A Conjecture of Sakellaridis-Venkatesh on the Unitary Spectrum of Spherical Varieties
aa r X i v : . [ m a t h . R T ] N ov A CONJECTURE OF SAKELLARIDIS-VENKATESHON THE UNITARY SPECTRUM OF SPHERICAL VARIETIES.
WEE TECK GAN AND RAUL GOMEZ to Nolan Wallach,with admiration and appreciation Introduction
The spectral decomposition of the unitary representation L ( H \ G ) when X = H \ G is a sym-metric space has been studied extensively, especially in the case when G is a real Lie group. Inparticular, through the work of many authors (such as [7], [17], [23], [3] and [1]), one now has thefull Plancherel theorem in this setting.In a recent preprint [20], Sakellaridis and Venkatesh considered the more general setting where X = H \ G is a spherical variety and G is a real or p-adic group. Motivated by the study of periodsin the theory of automorphic forms and the comparison of relative trace formulas, they formulatedan approach to this problem in the framework of Langlands functoriality. More precisely, led byand refining the work of Gaitsgory-Nadler [8] in the geometric Langlands program, they associatedto a spherical variety X = H \ G (satisfying some additional technical hypotheses) • a dual group ˇ G X ; • a natural map ι : ˇ G X × SL ( C ) −→ ˇ G The map ι induces a map from the set of tempered L-parameters of G X to the set of Arthurparameters of G , and if one is very optimistic, it may even give rise to a map ι ∗ : b G X −→ b G where G X is a (split) group with dual group ˇ G X and b G X and b G refer to the unitary dual of therelevant groups. Assuming for simplicity that this is the case, one has the following conjecture: Sakellaridis-Venkatesh Conjecture
One has a spectral decomposition L ( H \ G ) ∼ = Z b G X W ( π ) ⊗ ι ∗ ( π ) dµ ( π ) where µ is the Plancherel measure of b G X and W ( π ) is some multiplicity space.In particular, the class of the spectral measure of L ( H \ G ) is absolutely continuous with respectto that of the pushforward by ι ∗ of the Plancherel measure on b G X , and its support is contained inthose Arthur parameters of G which factor through ι . In addition, one expects that the multiplicityspace W ( π ) is related to the space of continuous H -invariant functionals on the representation ι ∗ ( π ) .The main purpose of this paper is to verify the above conjecture in many cases when H \ G ,or equivalently G X , has low rank, and to specify the multiplicity space W ( π ) . In particular, wedemonstrate this conjecture for many cases when G X has rank , and also some cases when G X hasrank or (see the tables in [20, §15 and §16]). More precisely, our main result is: heorem 1. The conjecture of Sakellaridis-Venkatesh holds for the spherical varieties H \ G listedin the following tables. H \ G GL n − \ GL n SO n − \ SO n Sp n − \ Sp n G X GL f SL SO (4) Table 1.
Classical cases H \ G SO \ SL Sp \ SL SL \ G ( J, ψ ) \ G G \ Spin G \ Spin Spin \ F F \ E G X f SL SL ˜ SL P GL SL SL / ∆ µ P GL SL Table 2.
Exceptional casesWe refer the reader to the main body of the paper for the precise statements and unexplainednotation.The theorem is proved using the technique of theta correspondence. More precisely, it turns outthat for the groups listed in the above table, one has a reductive dual pair G X × G ⊂ S for some larger group S . One then studies the restriction of the minimal representation of S to thesubgroup G X × G . In the context of theta correspondence in smooth representation theory, one cantypically show the following rough statement: A representation π of G has ψ -generic (and hence nonzero) theta lift to G X m π has nonzero H -period.Our main theorem is thus the L -manifestation of this phenomenon, giving a description of L ( H \ G ) in terms of L ( G X ) .This idea is not really new: a well known example of this kind of result is the correspondencebetween the irreducible components of the spherical harmonics on R n under the action of O ( n, R ) ,and holomorphic discrete series of the group f SL (2 , R ) , the double cover of SL (2 , R ) . Another ex-ample is given by the classical paper of Rallis and Schiffmann [18] where they used the oscillatorrepresentation to relate the discrete spectrum of L ( O ( p, q − \ O ( p, q )) with the discrete series rep-resentations of f SL (2 , R ) . Later, Howe [11] showed how these results can be inferred from his generaltheory of reductive dual pairs, and essentially provided a description of the Plancherel measure of L ( O ( p, q − \ O ( p, q )) in terms of the representation theory of f SL (2 , R ) . Then Ørsted and Zhang[27] proved a similar result for the space L ( U ( p, q − \ U ( p, q )) in terms of the representationtheory of U (1 , . We give a more steamlined treatment of these classical cases in Section 2, whichaccounts for Table 1. The rest of the paper is then devoted to the exceptional cases listed in Table2. Acknowledgments:
Both authors would like to pay tribute to Nolan Wallach for his guidance,encouragement and friendship over the past few years. It is an honor to be his colleague and studentrespectively. We wish him all the best in his retirement from UCSD, and hope to continue to interactwith him mathematically and personally for many years to come. he research of the first author is partially supported by NSF grant 0801071 and a startup grantfrom the National University of Singapore.2. Classical Dual Pairs
We begin by introducing the classical dual pairs.2.1.
Division algebra D . Let k be a local field, and let | · | denote its absolute value. Let D = k ,a quadratic field extension of k or the quaternion division k -algebra, and let x x be its canonicalinvolution. The case when D is the split quadratic algebra or quaternion algebra can also be includedin the discussion, but for simplicity, we shall stick with division algebras. We have the trace map T r ( x ) = x + x ∈ k and the norm map Q ( x ) = x · x ∈ k .2.2. Hermitian D -modules. Let V and W be two right D -modules. We will denote the set ofright D -module morphisms between V and W by Hom D ( V, W ) = { T : V −→ W | T ( v a + v b ) = T ( v ) a + T ( v ) b for all v , v ∈ V , a , b ∈ D } . In the same way, if V and W are two left D -modules, we set Hom D ( V, W ) = { T : V −→ W | ( av + bv ) T = a ( v ) T + b ( v ) T for all v , v ∈ V , a , b ∈ D } . If V = W , we will denote this set by End D ( V ) . Notice that for right D -module morphisms we areputting the argument on the right, while for left D -module morphisms we are putting it on the left.In general, for every statement involving right D -modules one can make an analogous one involv-ing left D -modules. From now on, we will focus on right D -modules, and we will let the reader withthe task of making the corresponding definitions and statements involving left D -modules. Set GL ( V, D ) = { T ∈ End D ( V ) | T is invertible } . When it is clear from the context what the division algebra is, we will just denote this group by GL ( V ) .Let V ′ be the set of right D -linear functionals on V . There is a natural left D -module structureon V ′ given by setting ( aλ )( v ) = aλ ( v ) , for all a ∈ D , v ∈ V , and λ ∈ V ′ .Observe that with this structure, W ⊗ D V ′ is naturally isomorphic to Hom D ( V, W ) as a k -vectorspace. Given T ∈ Hom D ( V, W ) , we will define an element in Hom D ( W ′ , V ′ ) , which we will alsodenote T , by setting ( λT )( v ) := λ ( T v ) . This correspondence gives rise to natural isomorphismsbetween End D ( V ) and End D ( V ′ ) and between GL ( V ) and GL ( V ′ ) . Definition 2.
Let ε = ± . We say that ( V, B ) is a right ε -Hermitian D -module, if V is a right D -module and B is an ε -Hermitian form, i.e B : V × V −→ D is a map such that(1) B is sesquilinear . That is, for all v , v , v ∈ V , a , b ∈ D , B ( v , v a + v b ) = B ( v , v ) a + B ( v , v ) b and B ( v a + v b, v ) = aB ( v , v ) + bB ( v , v ) . (2) B is ε - Hermitian . That is, B ( v, w ) = εB ( w, v ) for all v, w ∈ V .(3) B is non-degenerate . o define left ε -Hermitian D -modules ( V, B ) , we just have to replace the sesquilinear conditionby B ( av + bv , v ) = aB ( v , v ) + bB ( v , v ) and B ( v , av + bv ) = B ( v , v ) a + B ( v , v ) b, for all v , v , v ∈ V , a , b ∈ D .Given a right ε -Hermitian D -module ( V, B ) , we will define G ( V, B ) = { g ∈ GL ( V ) | B ( gv, gw ) = B ( v, w ) for all v , w ∈ V } , to be the subgroup of GL ( V ) preserving the ε -Hermitian form B . When there is no risk of confusionregarding B , we will denote this group just by G ( V ) . Usually, -Hermitian D -modules are simplycalled Hermitian, while − -Hermitian D -modules are called skew-Hermitian.Given a right ε -Hermitian D -module ( V, B ) , we can construct a left ε -Hermitian D -module ( V ∗ , B ∗ ) in the following way: as a set, V ∗ will be the set of symbols { v ∗ | v ∈ V } . Then wegive V ∗ a left D -module structure by setting, for all v , w ∈ V , a ∈ D , v ∗ + w ∗ = ( v + w ) ∗ and av ∗ = ( va ) ∗ .Finally, we set B ∗ ( v ∗ , w ∗ ) = B ( w, v ) for all v , w ∈ V .In an analogous way, if V is a left D -module, we can define a right D -module V ∗ , and V ∗∗ is naturallyisomorphic with V . Given T ∈ End D ( V ) , we can define T ∗ ∈ End D ( V ∗ ) by setting v ∗ T ∗ := ( T v ) ∗ .With this definition, it is easily seen that ( T S ) ∗ = S ∗ T ∗ , for all S , T ∈ End D ( V ) . Therefore themap g ( g ∗ ) − defines an algebraic group isomorphism between GL ( V ) and GL ( V ∗ ) .Now observe that the form B induces a left D -module isomorphism B ♭ : V ∗ −→ V ′ given by B ♭ ( v ∗ )( w ) = B ( v, w ) for v , w ∈ V . In what follows, we will make implicit use of this map toidentify this two spaces. With this identification we can think of T ∗ as a map in End D ( V ) definedby v ∗ ( T ∗ w ) := ( v ∗ T ∗ )( w ) , i.e, T ∗ is defined by the condition that B ( v, T ∗ w ) = B ( T v, w ) for all v , w ∈ V .
Observe that this agrees with the usual definition of T ∗ .A D -submodule X ⊂ V is said to be totally isotropic if B | X × X = 0 . If X is a totally isotropicsubmodule, then there exists a totally isotropic submodule Y ⊂ V such that B | X ⊕ Y × X ⊕ Y is non-degenerate. If we set U = ( X ⊕ Y ) ⊥ := { u ∈ V | B ( u, w ) = 0 for all w ∈ X ⊕ Y } , then V = X ⊕ Y ⊕ U , and B | U × U is non-degenerate. In this case we say that X and Y are totallyisotropic, complementary submodules. Observe that then B ♭ | Y ∗ : Y ∗ −→ X ′ is an isomorphism. Asbefore we will make implicit use of this isomorphism to identify Y ∗ with X ′ .2.3. Reductive dual pairs.
Let ( V, B V ) be a right ε V -Hermitian D -module and ( W, B W ) a right ε W -Hermitian D -module such that ε V ε W = − . On the k -vector space V ⊗ D W ∗ we can define asymplectic form B by setting B ( v ⊗ D λ , v ⊗ D λ ) = Tr( B W ( w , w ) B ∗ V ( λ , λ )) for all v , v ∈ V and λ , λ ∈ V ∗ .Let Sp ( V ⊗ D W ∗ ) = { g ∈ GL ( V ⊗ D W ∗ , k ) | B ( gv, gw ) = B ( v, w ) for all v , w ∈ V ⊗ D W ∗ } . Observe that Sp ( V ⊗ D W ∗ ) = G ( V ⊗ D W ∗ , B ) = G ( V ⊗ D W ∗ ) . oreover, there is a natural map G ( V ) × G ( W ) −→ Sp ( V ⊗ D W ∗ ) given by ( g , g ) · v ⊗ D λ = g v ⊗ λg ∗ . We will use this map to identify G ( V ) and G ( W ) with subgroups of Sp ( V ⊗ D W ∗ ) . These twosubgroups are mutual commutants of each other, and is an example of a reductive dual pair .2.4. Metaplectic cover.
The group Sp ( V ⊗ D W ∗ ) has an S - cover M p ( V ⊗ D W ∗ ) which is calleda metaplectic group. It is known that this S -cover splits over the subgroups G ( V ) and G ( W ) ,except when V is an odd dimensional quadratic space, in which it does not split over G ( W ) . In thisexceptional case, we shall simply redefine G ( W ) to be the induced double cover, so as to simplifynotation. We remark also that though the splittings (when they exist) are not necessarily unique,the precise choice of the splittings is of secondary importance in this paper.2.5. Siegel parabolic.
Assume in addition that there is a complete polarization W = E ⊕ F, where E , F , are complementary totally isotropic subspaces of W . We will use the ε W -Hermitian form B W to identify F ∗ with E ′ by setting f ∗ ( e ) = B W ( f, e ) . Observe that this identification induces anidentification between E ∗ and F ′ given by e ∗ ( f ) = f ∗ ( e ) = B W ( f, e ) = ε W B W ( e, f ) . In what follows, we will use this identifications between F ∗ and E ′ , and between E ∗ and F ′ .Let P = { p ∈ G ( W ) | p · E = E } be the Siegel parabolic subgroup of G ( W ) , and let P = M N be its Langlands decomposition. Togive a description of the groups M and N , we introduce some more notation.Let A ∈ End D ( E ) . We will define A ∗ ∈ End D ( F ) , by setting, for all e ∈ E , f ∈ F ,(1) B W ( e, A ∗ f ) = B W ( Ae, f ) . Now given T ∈ Hom D ( F, E ) , define T ∗ ∈ Hom D ( F, E ) by setting, for all f , f ∈ F ,(2) B W ( f , T ∗ f ) = ε W B W ( T f , f ) . Given ε = ± , set Hom D ( F, E ) ε = { T ∈ Hom D ( F, E ) | T ∗ = εT } . It is then clear that
Hom D ( F, E ) =
Hom D ( F, E ) ⊕ Hom D ( F, E ) − .Now we have: M = (cid:26)(cid:20) A ( A ∗ ) − (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) A ∈ GL ( E ) (cid:27) ∼ = GL ( E ) and N = (cid:26)(cid:20) X (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) X ∗ = − ε W X (cid:27) ∼ = Hom D ( F, E ) − ε W . Characters of N . Given Y ∈ Hom D ( E, F ) − ε W , define a character χ Y (cid:18)(cid:20) X (cid:21)(cid:19) = χ (Tr F ( Y X )) . Here Tr F is the trace of Y X : F −→ F seen as a map between k vector spaces. The map Y χ Y defines a group isomorphism between Hom D ( E, F ) − ε W and ˆ N .Observe that the adjoint action of M on N induces an action of M on ˆ N . Using the isomorphismsof M ∼ = GL ( E ) and ˆ N ∼ = Hom D ( E, F ) − ε W , we can describe the action of M on ˆ N by the formula A · Y = ( A ∗ ) − Y A − for all A ∈ GL ( E ) , Y ∈ Hom D ( E, F ) − ε W . iven Y ∈ Hom D ( E, F ) − ε W we can define a − ǫ W -Hermitian form on E , that we will also denoteY, by setting Y ( e , e ) = e ∗ ( Y e ) = ε W B W ( e , Y e ) . Hence the action of M on ˆ N is equivalent to the action of GL ( E ) on sesquilinear, − ε W -Hermitianforms on E .Let Ω be the set of orbits for the action of M on ˆ N . Given Y ∈ Hom D ( E, F ) − ε W , let O = O Y be its orbit under the action of GL ( E ) and set M χ Y = { m ∈ M | χ Y ( m − nm ) = χ Y ( n ) for all n ∈ N } . Using the identification of M with GL ( E ) , and of ˆ N with Hom D ( E, F ) − ε W , we see that M χ Y ∼ = { A ∈ GL ( E ) | ( A ∗ ) − Y A − = Y } = { A ∈ GL ( E ) | Y = A ∗ Y A } . Oscillator Representation
After the preparation of the previous section, we can now consider the theta correspondenceassociated to the dual pair G ( V ) × G ( W ) and use it to establish certain cases of the Sakellaridis-Venkatesh conjecture for classical groups.3.1. Oscillator representation and theta correspondence.
Fix a nontrivial unitary character χ of k . Associated to this character, there exists a very special representation of the metaplecticgroup, called the oscillator representation Π of M p ( V ⊗ D W ∗ ) . On restricting this representationto G ( V ) × G ( W ) , one obtains an injective map θ : A ⊂ G ( W ) ∧ −→ G ( V ) ∧ and a measure µ θ on e G ( W ) ∧ , such that(3) Π | G ( W ) × G ( V ) = Z A π ⊗ θ ( π ) dµ θ ( π ) , as a G ( W ) × G ( V ) -module.We may restrict Π further to P × G ( V ) . By Mackey theory, for a unitary representation π of G ( W ) ,(4) π | P = M O Y ∈ Ω Ind PM χY N W χ Y ( π ) , where W χ Y ( π ) is an M χ Y N -module such that n · λ = χ Y ( n ) λ , for all n ∈ N , λ ∈ W χ Y ( π ) . Therefore,from (3) and (4), we have:(5) Π = M O Y ∈ Ω Z A ⊂ b G ( W ) Ind PM χY N W χ Y ( π ) ⊗ Θ( π ) dµ θ ( π ) . The Schrödinger model.
On the other hand, we may compute the restriction of Π to P × G ( V ) using an explicit model of Π . The complete polarization W = E ⊕ F induces a completepolarization V ⊗ D W ∗ = V ⊗ D E ∗ ⊕ V ⊗ D F ∗ . With the identifications introduced above, V ⊗ D F ∗ = Hom D ( E, V ) , and the oscillator represen-tation Π can be realized on the Hilbert space L ( Hom D ( E, V )) ; this realization of Π is called theSchrodinger model. The action of P × G ( V ) in this model can be described as follows.Let B ♭V : V −→ ( V ∗ ) ′ be given by ( w ∗ )( B ♭V v ) = B V ( w, v ) . hen the action of P × G ( V ) on L ( Hom D ( E, V )) is given by the formulas (cid:20) X (cid:21) · φ ( T ) = χ (Tr F ( XT ∗ B ♭V T )) φ ( T ) , for all X ∈ Hom D ( F, E ) − ε W , (6) (cid:20) A ( A ∗ ) − (cid:21) · φ ( T ) = | det F ( A ) | − dim D ( V ) / φ ( T A ) , for all A ∈ GL ( E ) , (7) g · φ ( T ) = φ ( g − T ) , for all g ∈ G ( V ) . (8)Let Ω V = {O Y | O Y is open in Hom D ( E, F ) − ε W , and Y = T ∗ B ♭V T for some T ∈ Hom D ( E, V ) } . Given O Y ∈ Ω V , we will set Υ Y = { T ∈ Hom D ( E, V ) | T ∗ B ♭V T ∈ O Y } . Then [ O Y ∈ Ω V Υ Y ⊂ Hom D ( E, V ) is a dense open subset, and its complement in Hom D ( E, V ) has measure 0. Therefore(9) L ( Hom D ( E, V )) ∼ = M O Y ∈ Ω V L (Υ Y ) and each of these spaces is clearly P × G ( V ) -invariant, according to the formulas given in equations(6)–(8).We want to show that the spaces L (Υ Y ) are equivalent to some induced representation for P × G ( V ) . To do this, observe that the “geometric” part of the action of P × G ( V ) on L (Υ Y ) istransitive on Υ Y . In other words, under the action of P × G ( V ) on Hom D ( E, V ) given by (cid:18)(cid:20) A X ( A ∗ ) − (cid:21) , g (cid:19) · T = gT A − for all (cid:20) A X ( A ∗ ) − (cid:21) ∈ P , g ∈ G ( V ) and T ∈ Hom D ( E, V ) ,each of the Υ Y ’s is a single orbit. Fix T Y ∈ Υ Y such that T ∗ Y B ♭V T Y = Y . The stabilizer of T Y in P × G ( V ) is the subgroup ( P × G ( V )) T Y = (cid:26)(cid:18)(cid:20) A X ( A ∗ ) − (cid:21) , g (cid:19) ∈ P × G ( V ) (cid:12)(cid:12)(cid:12)(cid:12) gT Y = T Y A (cid:27) . Let g ∈ G ( V ) be such that gT Y = T Y A for some A ∈ GL ( E ) . Then by the definition of G ( V ) Y = T ∗ Y B ♭V T Y = T ∗ Y g ∗ B ♭V gT Y = A ∗ Y A, that is, A is an element in M χ Y .Define an equivalence relation in Hom D ( E, V ) by setting T ∼ S if T = SA for some A ∈ M χ Y .Given T ∈ Hom D ( E, V ) we will denote its equivalence class, under this equivalence relation, by [ T ] . Let P M χY ( Hom D ( E, V )) = { [ T ] | T ∈ Hom D ( E, V ) } . Since G ( V ) acts by left multiplication on Hom D ( E, V ) , there is natural action of G ( V ) on the space P M χY ( Hom D ( E, V )) . Set G ( V ) T Y = { g ∈ G ( V ) | gT Y = T Y } and G ( V ) [ T Y ] = { g ∈ G ( V ) | g [ T Y ] = [ T Y ] } . hen ( P × G ( V )) T Y ⊂ M χ Y × G ( V ) [ T Y ] , and according to equations (6)-(8), L (Υ Y ) ∼ = Ind P × G ( V )( P × G ( V )) TY χ Y (10) ∼ = Ind P × G ( V ) M χY N × G ( V ) [ TY ] Ind M χY N × G ( V ) [ TY ] ( P × G ( V )) TY χ Y (11)Now consider the short exact sequence −→ × G ( V ) T Y −→ ( P × G ( V )) T Y q −→ M χ Y N −→ , where q is the projection into the first component. Observe that the map q induces an isomorphism G ( V ) T Y \ G ( V ) [ T Y ] ∼ = M χ Y . From this exact sequence and equation (11), we get that L (Υ Y ) ∼ = Ind P × G ( V ) M χY N × G ( V ) [ TY ] L ( G ( V ) T Y \ G ( V ) [ T Y ] ) χ Y ∼ = Ind PM χY N L ( G ( V ) T Y \ G ( V )) χ Y . (12)The action of M χ Y N on L ( G ( V ) T Y \ G ( V ) [ T Y ] ) χ Y is given as follows: N acts by the character χ Y ,and M χ Y acts on L ( G ( V ) T Y \ G ( V ) [ T Y ] ) χ Y on the left using the isomorphism G ( V ) T Y \ G ( V ) [ T Y ] ∼ = M χ Y . Then according to equations (9) and (12)(13) L ( Hom D ( E, V )) ∼ = M O Y ∈ Ω V Ind PM χY N L ( G ( V ) T Y \ G ( V )) χ Y . But now, from equations (5), (13) and the uniqueness of the decomposition of the N -spectrum, weobtain: Proposition 3.
As an M χ Y N × G ( V ) -module, (14) L ( G ( V ) T Y \ G ( V )) χ Y ∼ = Z A ⊂ b G ( W ) W χ Y ( π ) ⊗ Θ( π ) dµ θ ( π ) , Our goal now is to give a more explicit characterization of the spaces W χ Y ( π ) and the measure µ θ appearing in this formula.3.3. Stable range.
Let ( V, B V ) and ( W, B W ) be as before. Assume now that there is a totallyisotropic D -submodule X ⊂ V such that dim D ( X ) = dim D ( W ) ; in other words, the dual pair ( G ( V ) , G ( W )) is in the stable range . In this case, the map θ : b G ( W ) −→ b G ( V ) can be understood in terms of the results of Jian-Shu Li [15]. The measure µ θ appearing in equation(3) is also known in this case: it is precisely the Plancherel measure of the group G ( W ) . In orderto make this paper more self-contained, we will include an alternative calculation of the measure µ θ using the so-called mixed model of the oscillator representation.3.4. Mixed model.
Let X , Y be a totally isotropic, complementary subspaces of V such that dim D ( X ) = dim D ( W ) , and let U = ( X ⊕ Y ) ⊥ . We will use B V to identify Y with ( X ∗ ) ′ by setting ( x ∗ ) y = B V ( x, y ) , for all x ∈ X , y ∈ Y .Given A ∈ GL ( X ) , we can use the above identification to define an element A ∗ ∈ GL ( Y ) in thefollowing way: given x ∈ X and y ∈ Y , we will set ( x ∗ )( A ∗ y ) := ( x ∗ A ∗ ) y , i.e., we will define A ∗ ∈ GL ( Y ) by requiring that B V ( x, A ∗ y ) = B V ( Ax, y ) , for all x ∈ X , y ∈ Y . bserve that the map A ( A ∗ ) − defines an isomorphism between GL ( X ) and GL ( Y ) . Further-more if x ∈ X , y ∈ Y and A ∈ GL ( X ) , then B V ( Ax, ( A ∗ ) − y ) = B V ( x, y ) . Therefore, we can define a map GL ( X ) × G ( U ) ֒ → G ( V ) that identifies GL ( X ) × G ( U ) with thesubgroup of G ( V ) that preserves the direct sum decomposition V = X ⊕ Y ⊕ U .Consider the polarization V ⊗ D W ∗ = ( X ⊗ W ∗ ⊕ U ⊗ F ∗ ) L ( Y ⊗ W ∗ ⊕ U ⊗ E ∗ ) . Then as avector space(15) L ( X ⊗ W ∗ ⊕ U ⊗ F ∗ ) ∼ = L ( Hom D ( W, X )) ⊗ L ( Hom D ( E, U )) . Let ( ω U , L ( Hom D ( E, U ))) be the Schrödinger model of the oscillator representation associated tothe metaplectic group f Sp ( U ⊗ D W ∗ ) . We will identify the space appearing on the right hand sideof equation (15) with the space of L functions from Hom D ( W, X ) to L ( Hom D ( E, U )) . This isthe so called mixed model of the oscillator representation.The action of G ( W ) × GL ( X ) × G ( U ) on this model can be described in the following way: If T ∈ Hom D ( W, X ) and S ∈ Hom D ( E, U ) , then g · φ ( T )( S ) = [ ω U ( g ) φ ( T g )]( S ) ∀ g ∈ G ( W ) (16) h · φ ( T )( S ) = φ ( T )( h − S ) ∀ h ∈ G ( U ) (17) A · φ ( T )( S ) = | det X ( A ) | dim W/ φ ( A − T )( S ) ∀ A ∈ GL ( X ) . (18)We now want to describe this space as an induced representation. To do this, observe thatthe set of invertible elements in Hom F ( W, X ) forms a single orbit under the natural action of G ( W ) × GL ( X ) . Furthermore this orbit is open and dense, and its complement has measure . Fix T ∈ Hom F ( W, X ) invertible, and define a ε W -Hermitian form B T on X , by setting B T ( x , x ) = B W ( T − x , T − x ) . The group that preserves this form is precisely G ( X, B T ) = { T gT − | g ∈ G ( W ) } ⊂ GL ( X ) . Let ( G ( W ) × GL ( X )) T = { ( g, T gT − ) | g ∈ G ( W ) } ∼ = G ( W ) be the stabilizer of T in G ( W ) × GL ( X ) . Then, according to equations (16)–(18), L ( W ⊗ X ) ⊗ L ( Hom D ( E, U )) ∼ = Ind G ( W ) × GL ( X )( G ( W ) × GL ( X )) T L ( Hom D ( E, U )) ∼ = Ind G ( W ) × GL ( X ) G ( W ) × G ( X,B T ) Ind G ( W ) × G ( X,B T )( G ( W ) × GL ( X )) T L ( Hom D ( E, U )) . Here ( G ( W ) × GL ( X )) T is acting on L ( Hom D ( E, U )) by taking projection into the first compo-nent, and then using the oscillator representation to define an action of G ( W ) on L ( Hom D ( E, U )) .But this representation is equivalent to taking projection into the second component and using theSchröridnger model of the oscillator representation of f Sp ( U ⊗ X ∗ ) (recall that X is equipped withthe form B T ) to define an action of G ( X, B T ) on L ( Hom D ( T ( E ) , U )) . Therefore L ( W ⊗ X ) ⊗ L ( Hom D ( E, U )) ∼ = Ind G ( W ) × GL ( X ) G ( W ) × G ( X,B T ) Ind G ( W ) × G ( X,B T )( G ( W ) × GL ( X )) T L ( Hom D ( T ( E ) , U )) ∼ = Ind G ( W ) × GL ( X ) G ( W ) × G ( X,B T ) R b G ( W ) π ∗ ⊗ ( π T ⊗ L ( Hom D ( T ( E ) , U ))) dµ G ( W ) ( π ) ∼ = R b G ( W ) π ∗ ⊗ Ind GL ( X ) G ( X,B T ) π T ⊗ L ( Hom D ( T ( E ) , U )) dµ G ( W ) ( π ) . (19) ere π ∗ is the contragredient representation of π , π T is the representation of G ( X, B T ) given by π T ( g ) = π ( T − gT ) , for all g ∈ G ( X, B T ) , and µ G ( W ) is the Plancherel measure of G ( W ) . Notethat the multiplicity space of π ∗ in (19) is nonzero for each π in the support of µ G ( W ) , i.e. as arepresentation of G ( W ) , Π is weakly equivalent to the regular representation L ( G ( W )) .Comparing (3) with (19), we obtain: Proposition 4. If ( G ( W ) , G ( V )) is in the stable range, with G ( W ) the smaller group, then inequations (3) and (14), A = b G ( W ) and µ θ = µ G ( W ) . The Bessel-Plancherel theorem.
Finally, we want to identify the multiplicity space W χ Y ( π ) in (14). Note that this is purely an issue about representations of G ( W ) ; a priori, it has nothing todo with theta correspondence. What we know is summarized in the following theorem. Theorem 5 (Bessel-Plancherel theorem) . Let ( W, B W ) be an ε W -Hermitian D -module, and assumethat W has a complete polarization W = E ⊕ F , where E , F are totally isotropic complementarysubspaces. Let P = { p ∈ G ( W ) | p · E = E } be a Siegel parabolic subgroup of G , and let P = M N beits Langlands decomposition. Given χ ∈ ˆ N , let O χ be its orbit under the action of M , and let M χ be the stabilizer of χ in M . Then(1) For µ G ( W ) -almost all tempered representation π of G ( W ) , π | P ∼ = M O χ ∈ Ω W Ind PM χ N W χ ( π ) . Here µ G ( W ) is the Plancherel measure of G ( W ) , Ω W = {O χ ∈ Ω | O χ is open in ˆ N } , and W χ ( π ) is some M χ N -module such that the action of N is given by the character χ .(2) If O χ ∈ Ω W , then there is an isomorphism of M χ × G ( W ) -modules: (20) L ( N \ G ( W ); χ ) ∼ = Z b G ( W ) W χ ( π ) ⊗ π dµ G ( W ) ( π ) . where W χ ( π ) is the same space appearing in (1).(3) If dim D ( W ) = 2 , then for O χ ∈ Ω W , dim W χ ( π ) < ∞ and W χ ( π ) ∼ = W h χ ( π ) = { λ : π ∞ −→ C | λ ( π ( n ) v ) = χ ( n ) λ ( v ) for all n ∈ N } as an M χ N -module. Here π ∞ stands for the set of C ∞ vectors of π .(4) If k is Archimedean, and M χ is compact, then W χ ( π ) ⊂ W h χ ( π ) as a dense subspace, and for any irreducible representation τ of M χ , one has an equality of τ -isotypic parts: W χ ( π )[ τ ] = W h χ ( π )[ τ ] . Moreover, this space is finite dimensional.Proof.
Part 2 follows from an argument analogous to the proof of the Whittaker-Plancherel measuregiven by Sakellaridis-Venkatesh [20, §6.3]. For the proof of part 1 observe that, by Harish-ChandraPlancherel theorem L ( G ( W )) | P × G ( W ) = Z b G ( W ) π ∗ | P ⊗ π dµ G ( W ) ( π ) . n the other hand L ( G ( W )) | P × G ( W ) = M O Y ∈ Ω W Ind PM χY N L ( N \ G ( W ); χ )= M O Y ∈ Ω W Ind PM χY N Z b G ( W ) W χ Y ( π ) ⊗ π dµ G ( W ) ( π )= Z b G ( W ) M O Y ∈ Ω W Ind PM χY N W χ Y ( π ) ⊗ π dµ G ( W ) ( π ) . Therefore π ∗ | P ∼ = M O Y ∈ Ω W Ind PM χ N W χ ( π ) for µ G ( W ) -almost all π . In the Archimedean case, this result has also been proved in the thesis ofthe second named author without the µ G ( W ) -almost all restriction, yielding an alternative proof ofpart 2 for the Archimedean case.Part 3 was proved by Wallach in the Archimedean case [24], and independently by Delorme,Sakellaridis-Venkatesh and U-Liang Tang in the p -adic case [4, 20, 22].Finally, Part 4 was shown by Wallach and the second named author in [10]. (cid:3) Spectral decomposition of generalized Stiefel manifolds.
We may now assemble all theprevious results together. For O Y ∈ Ω V , the space G ( V ) T Y \ G ( V ) is known as a generalized Stiefelmanifold . From equations (14) and (19), we deduce: Theorem 6.
Suppose that G ( V ) T Y \ G ( V ) is a generalized Stiefel manifold. If, in the notation ofequation (20) L ( N \ G ( W ); χ Y ) ∼ = Z b G ( W ) W χ Y ( π ) ⊗ π dµ G ( W ) ( π ) . then L ( G ( V ) T Y \ G ( V )) ∼ = Z b G ( W ) W χ Y ( π ) ⊗ Θ( π ) dµ G ( W ) ( π ) . In a certain sense, the last pair of equations says that the Plancherel measure of the generalizedStiefel manifold G ( V ) T Y \ G ( V ) is the pushforward of the Bessel-Plancherel measure of G ( W ) underthe θ -correspondence.3.7. The Sakellaridis-Venkatesh conjecture.
Using the previous theorem, we can obtain certainexamples of the Sakellaridis-Venkatesh conjecture: • Taking D = k , k × k or M ( k ) to be a split k -algebra and W to be skew-Hermitian with dim D W = 2 , we obtain the spectral decomposition of H \ G = O n − \ O n , GL n − \ GL n , Sp n − \ Sp n , in terms of the Bessel-Plancherel (essentially the Whittaker-Plancherel) decomposition for G X = GL , SL , ˜ SL or SO . This establishes the cases listed in Table 1 in Theorem 1. • Taking D to be a quadratic field extension of k or the quaternion division k -algebra, and W to be skew-Hermitian, we obtain the spectral decomposition of H \ G = U n − \ U n , Sp n − ( D ) \ Sp n ( D ) in terms of the Bessel-Plancherel decomposition of U and O ( D ) . This gives non-splitversion of the examples above. n addition, the multiplicity space W χ ( π ) = W h χ ( π ) should be describable in terms of thespace of H -invariant (continuous) functionals on Θ( π ) ∞ . Indeed, by the smooth analog of ourcomputation with the Schrodinger model in §3.2, one can show that there is a natural isomorphismof M χ -modules: W h χ ( π ) ∼ = Hom H (Θ ∞ ( π ∞ ) , C ) . Here Θ ∞ ( π ∞ ) refers to the (big) smooth theta lift of the smooth representation π ∞ , i.e. therepresentation π ∞ ⊠ Θ ∞ ( π ∞ ) is the maximal π ∞ -isotypic quotient of the smooth model Π ∞ = S ( Hom D ( E, V )) of the oscillator representation Π . One can show using the machinery developed inBernstein’s paper [2] that for µ θ -almost all π , one has the compatibility of L -theta lifts (consideredin this paper) with the smooth theta lifts: Θ( π ) ∞ ∼ = Θ ∞ ( π ∞ ) . With this compatibility, one will obtain W χ ( π ) ∼ = Hom H (Θ( π ) ∞ , C ) . Unstable range.
Though we have assumed that ( G ( W ) , G ( V )) is in the stable range from§3.3, it is possible to say something when one is not in the stable range as well. Namely, in §3.4,one would take X to be a maximal isotropic space in V (so dim X < dim W here), and considerthe mixed model defined on L ( Hom D ( W, X )) ⊗ L ( Hom D ( E, U )) . As an illustration, we note theresult for the case when W is a symplectic space of dimension and V is a split quadratic space ofdimension , so that G ( W ) × G ( V ) ∼ = ˜ SL × SO ∼ = ˜ SL × P GL . For a nonzero Y ∈ b N , the subgroup G ( V ) T Y of G ( V ) is simply a maximal torus A Y of P GL . Proposition 7.
We have L ( G ( V ) T Y \ G ( V )) = L ( A Y \ P GL ) ∼ = Z \ G ( W ) ( W χ ( σ ) ⊗ W χ Y ( σ )) ⊗ Θ χ ( π ) dµ G ( W ) ( π ) . We record the following corollary which is needed in the second half of this paper:
Corollary 8.
The unitary representation L ( sl ) associated to the adjoint action of P GL on itsLie algebra sl is weakly equivalent to the regular representation L ( P GL ) .Proof. Since the union of strongly regular semisimple classes are open dense in sl , we see that L ( sl ) is weakly equivalent to L A L ( A \ P GL ) , where the sum runs over conjugacy classes ofmaximal tori A in P GL . Applying Proposition 7, one deduces that M A L ( A \ P GL ) ∼ = Z \ G ( W ) M χ ( π ) ⊗ Θ χ ( π ) dµ G ( W ) ( π ) with M χ ( π ) = W χ ( π ) ⊗ M A W χ A ( π ) ! . One can show that the theta correspondence with respect to χ induces a bijection Θ χ : { π ∈ \ G ( W ) : W χ ( π ) = 0 } ←→ \ G ( V ) . Moreover, one can write down this bijection explicitly (in terms of the usual coordinates on theunitary duals of f SL and P GL ). From this description, one sees that (Θ χ ) ∗ ( µ G ( W ) ) = µ G ( V ) . his shows that Z \ G ( W ) M χ ( π ) ⊗ Θ χ ( π ) dµ G ( W ) ( π ) ∼ = Z \ G ( V ) M χ (Θ − χ ( σ )) ⊗ σ dµ G ( V ) ( σ ) , with M χ (Θ − χ ( σ )) = 0 . This proves the corollary. (cid:3) Exceptional Structures and Groups
The argument of the previous section can be adapted to various dual pairs in exceptional groups,thus giving rise to more exotic examples of the Sakellaridis-Venkatesh conjecture. In particular,we shall show that the spectral decomposition of L ( X ) = L ( H \ G ) can obtained from that of L ( G X ) , with X and G X given in the following table. X SO \ SL SL \ G ( J, ψ ) \ G Sp \ SL G \ Spin G \ Spin Spin \ F F \ E G X f SL SL P GL SL SL SL / ∆ µ P GL SL Table 3.
The unexplained notation will be explained in due course. Comparing with the tables in [20, §15 and§16], we see that these exceptional examples, together with the classical examples treated earlier,verify the conjecture of Sakallaridis-Venkatesh for almost all the rank spherical varieties (withcertain desirable properties), and also some rank or rank ones.Though the proof will be similar in spirit to that of the previous section, we shall need to dealwith the geometry of various exceptional groups, and this is ultimately based on the geometry ofthe (split) octonion algebra O and the exceptional Jordan algebra J ( O ) . Thus we need to recallsome basic properties of O and its automorphism group. A good reference for the material in thissection is the book [12]. One may also consult [16] and [25].4.1. Octonions and G . Let k be a local field of characteristic zero and let O denote the (8-dimensional) split octonion algebra over k . The octonion algebra O is non-commutative and non-associative. Like the quaternion algebra, it is endowed with a conjugation x ¯ x with an associatedtrace map T r ( x ) = x + ¯ x and an associated norm map N ( x ) = x · ¯ x . It is a composition algebra, inthe sense that N ( x · y ) = N ( x ) · N ( y ) .A useful model for O is the so-called Zorn’s model, which consists of × -matrices (cid:18) a vv ′ b (cid:19) , with a, b ∈ k , v ∈ V ∼ = k and v ′ ∈ V ′ ,with V a 3-dimensional k -vector space with dual V ′ . Note that there are natural isomorphisms ∧ V ∼ = V ′ and ∧ V ′ ∼ = V, and let h− , −i denote the natural pairing on V ′ × V . The multiplication on O is then defined by (cid:18) a vv ′ b (cid:19) · (cid:18) c ww ′ d (cid:19) = (cid:18) ac + h w ′ , v i aw + dv + v ′ ∧ w ′ cv ′ + bw ′ + v ∧ w bd + h v ′ , w i (cid:19) The conjugation map is (cid:18) a vv ′ b (cid:19) (cid:18) b − v − v ′ a (cid:19) o that T r (cid:18) a vv ′ b (cid:19) = a + b and N (cid:18) a vv ′ b (cid:19) = ab − h v ′ , v i . Any non-central element x ∈ O satisfies the quadratic polynomial x − T r ( x ) · x + N ( x ) = 0 .Thus, a non-central element x ∈ O generates a quadratic k -subalgebra described by this quadraticpolynomial. If this quadratic polynomial is separable, x is said to have rank . Otherwise, x is saidto have rank .The automorphism group of the algebra O is the split exceptional group of type G . The group G contains the subgroup SL ( V ) ∼ = SL which fixes the diagonal elements in Zorn’s model, andacts on V and V ′ naturally. Clearly, G fixes the identity element ∈ O , so that it acts on thesubspace O of trace zero elements. The following proposition summarizes various properties of theaction of G on O . Proposition 9. (i) Fix a ∈ k × , and let Ω a denote the subset of x ∈ O with N ( x ) = a , then Ω a is nonempty and G acts transitively on Ω a with stabilizer isomorphic to SU ( E a ) , where E a = k [ x ] / ( x − a ) .(ii) The automorphism group G acts transitively on the set Ω of trace zero, rank 1 elements. For x ∈ Ω , the stabilizer of the line k · x is a maximal parabolic subgroup Q = L · U with Levi factor L ∼ = GL and unipotent radical U a 3-step unipotent group. Now we note: • When a ∈ ( k × ) in (i), the stabilizer of an element in Ω a is isomorphic to SL ; this explainsthe 2nd entry in Table 3. • In (ii), the 3-step filtration of U is given by U ⊃ [ U, U ] ⊃ Z ( U ) ⊃ { } where [ U, U ] is the commutator subgroup and Z ( U ) is the center of U . Moreover, dim Z ( U ) = 2 and dim[ U, U ] = 3 , so that [ U, U ] /Z ( U ) ∼ = k . If ψ is a non-trivial character of k , then ψ gives rise to a nontrivialcharacter of [ U, U ] which is fixed by the subgroup [ L, L ] ∼ = SL . Setting J = [ L, L ] · [ U, U ] ,we may extend ψ to a character of J trivially across [ L, L ] . This explains the 3rd entry ofTable 3.Though the octonionic multiplication is neither commutative or associative, the trace form sat-isfies: T r (( x · y ) · z ) = T r ( x · ( y · z )) , (so there is no ambiguity in denoting this element of k by T r ( x · y · z ) ) and G is precisely thesubgroup of SO ( O , N ) satisfying T r (( gx ) · ( gy ) · ( gy )) = T r ( x · y · z ) for all x, y, z ∈ O . Exceptional Jordan algebra and F . Let J = J ( O ) denote the 27-dimensional vector spaceconsisting of all × Hermitian matrices with entries in O . Then a typical element in J has theform α = a z ¯ y ¯ z b xy ¯ x c , with a, b, c ∈ k and x, y, z ∈ O .The set J is endowed with a multiplication α ◦ β = 12 · ( αβ + βα ) here the multiplication on the RHS refers to usual matrix multiplication. With this multiplication, J is the exceptional Jordan algebra.The algebra J carries a natural cubic form d = det given by the determinant map on J , anda natural linear form tr given by the trace map. Moreover, every element in J satisfies a cubicpolynomial, by the analog of the Cayley-Hamilton theorem. An element α ∈ J is said to be of rank n if its minimal polynomial has degree n , so that ≤ n ≤ . For example, α ∈ J has rank if andonly if its entries satisfy N ( x ) = bc, N ( y ) = ca, N ( z ) = ab, xy = c ¯ z, yz = a ¯ x, zx = b ¯ y. More generally, the above discussion holds if one uses any composition k -algebra in place of O .Thus, if B = k , a quadratic algebra K , a quaternion algebra D or the octonion algebra O , one hasthe Jordan algebra J ( B ) . One may consider the group Aut( J ( B ) , det) of invertible linear maps on J ( B ) which fixes the cubic form det , and its subgroup Aut( J, det , e ) which fixes an element e with det( e ) = 0 . For the various B ’s, these groups are listed in the following table. B k K D O Aut( J ( B ) , det) SL SL ( K ) / ∆ µ SL ( D ) /µ = SL /µ E Aut( J ( B ) , det , e ) SO SL P GSp F Table 4.
Proposition 10. (i) For any a ∈ k × , the group Aut( J ( B ) , det) acts transitively on the set of e ∈ J with det( e ) = a , with stabilizer group Aut( J ( B ) , det , e ) described in the above table. If e is the unitelement of J ( B ) , then Aut( J ( B ) , det , e ) is the automorphism group of the Jordan algebra J ( B ) .(ii) The group F = Aut( J ( O )) acts transitively on the set of rank elements in J ( O ) of trace a = 0 . The stabilizer of a point is isomorphic to the group Spin of type B . In particular, the proposition explains the 1st, 4th, 7th and 8th entry of Table 3.4.3.
Triality and
Spin . An element α ∈ J = J ( O ) of rank generates a commutative separablecubic subalgebra k ( α ) ⊂ J . For any such cubic F -algebra E , one may consider the set Ω E of algebraembeddings E ֒ → J . Then one has: Proposition 11. (i) The set Ω E is non-empty and the group F acts transitively on Ω E .(ii) The stabilizer of a point in Ω E is isomorphic to the quasi-split simply-connected group Spin E of absolute type D .(iii) Fix an embedding j : E ֒ → J and let E ⊥ denote the orthogonal complement of the imageof E with respect to the symmetric bilinear form ( α, β ) = tr ( α ◦ β ) . The action of the stabilizer Spin E of j on E ⊥ is the 24-dimensional Spin representation, which on extending scalars to k , is thedirect sum of the three 8-dimensional irreducible representations of Spin ( k ) whose highest weightscorrespond to the 3 satellite vertices in the Dynkin diagram of type D . As an example, suppose that E = k × k × k , and we fix the natural embedding E ֒ → J whoseimage is the subspace of diagonal elements in J . Then E ⊥ is naturally O ⊕ O ⊕ O , and the splitgroup Spin acts on this, preserving each copy of O . This gives an injective homomorphism ρ : Spin −→ SO ( O , N ) × SO ( O , N ) × SO ( O , N ) whose image is given by Spin ∼ = { g = ( g , g , g ) : T r (( g x ) · ( g y ) · ( g z )) = T r ( x · y · z ) for all x, y, z ∈ O } . rom this description, one sees that there is an action of Z / Z on Spin given by the cyclic permu-tation of the components of g , and the subgroup fixed by this action is precisely G = Spin Z / Z . This explains the 6th entry of Table 3.More generally , the stabilizer of a triple ( x, y, z ) ∈ O with ( x · y ) · z ∈ k × is a subgroup of Spin isomorphic to G (see [25]). For example, the stabilizer in Spin of the vector (1 , , ∈ O is isomorphic to the group Spin which acts naturally on O ⊕ O ⊕ O . The action of Spin on O is via the standard representation of SO , whereas its action on the other two copies of O is via theSpin representation. From the discussion above, we see that the stabilizer in Spin of ( x, ¯ x ) ∈ O ,with N ( x ) = 0 , is isomorphic to the group G . In particular, this explains the 5th entry of Table 3.By the above discussion, it is not difficult to show: Proposition 12.
The group
Spin acts transitively on the set of rank elements in J ( O ) withdiagonal part ( a, b, c ) ∈ k × × k × × k × . Moreover, the stabilizer of a point is isomorphic to G . SL \ G and G \ Spin . From the discussion above, we see that there are isomorphisms ofhomogeneous varieties SL \ G ∼ = SO \ SO and G \ Spin ∼ = Spin \ Spin ∼ = SO \ SO . Since we have already determined the spectral decomposition of L ( SO \ SO ) and L ( SO \ SO ) in terms of the spectral decomposition of L ( ˜ SL ) and L ( SL ) respectively, we obtain the desireddescription for SL \ G and G \ Spin . Thus the rest of the paper is devoted to the remaining casesin Table 3. 5. Exceptional Dual Pairs
In this section, we introduce some exceptional dual pairs contained in the adjoint groups of type F , E , E and E . We begin with a uniform construction of the exceptional Lie algebras of thevarious exceptional groups introduced above. This construction can be found in [19] and will beuseful for exhibiting various reductive dual pairs. The reader may consult [16], [19], [21] and [25]for the material of this section.5.1. Exceptional Lie algebras.
Consider the chain of Jordan algebras k ⊂ E ⊂ J ( k ) ⊂ J ( K ) ⊂ J ( D ) ⊂ J ( O ) where E is a cubic k -algebra, K a quadratic k -algebra and D a quaternion k -algebra. Denoting suchan algebra by R , the determinant map det of J ( O ) restricts to give a cubic form on R . Now set(21) s R = sl ⊕ m R ⊕ ( k ⊗ R ) ⊕ ( k ⊗ R ) ′ , with m R = Lie( Aut ( R , det)) . One can define a Lie algebra structure on s R [19] whose type is given by the following table. R k E J ( k ) J ( K ) J ( D ) J ( O ) m R E sl sl ( K ) sl e s R g d f e e e We denote the corresponding adjoint group with Lie algebra s R by S R , or simply by S if R isfixed and understood. et { e , e , e } be the standard basis of k with dual basis { e ′ i } . The subalgebra of sl stabilizingthe lines k e i is the diagonal torus t . The nonzero wieghts under the adjoint action of t on s R forma root system of type G . The long root spaces are of dimension and are precisely the root spacesof sl , i.e. the spaces spanned by e ′ i ⊗ e j . We shall label these long roots by β , β and β − β , withcorresponding 1-parameter subgroups u β ( x ) = x
01 01 , u β ( x ) = x , u β − β ( x ) = x We also let w β = − denote the Weyl group element associated to β . The short root spaces, on the other hand, are e i ⊗ R and e ′ i ⊗ R ′ and are thus identifiable with R .5.2. Exceptional dual pairs.
We can now exhibit 2 families of dual pairs in S R . • From (21), one has sl ⊕ m R ⊂ s R . This gives a family of dual pairs(22) SL × Aut ( R , det) −→ S R . • For a pair of Jordan algebras R ⊂ R , we have s R ⊂ s R which gives a subgroup G R ⊂ S R ,where G R is isogeneous to S R . If G ′R , R = Aut ( R , R ) , then one has a second family ofdual pairs(23) G R × G ′R , R −→ S R . With R ⊂ R fixed, we shall simply write G × G ′ for this dual pair. For the various pairs R ⊂ R of interest here, we tabulate the associated dual pairs in the table below. R ⊂ R k ⊂ J ( K ) E ⊂ J ( D ) J ( k ) ⊂ J ( D ) G × G ′ G × P GL Spin × SL ( E ) / ∆ µ F × P GL Observe that in the language of Table 3, with X = H \ G , the dual pairs described above are precisely G X × G .5.3. Heisenberg parabolic.
The presentation (21) also allows one to describe certain parabolicsubalgebras of s R . If we consider the adjoint action of t = diag(1 , , − ∈ sl on s , we obtain a grading s = ⊕ i s [ i ] by the eigenvalues of t . Then s [0] = t ⊕ m ⊕ ( e ⊗ R ) ⊕ ( e ′ ⊗ R ′ ) s [1] = k e ′ ⊗ e ⊕ ( e ⊗ R ) ⊕ ( e ′ ⊗ R ′ ) ⊕ k e ′ ⊗ e s [2] = k e ′ ⊗ e , and p = ⊕ i ≥ s [ i ] is a Heisenberg parabolic subalgebra.We denote the corresponding Heisenberg parabolic subgroup by P S = M S · N S . In particular, itsunipotent radical is a Heisenberg group with 1-dimensional center Z S ∼ = u β ( k ) ∼ = s [2] and N S /Z S ∼ = s [1] = k ⊕ R ⊕ R ′ ⊕ k , The semisimple type of its Levi factor M S is given in the table below. F E E E M S C A D E The Lie bracket defines an alternating form on N S /Z S which is fixed by P S = [ P S , P S ] . This givesan embedding P S = M S · N s ֒ → Sp ( N S /Z S ) ⋉ N S . Intersection with dual pairs.
For a pair R ⊂ R , with associated dual pair given in (23),it follows by construction that ( G R × G ′R , R ) ∩ P S = P × G ′R , R , where P is the Heisenberg parabolic subgroup of G R . On the other hand, for the family of dualpairs given in (22), ( SL × Aut ( R , det)) ∩ P S = B × Aut ( R , det) where B is a Borel subgroup of SL .5.5. Siegel parabolic.
The group S of type E or E has a Siegel parabolic subgroup Q S = L S · U S whose unipotent radical U S is abelian; we call this a Siegel parabolic subgroup. The semisimpletype of L S and the structure of U S as an L S -module is summarized in the following table. S L S U S U S as L S -module E D O ⊕ O half spin representation of dimension E E J ( O ) miniscule representation of dimension Let Ω Q ⊂ U S be the orbit of a highest weight vector in U S . The following proposition describesthe set Ω Q : Proposition 13. (i) If S is of type E , then Ω Q = { ( x, y ) ∈ O : N ( x ) = N ( y ) = 0 = x · ¯ y } . (ii) If S is of type E , then Ω Q = { α ∈ J : rank ( α ) = 1 } . Intersection with dual pairs.
With R ⊂ R fixed, with associated dual pair G × G ′ as givenin (23), one may choose Q S so that ( G × G ′ ) ∩ Q S = G × Q with Q = L · U a Siegel parabolic subgroup of G ′ , so that U is abelian. The group Q and theembedding U ⊂ U S can be described by the following table. G ′ P GL SL ( E ) / ∆ µ P GL Q maximal parabolic Borel Borel U ⊂ U S k ⊂ O E ⊂ J ( O ) k ⊂ J ( O ) Identifying the opposite unipotent radical ¯ U with the dual space of U using the Killing form, onehas a natural projection τ : ¯ U S −→ U . This is simply given by the projection from U S to U along U ⊥ . . Generic Orbits
In this section, we consider an orbit problem which will be important for our applications. Namely,with the notation at the end of the last section, we have an action of L × G on the set Ω Q ⊂ U S .We would like to determine the generic orbits of this action. For simplicity, we shall consider thecase when S = E and E separately.6.1. Dual Pair in E . Suppose first that S = E so that G ′ × G = P GL × G . In this case, thenatural L × G -equivariant projection τ : ¯ U S −→ ¯ U is given by τ ( x, y ) = ( T r ( x ) , T r ( y )) . The nonzero elements in ¯ U ∼ = k are in one orbit of L ; we fix a representative (0 , ∈ k and notethat its stabilizer in L is the “mirabolic" subgroup P L of L ∼ = GL . Then the fiber over (0 , isgiven by { ( x, y ) ∈ O : N ( x ) = N ( y ) = T r ( x ) = 0 , T r ( y ) = 1 , x · ¯ y = 0 } , and carries a natural action of P L × G . We note: Lemma 14. (i) The group G acts transitively on the fiber τ − (0 , and the stabilizer of a point ( x , y ) is isomorphic to the subgroup [ L, L ] · Z ( U ) ⊂ J .(ii) If we consider the subset { ( x , y + λx ) : λ ∈ k } ⊂ τ − (0 , , then the subgroup of P L × G stabilizing this subset is isomorphic to ( P L × L · [ U, U ]) = { ( h, g · u ) : det h = det g } . The action of the element (cid:18) a b (cid:19) × g · u ∈ ( P L × L · [ U, U ]) is by ( x , y + λx ) ( x , y + a − · ( λ + b − p ( u )) x ) where p : J −→ k ∼ = J/ [ L, L ] · Z ( U ) is the natural projection. Thus, there is a unique generic L × G orbit on Ω Q given by ( L × G ) × ( P L × L · [ U,U ]) k . Dual Pairs in E . Now suppose that S = E . As above, we first determine the generic L -orbits on ¯ U . For each generic L -orbit in ¯ U , let us take a representative χ and let Z χ denoteits stabilizer in L . Then the fiber τ − ( χ ) is preserved by Z χ × G . In each case, it follows by Prop.10(ii) and Prop. 12 that G acts transitively on τ − ( χ ) . Denote the stabilizer in G of ˜ χ ∈ τ − ( χ ) by H χ . Then under the action of Z χ × G , the stabilizer group ˜ H χ of ˜ χ sits in a short exact sequence −−−−→ H χ −−−−→ ˜ H χ −−−−→ p Z χ −−−−→ . In fact, ˜ H χ is a direct product ˜ H χ ∼ = ∆ Z χ × H χ ⊂ Z χ × G Thus, the generic L × G -orbits are given by the disjoint union [ generic χ ( Z χ × G ) × ˜ H χ ˜ χ where the union runs over the generic L -orbits on ¯ U and ˜ χ is an element in τ − ( χ ) with stabilizer ˜ H χ . We summarize this discussion in the following table. × G ′ F × P GL Spin × SL ( E ) / ∆ µ generic L -orbits singleton ( a, b, c ) ∈ ( k × / k × ) / ∆ k × τ − ( χ ) α ∈ J ( O ) of rank and trace x, y, z ) ∈ O with T r ( xyz ) = abcZ χ trivial center of G ′ = µ × µ H χ Spin G Minimal Representation
In this section, we introduce the (unitary) minimal representation Π of S and describe somemodels for Π . Note that when S = F , Π is actually a representation of the double cover of F .When S is of type E , then Π is a representation of S .7.1. Schrodinger model.
Because the groups S = E and E have a Siegel parabolic subgroup,there is an analog of the Schrodinger model for the minimal representation Π of S . By [6], therepresentation Π can be realized on the space L (Ω Q , µ Q ) of square-integrable functions on Q withrespect to a L S -equivariant measure µ Q on Ω Q . This is analogous to the Schrodinger model of theWeil representation. In particular, we have the following action of Q S on Π : ( ( l · f )( χ ) = δ Q S ( l ) r · f ( l − · χ )( u · f )( χ ) = χ ( u ) · f ( χ ) , where r = 1 / (resp. / ) if S is of type E (resp. E ).7.2. Mixed model.
For general S = S R , one has the analog of the mixed model, on which theaction of the Heisenberg group P S is quite transparent. Recall that N S /Z S = k ⊕ R ⊕ R ′ ⊕ k andone has an embedding P S = [ P S , P S ] ֒ → Sp( N S /Z S ) ⋉ N S . Then by [13], the mixed model of the minimal representation is realized on the Hilbert space
Ind P S P S L ( R ′ ⊕ k ′ ) ∼ = L ( k × ⊕ R ⊕ k ) , where the action of P S on L ( R ⊕ k ) is via the Heisenberg-Weil representation (associated to anyfixed additive character ψ of k ). The explicit formula can be found in [19, Prop. 43].In fact, one can describe the full action of S on Π by giving the action of an extra Weyl groupelement. More precisely, if w β is the standard Weyl group element in SL associated to the root β (see §5.1), then by [19, Prop. 47], one has ( w β · f )( t, x, a ) = ψ (det( x ) /a ) · f ( − a/t, x, − a ) . Since S is generated by P S and the element w , this completely determines the representation Π .For example, one may work out the action of an element u − β ( b ) = w β u β ( b ) w − β (see §5.1). Ashort computation gives: ( u − β ( b ) · f )( t, x, a ) = ψ (cid:18) b det( x ) a − t (cid:19) · f ( t − abt , a − a bt , x ) . If f is continuous, then the above formula gives:(24) ( u − β ( b ) · f )(1 , x,
0) = ψ ( − b det( x )) · f (1 , x, . This formula will be useful in the last section. . Exceptional Theta Correspondences: G × G ′ Now we may study the restriction of the minimal representation Π to the dual pairs introducedearlier. In this section, we shall treat the family of dual pairs G × G ′ given in (23). For simplicity,we shall consider the case when S = E and E separately.8.1. Restriction to G × G ′ ⊂ E . Suppose first that S is of type E , so that Ω Q is the set of rank elements in J = J ( O ) . Consider the Schrodinger model for Π . On restricting Π to Q × G , wehave the following formulae: ( g · f )( α ) = f ( g − · α ) for g ∈ G ; ( u ( a ) · f )( α ) = ψ ( tr ( a · α )) · f ( α ) for u ( a ) ∈ U ; ( l · f )( α ) = | det( l ) | s · f ( l − · α ) for l ∈ L ,where s is a real number whose precise value will not be important to us here.From our description of generic L × G -orbits given in §6.2, we deduce as in the derivation of(12) that as a Q × G -module,(25) Π ∼ = M χ generic Ind Q × GU × ˜ H χ χ ⊠ ∼ = M χ generic Ind Q Z χ · U L ( H χ \ G ) . Here, G and Z χ act on L ( H χ \ G ) by right and left translation respectively, and U acts by χ .8.2. Abstract decomposition.
On the other hand, there is an abstract direct integral decompo-sition
Π = Z c G ′ π ⊠ Θ( π ) dν Θ ( π ) . Restricting to Q , we may write: π | Q ∼ = M χ Ind Q Z χ · U W χ ( π ) for some Z χ · U -module W χ ( π ) with U acting via χ . Thus,(26) Π ∼ = M χ Z c G ′ Ind Q Z χ · U W χ ( π ) ⊠ Θ( π ) dν Θ ( π ) . Comparison.
Comparing (25) and (26), we deduce that there is an isomorphism of G A -modules:(27) L ( H χ \ G ) ∼ = Z c G ′ W χ ( π ) ⊠ Θ( π ) dν Θ ( π ) . Since G ′ is isogenous to a product of SL , the space W χ ( π ) = W h χ ( π ) has been determined inTheorem 5(3) and is at most 1-dimensional.8.4. Mixed model.
To explicate the measure dν Θ ( π ) , we consider the mixed model of Π restrictedto P × G ′ . Since N/Z S = k ⊕ R ⊕ R ′ ⊕ k ⊂ N S /Z S . Under its adjoint action on
R ⊕ k , G ′ fixes R ⊕ k pointwise, and its action on R ⊥ is described inthe following table. G ′ R R ⊥ P GL J ( k ) adjoint ⊕ SL / ∆ µ k ⊕ i =1 std i ⊠ std ∨ i +1 hus as a representation of G ′ , we have: Π ∼ = L ( k × ) ⊗ L ( R ⊕ k ) ⊗ L ( R ⊥ ) where G ′ acts only on L ( R ⊥ ) and the action is geometric. Thus, Π is weakly equivalent to L ( R ⊥ ) as a representation of G ′ . By our description of the G ′ -module R ⊥ , we have: Lemma 15.
The representation L ( R ⊥ ) (and hence Π ) is weakly equivalent to the regular repre-sentation L ( G ′ ) Proof.
When G ′ = P GL , this follows from Corollary 8. When G ′ = SL / ∆ µ , the representationof G ′ on E ⊥ A is the restriction of a representation of ˜ G ′ = GL / ∆ k × (by the same formula). Nowthe action of ˜ G ′ on R ⊥ has finitely many open orbits with representatives (1 , , g ) ∈ GL with g regular semisimple, and the stabilizer of such a representative is ∆ T with T a maximal torus in P GL . Hence, as a representation of ˜ G ′ , L ( R ⊥ ) is weakly equivalent to M T Ind ˜ G ′ ∆ T C ∼ = M T Ind ˜ G ′ ∆ P GL L ( T \ P GL ) as T runs over conjugacy classes of maximal tori in P GL . By Corollary 8 and the continuityof induction, we deduce that L ( R ⊥ ) is weakly equivalent to L ( ˜ G ′ ) . Thus, on restriction to G ′ , L ( E ⊥ ) is weakly equivalent to L ( G ′ ) , as desired. (cid:3) Concluding, we have:
Theorem 16.
There is an isomorphism of G -modules: L ( H χ \ G ) ∼ = Z c G ′ W χ ( π ) ⊠ Θ( π ) dµ G ′ ( π ) , with W χ ( π ) = W h χ ( π ) as given in Theorem 5(3) and µ G ′ is the Plancherel measure. In addition, as we discussed in §3.7, the smoooth analog of our argument in this section impliesthat W χ ( π ) = W h χ ( π ) ∼ = Hom H χ (Θ ∞ ( π ∞ ) , C ) = Hom H χ (Θ( π ) ∞ , C ) . Restriction to
P GL × G . We now treat the dual pair
P GL × G in S = E , which canbe done by a similar analysis. In this case, Ω Q ⊂ O . If we restrict the action of S to Q × G , wededuce by Lemma 14(ii) that as a representation of Q × G , Π ∼ = Ind Q × G ( P L × L · [ U,U ]) · U L ( k ) where the action of ( P L × L · [ U, U ]) on L ( k ) is given through the geometric action described inLemma 14(ii) and the action of U is by a nontrivial character fixed by P L .By using the Fourier transform on L ( k ) , we deduce that as a representation of ( P L × L · [ U, U ]) , L ( k ) ∼ = Ind ( P L × L · [ U,U ]) U L × J ψ − ⊠ ψ. Hence, as a representation of Q × G (28) Π ∼ = Ind Q N χ ⊠ Ind G J ψ where N = U L · U is the unipotent radical of a Borel subgroup of P GL and χ is a genericcharacter of N .On the other hand, we have abstractly(29) Π ∼ = Z \ P GL π | Q ⊗ Θ( π ) dν Θ ( π ) . e note that if π is tempered, then π | Q ∼ = Ind Q N χ, in which case we deduce on comparing (28) and (29) that(30) L (( J, ψ ) \ G ) = Ind G J ψ ∼ = Z \ P GL Θ( π ) dν Θ ( π ) . For (30) to hold, we thus need to show that ν Θ is absolutely continuous with respect to the Plancherelmeasure of P GL .For this, we examine the mixed model of Π which is realized on L ( k × × J ( k ) × k ) . Notingthat J ( k ) ∼ = gl as P GL -module [16], we deduce that as a representation of P GL , Π is weaklyequivalent to the representation on L ( sl ) associated to the adjoint action on sl . As in Corollary8, we know that L ( sl ) is weakly equivalent to L T L ( T \ P GL ) , with T running over conjugacyclasses of maximal tori in P GL .Using the same argument as in [20, §6], one can show that for each T , the spectral measure for L ( T \ P GL ) is absolutely continuous with respect to the Plancherel measure of P GL , and henceso is the spectral measure of L ( sl ) ; this justifies (30) and shows that L (( J, ψ ) \ G ) = Ind G J ψ ∼ = Z \ P GL W ( π ) ⊗ Θ( π ) dµ P GL ( π ) for some multiplicity space W ( π ) of dimension ≤ .It is natural to state: Conjecture 17.
For an adjoint simple algebraic group G , the representation L ( g ) of G is weaklyequivalent to the regular representation L ( G ) . Corollary 8 verifies this conjecture for
P GL . If the conjecture holds for P GL , one can thentake W ( π ) to be C for all π .9. Exceptional Theta Correspondence: SL × Aut ( R , det) Finally we come to the family of dual pairs SL × Aut ( R , det) ⊂ S = S R given by (22). What isinteresting about this situation is that the group S may have no Siegel parabolic subgroup, so thatthe argument below is not the analog of that in the classical cases of §3. To simplify notation, weshall set G = Aut ( R , det) . Note that in the case of F , S is the double cover of F and the dualpair is ˜ SL × G = ˜ SL × SL .Let Q = L · U ⊂ SL be the maximal parabolic subgroup stabilizing the subspace k e + k e ,so that L ∼ = GL and U = u β − β ( k ) × u β ( k ) . Let χ be a generic character of U trivial on u β − β ( k ) . The stabilizer in L of χ is a subgroup ofthe form T ⋉ U L with T ∼ = k × contained in the diagonal torus and U L = u − β ( k ) . On restrictingthe minimal representation Π to Q × G , we may write Π ∼ = Ind Q × GP L U × G Π χ for some representation Π χ of P L U × G with U acting by χ . Here, we have used the theorem ofHowe-Moore which ensures that the trivial character of U does not intervene.Now we can describe the P L U × G -module Π χ using the mixed model of Π . Recall that thismixed model of Π is realized on L ( k × × R × k ) . Moreover, the action of U = u β − β ( k ) × u β ( k ) in his model is: ( ( u β ( z ) f )( t, x, a ) = ψ ( tz ) · f ( t, x, a )( u β − β ( y ) f )( t, x, a ) = ψ ( ay ) · f ( t, x, a ) . As such, Π χ is the representation obtained from Π by specializing (continuous) functions f ∈ Π tothe function x f (1 , x, of R . Thus Π χ = L ( R ) where the action of T × G is geometric, with T acting by scaling on R . Moreover, it follows by(24) that the action of u − β ( b ) ∈ U L is: ( u − β ( b ) · f )( x ) = ψ ( − b · det( x )) · f ( x ) . Now the set { x ∈ R : det( x ) = 0 } is open dense and by Proposition 10(i), it is the union offinitely many generic orbits of T × G indexed by k × / ( k × ) . For each a ∈ k × / ( k × ) , let H a be thecorresponding stabilizer group whose type is described in Table 4 in §4.2. Then Π ∼ = M a Ind Q × GN × H a χ a ⊠ C ∼ = Ind Q N χ a ⊠ L ( H a \ G ) . On the other hand, one has abstractly Π ∼ = Z d SL π | Q ⊗ Θ( π ) dν θ ( π ) . Now we note:
Lemma 18.
As a representation of SL , Π is weakly equivalent to L ( SL ) .Proof. If S is of type E , the group SL is contained in a conjugate of the Heisenberg parabolicsubgroup P S . Indeed, after an appropriate conjugation, we may assume that SL ⊂ Aut ( J ( k ) , det) = SL × µ SL ⊂ Aut( J ( B ) , det) , where B = k , M ( k ) or the split octonion algebra O in the respective case. From the description ofthe mixed model, one sees that Π is nearly equivalent to the representation of SL on L ( J ( B )) = L ( J ( k )) ⊗ L ( J ( k ) ⊥ ) . Since J ( k ) ∼ = M ( k ) with SL acting by left multiplication, we see that J ( k ) is weakly equivalent to the regular representation of SL . This implies that Π is weaklyequivalent to the regular representation of SL .The case when S = F is a bit more intricate; we omit the details here. (cid:3) Thus ν θ = µ SL and every π in the support of ν θ is tempered, so that π | Q = M a ∈ k × / ( k × ) W h χ a ( π ) ⊗ Ind Q N χ a . Comparing, we see that L ( H a \ G ) ∼ = Z d SL W h χ a ( π ) ⊗ Θ( π ) dµ SL ( π ) , as desired. eferences [1] van den Ban E. and Schlichtkrull H. The Plancherel theorem for a reductive symmetric space I. Spherical func-tions, and II. Representation theory.
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Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent RidgeRoad, Singapore 587628
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