The results of the previous section imply the formula˜∆ rel (( δ a , δ b ) , δ ) = η E/F ( C a,b,ν ( x, δ )) | C a,b,ν ( x, δ ) | F ∆ rel (( x a , x b ) , x ) . (15)Since E/F is unramified and the residue characteristic is odd, these transfer factors agreewhenever C a,b,ν ( x, δ ) is a unit. In this section, we define certain open subsets of Q n ( O F ) forwhich this is the case.Identifying u (Λ n ⊕ Λ n ) = End(Λ n ), we define the very regular locus of End(Λ n )End(Λ n ) ♥ = { δ ∈ End(Λ n ) rss − D ( O F ) : δ ∈ End(Λ /̟ Λ) − D ( k ) } , where δ ∈ End(Λ /̟ Λ) denotes the image of δ under the modular reduction map. Similarly, wedefine the ν - very regular locus of Q n ( O F ) Q ♥ ,νn ( O F ) = { x ∈ Q rssn ( O F ) : x ∈ Q n ( k ) − D ν ( k ) } . Lemma 6.7.
Suppose that x ∈ Q n ( F ) and suppose x = c ν ( δ ) for δ ∈ u ( W ) . Then x ∈Q ♥ ,νn ( O F ) if and only if δ ∈ End(Λ n ) ♥ .Proof. We begin by assuming x ∈ Q ♥ ,νn ( O F ) . This implies that if λ ∈ O × F alg is a root of thecharacteristic polynomial of x , then ν − λ ∈ O × F alg . In particular, ν − x ∈ GL(Λ n ). Clearly, wealso have ν + x ∈ End(Λ), implying that δ = c − ν ( x ) = − ( ν + x )( ν − x ) − ∈ End(Λ n ) . In particular, δ = c − ν ( x ) and Lemma 6.4 implies that 1 is not a root of car δ ( t ).Conversely, assume δ ∈ End(Λ n ) ♥ with x = c ν ( δ ) . Similar elementary considerations implythat 1 − δ ∈ GL(Λ n ), so that x = ν (1 + δ )(1 − δ ) − ∈ Q ♥ ,νn ( O F ) . (cid:3) This shows that Q ♥ ,νn ( O F ) = c ν (cid:16) End(Λ n ) ♥ (cid:17) . In particular, for any x ∈ Q ♥ ,νn ( O F ), the reduction x ∈ Q n ( k ) is in the image of the Cayleytransform.The following lemma is immediate from the definitions. Lemma 6.8.
Let x ∈ Q ♥ ,νn ( O F ) and suppose x ′ ∈ Q n ( O F ) lies in the stable orbit of x . Then x ′ ∈ Q ♥ ,νn ( O F ) Proof.
We need to show that x ′ / ∈ D ν ( k ) . But D ν ( k ) is closed under the stable action of H ( k alg )and x / ∈ D ν ( k ) . (cid:3) Define Q ♥ ,νn ( F ) to be the open subset of Q rssn ( F ) consisting of elements in the same stableorbit as an element in Q ♥ ,νn ( O F ); we similarly define End( V n ) ♥ . The next proposition showsthat the fundamental lemma holds for these open sets of Q n ( O F ). Proposition 6.9.
Fix an elliptic endoscopic datum Ξ a,b . Suppose that x ∈ Q ♥ ,νn ( F ) and that ( x a , x b ) ∈ Q a,α ( F ) × Q b,β ( F ) match. Then the fundamental lemma holds for ( x, ( x a , x b )) . Thatis, if κ is the character associated to the endoscopic datum, we have ∆ rel (( x a , x b ) , x ) O κ ( x, Q n ( O F ) ) = ( SO(( x a , x b ) , Q a ( O F ) ⊗ Q b ( O F ) ) : ( α, β ) = ( I a , I b ) , α, β ) = ( I a , I b ) . Proof.
We first consider the transfer factor. First assume that x and ( x a , x b ) are a nice matchingpair. Then we have an embedding φ a,b : H erm ( V a ) ⊕ H erm ( V b ) ֒ → H erm ( V )satisfying that φ a,b ( y a , y b )) = y, where R ( x ) = y and similarly for ( y a , y b ) and∆ rel (( x a , x b ) , x ) = η E/F ( D a,b ( x )) | D a,b ( x ) | F . The assumption x ∈ Q ♥ ,νn ( F ) implies that x = c ν ( δ ) for some δ ∈ End( V n ) rss . Moreover, thesame holds for the matching pair ( x a , x b ) = ( c ν ( δ a ) , c ν ( δ b )). Lemma 6.5 implies that x and( x a , x b ) are a good match if and only if δ and ( δ a , δ b ) are.Combining the calculation of Lemma 6.6 with the fact that C a,b,ν ( δ, x ) is a unit when x ∈Q ♥ ,νn ( F ), we have ∆ rel (( x a , x b ) , x ) = ˜∆ rel (( δ a , δ b ) , δ ) . (16)In general, suppose that x and x ′ are in the same stable orbit with inv( x, x ′ ) ∈ H ( F, H x )the corresponding invariant. Writing x ′ = c ν ( δ ′ ), the equivariance of c ν implies that δ ′ is inthe same stable class as δ and that under the induced isomorphism H x ∼ −→ H δ , we have theidentification inv( δ, δ ′ ) = inv( x, x ′ ) . Thus, the identity (16) holds for any matching pair x = c ν ( δ ) and ( x a , x b ) = ( c ν ( δ a ) , c ν ( δ b )).We now note that Lemma 6.7 implies immediately that for any x = c ν ( δ ) ∈ Q ♥ ,νn ( F ),O( x, Q n ( O F ) ) = O( δ, End(Λ n ) ) . This shows that∆ rel (( x a , x b ) , x ) O κ ( x, Q n ( O F ) ) = ˜∆ rel (( δ a , δ b ) , δ ) O κ ( δ, End(Λ n ) ) . (17)If ( α, β ) = ( I a , I b ), the proposition follows from the corresponding vanishing of orbital integralsin Theorem 6.1.Assuming now that ( α, β ) = ( I a , I b ), we further claim thatSO(( x a , x b ) , Q a ( O F ) ⊗ Q b ( O F ) ) = SO(( δ a , δ b ) , End(Λ a ) ⊗ End(Λ b ) ) . (18)where ( δ a , δ b ) ∈ End( V α ) ⊕ End( V β ) . To see this, suppose ( x ′ a , x ′ b ) ∈ Q a ( O F ) × Q b ( O F ) lies inthe stable orbit of ( x a , x b ) . Since x ∈ Q ♥ ,νn ( F ), if λ is a root of car x ( t ), then λ ∈ O × F alg and λ = ν ∈ k alg . By the definition of matching of orbits and Lemma 3.3, the same is true of the roots of thecharacteristic polynomials of x ′ a and x ′ b . In particular,( x ′ a , x ′ b ) ∈ Q ♥ ,νa ( O F ) × Q ♥ ,νb ( O F ) . The equality (18) now follows from Lemma 18. This proves the proposition by combining (17)and (18) with the matching of orbital integrals in Theorem 6.1. (cid:3)
Remark . Combining Lemma 5.10 and Proposition 6.9, the fundamental lemma is nowestablished unless x ∈ Q rssn ( O F ) has the property that x ∈ ( D ∩ D − )( k ) . N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 29 Descent to the very regular case
In this final section, we use the results of Section 4.1 and Proposition 4.9 to prove the finalcases of Theorem 5.9. More precisely, we prove the following proposition.
Proposition 7.1.
Suppose x ∈ Q rssn ( O F ) such that x ∈ ( D ∩ D − )( k ) . and that ( x a , x b ) ∈ Q a,α ( F ) × Q b,β ( F ) are smooth transfers. Then the fundamental lemma holdsfor ( x, ( x a , x b )) . That is, if κ is the character associated to the endoscopic datum, we have ∆ rel (( x a , x b ) , x ) O κ ( x, Q n ( O F ) ) = ( SO(( x a , x b ) , Q a ( O F ) ⊗ Q b ( O F ) ) : ( α, β ) = ( I a , I b ) , α, β ) = ( I a , I b ) . The proof of this Proposition is contained in Sections 7.2-7.4. More specifically, Lemma7.5 allows us to apply Proposition 6.9 to the descended orbital integrals and transfer factorsappearing in Lemmas 7.6, 7.7, and 7.8. The resulting matching of orbital integrals gives theidentity.7.1.
Descent of orbital integrals.
Suppose that x = x as x tu ∈ Q rssn ( O F ) is the topologicalJordan decomposition. Proposition 4.5 implies that x as , x tu ∈ Q n ( O F ) . Set γ = x as and Q γ =G γ / H γ . Proposition 4.9 implies that Q γ is a connected smooth scheme over O F .Consider the map s γ : G γ −→ Q given by s γ ( g ) = gγσ ( g ) = γgσ ( g ) = γs ( g ) . By Proposition 4.9 and Corollary 4.12, this gives a closed embedding of O F -schemes s γ : Q γ −→ Q x γx = xγ. In particular, s γ ( x tu ) = x . In this and the next section, we will relate orbital integrals on Q ( F )with orbital integrals on Q γ via descent. For simplicity, we always identify x = s γ ( x tu ) with itspreimage in Q γ ( F ) and say that x ∈ Q γ ( O F ). This should cause no confusion. Recall that we have fixed measures dh on H( F ), dt on H x ( F ), and dh γ on H γ ( F ) so that ourchosen hyperspecial maximal compact subgroups have volume 1. In particular, both Q n ( O F )and Q γ ( O F ) have volume one with these choices. Let κ γ denote the restriction of the character κ to ker (cid:2) H ( F, H x ) −→ H ( F, H γ ) (cid:3) ⊂ D (H x /F ) = ker (cid:2) H ( F, H x ) −→ H ( F, H) (cid:3) . Proposition 7.2.
Under the above assumptions, we have the identity O κ ( x, Q n ( O F ) ) = O κ γ ( x, Q γ ( O F ) ) . Proof.
Our choice of x ∈ Q n ( O F ) fixes a bijection { H( F )-orbits in (H x \ H)( F ) } ∼ −→ D (H x /F ) x ′ inv( x, x ′ ) . For each rational orbit H( F ) · x ′ , we decompose this with respect to the action of H ( O F )H( F ) · x ′ = G y H ( O F ) · y with y ∈ H( F ) · x ′ running over a set of H ( O F )-orbit representatives and set H ( O F )[ x ′ ] := { y : y ∈ Q n ( O F ) } . In order to be completely correct, we would need to write our orbital integrals in terms of x tu (or the element y in Section 7.2), and remark that since γ ∈ Q n ( O F ) the values of the orbital integrals do not depend on whetherwe compute them on Q γ or its image under s γ . With our choices of measure, we find thatO κ ( x, Q n ( O F ) ) = X x ′ κ (inv( x, x ′ )) H ( O F )[ x ′ ] . Considering the absolutely semi-simple part γ = x as and applying Proposition 4.9, if x ′ = x ′ as x ′ tu ∈ Q n ( O F ) lies in the same stable orbit, then γ = x ′ as . This implies that x ′ ∈ G γ ( O F ),and x ′ may be stably conjugated to x by an element of H γ ( F alg ), showing thatinv( x, x ′ ) ∈ ker (cid:2) H ( F, H x ) −→ H ( F, H γ ) (cid:3) . Additionally, Proposition 4.9 forces H ( O F )[ x ′ ] = H γ ( O F )[ x ′ ] . Using our normalization of measures, we deduce the result. (cid:3)
Remark . It is useful to note that the assumption that the decomposition x = γx tu is thetopological Jordan decomposition may be loosened to the assumption that(1) x = γx tu ∈ Q rssn ( O F ),(2) γ is absolutely semi-simple,(3) x tu ∈ G γ ( O F ) ∩ Q n ( F ) . Descent on Q n . We now use Proposition 7.2 to descend our κ -orbital integral to thecontext of Section 6.5.Suppose now that x ∈ Q rssn ( O F ) be as in Proposition 7.1, and let x = x as x tu be the topologicalJordan decomposition. Corollary 4.12 tells us that there exists g ∈ G x as ( O F ) such that s ( g ) = x as . If W = V ′ ⊕ V ⊕ V − is the eigenspace decomposition of W for x as discussed in Lemma 3.9, wehave G x as = G ′ x as × U( V ) × U( V − )where the assumption of Proposition 7.1 implies dim( V ) > V − ) >
0. We may write g = ( g ′ , g , g − ) ∈ G x as ( O F ) . We need the following characterization of the eigenvalues that canoccur in V ′ . Lemma 7.4.
Let e be the ramification degree of F/ Q p and assume that p > max { e + 1 , } . Let ν ∈ {± } . Suppose that x = x as x tu ∈ Q rssn ( O F ) is is the topological Jordan decomposition. If µ ∈ O × F alg is an eigenvalue of x as such that λ = ν ∈ k alg , then λ = ν. Proof.
Suppose that λ as ∈ O × F alg is as in the statement. Then there exists some t ∈ Q > suchthat λ as ≡ ν (mod ̟ t ) . Recall now the construction of x as : recall c G as in Section 4.1, and let l ∈ Z > be such that q l ≡ c G ). Then we have x as = lim m →∞ x q lm . Thus, the eigenvalues of x as are n lim m →∞ λ q lm : car x ( λ ) = 0 o . In particular, there exists λ ∈ Ω( x ) (see Lemma 3.3) such that λ as = lim m →∞ λ q lm . (19)We claim that λ = ν ∈ k alg . Indeed, since x as x tu = x tu x as we may simultaneously diagonalizethese semi-simple elements of GL( W ) over F alg . A fortiori, this diagonalizes x = x as x tu andshows that λ decomposes λ = λ as λ tu where λ as (resp. λ tu ) is the associated eigenvalue of x as (resp. x tu ). Since x tu is topologicallyunipotent, we see that λ tu = 1 so that λ = λ as λ tu = λ as ∈ k alg . By assumption, we see λ = ν . N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 31
We now claim the limit (19) is ν . Let q = p f and note that h ̟ e i = h p i . To see this, we maywrite λ = ν + V with V ∈ O F alg such that v := val( V ) >
0. For m ≥
1, set D = q lm . We have λ D = ν + D X j =1 (cid:18) Dj (cid:19) ν ( D − j ) V j . By our assumption that p > e + 1 and a simple argument on the divisibility of binomialcoefficients (see [Hal93, Lemma 3.1]), we haveval (cid:18)(cid:18) Dj (cid:19) ν ( D − j ) V j (cid:19) ≥ val ( DV ) = lmf + v. Thus, (cid:12)(cid:12)(cid:12)(cid:12) λ q lm − ν (cid:12)(cid:12)(cid:12)(cid:12) F −→ m −→ ∞ . (cid:3) Since we need only reduce to the very regular locus, descending all the way to G x as is notnecessary. Instead, we decompose x as = γ · y as , where γ = I W ′ ⊕ I W ⊕ − I W − , and y as = x as | W ′ ⊕ I W ⊕ I W − . If ι : G x as ֒ → G is the natural inclusion of the centralizer, then( s ◦ ι )(1 , , g − ) = γ and ( s ◦ ι )( g ′ , g ,
1) = y as both lie in Q n ( O F ) by Proposition 4.9. If we set y = γ − x, then y = y as x tu ∈ G γ ( O F ) is thetopological Jordan decomposition. Lemma 4.4 and Proposition 4.5 also imply that y ∈ Q n ( F ).Following Remark 7.3, we apply Proposition 7.2 to the decomposition x = γy . By abuse ofnotation, we now decompose W with respect to the action of γ , writing W = V ⊕ V − , where V ν is the ν -eigenspace for γ . The descendant (G γ , H γ ) decomposes as a product(G γ , H γ ) = (G γ, , H γ, ) × (G γ, − , H γ, − )where for both ν = ±
1, (G γ,ν , H γ,ν ) = (U( V ν ) , U( V ν, ) × U( V ν, − )) . Here V ν,ν ′ = { w ∈ W : γw = νw, ǫw = ν ′ w } . We have the associated symmetric spaces Q = U( V ) / U( V , ) × U( V , − )and Q − = U( V − ) / U( V − , ) × U( V − , − ) . Proposition 4.9 implies that each of these Hermitian spaces are split. In fact, we can identifythe self-dual lattices directly. For the factors of H γ , we have the decompositionΛ = (Λ ∩ V , ) ⊕ (Λ ∩ V − , )and Λ − = (Λ − ∩ V , − ) ⊕ (Λ − ∩ V − , − ) . are both self-dual lattice giving rise to a hyperspecial subgroup H γ ( O F ). Lemma 7.5.
Writing x = γy = ( y , − y − ) ∈ Q ( O F ) × Q − ( O F ) , we have − y − ∈ Q ♥ , − ( O F ) and y ∈ Q ♥ , − ( O F ) . Proof.
The first assertion is immediate as y − is topologically unipotent, so that − y − / ∈ D ( k ).Likewise, our analysis of eigenvalues of y as in Lemma 7.4 tells us that no eigenvalue of y iscongruent to −
1, so that y ∈ Q ♥ , − ( O F ) . (cid:3) The product decomposition of (G γ , H γ ) induces a decomposition of the centralizerH x = H x, × H x, − ⊂ U( V ) × U( V − ) (20)and a resulting decomposition κ γ = ( κ , κ − ) : D (H x, /F ) × D (H x, − /F ) −→ C × . Proposition 7.2 now implies the following lemma.
Lemma 7.6.
With the notation as above, O κ ( x, Q n ( O F ) ) = O κ ( y , Q ( O F ) ) · O κ − ( − y − , Q − ( O F ) ) . Descent on the endoscopic side.
Suppose now that ( x a , x b ) ∈ Q a,α ( F ) × Q b,β ( F )matches x . Comparing characteristic polynomials, it follows from the definition of matching oforbits, Lemma 3.3, and the definition of strongly compact elements that x a ∈ U ( V a ⊕ V α ) and x b ∈ U ( V b ⊕ V β ) are strongly compact. In particular, there exist topological Jordan decompo-sitions x a = x a,as x a,tu and x b = x b,as x b,tu . Running the above argument in each case gives the descendants(G γ a , H γ a ) and (G γ b , H γ b )where we have ( x a , x b ) = ( γ a y a , γ b y b ) , with the obvious meaning for the notation. Write W a = V a ⊕ V α and W b = V b ⊕ V β . The action of γ a on W a and γ b on W b induce analogous decompositions W a = V a, ⊕ V a, − and W b = V b, ⊕ V b, − and the centralizers are of the formG γ a = U( V a, ) × U( V a, − ) and G γ b = U( V b, ) × U( V b, − ) , and the groups H γ a and H γ b are the appropriate products of unitary subgroups as above. Lemma 7.7. If ( α, β ) = ( I a , I b ) , then O κ ( x, Q n ( O F ) ) = 0 . Otherwise, the stable orbital integral
SO(( x a , x b ) , Q a ( O F ) ⊗ Q b ( O F ) ) equals SO (cid:16) ( y a, , y b, ) , Q a, ( O F ) ⊗ Q b, ( O F ) (cid:17) · SO (cid:16) ( − y a, − , − y b, − ) , Q a, − ( O F ) ⊗ Q b, − ( O F ) (cid:17) , where Q a, ± and Q b, ± are the descent symmetric spaces associated to ( γ a , γ b ) ∈ Q a ( F ) ×Q b ( F ) . Proof.
If ( α, β ) = ( I a , I b ) , then at least one of the forms on the four Hermitian spaces V a, , V a, − or V b, , V b, − is not split. Combining Lemma 7.6 with the vanishing statement in Proposition 6.9 implies inthis case that O κ ( x, Q n ( O F ) ) = 0 . Thus, we assume may that ( α, β ) = ( I a , I b ), and the result follows from Proposition 7.2. (cid:3) N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 33
Descent of transfer factors.
Finally, we consider the transfer factor.
Lemma 7.8.
Suppose that x ∈ Q n ( O F ) and ( x a , x b ) ∈ Q a ( F ) × Q b ( F ) are as in the previoussection. Then ∆ rel (( x a , x b ) , x ) = ∆ rel (( y a, , y b, ) , y ) · ∆ rel (( − y a, − , − y b, − ) , − y − ) . Proof.
The matching induces a partition of multi-sets R = R a G R b , where R ⊂ F alg × are the roots of χ x (with multiplicity) and R a (resp., R b ) are the roots of theinvariant of x a (resp., x b ). On the other hand, the decomposition W = V ⊕ V − induces a partition of multi-sets R = R G R − . The actions of γ a on W a and γ b on W b induce analogous decompositions R a = R a, G R a, − and R b = R b, G R b, − . It is immediate from the definition of matching that these are compatible.Recalling the formulas for the transfer factor in Appendix A.2.2, consider the relative dis-criminant D a,b ( x ) = Y ( z a ,z b ) ∈ R a × R b ( z a − z b ) . If z a ∈ R ⊥ a and z b ∈ R b, − , then z a − z b is clearly a unit in O F alg , and similarly if z a ∈ R a, − and z b ∈ R ⊥ b . Thus, recalling x = ( y , − y − ) ∈ Q γ ( O F ) we compute that | D a,b ( x ) | F = | D a,b ( y ) | F · | D a,b ( − y − ) | F and η E/F ( D a,b ( x )) = η E/F ( D a,b ( y )) · η E/F ( D a,b ( − y − ))Comparing with the definitions of the transfer factors in Section A.2.2, this proves the lemmawhen ( x, ( x a , x b )) is a good matching pair.To conclude, we study the compatibility of invariants and descent. Since we are able toassume that ( α, β ) = ( I a , I b ) , we may fix an embedding ϕ a,b : H erm ( V a ) ⊕ H erm ( V b ) ֒ → H erm ( V n ) , and set v := ϕ a,b ( R a ( x a ) , R b ( x b )). Then v is stably conjugate to R ( x ) ∈ H erm ( V n ), so Lemma5.3 tells us that there exists x ′ ∈ Q ( F ) in the same stable orbit as x such that R ( x ′ ) = v. Then x ′ and ( x a , x b ) are a good matching pair.Letting x ′ = γ ′ y ′ be the analogous decomposition, we similarly may write x ′ = ( y ′ , − y ′− ) ∈Q γ ( F ). Consider the invariant inv( x, x ′ ) ∈ H ( F, H x ). We need to check that under theisomorphism induced by the decomposition (20) we have H ( F, H x ) ∼ = H ( F, H x, ) × H ( F, H x, − )inv( x, x ′ ) (inv( y , y ′ ) , inv( − y − , − y ′− )) . Suppose that x ′ = hxh − for h ∈ H ( F alg ) . Uniqueness of the topological Jordan decompositionforces x ′ as = hx as h − and x ′ tu = hx tu h − . From this, it is immediate that γ ′ = hγh − so that y ′± = hy ± h − . This proves the claim as itshows that κ (inv( x, x ′ )) = κ (inv( y , y ′ )) · κ − (inv( − y − , − y ′− )) . (cid:3) Appendix A. Endoscopy for unitary Lie algebras
We recall the necessary facts from the theory of endoscopy for unitary Lie algebras. Wefollow [Les19b] closely. Assume
E/F is a quadratic extension of p -adic fields and let W be a n -dimensional Hermitian space over E . As previously noted, we will work with the twisted Liealgebra H erm ( W ) = { x ∈ End( W ) : h xu, v i = h u, xv i} . Let δ ∈ H erm ( W ) be regular and semi-simple. Recalling that the set of rational conjugacyclasses O st ( δ ) in the stable conjugacy class of y form a D ( T δ /F )-torsor, we have a mapinv( δ, − ) : O st ( δ ) ∼ −→ D ( T δ /F ) (21)trivializing the torsor by fixing the orbit [ δ ] as the base point. This map is given by[ δ ′ ] inv( δ, δ ′ ) := [ σ ∈ Gal(
E/F ) g − σ ( g )] , where g ∈ GL( W ) such that δ ′ = Ad( g )( δ ) . A.1.
Endoscopy for unitary Lie algebras.
An elliptic endoscopic datum for H erm ( W ) isthe same as a datum for the group U ( W ) , namely a triple(U( V a ) × U( V b ) , s, η ) , where a + b = n . Here s ∈ ˆ U ( W ) a semi-simple element of the Langlands dual group of U ( W ),and an embedding η : ˆ U ( V a ) × ˆ U ( V b ) ֒ → ˆ U ( W )identifying ˆ U ( V a ) × ˆ U ( V b ) with the neutral component of the centralizer of s in the L -group L U ( W ). Fixing such a datum, we consider the endoscopic Lie algebra H erm ( V a ) ⊕ H erm ( V b ).Let δ ∈ H erm ( W ) and ( δ a , δ b ) ∈ H erm ( V a ) ⊕ H erm ( V b ) be regular semi-simple.Denote W a,b = V a ⊕ V b . In the non-archimedean case, the isomorphism class of W a,b isuniquely determined by those of V a and V b [Jac62, Theorem 3.1.1].A.2. Matching of orbits.
We first recall the notion of Jacquet–Langlands transfer betweentwo non-isomorphic Hermitian spaces W and W ′ . If we identify the underlying vector spaces(but not necessarily the Hermitian structures) W ∼ = E n ∼ = W ′ , (22)we have embeddings H erm ( W ) , H erm ( W ′ ) ֒ → g l n ( E ) . Then δ ∈ H erm ( W ) and δ ′ ∈ H erm ( W ′ ) are said to be Jacquet–Langlands transfers if theyare GL n ( E )-conjugate in g l n ( E ) . This is well defined since the above embeddings are determinedup to GL n ( E )-conjugacy. Note that if δ and δ ′ are Jacquet–Langlands transfers, then δ ′ = Ad( g )( δ )for some g ∈ GL( W ) and we obtain a well-defined cohomology classinv( δ, δ ′ ) = [ σ ∈ Gal(
E/F ) g − σ ( g )] ∈ H ( F, T δ )extending the invariant map on D ( T δ /F ). Definition A.1.
In the case that W ′ = W a,b , we say that y and ( δ a , δ b ) are transfers (or aresaid to match) if they are Jacquet–Langlands transfers in the above sense.For later purposes, note that we have an embedding φ a,b : H erm ( V a ) ⊕ H erm ( V b ) ֒ → H erm ( W a,b ) , well defined up to conjugation by U ( W a,b ). If W ∼ = W a,b , we say that a matching pair y and( δ a , δ b ) are a nice matching pair if we may choose φ a,b so that φ a,b ( δ a , δ b ) = δ. N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 35
A.2.1.
Orbital integrals.
For δ ∈ H erm ( W ) rss and f ∈ C ∞ c ( H erm ( W )), we define the orbitalintegral O( δ, f ) = Z T δ \ U ( W ) f ( g − δg ) d ˙ g, where dg is a Haar measure on U ( W ), dt is the unique normalized Haar measure on the torus T δ , and d ˙ g is the invariant measure such that dtd ˙ g = dg. To an elliptic endoscopic datum (U( V a ) × U( V b ) , s, η ) and regular semi-simple element δ ∈H erm ( W ), there is a natural character κ : D ( T δ /F ) −→ C × , which may be computed as follows. For matching elements δ and ( δ a , δ b ), H ( F, T δ ) = Y S Z / Z = Y S ( a ) Z / Z × Y S ( b ) Z / Z = H ( F, T δ a × T δ b ) , (23)where the notation indicates which elements of S arise from the torus T δ a or T δ b . Lemma A.2. [Xia18, Proposition 3.10]
Consider the character ˜ κ : H ( F, T δ ) → C × such thaton each Z / Z -factor arising from S ( a ) , ˜ κ is the trivial map, while it is the unique non-trivialmap on each Z / Z -factor arising from S ( b ) . Then κ = ˜ κ | D ( T δ /F ) . Recalling the invariant map inv( δ, − ) : O st ( δ ) ∼ −→ D ( T δ /F ) , we form the κ -orbital integral of f ∈ C ∞ c ( H erm ( W ))O κ ( δ, f ) = X δ ′ ∼ st δ κ (inv( δ, δ ′ )) O( δ ′ , f ) . When κ = 1 is trivial, write SO = O κ . A.2.2.
Transfer factors.
We now recall the transfer factor of Langlands-Sheldstad and Kottwitz.This is a function ∆ : [ H erm ( V a ) ⊕ H erm ( V b )] rss × H erm ( W ) rss → C . The two important properties are(1) ∆(( δ a , δ b ) , δ ) = 0 if δ does not match ( δ a , δ b ) , and(2) if δ is stably conjugate to δ ′ , then∆(( δ a , δ b ) , δ ) O κ ( δ, f ) = ∆(( δ a , δ b ) , δ ′ ) O κ ( δ ′ , f ) . While the general definition, given in [LS87] for the group case and [Kot99] in the quasi-splitLie algebra setting, is subtle, our present setting enjoys the following simplified formulation.When δ ∈ H erm ( W ) and ( δ a , δ b ) ∈ H erm ( V a ) ⊕ H erm ( V b ) do not match, we set∆(( δ a , δ b ) , δ ) = 0 . Now suppose that δ and ( δ a , δ b ) match. We define the relative discriminant D a,b ( δ ) = Y x a ,x b ( x a − x b ) , where x a (resp. x b ) ranges over the eigenvalues of δ a (resp. δ b ) in F alg . Remark
A.3 . This is precisely the quotient of the standard Weyl discriminants that occurs inthe factor ∆ IV in [LS87]. It is well known that the magnitude of D a,b ( δ ) measures the differencein asymptotic behavior of orbital integrals on H erm ( W ) and H erm ( V a ) ⊕ H erm ( V b ). Recall our notation W a,b = V a ⊕ V b and first assume that W ∼ = W a,b and that δ and ( δ a , δ b )are a nice matching pair. In this case, the transfer factor is then given by∆(( δ a , δ b ) , δ ) := η E/F ( D a,b ( δ )) | D a,b ( δ ) | F , (24)where η E/F is the quadratic character associated to
E/F .Now for any matching pair δ and ( δ a , δ b ), let δ ′ = φ a,b ( δ a , δ b ) ∈ H erm ( W a,b ) . As discussed in Section A.1, δ and δ ′ are Jacquet–Langlands transfers of each other and we set∆(( δ a , δ b ) , δ ) = κ (inv( δ, δ ′ )) η E/F ( D a,b ( δ )) | D a,b ( δ ) | F , where κ : H ( F, T δ ) → C × is the character arising from the datum (U( V a ) × U( V b ) , s, η ) and invis the extension of the invariant map discussed in Section A.1.A.3. Smooth transfer.
A pair of functions f ∈ C ∞ c ( H erm ( W )) and f a,b ∈ C ∞ c ( H erm ( V a ) ⊕ H erm ( V b ))are said to be smooth transfers (or matching functions) if the following conditions are satisfied:(1) for any matching elements regular semi-simple elements δ and ( δ a , δ b ),SO(( δ a , δ b ) , f a,b ) = ∆(( δ a , δ b ) , δ ) O κ ( δ, f );(2) if there does not exist y matching ( δ a , δ b ), thenSO(( δ a , δ b ) , f a,b ) = 0 . The existence of smooth transfer follows by combining [LN08], [Wal06], and [Wal97].
N THE STABILIZATION OF A RELATIVE TRACE FORMULA: THE FUNDAMENTAL LEMMA 37
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