Howe duality and dichotomy for exceptional theta correspondences
aa r X i v : . [ m a t h . R T ] J a n HOWE DUALITY AND DICHOTOMYFOR EXCEPTIONAL THETA CORRESPONDENCES
WEE TECK GAN AND GORDAN SAVIN
Abstract.
We study three exceptional theta correspondences for p -adic groups, where onemember of the dual pair is the exceptional group G . We prove the Howe duality conjecturefor these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of allnon-cuspidal representations. Introduction
Let F be a non-archimedean local field of characteristic 0 and residue characteristic p . Inthis paper, we study the local theta correspondence furnished by the following diagram ofdual pairs: PGSp G ❣❣❣❣❣❣❣❣❣❦❦❦❦❦❦ ❱❱❱❱❱❱ PD × PGL ⋊ Z / Z where D denotes a cubic division F -algebra, so that P D × is the unique inner form of PGL .More precisely, one has the dual pairs (PGL ⋊ Z / Z ) × G ⊂ E ⋊ Z / Z P D × × G ⊂ E D G × PGSp ⊂ E where the exceptional groups of type E are all of adjoint type. In each of the three cases, thecentralizer of G is a group H J = Aut( J ), where J is a Freudenthal-Jordan algebra of rank3. One can thus consider the restriction of the minimal representation Π [GS05] of E to therelevant dual pair and obtain a local theta correspondence.More precisely, if π ∈ Irr( G ) is an irreducible smooth representation of G , then themaximal π -isotypic quotient of Π Π / ∩ φ ∈ Hom G (Π ,π ) ker( φ ) . can be expressed as π ⊗ Θ( π ) for some smooth representation Θ( π ) of H J [MVW, LemmeIII.4]. The representation Θ( π ) is called the big theta lift of π , and its maximal semi-simplequotient (cosocle) is denoted θ ( π ). We say that π has nonzero theta lift to H J if Θ( π ) = 0,or equivalently Hom G (Π , π ) = 0. Similarly, one can consider the theta lift from H J to G and have the analogous notions. Mathematics Subject Classification.
The first main result of this paper is the following dichotomy theorem:
Theorem 1.1.
Let π ∈ Irr( G ( F )) . Then π has nonzero theta lift to exactly one of P D × or PGSp ( F ) . The group PGL ⋊ Z / Z is not featured in the dichotomy theorem, but it is needed forsome finer aspects of the theta correspondences. For example, every irreducible discrete seriesrepresentation of G lifts to a discrete series representation of precisely one of the three groups.After the above dichotomy theorem, we consider the problem of understanding these thetacorrespondences more precisely. These local theta correspondences have all been studied tosome extent by Maggard-Savin [MS], Gross-Savin [GrS], Gan [G99], Savin [Sa], Gan-Savin[GS99, GS04] and Savin-Weissman [SWe]. Though various neat results were obtained in thevarious cases, they fall short of determining the theta correspondences completely. One ofthe main results of this paper is the completion of the analysis begun in these papers.The main difficulty in studying these exceptional theta correspondences is that, unlike theclassical theta correspondence, one does not know a priori the analog of the Howe dualityconjecture. Namely, one does not know that Θ( π ) has finite length with unique irreduciblequotient (that is, θ ( π ) is irreducible if Θ( π ) is nonzero). In this paper, we show that theanalog of the Howe duality conjecture holds for these dual pairs. To summarize, we have: Theorem 1.2.
The Howe duality conjecture holds for the three dual pairs considered here.Namely, for π , π ∈ Irr( G ( F )) , Θ( π i ) has finite length and dim Hom H J ( θ ( π ) , θ ( π )) ≤ dim Hom G ( π , π ) . Likewise, for τ ∈ Irr( H J ) , Θ( τ ) has finite length with unique irreducible quotient (if nonzero).More precisely, we have:(i) The theta correspondence for P D × × G defines an injective map θ D : Irr ♥ ( P D × ) ֒ → Irr( G ( F )) , where Irr ♥ ( P D × ) ⊂ Irr(
P D × ) is the subset of representations which have nonzero theta liftto G . If p = 3 , then Irr ♥ ( P D × ) = Irr( P D × ) , so that one has an injective map: θ D : Irr( P D × ) ֒ → Irr( G ( F )) (ii) The theta correspondence for (PGL ( F ) ⋊ Z / Z ) × G defines an injective map θ B : Irr ♥ (PGL ( F ) ⋊ Z / Z ) ֒ → Irr( G ( F )) , where Irr ♥ (PGL ( F ) ⋊ Z / Z ) ⊂ Irr(PGL ( F ) ⋊ Z / Z ) is the subset of representations whichhave nonzero theta lift to G . Moreover, one can determine the subset Irr ♥ (PGL ( F ) ⋊Z / Z ) explicitly, and the image of θ B is disjoint from that of θ D by the dichotomy theorem.(iii) The theta correspondence for G × PGSp defines an injection θ : Irr( G ( F )) r Im( θ D ) ֒ → Irr(PGSp ( F )) . For an irreducible representation τ of P D × , the non-vanishing of θ D ( τ ) is equivalent toexistence of non-zero vectors in τ fixed by a maximal torus in P D × . The existence of such OWE DUALITY AND DICHOTOMY 3 vectors has been checked by Lonka and Tandon [LT] in the tame case, where p = 3. Thus, if p = 3, we do not know that all irreducible representations of P D × lifts to G , but the lift isstill one-to-one on the subset of those representations that have nonzero lift.In fact, for the three dual pairs, we determine the theta lift of all non-supercuspidal repre-sentations of G , and the lift of supercuspidal representations whose lift is not supercuspidal.The detailed statements are in the main text, and we simply state the following qualitativeresult here: Theorem 1.3.
The three theta correspondences satisfy the following properties:(i) The correspondences preserve tempered representations.(ii) Any discrete series representation of G lifts to a discrete series representation of preciselyone of the three groups.(iii) The correspondences are functorial for non-tempered representations. As an illustration of our results, let π gen be a Whittaker generic (henceforth simply generic)supercuspidal representation of G . Then we have the following description of Θ( π gen ), thebig theta lift to PGSp , completing the results of [SWe]:(a) If π gen is not a theta lift from PGL ( F ) ⋊ Z / Z , then Θ( π gen ) is an irreducible genericsupercuspidal representation of PGSp .(b) Otherwise, π gen is a lift of a supercuspidal representation of PGL ( F ) ⋊ Z / Z . Now theirreducible representations of PGL ( F ) ⋊ Z / Z can be described in terms of those of PGL as follows: • Ind
PGL ⋊Z / Z PGL ( τ ) ∼ = Ind PGL ⋊Z / Z PGL ( τ ∨ ) where τ ≇ τ ∨ ∈ Irr(PGL ); • τ + ≇ τ − , two extensions of a self dual ( τ ∼ = τ ∨ ) representation of PGL .Accordingly, we have the two following two cases:(b ) If θ B (Ind PGL ⋊Z / Z PGL ( τ )) = π gen , thenΘ( π gen ) ∼ = Ind PGSp P ( τ ) ∼ = Ind PGSp P ( τ ∨ )where P is a maximal parabolic of PGSp with Levi factor GL .(b ) If θ B ( τ + ) = π gen , then θ B ( τ − ) = π deg is an irreducible degenerate (i.e. nongeneric) su-percuspidal representation of G . Since τ is self-dual, the induced representation Ind PGSp P ( τ )is reducible. It is a direct sumInd PGSp P ( τ ) = Ind PGSp P ( τ ) gen ⊕ Ind
PGSp P ( τ ) deg of two summands, with the first summand generic and the second degenerate. We have:Θ( π gen ) ∼ = Ind PGSp P ( τ ) gen and Θ( π deg ) ∼ = Ind PGSp P ( τ ) deg . To conclude the introduction, we would like to explain the general idea and strategy forproving the Howe duality theorem. We begin with a discussion of the statement:
WEE TECK GAN AND GORDAN SAVIN (a) Θ( π ) has finite length.This finiteness result is fundamental and it was shown by Kudla [K] for the classical thetacorrespondence. The main tools used are his computation of the Jacquet modules of theWeil representation (relative to maximal parabolic subgroups of the two members of the dualpair) and his exploitation of the doubling see-saw identity. One key consequence of the finitelength of Θ( π ) is(b) If Θ( π ) = 0, then it has an irreducible quotient.For the dual pairs considered in this paper, we will in fact first prove statement (b) and thenuse it with other inputs to show (a).Let us elaborate on this slightly subtle point and our strategy of proof. By Bernstein’sdecomposition, we may decomposeΘ( π ) = Θ( π ) c ⊕ Θ( π ) nc as the sum of its cuspidal part and non-cuspidal part. If Θ( π ) c is nonzero, then it certainlyhas an irreducible quotient, since it is semisimple. On the other hand, we shall show usingJacquet module computations that(c) Θ( π ) nc has finite length and hence has an irreducible quotient if it is nonzero.The necessary Jacquet module computations are already available in the literature [MS, Sa]when H J = P D × or PGL and are partially available [MS, GrS] for H J = PGSp . In §
13, wecomplete the remaining Jacquet module computations. We stress that the material in §
13 isindependent of the rest of the paper and could have been discussed earlier in the paper; wehave refrained from doing so, as the computations are rather technical. Consequences of theresults of §
13 are then discussed in § § §
6. For nontempered representations, the Jacquet modulecomputations (of [MS, GrS] and § π )), it remains to show that Θ( π ) c is of finitelength. We shall show this together with the Howe duality conjecture, by showing that Θ( π ) c is either irreducible or 0. This part of the argument may be considered the analog of thedoubling see-saw argument, though one would legitimately question what that means in thesetting of exceptional dual pairs.It will be instructive to first recall the argument for a classical dual pair Sp( W ) × O( V ),where W is a symplectic space and V a quadratic space. The Howe duality theorem wasshown by examining the so-called doubling see-saw diagram:O( V (cid:3) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ Sp( W ) × Sp( W )O( V ) × O( V ) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Sp( W ) ∆OWE DUALITY AND DICHOTOMY 5 where V (cid:3) = V + V − is the doubled quadratic space. Starting from π, π ′ ∈ Irr(O( V )), theresulting see-saw identity givesdim Hom Sp( W ) ( θ ( π ′ ) , θ ( π )) ≤ dim Hom O( V ) × O( V ) (Θ(1) , π ′ ⊗ π ∨ ) . where Θ(1) is the big theta lift of the trivial representation of Sp( W ) ∆ to O( V (cid:3) ). By thelocal analog of the Siegel-Weil formula, one identifies Θ(1) with a submodule of a certaindegenerate principal series representation I on O( V (cid:3) ). This implies that, for π outside asmall family of representations,dim Hom O( V ) × O( V ) (Θ(1) , π ′ ⊗ π ∨ ) ⊂ dim Hom O( V ) × O( V ) ( I, π ′ ⊗ π ∨ ) . Using Mackey theory, one can analyze the latter space and show that, for π outside anothersmall family of representations,dim Hom O( V ) × O( V ) ( I, π ′ ⊗ π ∨ ) ≤ dim Hom O( V ) ( π ′ , π ) . Taken together, one obtains the desired inequalitydim Hom
Sp( W ) ( θ ( π ′ ) , θ ( π )) ≤ dim Hom O( V ) ( π ′ , π )for π outside a small family of representations. For this small family of representations, oneneeds to do a separate argument.Now for the exceptional dual pairs G × H studied in this paper, there is no analog of thedoubling see-saw; this is ultimately tied to the sporadic nature of the geometry underlyingexceptional groups. There is thus no direct analog of the above argument. However, theabove argument is a particular manifestation of a general principle: Theta correspondence typically relates or transfers a period on G to a period on H .More precisely, given a subgroup H ⊂ H and τ ∈ Irr( H ), we may consider the spaceHom H ( τ, χ ) for some one-dimensional character χ of H , typically trivial; let us call thisHom space the H -period for τ . Now, given- π ∈ Irr( G ), with big theta lift Θ( π ) on H ,- τ ∈ Irr( H ) an irreducible quotient of Θ( π ),one typically obtains a statement of the form H -period of τ ⊂ H -period of Θ( π ) ∼ = G -period of π for some subgroup G of G .Now one can turn the table around. For an irreducible quotient τ of Θ( π ), one can considerthe G -period of Θ( τ ), which has π as an irreducible quotient. One typically gets a statement G -period of π ⊂ G -period of Θ( τ ) ∼ = H -period of τ for some subgroup H on H .Iterating this process, one obtains a family of periods relative to subgroups G i on G and H i on H such that H i -period of τ ⊂ H i -period of Θ( π ) ∼ = G i -period of π and G i -period of π ⊂ G i -period of Θ( τ ) ∼ = H i +1 -period of τ , WEE TECK GAN AND GORDAN SAVIN leading to a chain of containment of periods of π and τ . One may call this a game of ping-pongwith periods. Now an (empirical) observation is that the subgroups G i and H i become moreand more reductive (as i increases) and one ultimately obtains a reductive period. When thathappens, the next iteration will result in a seesaw diagram analogous to that in the classicalcase above and the consideration of an appropriate degenerate principal series representation.Now the miracle is that a Mackey theory argument with this degenerate principal seriesrepresentation then returns us the initial H -period! In other words, for some i >
1, one has H i = H , and this allows one to complete the chain of containment of periods into a cycle. Inparticular, if one of these period spaces is finite-dimensional, then this cycle of containmentis a cycle of equalities. This is the key step in our proof of the Howe duality theorem for thedual pairs treated here. We shall play this game of ping-pong with periods on two occasions,in § §
12. This seems to us to be a rather robust method for proving the Howe dualityconjecture and should be applicable to other exceptional dual pairs, though the precise detailswill undoubtedly be different in each case.To be honest, just as in the classical case, this last step will hold for representations of G outside a small family, which needs to be treated separately. One can characterize thisexceptional family precisely, but we prefer not to do it, and simply observe that temperedirreducible representations of G do not lie in this exceptional family. As mentioned earlier,the theta lifts of nontempered representations can be explicitly determined using the Jacquetmodule computations of [MS, GrS] and § G . We will treat this application in asequel to this paper. 2. The Group G We begin by introducing the algebraic group G over F .2.1. Octonion algebra.
Let O be the split octonion algebra over F . Thus, O is an 8-dimensional non-associative and non-commutative F -algebra. It comes equipped with a con-jugation map x x with associated norm N ( x ) = x · x = x · x and trace T r ( x ) = x + x .Moreover, N : O → F is a nondegenerate quadratic form.Every element x of O satisfies its characteristic polynomial t − T r ( x ) t + N ( x ). A nonzeroelement x ∈ O is said to be of rank 1 if N ( x ) = 0. Otherwise it is of rank 2, in which casethe subalgebra F [ x ] of O generated by x over F is isomorphic to the separable quadratic F -algebra F [ t ] / ( t − T r ( x ) t + N ( x )). We denote by O the 7-dimensional subspace of trace0 elements in O .2.2. Automorphism group.
The group G is the automorphism group of the F -algebra O . It is a split simple linear algebraic group of rank 2 which is both simply connected andadjoint. If we fix a maximal torus T contained in a Borel subgroup B , then we obtain asystem of simple roots { α, β } of G relative to ( T, B ), with α short and β long. The resultingroot system is given by the following diagram. OWE DUALITY AND DICHOTOMY 7 ✲✛ ✻❄✑✑✑✑✑✑✸◗◗◗◗◗◗❦ ✑✑✑✑✑✑✰ ◗◗◗◗◗◗s❏❏❏❏❫✡✡✡✡✣❏❏❏❏❪ ✡✡✡✡✢ α β
The highest root is β = 3 α + 2 β .2.3. Maximal torus.
Following Mui´c, we will fix the isomorphism T ∼ = G m by t ((2 α + β )( t ) , ( α + β )( t )) . Any pair of characters ( χ , χ ) of F × thus define a character χ × χ of T by compositionwith the above isomorphism.2.4. Parabolic subgroups.
Up to conjugation, G has 2 maximal parabolic subgroupswhich may be described as follows. Let V ⊂ V ⊂ O be subspaces of dimension 1 and2 respectively on which the octonion multiplication is identically zero. Let P and Q be thestabiliser of V and V respectively. Then P = M N and Q = LU are the two maximalparabolic subgroups of G . Moreover, their intersection B = P ∩ Q is a Borel subgroup of G .The Levi factor M of P is given by M ∼ = GL( V ) ∼ = GL . The isomorphism of M with GL can be fixed so that the modulus character of M is δ P =det . Its unipotent radical N is a 5-dimensional Heisenberg group with 1-dimensional center Z = U β . The action of M on N/Z is isomorphic to
Sym ( F ) ⊗ det − . Moreover, the generic M ( F )-orbits on N ( F ) /Z ( F ) is naturally parametrized by the set of isomorphism classes ofseparable cubic F -algebras.The Levi factor L of Q is given by L ∼ = GL( V /V ) ∼ = GL where V = { x ∈ O : x · y = 0 for all y ∈ V } . WEE TECK GAN AND GORDAN SAVIN
The isomorphism of L with GL can be fixed so that the modulus character of L is δ Q = det .The unipotent radical U is a 5-dimensional 3-step nilpotent group: U = U ⊃ U ⊃ U ⊃ U = { } , such that U /U = U α × U α + β , U /U ∼ = U α + β U /U = U β × U β − β . As representation of L , one has U /U ∼ = F , U /U ∼ = det U /U ∼ = F ⊗ det . The subgroup SL . The subgroup of G generated by the long root subgroups isisomorphic to SL . The normaliser of SL in G is a semidirect product SL ⋊ Z / Z , with thenontrivial element of Z / Z acting on SL as a pinned outer automorphism. The subgroupSL is the pointwise stabilizer of a quadratic subalgebra of O which is isomorphic to F × F ,whereas the setwise stabilizer of such a subalgebra is SL ⋊ Z / Z .More generally, given a subalgebra of O which is isomorphic to a quadratic field extensionof F , the pointwise stabilizer of this subalgebra is isomorphic to the quasi-split special unitarygroup SU K ; the setwise stabilizer of this subalgebra is SU K ⋊ Z / Z .2.6. The dual group.
The Langlands dual group of G is the complex Lie group G ( C ). Inparticular, one has the subgroupsSO ( C ) ⊂ SL ( C ) ⊂ SL ( C ) ⋊ Z / Z ⊂ G ( C ) . The centralizer of SL ( C ) in G ( C ) is µ , and the centralizer of SO ( C ) in G ( C ) is S = µ ⋊ Z / Z . Let SL ,l ( C ) be a long root SL . Then the centralizer of SL ,l ( C ) in G ( C ) isSL ,s ( C ), a short root SL , and vice versa. Thus we also have the subgroupSL ,l ( C ) × µ SL ,r ( C ) ∼ = SO ( C ) ⊂ G ( C ) . Representations of G In this section, we introduce some notations for the representations of G ( F ). In particular,we shall describe all non-supercuspidal representations. The results in this section are almostentirely due to Mui´c [Mu].3.1. Principal series representations for P . We first consider the principal series repre-sentations for the Heisenberg parabolic subgroup P = M N , where M ∼ = GL . Let τ be anirreducible representation of M with central character ω τ and set I P ( τ ) = Ind G P τ and I P ( s, τ ) = Ind G P ( | det | s · τ )if we need to consider a family of induced representations. If I P ( s, τ ) is a standard module,we will denote its unique Langlands quotient by J P ( s, τ ). Now we have: Proposition 3.1. (i) If τ is a unitary supercuspidal representation, then I P ( s, τ ) is reducibleif and only if τ ∨ ∼ = τ (so ω τ = 1 ) and one of the following holds: OWE DUALITY AND DICHOTOMY 9 • ω τ = 1 and s = 1 / , in which case there is a non-split short exact sequence of length , −−−−→ δ P ( τ ) −−−−→ I P (1 / , τ ) −−−−→ J P (1 / , τ ) −−−−→ , where δ P ( τ ) is a generic discrete series representation. • ω τ = 1 and s = 0 , in which case I P (0 , τ ) = I P ( τ ) gen ⊕ I P ( τ ) deg where I P ( τ ) gen is generic.(ii) If τ = st χ is a twisted Steinberg representation, then I P ( s, τ ) is irreducible except for thefollowing cases: • χ = 1 and s = ± / or ± / , in which case one has: −−−−→ St G −−−−→ I P (3 / , st) −−−−→ J P (3 / , st) −−−−→ , with St G the Steinberg representation. On the other hand, I P (1 / , st) has length ,with a unique irreducible submodule π gen [1] which is a generic discrete series rep-resentation, a unique irreducible Langlands quotient J P (1 / , st) and a subquotient J Q (1 / , st) . • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ π gen [ χ ] −−−−→ I P (1 / , st χ ) −−−−→ J P (1 / , st χ ) −−−−→ where π gen [ χ ] is a generic discrete series representation. • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ π gen [ χ ] −−−−→ I P (1 / , st χ ) −−−−→ J P (1 / , st χ ) −−−−→ . where π gen [ χ ] ∼ = π gen [ χ − ] is a generic discrete series representation.(iii) If τ = χ is 1-dimensional unitary, then I P ( s, τ ) is irreducible except in the followingcases: • χ = 1 and s = ± / or ± / , in which case one has: −−−−→ J Q (5 / , st) −−−−→ I P (3 / , −−−−→ G −−−−→ , whereas I P (1 / , is of length , with a unique irreducible submodule π deg [1] which is anongeneric discrete series representation, a unique irreducible quotient J Q (1 , π (1 , and a subquotient J Q (1 / , st) . • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ J Q (1 / , st χ ) −−−−→ I P (1 / , χ ) −−−−→ J Q (1 / , π (1 , χ )) −−−−→ . • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ J P (1 / , st χ − ) −−−−→ I P (1 / , st χ ) −−−−→ J Q (1 / , π ( χ, χ − )) −−−−→ . Principal series representations for Q . Now we consider the principal series repre-sentations for the 3-step parabolic subgroup Q = LU , where L ∼ = GL . Let τ be an irreducibleunitary representation of L with L-parameter φ τ and set I Q ( τ ) = Ind G Q τ and I Q ( s, τ ) = Ind G Q | det | s · τ if we need to consider a family of induced representations. As before, we let J Q ( s, τ ) denotethe unique Langlands quotient of I Q ( s, τ ) if the latter is a standard module. Then we have: Proposition 3.2. (i) If τ is unitary supercuspidal, then I Q ( s, τ ) is reducible if and only if τ ∨ ∼ = τ (so ω τ = 1 ) and one of the following holds: • ω τ = 1 and s = ± / , in which case one has: −−−−→ δ Q ( τ ) −−−−→ I Q (1 / , τ ) −−−−→ J Q (1 / , τ ) −−−−→ , where δ Q ( τ ) is a generic discrete series representation. • ω τ = 1 (so τ is dihedral), Im( φ τ ) = S and s = ± ,in which case one has: −−−−→ π gen [ τ ] −−−−→ I Q (1 , τ ) −−−−→ J Q (1 , τ ) −−−−→ , where π gen [ τ ] is a generic discrete series representation. • ω τ = 1 , Im( φ τ ) = S and s = 0 , in which case one has: I Q (0 , τ ) = I Q ( τ ) gen ⊕ I Q ( τ ) deg where I Q ( τ ) gen is generic.(ii) If τ = st χ is a twisted Steinberg representation, the I Q ( s, τ ) is irreducible except for thefollowing cases: • χ = 1 and s = ± / or ± / , in which case one has −−−−→ St G −−−−→ I Q (5 / , st) −−−−→ J Q (5 / , st) −−−−→ , and −−−−→ π gen [1] ⊕ π deg [1] −−−−→ I Q (1 / , st) −−−−→ J Q (1 / , st) −−−−→ . Here π gen [1] is the generic discrete series representation already defined in Proposition3.1(ii) (first bullet point) and π deg [1] is the nongeneric discrete series representationalready defined in Proposition 3.1(iii) (first bullet point). • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ π gen [ χ ] −−−−→ I Q (1 / , st χ ) −−−−→ J Q (1 / , st χ ) −−−−→ . Here, π gen [ χ ] is the generic discrete series representation defined in Proposition 3.1(ii)(second bullet point).(iii) If τ = χ is 1-dimensional unitary, then I Q ( s, τ ) is irreducible except in the followingcases: OWE DUALITY AND DICHOTOMY 11 • χ = 1 and s = ± / or ± / , in which case one has: −−−−→ J P (3 / , st) −−−−→ I Q (5 / , −−−−→ G −−−−→ , whereas I Q (1 / , is of length , with unique irreducible submodule J Q (1 / , st) , aunique irreducible quotient J Q (1 , π (1 , and subquotient J P (1 / , st) . • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ J P (1 / , st χ ) −−−−→ I Q (1 / , χ ) −−−−→ J Q (1 / , π (1 , χ )) −−−−→ . Principal series representations for B . We now consider the principal series repre-sentations induced from the Borel subgroup B . More precisely, suppose that π is a Langlandsquotient of a standard module I ( s , s , χ , χ ) ։ π with s ≥ s ≥ χ i unitary characters of F × . Here, recall the convention about characters of T which wehave fixed in § π ֒ → I ( − s , − s , χ − , χ − ) = I P ( π ( χ − | − | − s , χ − | − | − s )) . Now the representation π ( χ − | − | − s , χ − | − | − s ) of M ∼ = GL is reducible if and only if χ /χ · | − | s − s = | − | − , i.e. χ = χ and s = s + 1 ≥ π ֒ → I P ( − s + 12 , χ − ) , with s ≥ . There is another, convenient, way to bookkeep the principal series Ind G B ( χ ). Let β , β , β be three long roots such that β + β + β = 0. This triple is unique up the action of theWeyl group of G . Then the corresponding co-roots β ∨ i : F × → T generate T , in particular,the character χ defines three characters of F × by χ i = χ ◦ β ∨ i (and is determined by them).Clearly, these characters satisfy χ · χ · χ = 1. Proposition 3.3.
The induced representation
Ind G B ( χ ) is irreducible unless one of the fol-lowing two conditions hold: • χ i = | · | ± for some i or χ i /χ j = | · | ± for a pair i = j . • The three characters χ i are quadratic, non-trivial and pairwise different. Then Ind G B ( χ ) = Ind G B ( χ ) gen ⊕ Ind G B ( χ ) deg where Ind G B ( χ ) gen is generic. Conjectural L-packets of G . The above results allow one to give an enumerationof the non-cuspidal representations of G . Using the desiderata of the conjectural localLanglands correspondence (LLC) for G , we explain how one can assign L-parameters to thenoncuspidal representations of G , and hence partition them into L-packets. Recall that an L -parameter of G is an admissible homomorphism ϕ : W D F −→ G ∨ = G ( C )of the Weil-Deligne group W D = W F × SL ( C ) to the dual group G ( C ), taken up toconjugacy by G ( C ). Let A ϕ = π ( Z G ( ϕ ))be the associated component group of ϕ . Then one expects that there should be an L-packetΠ ϕ = { π ( ρ ) : ρ ∈ ˆ A ϕ } ⊂ Irr( G )associated to each ϕ , whose members are indexed by the characters of A ϕ , such thatIrr( G ) = [ ϕ Π ϕ . The non-tempered irreducible representations of G are uniquely realized as Langlandsquotients of standard modules, so have the form J P ( τ ), J Q ( τ ) or J B ( χ ). The Levi factors ofthe parabolic subgroups P , Q aand B are isomorphic to GL and GL × GL . Since the LLCfor these groups are known, one can assign L-parameters to the nontempered representations.For example, if π = J P ( τ ), and ϕ τ : W D F −→ M ∨ = GL ( C ) is the L-parameter of τ , thenthe L-parametrer of π = J P ( τ ) is the composite ϕ π : W D F −→ M ∨ ֒ → G ∨ = G ( C ) . Since the L-packets on the Levi subgroups are singletons, we see also that the nontemperedL-packets of G are singletons, and A ϕ π is correspondingly trivial.In other words, the non-tempered irreducible representations of G are naturally parametrizedby the nontempered L-parameters of G ; these are the L-parameters ϕ such that ϕ ( W F ) isunbounded. In the following, we will use this partial LLC to describe the effect of the varioustheta correspondences on nontempered representations.By the same token, since irreducible tempered representations which are not square-integrable are uniquely realized as summands of principal series representations inducedfrom unitary square-integrable representations of Levi factors, one can attach L-parameters tothese tempered (but not square-integrable) representations of G . The resulting L-parameters ϕ have the property that ϕ ( W F ) is bounded but ϕ ( W D F ) is contained in a proper Levi sub-group. The size of such a tempered L-packet now depends on the number of irreduciblesummands in the corresponding parabolically induced representations. From the results re-called in this section, one sees that the size of a tempered L-packet Π ϕ is 1 or 2. One canverify that this is the same as the size of A ϕ . Moreover, in each tempered L-packet, there isa unique generic representation, and this is assigned to the trivial character of A ϕ . Thus, theLLC for tempered non-discrete series representations of G is also known, and we may referto this partial LLC for describing these representations. OWE DUALITY AND DICHOTOMY 13
Hence, the main issue with the LLC for G comes down to the classification of the square-integrable or discrete series representations by discrete series L-parameters; these are the L-parameters ϕ which do not factor through any proper Levi subgroup, or equivalently whosecentralizer C ϕ = Z G ( ϕ ) is finite. Guided by the desiderata of the LLC, we can now describethe various families of discrete series L-parameters, according to ϕ (SL ), and list all non-supercuspidal members.(1) ϕ (SL ) is the principal SL . Then A ϕ = 1 and the packet consists of the Steinbergrepresentation: Π ϕ = { St G } (2) ϕ (SL ) = SO ⊆ SL ⊆ G ; this is the subregular SL . The centralizer of SO in G is the finite symmetric group S , so that ϕ gives by restriction a map φ : W F → S .There are four cases to discuss: • φ ( W F ) = 1. Then A ϕ = S . Let 1 , r, ǫ be the three irreducible representationsof S : the trivial, 2-dimensional and the sign character respectively. ThenΠ ϕ = { π (1) = π gen [1] , π ( r ) = π deg [1] , π ( ǫ ) = π sc [1] } where π gen [1] is defined in Proposition 3.1(ii) (first bullet point) and π deg [1] isgiven in Proposition 3.1(iii) (first b.p.). The representation π ( ǫ ) is a depth 0supercuspidal representation induced from a cuspidal unipotent representationof G ( F q ), inflated to a hyperspecial maximal compact group. The cuspidalunipotent representation is denoted in the literature by G [1] and hence ournotation π sc [1]. • φ ( W F ) = µ . Then, by the local class field theory, φ defines a quadratic character χ of F × . Let 1 and − A ϕ = µ .Then Π ϕ = { π (1) = π gen [ χ ] , π ( − } , where π gen [ χ ] is as defined in Proposition 3.1(ii) (second b.p.). If the character χ is unramified, then π ( −
1) = π sc [ −
1] is a depth 0 supercuspidal representation.It is induced from a cuspidal unipotent representation of G ( F q ), denoted by G [ − • φ ( W F ) = µ . Then, by local class field theory, φ defines a cubic character χ of F × . Let 1, ω and ω denote the characters of A ϕ = µ . ThenΠ ϕ = { π (1) = π gen [ χ ] , π ( ω ) , π ( ω ) } , where π gen [ χ ] is as defined in Proposition 3.1(ii) (third b.p.). If the character χ isunramified, then π ( ω ) = π sc [ ω ] and π ( ω ) = π sc [ ω ] are induced from a cuspidalunipotent representations of G ( F q ), denoted by G [ ω ] and G [ ω ], inflated to ahyperspecial maximal compact group. • φ ( W F ) = S . Then r ◦ φ corresponds to a supercuspidal representation τ of GL (where we recall that r denotes the two-dimensional irreducible representationof S ). In this case A ϕ is trivial andΠ ϕ = { π gen [ τ ] } , where π gen [ τ ] is as defined in Proposition 3.2(i) (second b.p.).(3) ϕ (SL ) = SL ,s , a short root SL . The centralizer of SL ,s in G is SL ,l , a longroot SL . Then ϕ gives, by restriction, a map form the Weil group φ : W F → SL ,l ,that corresponds to supercuspidal representation τ of GL with the trivial centralcharacter (and hence τ ∼ = τ ∨ ). In this case A ϕ = µ , andΠ ϕ = { π (1) = δ P ( τ ) , π ( − } , where δ P ( τ ) is as defined in Proposition 3.1(i) (first b.p.) and π ( −
1) is supercuspidal.(4) ϕ (SL ) = SL ,l , a long root SL . The centralizer of SL ,l in G is SL ,s , a long rootSL . Then ϕ gives, by restriction, a map form the Weil group φ : W F → SL ,s ,that corresponds to supercuspidal representation τ of GL with the trivial centralcharacter (and hence τ ∼ = τ ∨ ). In this case A ϕ = µ , andΠ ϕ = { π (1) = δ Q ( τ ) , π ( − } , where δ Q ( τ ) is as defined in Proposition 3.2(i) (first b.p.) and π ( −
1) is supercuspidal.(5) ϕ ( SL ) = 1. Then ϕ : W F → G ( C ) gives rise to an L-packet consisting entirely ofsupercuspidal representations of G .There has been some work towards the above conjectural LLC for G , most notably [SWe]and [HKT]. At the moment, we simply wish to point out that all the noncuspidal discreteseries representations are fully accounted for by the above classification scheme.3.5. Local Fourier coefficients.
It will be useful to consider the twisted Jacquet modules ofa representation π of G along the unipotent radical N of P . The M -orbits of 1-dimensionalcharacters of N are naturally indexed by cubic F -algebras, with the generic orbits corre-sponding to ´etale cubic F -algebras. For any such cubic F -algebra E , we shall write ψ E for acharacter of N in the corresponding M -orbit. Then one may consider π N,ψ E . In particular,we note: Lemma 3.4.
For any irreducible, nongeneric, infinite dimensional representation π of G ,there exists an ´etale cubic F -algebra E such that π N,ψ E = 0 . Moreover, π N,ψ E is finite-dimensional for any ´etale E .Proof. Since π is not Whittaker generic, and not the trivial representation, its wave-front setis supported on subregular nilpotent orbits, which are paramterized by cubic ´etale algebras,see [HMS] and [LS]. The non-vanishing of π N,ψ E for some E and the finite-dimensionality ofthese spaces is the main result of [MW] and [Va]. (cid:3) Exceptional Dual Pairs
In this section, we briefly describe the dual pairs which intervene in this paper and somestructural results which will be important in the study of the associated theta correspon-dences.
OWE DUALITY AND DICHOTOMY 15
The group M J . Let J be a Freudenthal-Jordan F -algebra [KMRT]. The algebra J comes equipped with a cubic norm form N J , and we let M J = { g ∈ GL( J ) : N J ◦ g = N J } . It contains the automorphism group Aut( J ) as a subgroup. Now we consider the F -vectorspace g J = sl ⊕ Lie ( M J ) ⊕ ( F ⊗ J ) ⊕ ( F ⊗ J ) ∗ Then g J can be given the structure of a simple exceptional Lie algebra (see, for example,[GS05]). We have the following cases of interest:dim J g J G D E E We observe: • If dim J = 3, then J is a cubic ´etale F -algebra E . • If dim J = 9 ,then J corresponds to a pair ( B K , τ ) where B K is a central simplealgebra over an ´etale quadratic F -algebra K and ι is an involution of the secondkind. Thus, J = B ιK is the subspace of ι -symmetric elements. The split versionof this algebra is M ( F ) = M ( F ) ⊕ M ( F ) where ι permutes two factors. Thus J = M ( F ), and Aut( M ) = PGL ⋊ Z / Z . • If dim J = 15, then J is H ( B F ) is the space of all 3 × B F is a quaternion algebra over F . The split version is when B F = M ( F ).Let G J be the identity component of Aut( g J ). If dim J = 9 then1 −−−−→ G J −−−−→ Aut( g J ) −−−−→ Z / Z −−−−→ . This short exact sequence need not split in general.4.2.
Dual pair G × Aut( J ) . We can now describe some dual pairs in G J or rather inAut( g J ). It will be easier to do this on the level of Lie algebras.The centralizer of Aut( J ) in g J is sl ⊕ F ⊗ J ⊕ F ⊗ J which one recognizes to be g F (i.e. taking J = F ). Thus this is a Lie subalgebra of type G ,and we have a dual pair G × Aut( J ) ⊂ Aut( G J ) . If dim J = 9, we recall that Aut( J ) sits in a short exact sequence1 −−−−→ Aut( J ) −−−−→ Aut( J ) −−−−→ Z / Z −−−−→ . If J is associated to a pair ( B K , τ ), then Aut( J ) = P GU ( B K , τ ) is an adjoint group of type A . Dual pair
Aut( i : E → J ) × G E . Now we fix an embedding i : E −→ J , where E is acubic ´etale F -algebra. We have the subgroupAut( i : E → J ) ⊂ Aut( J ) . If dim J = 9 a detailed description of this group is in [GS14]. Its identity component is a2-dimensional torus. The centralizer of Aut( i : E → J ) in g J contains g E = sl ⊕ t E ⊕ F ⊗ E ⊕ ( F ⊗ E ) ∗ where E ֒ → J via i and t E ∼ = E is the toral Lie subalgebra of trace 0 elements in E . ThisLie algebra is isomorphic to Lie ( G E ) (where G E is the simply connected quasi-split group Spin E ), and we have the dual pairAut( i : E → J ) × G E −→ Aut( G J ) . Note that this map is not injective.4.4.
A see-saw diagram.
The two dual pairs we described above fit together into a see-sawdiagram:(4.1) G E := Spin E ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ Aut( J ) =: H J G ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Aut( i : E → J ) =: H J,E in Aut( G J ). The various J ’s of interest in this paper, and the corresponding groups H J =Aut( J ) and H J,E = Aut( i : E → J ) are given in the table below. J D + M ( F ) + H ( M ( F )) H J P D × PGL ⋊ Z / Z PGSp H J,E
P E × P E × ⋊ Z / Z SL ( E ) /µ Here, note that D + denotes the Jordan algebra associated to a cubic division F -algebra D ,in which case E is necessarily a field.5. The See-Saw Argument
In this section, we shall consider the see-saw identity arising from the seesaw diagram (4.1)and pursue some of its consequences.5.1.
See-saw identity.
Suppose that π ∈ Irr( G ). Then we have the see-saw identity asso-ciated with the seesaw (4.1):(5.1) Hom H J,E (Θ( π ) , C ) ∼ = Hom G ( R J ( E ) , π )where R J ( E ) := Θ(1)is the big theta lift of the trivial representation of H J,E . To make use of this see-saw identity,we need to understand the representation R J ( E ) of Spin E . This has been studied in [GS15]and we recall the relevant results there. OWE DUALITY AND DICHOTOMY 17
Degenerate principal series of
Spin E . Let P E = M E · N E ⊂ Spin E be the Heisenbergparabolic subgroup, so that its Levi factor is M E ∼ = GL ( E ) . Then the determinant map defines an algebraic character M E → G m which is a basis elementof Hom( M E , G m ). We may consider the degenerate principal series representation I E ( s ) = Ind Spin E P E | det | s . In [GS15, S], the module structure of this family of degenerate principal series representationhas been determined. In particular, we have:
Proposition 5.2. R J ( E ) ֒ → I E ( s J ) where s J = ( − / , if J = D + or M ( F ) + ; / , if J = H ( M ( F )) .The representation I E (1 / has length when E is a field and has length otherwise. Moreprecisely, it has a unique irreducible submodule V with quotient isomorphic to R M ( F ) ( E ) ⊕ R D ( E ) (where R D ( E ) is interpreted to be when E is not a field). Indeed, one has the shortexact sequence: −−−−→ R H ( M ( F )) ( E ) −−−−→ I E (1 / −−−−→ R D ( E ) −−−−→ . and −−−−→ V −−−−→ R H ( M ( F )) ( E ) −−−−→ R M ( F ) ( E ) −−−−→ . In particular, when E is not a field, I E (1 /
2) = R H ( M ( F )) ( E ) . As a consequence of the above discussion, we see that it is useful to understand the Homspace Hom G ( I E ( s ) , π ) for π ∈ Irr( G ).We shall study this in two ways.5.3. Vanishing of an
Ext . In view of the proposition, we see that there is an exact sequence0 −−−−→
Hom G ( R D ( E ) , π ) −−−−→ Hom G ( I E (1 / , π ) −−−−→ Hom G ( R H ( M ( F )) ( E ) , π ) y Ext G ( R D ( E ) , π ) . Now we have the following useful fact:
Proposition 5.3.
Assume that E is a field. If π ∈ Irr( G ) is tempered or has cuspidalsupport different from π deg [1] , then Ext G ( R D ( E ) , π ) = 0 , so that one has a short exact sequence → Hom G ( R D ( E ) , π ) −−−−→ Hom G ( I E (1 / , π ) −−−−→ Hom G ( R H ( M ( F )) ( E ) , π ) → . Proof.
One needs to understand R D ( E ) as a representation of G , and this is essentiallydone in [Sa], where the dual pair correspondence for P D × × G was studied. We shallrecall the results of [Sa] in greater detail later on. At this point, we simply note that as arepresentation of G , R D ( E ) is the direct sum of a supercuspidal representation (of infinitelength) and the irreducible discrete series representation π deg [1], which is a constituent of I Q (1 / , st). From this, the vanishing of Ext G ( R D ( E ) , π ) for those π with different cuspidalsupport from π deg [1] follows immediately. On the other hand, if π is tempered, then one alsohas Ext ( π deg [1] , π ) = 0 since discrete series representations are projective in the category oftempered representations. (cid:3) I E ( s ) as G -module. On the other hand, we may understand the restriction of I E ( s )to G using Mackey theory. The following is a key technical result: Proposition 5.4.
As a representation of G , I E ( s ) admits an equivariant filtration ⊂ I ⊂ I ⊂ I ⊂ I ⊂ I with successive quotients described as follows: • I ∼ = ind G N ¯ ψ E ; • J := I /I ∼ = I P ( s + , C ∞ c (PGL )) . • J := I /I ∼ = m E · I P ( s + , ind PGL N ψ ) • J := I /I ∼ = m E · I Q ( s + 1) . • J := I /I ∼ = I P ( s + 1) .Here, m E = , if E = F ; , if E = F × K ; , if E is a field. The proposition implies that one has a short exact sequence0 −−−−→ ind G N ¯ ψ E −−−−→ I E ( s ) −−−−→ Σ E ( s ) −−−−→ ., from which one deduces an exact sequence:0 → Hom G (Σ E ( s ) , π ) → Hom G ( I E ( s ) , π ) → Hom N ( π ∨ , ψ E ) → Ext G (Σ E ( s ) , π ) . We now specialize to s = 1 /
2, where we need to be more precise.
Proposition 5.5.
Suppose that π ∈ Irr( G ) is tempered or has cuspidal support along Q .Then Hom G ( I E (1 / , π ) ∼ = Hom N ( π ∨ , ψ E ) . Proof.
We need to proveHom G (Σ E (1 / , π ) = 0 = Ext G (Σ E (1 / , π ) . To that end, it suffices to prove the following lemma:
Lemma 5.6.
For all i and j , Ext iG ( J j , π ) = 0 , with π as in Proposition 5.5. OWE DUALITY AND DICHOTOMY 19
Proof.
Consider J firstly. By the Frobenius reciprocity,Ext iG ( J , π ) = Ext iM ( | det | / · C ∞ c (PGL ) , r P ( π )) . Since π is tempered, the center of M = GL acts on R P ( π )) by characters χ such that | χ ( z ) | = | z | t where t ≤
0. On the other hand, the center of M acts on | det | / · C ∞ c (PGL )by | z | . Thus the right hand side is 0. The other cases are dealt with in the same way. (cid:3) This completes the proof of Proposition 5.5. (cid:3) Dichotomy
The goal of this section is to prove the following theorem:
Theorem 6.1.
For any representation π ∈ Irr( G ) , π has nonzero theta lift to exactly oneof P D × or PGSp . To prove this dichotomy theorem, we need some preliminary results which are consequencesof the computation of the Jacquet modules of the minimal representation Π J with respectto the various maximal parabolic subgroups of G and H J . The required Jacquet modulecomputations were carried out in [Sa] when H J = P D × and in [MS, GS04] when H J = PGL .For H J = PGSp , the Jacquet module computations for some parabolic subgroups werecarried out in [MS]. The remaining ones will be done in §
13 and some implications of thesecomputations are discussed in §
14. We note that §
13 is a self-contained section independentof the rest of this paper. Hence, we first record some results from § Consequences of Jacquet module computations.
We first note:
Lemma 6.2.
Consider the theta correspondence for G × H J for the 3 cases of J .(i) Let π ∈ Irr( G ) and write Θ J ( π ) = Θ J ( π ) c ⊕ Θ J ( π ) nc as a sum of its cuspidal and noncuspidal components. Then Θ J ( π ) nc has finite length. Inparticular, if Θ J ( π ) = 0 , then it has an irreducible quotient.(ii) Likewise, let τ ∈ Irr( H J ) and write Θ J ( τ ) = Θ J ( τ ) c ⊕ Θ J ( τ ) nc . Then Θ J ( τ ) nc has finite length. In particular, if Θ J ( τ ) = 0 , then it has an irreduciblequotient.Proof. The case of J = D + follows from results of [Sa], whereas that for J = M ( F ) + followsfrom [GS04]. The case of J = H ( M ( F )) is shown in §
14, based on the results of § (cid:3) In fact, the Jacquet module computations allow one to determine the theta lift of non-tempered representations explicitly (see Theorem 14.1). We simply note the following here:
Lemma 6.3. (i) Let π ∈ Irr( G ) and τ ∈ Irr( H J ) be such that π ⊗ τ is a quotient of Π J .Then π is tempered ⇐⇒ τ is tempered . (ii) Let π ∈ Irr( G ) be non-tempered. Then π has nonzero theta lifting to PGSp .Proof. For H J = P D × or PGL ⋊ Z / Z , the desired results have been verified in [Sa] and[GS04] respectively. For the case when H J = PGSp , this is shown in Theorem 14.1 in § (cid:3) Reduction to non-generic tempered case.
With the above inputs in place, we cannow begin the proof of the dichotomy theorem. We note: • The dichotmy theorem holds for nontempered π . Indeed, if π is non-tempered, thenLemma 6.3(ii) says that π has nonzero theta lift to PGSp , whereas [Sa] shows that π has zero theta lift to P D × . • The dichotmy theorem holds for generic π . Indeed, it was shown in [GS04, Cor. 20]that a generic π has nonzero theta lift to PGSp (see also Cor. 11.2 below) and itwas shown in [Sa] that π has zero theta lift to P D × .Thus, to prove the dichotomy theorem, it remains to deal with non-generic tempered π .6.3. Weak dichotomy.
We first prove that a non-generic tempered π has nonzero thetalift to one of P D × or PGSp . Since π is non-generic and infinite-dimensional, there existsan ´etale cubic F -algebra E such that Hom N ( π ∨ , ψ E ) = 0. By Proposition 5.5, we have anisomorphism Hom G ( I E (1 / , π ) ։ Hom N ( π ∨ , ψ E ) = 0 . This implies, by Proposition 5.3,Hom G ( R D ( E ) , π ) = 0 or Hom G ( R H ( M ( F ) ( E ) , π ) = 0 . By the see-saw identity (5.1), we deduce thatHom H E (Θ J ( π ) , C ) = 0for J = D + or H ( M ( F )). In particular, Θ J ( π ) = 0 for J = D + or H ( M ( F )). We havethus verified that π has nonzero theta lift to at least one of P D × or PGSp .6.4. Curious chain of containments.
It remains to show that a nongeneric tempered π cannot lift to both P D × and PGSp . Let ¯ π ∼ = π ⊗ C C be the complex conjugate of π . If π isunitarizable (e.g. if π is tempered), then ¯ π ∼ = π ∨ . Thus π ¯ ψ E ∼ = ¯ π ψ E ∼ = ( π ∨ ) ψ E where, in the second isomorphism, we assume that π is unitarizable. Since the minimalrepresentation Π J used in this paper is defined over R , we have a canonical isomorphism¯Π J ∼ = Π J . It follows that Θ J (¯ π ) is the complex conjugate of Θ J ( π ).We shall make use of the curious chain of containment given in the following lemma;this is the first instance of the game of ping-pong with periods discussed at the end of theintroduction. OWE DUALITY AND DICHOTOMY 21
Lemma 6.4.
Let π ∈ Irr( G ) be tempered. For J = D + , M ( F ) or H ( M ( F )) , let τ ∈ Irr( H J ) be tempered and such that Hom G × H J (Π J , π ⊠ τ ) = 0 . Then we have the following natural inclusions
Hom N ( π, ψ E ) ⊆ Hom N (Θ( τ ) , ψ E ) ∼ = Hom H J,E ( τ ∨ , C ) ⊆ Hom H J,E (Θ( π ∨ ) , C ) ∼ = Hom G ( R J ( E ) , π ∨ ) . If one of these spaces is finite-dimensional, then all inclusions are isomorphisms.Proof.
The first inclusion arises from Θ( τ ) ։ π . The second follows fromHom N (Θ( τ ) , ψ E ) ∼ = Hom H J ((Π J ) N,ψ E , τ )combined with (see Lemma 2.9 on page 213 in [GrS])(Π J ) N,ψ E ∼ = ind H J H J,E (1)and the Frobenius reciprocity. For the third, observe that Θ J (¯ π ) is the complex conjugateof Θ J ( π ). Since ¯ π ∼ = π ∨ and ¯ τ ∼ = τ ∨ and we have Θ( π ∨ ) ։ τ ∨ . The fourth is the see-sawidentity (5.1).If any of the spaces is finite-dimensional, then Hom N ( π, ψ E ) is finite-dimensional. If thisspace is finite dimensional then, since π is tempered, by Propositions 5.3 and 5.5, one has(6.5) dim Hom G ( R J ( E ) , π ∨ ) ≤ dim Hom G ( I E (1 / , π ∨ ) = dim Hom N ( π, ψ E ) . It follows that all spaces have the same dimension and the lemma is proved. (cid:3)
Conclusion of proof.
Using the lemma, we can now conclude the proof of Theorem6.1.Assume π is tempered nongeneric and has nonzero theta lift to P D × . Since P D × iscompact, one can find τ ∈ Irr(
P D × ) such that τ is an irreducible quotient of Θ D ( π ). Moreover τ is tempered. Choose E so that Hom N ( π, ψ E ) = 0. We may now apply Lemma 6.4 with thechosen τ and E to deduce that d := dim Hom G ( I E (1 / , π ) = dim Hom G ( R D ( E ) , π ) = dim Hom N ( π, ψ E ) = 0Similarly, if π has nonzero theta lift to PGSp , then we may find a tempered τ ∈ Irr(PGSp )such that τ is an irreducible quotient of Θ( π ) (by Lemma 6.2 and Lemma 6.3(i)). With E as above, an application of Lemma 6.4 shows that d = dim Hom G ( I E (1 / , π ) = dim Hom G ( R H ( M ( F )) ( E ) , π ) = dim Hom N ( π, ψ E ) = 0Moreover, since all these dimensions are finite, one deduces by Proposition 5.3 that d = dim Hom G ( I E (1 / , π ) = dim Hom G ( R D ( E ) , π ) + dim Hom G ( R H ( M ( F )) ( E ) , π ) = 2 d. This gives the desired contradiction and completes the proof of Theorem 6.1.
Uniqueness results.
As further applications of Lemma 6.4, we may now derive twomultiplicity one statements which will play a key role in the reminder of the paper. Thesestatements are the first steps towards the proof of the Howe duality theorem.
Proposition 6.6.
Let τ ∈ Irr( H J ) be tempered. Let π ∈ Irr( G ) be a tempered, non-genericquotient of Θ J ( τ ) . Then π is the unique irreducible tempered quotient of Θ J ( τ ) .Proof. Since π is non-generic, there exists E such that d = dim Hom N ( π, ψ E ) is finite andnon-zero. By Lemma 6.4, dim Hom N (Θ( τ ) , ψ E ) = d . Let π ′ be another tempered irreduciblequotient of Θ J ( τ ). We apply Lemma 6.4 to π ′ and τ to deduce that dim Hom N ( π ′ , ψ E ) = d .Since π ⊕ π ′ is a quotient of Θ J ( τ ), d = dim Hom N (Θ J ( τ ) , ψ E ) ≥ dim Hom N ( π, ψ E ) + dim Hom N ( π ′ , ψ E ) = 2 d, a contradiction. (cid:3) Proposition 6.7.
Let π ∈ Irr( G ) be tempered and non-generic. Then Θ J ( π ) cannot havetwo tempered irreducible quotients. In particular, the cuspidal representation Θ J ( π ) c is irre-ducible or .Proof. Let τ , τ ∈ Irr( H J ), irreducible tempered, such that Θ J ( π ) ։ τ ⊕ τ . Since π is non-generic there exists E such that d = dim Hom N ( π ∨ , ψ E ) is finite and non-zero. By Lemma6.4, applied to π ∨ , τ ∨ and then to π ∨ , τ ∨ , d = dim Hom H J,E ( τ , C ) = dim Hom H J,E (Θ J ( π ) , C ) = dim Hom H J,E ( τ , C ) . Since τ ⊕ τ is a quotient of Θ( π ), d = dim Hom H J,E (Θ J ( π ) , C ) ≥ dim Hom H J,E ( τ , C ) + dim Hom H J,E ( τ , C ) = 2 d, a contradiction. (cid:3) Combining Proposition 6.6 and 6.7 with Lemmas 6.2(i) and 6.3(i), we deduce the followingcorollary which may be considered as a first step towards the Howe duality theorem.
Corollary 6.8.
Let π ∈ Irr( G ) be tempered and non-generic. Then Θ J ( π ) has finite length.If Θ J ( π ) = 0 , then it has a unique irreducible quotient θ ( π ) and θ ( π ) is tempered. Moreover,for π , π ∈ Irr( G ) tempered and non-generic, θ ( π ) ∼ = θ ( π ) = ⇒ π ∼ = π . Proof.
Writing Θ J ( π ) = Θ J ( π ) c ⊕ Θ J ( π ) nc , Proposition 6.7 tells us that Θ J ( π ) c is irreducibleor 0, whereas Lemma 6.2(i) tells us that Θ J ( π ) nc has finite length. Hence Θ J ( π ) has finitelength, so that its cosocle θ J ( π ) is a finite sum of irreducible representations. Moreover,Lemma 6.3(i) says that θ J ( π ) is tempered, and Proposition 6.7 then shows the irreducibilityof θ J ( π ) if it is nonzero. The final implication now follows by Proposition 6.6. (cid:3) In the rest of the paper, we shall examine each of the 3 dual pairs G × H J in turn andcomplete the proof of the Howe duality conjecture. OWE DUALITY AND DICHOTOMY 23 Theta Correspondence for
P D × × G In this section, we discuss the theta correspondence for the dual pair
P D × × G . Apreliminary study of this dual pair correspondence has been carried out by the second authorin [Sa]. We first recall the results established there.Let Π D be the minimal representation of P D × × G . Then we haveΠ D = M τ ∈ Irr(
P D × ) τ ⊠ Θ( τ ) . The following was shown in [Sa]:
Proposition 7.1. (i) If τ = 1 is the trivial representation of P D × , then Θ(1) = π deg [1] , the unipotent discrete series representation introduced in Proposition 3.1(iii) (first bulletpoint).(ii) If τ is not the trivial representation, then Θ( τ ) is nongeneric supercuspidal of finite length(possibly zero).(iii) If τ = χ is a nontrivial unramified cubic character, then Θ( χ ) = π sc [ χ ] and Θ( χ ) = π sc [ χ ] the two depth 0 supercuspidal representations introduced in § E/F , Hom N (Θ( τ ) , ψ E ) ∼ = Hom P E × ( τ, C ) . We can now easily extend the above results. More precisely,
Theorem 7.2. (i) For any τ ∈ Irr(
P D × ) , Θ( τ ) is an irreducible representation of G if itis nonzero.(ii) If τ , τ ∈ Irr(
P D × ) are such that Θ( τ ) ∼ = Θ( τ ) = 0 , then τ ∼ = τ .(ii) If p = 3 , then the map τ Θ( τ ) defines an injection Irr(
P D × ) ֒ → Irr( G ) . Hence, the Howe duality theorem holds for
P D × × G , so that dim Hom G ( θ ( τ ) , θ ( τ )) ≤ dim Hom P D × ( τ , τ ) for any τ , τ ∈ Irr(
P D × ) . In particular, for any π ∈ Irr( G ) , the representation Θ( π ) of P D × is irreducible or zero.Proof. The first two parts follow from Propositions 6.6 and 6.7. As for (iii) we useHom N (Θ( τ ) , ψ E ) ∼ = Hom P E × ( τ, C ) , so it suffices to show that there exists a field E such that Hom P E × ( τ, C ) = 0. If p = 3, thiswas proved for all irreducible τ by [LT], Theorem 2.4. (cid:3) Theta Correspondence for (PGL ⋊ Z / Z ) × G In this section, we consider the theta correspondence for dual pair (PGL ⋊Z / Z ) × G andprove various results analogous to those in the last section. In fact, the theta correspondencefor PGL × G was almost completely studied in [GS04]. But the treatment there ignores theouter automorphism group of PGL ; this is akin to working with special orthogonal groupsinstead of orthogonal groups in classical theta correspondence and is of course undesirable.Thus, we shall complete the results of [GS04] in their natural setting here. We extend theminimal representation of E to E ⋊ Z / Z so that Z / Z fixes the spherical vector.8.1. Representations of H = PGL ⋊ Z / Z . We realize Z / Z , acting on GL as a pinnedautomorphism preserving the standard pinning, i.e. acting via A w · t A − · w − with − Let U ⊂ GL be the maximal unipotent subgroup of upper triangular matrices and let ψ be a Z / Z -invariant Whittaker character of U . Then ψ extends to two characters of U ⋊Z / Z . Let ψ ⊗ Z / Z acts trivially, and let ψ ⊗ sign be the other extension.If τ ∈ Irr(PGL ), then there are two possibilities: • if τ ∨ ≇ τ , then τ + := Ind H PGL τ ∼ = Ind H PGL τ ∨ is irreducible. If τ is generic thendim Hom U ⋊Z / Z ( τ + , ψ ⊗
1) = dim Hom U ⋊Z / Z ( τ + , ψ ⊗ sign) = 1 . • if τ ∼ = τ ∨ , then τ has two extensions to H , which differ from each other by twistingwith the unique quadratic character sign : H −→ h± i of H . When τ is generic (forexample when τ is tempered), we let τ + denote the unique extension of τ such thatdim Hom U ⋊Z / Z ( τ + , ψ ⊗
1) = 1 , and let τ − denote the other extension.The only nongeneric and self-dual representations of PGL are the Langlands quo-tients J B ( µ ), where B = T U is the normalizer of U and µ = χ | − | / ⊗ ⊗ χ | − | − / is a character of T such that χ = 1. In this case, θ ( τ ) is irreducible by [GS04], andwe define τ + by setting θ ( τ + ) = θ ( τ ) and θ ( τ − ) = 0.It follows from the above discussion that any irreducible representation of H is self-dual.8.2. Whittaker models.
The following lemma summarizes some basic computations.
Lemma 8.1.
Let Π be the minimal representation of split E ⋊ Z / Z .(i) Let ψ V : V → C × be a Whitaker character for G (so V is a maximal unipotent subgroupof G ). Then Π V,ψ V ∼ = ind HU ⋊Z / Z ψ ⊗ . OWE DUALITY AND DICHOTOMY 25
In particular, for any τ ǫ ∈ Irr( H ) , Hom V (Θ( τ ǫ ) , ψ V ) ∼ = Hom U ⋊Z / Z ( τ ǫ , ψ ⊗ (ii) For any ´etale cubic F -algebra, we have: Hom N (Θ( τ ǫ ) , ψ E ) ∼ = Hom P E × ⋊Z / Z ( τ ǫ , C ) . Our earlier results.
The following is a simple combination of results [GS04] and theprevious discussion:
Theorem 8.2.
For τ ∈ Irr(PGL ) , let φ τ : W D F −→ SL ( C ) denote the L-parameter of τ .If τ is non-supercuspidal, then Θ( τ ) has finite length. If τ ≇ τ ∨ is supercuspidal, then Θ( τ ) is irreducible supercuspidal. (This covers all τ ∈ Irr(PGL ) if p = 2 ). In these cases, set θ ( τ ǫ ) to be the maximal semisimple quotient of Θ( τ ǫ ) for ǫ = ± .More precisely, we have:(i) If τ = τ ∨ , then θ ( τ + ) is irreducible and nonzero. If τ is generic, or supercuspidal, or adiscrete series representation, or tempered, so is θ ( τ + ) . When τ is not supercuspidal, then θ ( τ + ) is not supercuspidal and its L-parameter is obtained by composing φ τ with the inclusion SL ( C ) ⊂ G ( C ) .(ii) If τ = τ ∨ and the parameter φ τ contains the trivial representation, then θ ( τ − ) = 0 and θ ( τ + ) is nonzero irreducible, non-discrete-series and its L-parameter is obtained by composing φ τ with the inclusion SL ( C ) ⊂ G ( C ) .(iii) If τ = τ ∨ and the parameter φ τ does not contain the trivial representation, then we havethe following cases: • τ = St , the Steinberg representation. Then θ (St + ) ⊕ θ (St − ) = π gen [1] ⊕ π s c [1] where π gen [1] is the generic discrete series representation introduced in Proposition3.1(ii) and π sc [1] is the depth 0 supercuspidal representation introduced in § • τ is a tempered representation induced from a supercuspidal representation σ ∼ = σ ∨ of GL with a non-trivial central character. Then θ ( τ + ) ⊕ θ ( τ − ) = Ind G P ( σ ) = Ind G P ( σ ) gen ⊕ Ind G P ( σ ) deg • τ is a tempered principal series induced from a triple of non-trivial quadratic charac-ters ( χ , χ , χ ) such that χ · χ · χ = 1 then θ ( τ + ) ⊕ θ ( τ − ) = Ind G B ( χ ) = Ind G B ( χ ) gen ⊕ Ind G B ( χ ) deg where χ is the quadratic character of T determined by ( χ , χ , χ ) as in § • τ is a self-dual supercuspidal representation (so p = 2 ). Then Θ( τ ǫ ) is supercuspidaland Θ( τ + ) ⊕ Θ( τ − ) = π gen ⊕ π deg where π gen is a generic irreducible supercuspidal representation, while π deg is a non-generic supercuspidal representation of unknown length. Observe that the only case for which we do not know that Θ( τ ǫ ) has finite length (andhence θ ( τ ǫ ) is defined) is when τ is a self-dual supercuspidal representation (so p = 2). In thiscase, however, the last bullet point states that Θ( τ ǫ ) is supercuspidal and hence semisimple.Hence, even in this exceptional case, we may set θ ( τ ǫ ) = Θ( τ ǫ ). Moreover, observe that if τ ǫ is nontempered, then θ ( τ ǫ ) is irreducible nontempered and is completely determined byTheorem 8.2. On the other hand, when τ ǫ is tempered, then so is every irreducible summandof θ ( τ ǫ ). In particular, the results highlighted in Lemma 6.2 and 6.3 hold in this case.In the rest of the section, we shall complete the results above by completing the unresolvedparts of the theorem.8.4. A miracle of Oberwolfach.
Let τ ∈ Irr(PGL ) be a self-dual supercuspidal repre-sentation. The goal here is to show that Θ( τ − ) = 0. Let Q = LU be the 3-step maximalparabolic subgroup of G . Recall that the group U has the 3-step filtration U ⊃ [ U, U ] ⊃ Z U where Z U is the 2-dimensional center of U and U/Z U is a 3-dimensional Heisenberg groupwith the center [ U, U ] /Z U . Let ψ be a non-trivial character of [ U, U ], trivial on Z U . ThenΠ [ U,U ] ,ψ is naturally a (PGL ⋊ Z / Z ) × SL -module, where SL = [ L, L ]. In order to describeΠ [ U,U ] ,ψ , we need some additional notation.Consider the action of the group GL ⋊ Z / Z on M ( F ), with elements in GL ( F ) actingby conjugation and the nontrivial element of Z / Z acting via: X (cid:18) (cid:19) · X t · (cid:18) (cid:19) . This action preserves the determinant (quadratic) form on M ( F ) and descends to the quo-tient group PGL ( F ) × Z / Z ∼ = {± } × SO = O . On the space C c ( M ( F )), we have a Weil representation of O × SL , which we may regardas a representation of GL ( F ) ⋊ Z / Z . Then the following lemma follows by a standardcomputation: Lemma 8.3.
We have an isomorphism of (PGL ⋊ Z / Z ) × SL -modules: Π [ U,U ] ,ψ ∼ = ind PGL ⋊Z / Z GL ⋊Z / Z ( C c ( M ( F )) where GL is embedded in PGL via (cid:18) a bc d (cid:19) a b c d . Using the lemma, we can now prove:
Proposition 8.4.
Let τ ∈ Irr(PGL ) be a self-dual supercuspidal representation. Then Θ( τ − ) = 0 . OWE DUALITY AND DICHOTOMY 27
Proof.
It suffices to show that τ − is a quotient of Π [ U,U ] ,ψ , in fact we shall show that τ − isa quotient of SL -coinvaraints of Π [ U,U ] ,ψ . Decompose M ( F ) = M ◦ ( F ) ⊕ F , where M ( F ) ◦ is the subspace of trace 0 elements and F is the center. Accordingly, the Weil representationof O × f SL on C c ( M ( F )) decomposes as a tensor product C c ( M ( F )) = C c ( M ◦ ( F )) ⊗ C c ( F )where O acts trivially on C c ( F ) and ˜SL acts by the Weil representations ρ ψ . Recall thatas an f SL -module, ρ ψ decomposes as a sum ρ ψ = ρ + ψ ⊕ ρ − ψ of even and odd Weil representations. Let Θ( ρ +¯ ψ ) and Θ( ρ − ¯ ψ ) be the theta lifts of theircontragredients to O , via the Weil representation on C c ( M ◦ ( F )) with respect to ψ . Thusthe SL -coinvariant of Π [ U,U ] ,ψ is given byind PGL ⋊Z / Z GL ⋊Z / Z (Θ( ρ +¯ ψ )) ⊕ ind PGL ⋊Z / Z GL ⋊Z / Z (Θ( ρ − ¯ ψ )) . Let st be the Steinberg representation of SO(3) ∼ = PGL . We extend st to two representationsst + and st − of O(3) by letting − ∈ O(3) act by 1 and − ρ − ψ ) ∼ = st − while Θ( ρ + ψ ) is the principal series representation with the trivial representation as a quotientand st + as a submodule. Since τ − is a supercuspidal representation, it suffices to show thatit is a quotient of ind PGL ⋊Z / Z GL ⋊Z / Z (st + ) ⊕ ind PGL ⋊Z / Z GL ⋊Z / Z (st − ) . It is known that any generic representation of GL , in particular st, is a quotient of τ . Henceeither st + or st − is a quotient of τ − . Now the proposition follows from Frobenius reciprocity. (cid:3) Main result.
We shall now strengthen Theorem 8.2.
Theorem 8.5.
For any τ ∈ Irr(PGL ) , let φ τ : W D F −→ SL ( C ) denote the L-parameterof τ .(i) The representation Θ( τ ǫ ) is zero if and only if φ τ contains the trivial representation(so τ ∼ = τ ∨ ) and ǫ = − .(ii) For any ǫ = ± , Θ( τ ǫ ) has finite length with unique irreducible quotient θ ( τ ǫ ) (if it isnonzero).(iii) θ ( τ ǫ ) is generic if and only if τ is generic and ǫ = + .(iv) Suppose that θ ( τ ǫ ) = 0 . If τ is a discrete series (resp. tempered) representation, so is θ ( τ ǫ ) . Moreover, θ ( τ ǫ ) is supercuspidal if and only if τ is supercuspidal or τ ǫ = St − .(v) If θ ( τ ǫ ) ∼ = θ ( τ ǫ ) = 0 , then τ ǫ = τ ǫ .In particular, the Howe duality theorem holds for the dual pair (PGL ⋊ Z / Z ) × G : dim Hom G ( θ ( τ ǫ ) , θ ( τ ǫ )) ≤ dim Hom PGL ⋊Z / Z ( τ ǫ , τ ǫ ) for any τ ǫ , τ ǫ ∈ Irr(PGL ⋊ Z / Z ) . Moreover, for π ∈ Irr( G ) , Θ( π ) is a finite lengthrepresentation of PGL with a unique irreducible quotient (if nonzero). Proof. (i) From Theorem 8.2, it remains to show that Θ( τ ǫ ) is nonzero for those represen-tations τ as in Theorem 8.2(iii) and any ǫ = ± . Consider first the Steinberg representation.Recall that π gen [1] is generic while π sc [1] is not. It follows, from Lemma 8.1 part (ii), that π gen [1] is a summand of θ (St + ). Furthermore, by Proposition 6.6, π sc [1] cannot be a summandof θ (St + ). Hence θ (St + ) = π gen [1] and θ (St − ) = π sc [1] . The same argument works in the other three cases to show that θ ( τ + ) is the generic G summand and θ ( τ − ) is the degenerate summand. Moreover, in the last case of Theorem 8.2,where τ is a self-dual supercuspidal representation (so p = 2), we deduce by Proposition 6.6again that π deg is nonzero irreducible.(ii) This follows from Theorem 8.2 and the irreducibility of π deg in the proof of (i) above.(iii) and (iv): These summarize what we already know from Theorem 8.2.(v) Suppose that π := θ ( τ ǫ ) ∼ = θ ( τ ǫ ) = 0 . If π is non-supercuspidal, then τ and τ are both non-supercuspidal. The desired equal-ity τ ǫ ∼ = τ ǫ follows from the results of [GS04] and our new understanding in (i) (whichdetermines θ ( τ ǫ ) for those τ in Theorem 8.2(iii))Suppose that π is supercuspidal. Then τ ǫ i i is either supercuspidal or St − , in which caseboth τ and τ are generic discrete series representations. By (iii), we deduce that ǫ = ǫ .Hence, it remains to show that τ = τ or τ ∨ . We now consider the following two cases:(a) Suppose τ ≇ τ ∨ and τ ≇ τ ∨ . Then ǫ = ǫ = + and π is an irreducible genericsupercuspidal representation.Consider, for i = 1 or 2, the induced representation Ind PGSp P ( τ i ), where P isthe Siegel parabolic subgroup. Its normalized Jacquet functor with respect to P is τ i ⊕ τ ∨ i . Since τ i = τ ∨ i , it follows that Ind PGSp P ( τ i ) is an irreducible generic temperedrepresentation.By the computation of the Jacquet module of the minimal representation Π ′ of G × PGSp along P given in [MS], we deduce that π ⊗ Ind
PGSp P ( τ i ) is an irreduciblequotient of Π ′ . By [SWe], a generic representation cannot lift to two different genericrepresentaitons of PGSp . Hence, we must haveInd PGSp P ( τ ) ∼ = Ind PGSp P ( τ ) . By consideration of the Jacquet modules with respect to P , we see that τ ∼ = τ or τ ∨ , as desired.(b) Assume now that τ = τ ∨ . In this case, we know that θ ( τ +1 ) = π gen and θ ( τ − ) = π deg . Moreover, the tempered representation Ind
PGSp P ( τ ) is the sum of two representa-tions, one of which is generic and the other degenerate (see Proposition 10.3(i)). Bythe Jacquet module of Π ′ again, we see that both π gen and π deg lifts to irreduciblesummands of Ind PGSp P ( τ ). Moreover, π deg cannot lift to a generic representation of OWE DUALITY AND DICHOTOMY 29
PGSp and hence must lift to the nondegenerate summand [SWe]. By Proposition6.6, it follows that π gen cannot lift to the degenerate summand and thus must lift tothe generic summand.Now suppose that ǫ = ǫ = +, so that π = θ ( τ +1 ) = θ ( τ +2 ) is generic. Thenas before, we see that π lifts to the generic summand of Ind PGSp P ( τ i ) (regardless ofwhether τ is self-dual or not). By Jacquet module consideration, we see that τ ∼ = τ .On the other hand, if ǫ = ǫ = − , so that π = θ ( τ − ) = θ ( τ − ) is nongeneric, thenProposition 6.7 implies that the nongeneric summand of Ind PGSp P ( τ ) is contained inInd PGSp P ( τ ). Again, Jacquet module considerations shows that τ ∼ = τ .The inequality at the end of the theorem is simply a restatement of (v). Finally, given π ∈ Irr( G ), we write Θ( π ) = Θ( π ) c ⊕ Θ( π ) nc as a sum of its cuspidal and noncuspidal component. As we noted in Lemma 6.2, the resultsof [GS04] imply that Θ( π ) nc has finite length. The result in (v) shows that Θ( π ) has aunique irreducible quotient if it is nonzero, implying in particular that Θ( π ) c is either 0 orirreducible, and hence Θ( π ) has finite length. (cid:3) The group
PGSp Before discussing the last dual pair G × PGSp , we need to devote the next few sectionsto a discussion of the structure and representations of PGSp , as well as certain particularperiods on G and PGSp .Let e , . . . , e be the standard basis of F , where we have a symplectic form defined by ω ( e , e ) = ω ( e , e ) = ω ( e , e ) = 1and all other ω ( e i , e j ) = 0 with i < j . Let GSp be the group of linear transformations g of F , such that for some ν ( g ) ∈ F × ω ( gv, gw ) = ν ( g ) · ω ( v, w )for all v, w ∈ F . Then ν : GSp → F × is the similitude character.Let ˜ P , ˜ P and ˜ P be maximal parabolic subgroups of GSp defined as the stabilizers ofsubspaces h e i ⊂ h e , e i ⊂ h e , e , e i respectively. For i = 1 , ,
3, let P i ⊆ PGSp be the quotient of ˜ P i by the center of GSp . Thegroup PGSp acts faithfully on J = ∧ F ⊗ i − , and we shall (partially) describe how theparabolic subgroups act on this module.The group PGSp can be explicitly described in terms of its action on J as follows. Let x ij = e i ∧ e j ∈ J for i = j . On J , we have a natural trilinear form ( x, y, z ) ∧ F × ∧ F × ∧ F → ∧ F ∼ = F. The group of linear transformations of J preserving this form is SL /µ and PGSp = Sp /µ is the subgroup fixing e = x + x + x . The Levi factor M of P , as an algebraic group, is isomorphic to GL /µ . Observe thatgroup acts faithfully on ∧ F , and since the latter is a three dimensional vector space, thisaction gives an isomorphism GL /µ ∼ = GL . Thus we have an identification M = GL( h x , x , x i ) . Under this identification, the maximal torus is given by diagonal matrices ( t , t , t ). Thethree simple co-roots of PGSp are, respectively, α ∨ ( t ) = (1 , t, t − ) , α ∨ ( t ) = ( t, t − , , α ∨ ( t ) = (1 , t, t ) . An unramified character χ of the maximal torus is given by a triple of complex numbers( s , s , s ) χ ( t , t , t ) = | t | s | t | s | t | s . The Weyl group action on the characters is somewhat different in this picture. The simplereflections corresponding to the first two roots α and α are the usual permutations of entriesof ( s , s , s ), however, the simple reflection corresponding to the third simple root α is givenby ( s , s , s ) ( s + s + s , − s , − s ) . Thus the root hyperplanes are s i − s j = 0 and s i + s j = 0 for short and long roots, respectively.This looks like a D root system; however, the Weyl-invariant quadratic form in this case is q ( s , s , s ) = s + s + s −
14 ( s + s + s ) rather than the usual dot product, and with this form, we have a realization of the C rootsystem with simple roots α = (0 , , − , α = (1 , − , , α = (0 , , . This somewhat unconventional description of the C root system is a source of potentialconfusion, as one has the tendency to confound it with the more familiar description of theroot system of Sp , but what we have done here is definitely the natural way to set thingsup for PGSp .The character χ is in the positive chamber if for every positive root α , χ ( α ∨ ( t )) = | t | s forsome s ∈ C such that ℜ ( s ) > χ is positive if ℜ ( s ) > ℜ ( s ) > |ℜ ( s ) | . The modulus character of M ∼ = GL is δ P ( m ) = | det( m ) | . It follows that the Levi factor M of P = P ∩ P is M = GL( h x , x i ) × GL( h x i ) . The group P is the stabilizer of the amber space V = h x , x i .Consider now the group P and its Levi factor M . The standard Levi factor of ˜ P isGL × GL where the first GL acts on h e , e i in the standard way, fixes h e , e i and acts by OWE DUALITY AND DICHOTOMY 31 transpose-inverse on h e , e i . The second GL acts on h e , e i in the standard way, by det on h e , e i and fixes h e , e i . The group P is the stabilizer of the singular line V = h x i , andthe Levi factor M acts faithfully on the 4-dimensional subspace V = h x , x , x , x i preserving the quadratic form x ( x, x, x ). If we identify x = ax + bx + cx + d with the matrix (cid:18) a bc d (cid:19) then ( x, x, x ) = 2 det( x ). Thus, with V identified with the set of 2 × M ∼ = GL × GL / GL ∇ where GL ∇ = { ( t, t − ) : t ∈ GL } , so that ( α, β ) ∈ M acts on x ∈ V by x αx ¯ β where ¯ β is the transpose of β . The element( α, β ) acts on the line h x i by det( αβ ). The modulus character is δ P (( α, β )) = | det( αβ ) | . This sets up the necessary notation to discuss the representations of PGSp .10. Representations of
PGSp In this section we list some irreducible non-supercuspidal representations of PGSp ( F ),relevant to this work. Observe that Langlands parameterization of irreducible representationsis known for all of Levi factors of parabolic subgroups of PGSp ( F ) (by Gan and Takeda [GT]for M ∼ = GSp ). Thus, following Shahidi [Sh], reducibility points of generalized principalseries can be computed using L -functions of Langlands parameters.10.1. Principal series representations for P . We first consider certain principal seriesrepresentations for the parabolic subgroup P = M N , where M ∼ = GL × GL / GL ∇ . Let τ be an irreducible representation of GL with L-parameter φ τ and the central character ω τ .Set I ( τ ⊗ τ ) = Ind PGSp P τ ⊗ τ and I ( s, τ ⊗ τ ) = Ind G P ( | det | s τ ) ⊗ ( | det | s τ )if we need to consider a family of induced representations. Then we have: Proposition 10.1. (i) If τ is unitary supercuspidal, then I ( s, τ ⊗ τ ) is reducible if and onlyif τ ∨ ∼ = τ (so ω τ = 1 ) and one of the following holds: • ω τ = 1 and s = ± / , in which case one has: −−−−→ δ ( τ ) −−−−→ I (1 / , τ ⊗ τ ) −−−−→ J (1 / , τ ⊗ τ ) −−−−→ , where δ ( τ ) is a generic discrete series representation. • ω τ = 1 (so τ is dihedral), Im( φ τ ) = S and s = ± ,in which case one has: −−−−→ σ gen [ τ ] −−−−→ I (1 , τ ⊗ τ ) −−−−→ J (1 , τ ⊗ τ ) −−−−→ , where σ gen [ τ ] is a generic discrete series representation. • ω τ = 1 , Im( φ τ ) = S and s = 0 , in which case one has: I (0 , τ ⊗ τ ) = I ( τ ⊗ τ ) gen ⊕ I ( τ ⊗ τ ) deg where I ( τ ⊗ τ ) gen is generic.(ii) If τ = st χ is a twisted Steinberg representation, then I ( s, τ ⊗ τ ) is irreducible except forthe following cases: • χ = 1 and s = ± / or ± / , in which case one has −−−−→ St PGSp −−−−→ I (5 / , st ⊗ st) −−−−→ J (5 / , st ⊗ st) −−−−→ , and −−−−→ Ind
PGSp P (St) gen −−−−→ I (1 / , st ⊗ st) −−−−→ J (1 / , st ⊗ st) −−−−→ . • χ = 1 but χ = 1 and s = ± / , in which case one has: −−−−→ σ gen [ χ ] −−−−→ I (1 / , st χ ⊗ st χ ) −−−−→ J (1 / , st χ ⊗ st χ ) −−−−→ , where σ gen [ χ ] is a generic discrete series representation. Principal series representations for P . Now we consider certain principal seriesrepresentations for the parabolic subgroup P = M N , where M ∼ = GL × GL . Let τ be an irreducible representation of GL with the central character ω τ . Set I ( τ ⊗
1) = Ind
PGSp P τ ⊗ I ( s, τ ⊗
1) = Ind G P ( | det | s τ ) ⊗ , the restriction of I ( s, τ ⊗
1) to Sp is a principal series inducedfrom | · | s ω τ ⊗ | det | s τ . In particular, if τ is unitary tempered and s >
0, this is a standardmodule. We have:
Proposition 10.2. If τ is unitary supercuspidal, then I ( s, τ ⊗ is reducible if and only if τ ∨ ∼ = τ (so ω τ = 1 ) and one of the following holds: • ω τ = 1 and s = ± / , in which case I (1 / , τ ⊗ has length 4 and has a uniqueirreducible submodule δ ( τ ) , which is a generic discrete series representation. • ω τ = 1 , Im( φ τ ) = S and s = 0 , in which case one has: I (0 , τ ⊗
1) = I ( τ ⊗ gen ⊕ I ( τ ⊗ deg where I ( τ ⊗ gen is generic. Principal series representations for P . Now we consider certain principal seriesrepresentations for the parabolic subgroup P = M N , where M ∼ = GL . Let τ be anirreducible representation of GL . We set I ( τ ) = Ind PGSp P τ. Then we have [Ta, Example 7.7 and Theorem 7.9:]:
OWE DUALITY AND DICHOTOMY 33
Proposition 10.3. (i) Assume that τ is discrete series representation with trivial centralcharacter. Then we have two cases: • If τ = τ ∨ then I ( τ ) ∼ = I ( τ ∨ ) is irreducible. • If τ ∼ = τ ∨ then I ( τ ) = I ( τ ) gen ⊕ I ( τ ) deg where I ( τ ) gen is generic.(ii) Let χ , χ , χ be three characters of F × such that χ · χ · χ = 1 , and let τ = τ ( χ , χ , χ ) be the associated principal series representation of GL ( F ) (which is possiblyreducible). Then the induced representation I ( τ ) is irreducible unless one of the followingtwo conditions hold: • χ i = | · | ± for some i or χ i /χ j = | · | ± for a pair i = j . • The three characters χ i are quadratic, non-trivial and pairwise different. Then I ( τ ) = I ( τ ) gen ⊕ I ( τ ) deg where I ( τ ) gen is generic. Principal series representations for P . Now we consider certain principal seriesrepresentations for the parabolic subgroup P = M N , where M ∼ = GSp . Let τ be anirreducible representation of GSp . We set I ( τ ) = Ind PGSp P τ and I ( s, τ ) = Ind PGSp P | ν | s τ where ν is the similitude character of GSp . Let τ be an irreducible supercuspidal repre-sentation of GSp ( F ) with trivial central character. Let ϕ τ : W D F → Spin ∼ = Sp(4) be itsLanglands parameter [GT]. Proposition 10.4.
Assume that τ is a supercuspidal representation of GSp ( F ) with trivialcentral character such that the parameter std ◦ ϕ τ contains the trivial representation, where std denotes the 5-dimensional standard representation of Spin . Then I ( s, τ ) is reducible ifand only if s = ± / , in which case one has: −−−−→ δ ( τ ) −−−−→ I (1 / , τ ) −−−−→ J (1 / , τ ) −−−−→ , where δ ( τ ) is a discrete series representation. Fourier-Jacobi and Shalika periods
In this section, we introduce and study a Fourier-Jacobi-type model for the group G anda Shalika period for PGSp . These are some of the periods that will appear when we considera game of ping-pong with periods for the dual pair G × PGSp , as discussed at the end ofthe introduction. Whittaker periods.
We begin by recalling the following results about Whittakerperiods from [GS04, Prop. 19 and Cor. 20], see also the appendix of [HKT].
Proposition 11.1.
Let Π be the minimal representation of E and let ( V ′ , ψ V ′ ) be a Whit-taker datum for PGSp (so V ′ is a maximal unipotent subgroup and ψ V ′ a generic characterof V ′ ). Then we have an isomorphism of G -modules: Π V ′ ,ψ ′ ∼ = ind G V ψ V where ( V, ψ V ) is a Whittaker datum for G . Corollary 11.2. (i) If π ∈ Irr( G ) is generic and Θ( π ) is its big theta lift to PGSp , then dim Hom V ′ (Θ( π ) , ψ ) = 1 so that Θ( π ) contains a unique irreducible generic subquotient and thus is nonzero.(ii) If π ∈ Irr( G ) is non-generic and τ ∈ Irr(PGSp ) is generic, then Hom G × PGSp (Π , π ⊗ τ ) = 0 . Fourier-Jacobi period of G . Let Q = LU be the 3-step maximal parabolic subgroupof G . Recall that [ L, L ] ∼ = SL corresponds to the long simple root β . Thus V = U β U isthe unipotent radical of the standard Borel subgroup of G . If we set J = [ L, L ] U , thenthe quotient of J by the center of U is the Jacobi group. Let ρ ψ be the unique irreduciblerepresentation of the 2-fold cover ˜ J , such that the center of U/ [ U, U ] ∼ = U α + β acts by ψ . If σ is a genuine representation of f SL , we have a representation of J on σ ⊗ ρ ψ . For π ∈ Irr( G )and genuine σ ∈ Irr( ˜SL ), the Fourier-Jacobi period of π with respect to σ is the spaceHom J ( π, σ ⊗ ρ ψ ) ∼ = Hom G ( π, Ind G J σ ⊗ ρ ψ ) . Recall that f SL has a family of principal series representations I ψ ( s ) such that when s = 1 /
2, one has a short exact sequence:0 → St ψ → I ψ (1 / → ρ + ψ → . Moreover, the contragredient of I ψ (1 /
2) is I ¯ ψ ( − / Lemma 11.3. If π ∈ Irr( G ) is generic and tempered (or has cuspidal support on Q ), then Hom G (ind G J ( I ψ ( − s ) ⊗ ρ ¯ ψ ) , π ∨ ) ∼ = Hom G ( π, Ind G J ( I ¯ ψ ( s ) ⊗ ρ ψ )) ∼ = C . for all s < / .Proof. Let B = T β U β ⊂ [ L, L ] ∼ = SL be the Borel subgroup, where T β is the one-dimensionaltorus generated by the image of the simple coroot β ∨ : GL −→ T . Observe that for anygenuine character χ of e B , we have an isomorphism of J -modulesInd f SL e B ( χ ) ⊗ ρ ψ ∼ = Ind JBU ( χ · ρ ψ ) OWE DUALITY AND DICHOTOMY 35 where f ⊗ v is mapped to a function on SL given by g f ( g ) · ρ ψ ( g )( v ). Since ρ ψ restrictedto e BU is induced, using the transitivity of induction, it is easy to check that I ¯ ψ ( s ) ⊗ ρ ψ ∼ = ind JT β N ( | · | s +3 / · ψ α + β ) , where N is the unipotent radical of the maximal parabolic P (so V = U α N ) and ψ α + β is acharacter of N nontrivial only on the root space U α + β ⊂ N as the subscript indicates. Herethe induction is not normalized.Thus, it follows by Frobenius reciprocity thatHom G ( π, Ind G J ( I ¯ ψ ( s ) ⊗ ρ ψ )) ∼ = Hom T β N ( π, | · | s +3 / · ψ α + β )and to prove the lemma, it suffices to show thatdim Hom T β N ( π, | · | s +3 / · ψ α + β ) = 1 if s < / ψ α be the character of U obtained by restricting a Whittaker character of V . As thenotation indicates, ψ α is only supported on the root space U α ⊂ U . Let γ = 3 α + 2 β be thehighest root. Observe that γ is perpendicular to α and therefore, π U,ψ α is naturally a modulefor T γ U β , a mirabolic subgroup of L ∼ = GL . Recall that the mirabolic subgroup has a uniqueirreducible infinite-dimensional (generic) representation, and all others are one-dimensionalwith the trivial action of U β . The infinite-dimensional representation is realized on C c ( T γ ),so it is a regular representation of T γ ∼ = GL . Since π is Whittaker generic, C c ( T γ ) has toappear with multiplicity one in π U,ψ α . Thus we have an exact sequence of T γ U β -modules0 → C c ( T γ ) → π U,ψ α → π V,ψ α → U β -coinvariants. Next, observe that π V,ψ α is a quotient of π N .Since we have assumed that π is tempered, the group T γ acts on π V,ψ α by characters whichare, after taking the absolute value, equal to | · | t +3 where t ≥
0. It follows thatdim Hom T γ U ( π, | · | s +3 · ψ α ) = 1 if s < T γ U is not even conjugate to T β N . Thelast step requires the technique of root exchange.Let N ′ be obtained by adding U − β to U and removing U α + β , so that N ′ is conjugate to N (by the simple Weyl reflection w β ). Now we claim that there is an isomorphismHom T γ U ( π, | − | s +3 · ψ α ) ∼ = Hom T γ N ′ ( π, | − | s +2 · ψ α ) . which sends ℓ on the LHS to an element ℓ ′ on the RHS defined by the convergent integral ℓ ′ ( v ) = Z U − β ℓ ( π ( u ) · v ) du Conversely, we can recover ℓ from ℓ ′ by integrating over U α + β . Assuming the claim, wecomplete the proof of the lemma by observing that the pair ( T β N, |−| s +2 · ψ α + β ) is conjugateto the pair ( T γ N ′ , | − | s +2 · ψ α ) (by the Weyl element w α · w β ). To justify the root exchange argument in the claim, we observe that U − β and U α + β generatea Heisenberg group with center U α , modulo higher order commutators. More precisely,consider the group V ′ = U · U β = N ′ · U α + β , which is a maximal unipotent subgroup of G and hence conjugate to V (by the simplereflection w β ). If we consider the lower central series of the unipotent group V ′ : V ′ ⊃ [ V ′ , V ′ ] = V ′ ⊃ V ′ = [ V ′ , V ′ ] ⊃ V ′ ⊃ { } , then V /V ′ is the Heisenberg group in question with center V ′ /V ′ = U α . Note moreover thatthe elements ℓ and ℓ ′ in the two Hom spaces in the claim both factors through π V ′ (whichis a module for the Heisenberg group V ′ /V ′ ). With this observation, the justification of theclaim is given by the following lemma, included as a convenience to the reader. (cid:3) Lemma 11.4.
Let H be a Heisenberg group. Let π be a smooth H -module. Let X and Y betwo maximal abelian subgroups of H . Let ψ X and ψ Y be characters of X and Y , agreeing onthe intersection X ∩ Y , and non-trivial on the center of H . Then we have an isomorphism Hom X ( π, ψ X ) ∼ = Hom Y ( π, ψ Y ) , ℓ ℓ ′ , defined by ℓ ′ ( v ) = Z Y/X ∩ Y ℓ ( π ( y ) v ) dy. Proof.
By the Frobenius reciprocity, we haveHom X ( π, ψ X ) ∼ = Hom H ( π, Ind HX ψ X ) and Hom Y ( π, ψ Y ) ∼ = Hom H ( π, Ind HY ψ Y ) . We also have an isomorphism Ind HX ψ X ∼ = Ind HY ψ Y , where every f ∈ Ind HX ψ X goes to f ′ ∈ Ind HY ψ Y defined by f ′ ( h ) = Z Y/X ∩ Y f ( yh ) ¯ ψ Y ( y ) dy. The lemma follows by combining this isomorphism with the two Frobenius reciprocity iso-morphisms. (cid:3)
Shalika period on
PGSp . We shall now discuss a Shalika period on PGSp .Recall the maximal parabolic subgroup P = M N of PGSp with Levi factor M ∼ =(GL × GL ) / GL ∇ and unipotent radical N . Let ψ be a generic character of N ( F ). Thestabilizer of ψ in the Levi group M is the diagonally embedded PGL ∆2 . The Shalika subgroupof PGSp is the semi-direct product S = P GL ∆2 ⋉ N and the Shalika character ψ S is the character ψ extended to S ( F ) (trivially on PGL ( F )).For any smooth representation π of PGSp ( F ), the Shalika period of π is the coinvariantspace π S,ψ S .This Shalikla period has already been exploited in [SWe]. Indeed, the following was shownin [SWe, Lemma 4.5]: OWE DUALITY AND DICHOTOMY 37
Proposition 11.5.
Let Π be the minimal representation of E and ( V, ψ V ) a Whittakerdatum for G . Then Π V,ψ V ∼ = ind PGSp S ψ S as PGSp -modules. Shalika period of Π . We now consider the minimal representation Π of the dual pair G × PGSp and determine its Shalika period Π S,ψ S as a representation of G ( F ). To describethe answer, we need to introduce some more notations.The group PGL acts by conjugation on the space M ( F ) of 2 × § × SL on C c ( M ( F )) which decomposes as a tensor product C c ( M ( F )) = C c ( M ◦ ( F )) ⊗ ρ ψ where M ◦ ( F ) is the space of trace zero matrices. We view C c ( M ◦ ( F )) ⊗ ρ ψ as a representationof the group J = [ L, L ] U ⊂ G introduced in § f SL and ρ ψ is the irreducible representation of ˜ J introduced in § acting trivially on ρ ψ , we see that C c ( M ( F )) becomes a representation of PGL × J .We are now ready to compute Π S,ψ S . Firstly, we haveΠ N ,ψ ∼ = ind G J ( C c ( M ◦ ( F )) ⊗ ρ ψ )as G × PGL -modules. It remains to compute the PGL -coinvariant of the RHS. We needthe following: Lemma 11.6.
Let H ⊂ G and L be three p -adic groups. Let W be a smooth H × L -module,and τ an irreducible representation of L . Let Θ( τ ) ⊗ τ be the maximal τ -isotypic quotient of W . If Ext L ( τ, τ ) = 0 then ind GH Θ( τ ) ⊗ τ is the maximal τ -isotypic quotient of ind GH W . Here ind stands for induction with compactsupport.Proof. Since Ext L ( τ, τ ) = 0, the kernel of the projection of W on Θ( τ ) ⊗ τ does not have τ as a quotient. Thus, it suffices to prove that if Hom L ( W, τ ) = 0, then Hom L (ind GH W, τ ) = 0.We shall prove that Hom L ((ind GH W ) K , τ ) = 0for any open compact subgroup K of G . Write G = ∪ i ∈ I Hg i K where I is an index set, andset K i = H ∩ g i Kg − i for every i ∈ I . Then, as an L -module, (ind GH W ) K is a direct sum of W K i . Since W K i is a direct summand of the L -module W , it follows that Hom L ( W K i , τ ) = 0,and this proves the lemma. (cid:3) We apply Lemma 11.6 taking H ⊂ G to be J ⊂ G and L = PGL . Since Ext (1 ,
1) =0, the lemma implies that computing PGL -coinvariants of Π N ,ψ boils down to computingthe PGL -coinvariant of C c ( M ◦ ( F )), where it is the full degenerate principal series I ¯ ψ (1 / Proposition 11.7.
As a representation of G ( F ) , one has: Π S,ψ S ∼ = (Π N ,ψ ) PGL ∼ = ind G J ( I ¯ ψ (1 / ⊗ ρ ψ ) . Howe Duality for G × PGSp : Tempered Case After the preparation of the previous 3 sections, we are now in a position to begin ourstudy of the theta correspondence for the dual pair G × PGSp . In this section, we shallshow the Howe duality theorem for tempered representations of G . The key is to show theanalog of Propositions 6.6 and 6.7 for generic representations of G . This will rely on anothercurious chain of containment given in the following lemma, which comes from a considerationof a game of ping-pong with periods. Lemma 12.1.
Let Π be the minimal representation of E . Let π ∈ Irr( G ) be tempered andlet ψ V : V → C × be a Whittaker character for G . Let H = PGSp and τ ∈ Irr( H ) betempered such that Hom G × H (Π , π ⊠ τ ) = 0 . Then we have the following natural inclusions
Hom V ( π, ψ V ) ⊆ Hom V (Θ( τ ) , ψ V ) ∼ = Hom S ( τ ∨ , ¯ ψ S ) ⊆ Hom S (Θ( π ∨ ) , ¯ ψ S ) ∼ = Hom G (ind G J I ψ (1 / ⊗ ρ ¯ ψ , π ∨ ) . If π is generic, then all of these spaces are one-dimensional.Proof. We examine each containment in turn: • The first inclusion arises from the surjection Θ( τ ) ։ π . • The second follows from the identityHom V (Θ( τ ) , ψ V ) ∼ = Hom V × H (Π , ψ V ⊠ τ ) ∼ = Hom H (Π V,ψ V , τ )combined with Proposition 11.5 (i.e. [SWe, Lemma 4,.5]):Π V,ψ V ∼ = ind HS ψ S and the Frobenius reciprocity. • For the third, observe that Θ(¯ π ) is the complex conjugate of Θ( π ). Since ¯ π ∼ = π ∨ and¯ τ ∼ = τ ∨ and we have Θ( π ∨ ) ։ τ ∨ . • The fourth follows from the identity,Hom S (Θ( π ∨ ) , ¯ ψ S ) ∼ = Hom S × G (Π , ¯ ψ S ⊠ π ∨ ) ∼ = Hom G (Π S, ¯ ψ S , π ∨ )combined with Proposition 11.7:Π S, ¯ ψ S ∼ = ind G J I ψ (1 / ⊗ ρ ¯ ψ and Frobenius reciprocity.If π is generic, then the first and the last spaces are one-dimensional, with the latter byLemma 11.3 applied to s = − / <
0. Hence, all spaces in the chain are one-dimensional. (cid:3)
We can now obtain the following two propositions as consequences of Lemma 12.1.
OWE DUALITY AND DICHOTOMY 39
Proposition 12.2.
Let τ ∈ Irr(PGSp ) be tempered. Then Θ( τ ) cannot have two irreducibletempered and generic quotients.Proof. Let π , π ∈ Irr( G ) be tempered and generic such that Θ( τ ) ։ π ⊕ π . Thendim Θ( τ ) V,ψ V ≥ . On the other hand, dim Θ( τ ) V,ψ V = 1 by Lemma 12.1, which is a contradiction. (cid:3) Remark:
This proposition is proved in [SWe] using the uniqueness of Shalika functional,however, the proof of this uniqueness is difficult. The proof here is based on Lemma 11.3whose proof is very elementary, based on the Frobenius reciprocity and the root exchangetechnique.
Proposition 12.3.
Let π ∈ Irr( G ) be tempered and generic. Then Θ( π ) cannot have twotempered irreducible quotients. In particular, its cuspidal component Θ( π ) c is irreducible or .Proof. Let τ , τ ∈ Irr(PGSp ) be irreducible tempered and such that Θ( π ) ։ τ ⊕ τ . ByLemma 12.1, applied to π ∨ , τ ∨ and then to π ∨ , τ ∨ , one has:1 = dim Hom S ( τ , ψ S ) = dim Hom S (Θ( π ) , ψ S ) = dim Hom S ( τ , ψ S ) . Since τ ⊕ τ is a quotient of Θ( π ),1 = dim Hom S (Θ( π ) , ψ S ) ≥ dim Hom S ( τ , ψ S ) + dim Hom S ( τ , ψ S ) = 2 , which is a contradiction. (cid:3) Combining Propositions 12.2 and 12.3 with the results of §
6, we can now show the Howeduality theorem for tempered representations of G : Theorem 12.4.
Let π ∈ Irr( G ) be tempered and consider its big theta lift Θ( π ) on PGSp .Then(i) Θ( π ) has finite length and a unique irreducible quotient θ ( π ) (if nonzero), which istempered.(ii) Moereover, for tempered π , π ∈ Irr( G ) , = θ ( π ) ∼ = θ ( π ) = ⇒ π ∼ = π . Proof. (i) We have seen (i) for non-generic π in Corollary 6.8. The proof for generic π is thesame, using Lemmas 6.2(i) and 6.3(ii), as well as Proposition 12.3.(ii) If one of π or π is nongeneric, then the desired result follows by Proposition 6.6. If π and π are both generic, then the desired result follows by Proposition 12.2. (cid:3) We also point out the following corollary:
Corollary 12.5.
Let π ∈ Irr( G ) be generic, supercuspidal and not a theta lift from PGL .Then Θ( π ) is generic, supercuspidal and irreducible.Proof. By [SWe], we have known that Θ( π ) is generic and supercuspidal (hence temperedand semisimple), but now we know by Proposition 12.3 that it is also irreducible. (cid:3) Jacquet Modules
The purpose of this section is to compute the various Jacquet modules of the minimalrepresentation of E with respect to the maximal parabolic subgroups of G and PGSp . Wenote that the results of this section are entirely self-contained, and do not depend on anyprior results in this paper. As consequences of the results here, we deduce Lemmas 6.2 and6.3 for the dual pair G × PGSp . Indeed, we shall determine in Theorem 14.1 the theta liftsof all non-tempered representations of G and PGSp precisely.13.1. Jacquet functors for G . Recall that P = M N and Q = LU are the two maximalparabolic subgroups of G as before, in standard position relative to a maximal split torus T in G and a choice of positive roots, so that P ∩ Q is a Borel subgroup. In particular,their Lie algebras arise from Z -gradings given by two fundamental co-characters. Since G is contained in E , as a memeber of the dual pair, the two co-characters give Z -gradings ofthe Lie algebra of E , defining parabolic subgroups P = MN and Q = LU of E , whoseintersections with G are P and Q , respectively. The Lie types of the Levi factors M and L are D and A × A , as explained in [GS99]. In the rest of the paper, we shall fix thefollowing: • The group P acts on C c (GL ) and C c (GL ) by left translation by g and det( g ),respectively, via the identification M ∼ = GL . • The group Q acts on C c (GL ) and C c (GL ) by left translation by g and det( g ),respectively, via the identification L ∼ = GL . • Let ¯ B be the group of lower-triangular matrices in GL . Then ¯ B acts on C c (GL ) byright translation by the (1 ,
1) matrix entry of g ∈ ¯ B .We identify M ∼ = GL such that the action of M on N/ [ N, N ] is the symmetric cube ofthe standard representation of GL twisted by determinant inverse. In particular, a scalarmatrix ( z, z ) in GL acts by z . We have [MS, Theorem 6.1], Proposition 13.1.
Let H = PGSp . As a GL × H -module, r P (Π) (the normalized Jacquetfunctor) has a filtration with three successive sub quotients (top to bottom): (1) δ − / P · Π N = Π D · | det | / ⊕ Π ∅ · | det | / . (2) Ind GL × H ¯ B × P ( δ · C c (GL )) . (3) Ind HP C c (GL ) .Here, note that: - In (1), the center of M ∼ = GL acts trivially on both Π D and Π ∅ , the minimal andthe trivial representation of the Levi M . - In (2), δ = | · | − / × | · | is a character of the group ¯ B of lower triangular matrices in GL . Let W be the Weil representation for the similitude dual pair GL × GSO ; see [Ro] wheretheta correspondences for similitude groups are treated in detail. Observe that GSO ∼ =(GL × GL ) / GL ∇ , with the isomorphism realized by latter acting on the space M ( F ) of 2 × with L so that the action of L on U/ [ U, U ] is the standard
OWE DUALITY AND DICHOTOMY 41 representation of GL . The irreducible quotients of W are π ∨ ⊗ π ⊗ π , where π is an irreduciblerepresentation of GL . We need a slight refinement of this to the big theta lifts. Lemma 13.2.
Consider the similitude theta correspondence for the dual pair GL × GSO on W . Let π be an irreducible generic representation of GL . Then Θ( π ∨ ) = π ⊗ π and Θ( π ⊗ π ) = π ∨ .Proof. Let V ∼ = F be a maximal unipotent subgroup of GL and ψ a non-trivial character of V . We shall use that W V,ψ = C c (GL ), the regular representation of GL , where V is in anyof the three GL .We have Θ( π ⊗ π ) ⊗ ( π ⊗ π ) as a quotient of W . Apply the functor of ( V, ψ )-coinvariants,with V sitting in one of GL factors of GSO , to conclude that Θ( π ⊗ π ) ⊗ π is a quotient ofthe regular representation of GL . This implies that Θ( π ⊗ π ) ∼ = π ∨ , as desired. In the otherdirection, similar arguing shows that Θ( π ∨ ) cannot be an extension of π ⊗ π by π ⊗ π . Thusthe lemma holds except perhaps when π is a character twist of the Steinberg representationst. For example, Θ(st) could be a non trivial extension of st ⊗ st by 1 ⊗
1. ButExt (1 ⊗ , st ⊗ st) = 0thus one can have a non-trivial extension of these two representations of GSO only if thecenter of GSO does not act by scalars on Θ(st). But it does, since the centers of the threeGL are identified, the action on Θ(st) is equal to the action on st where it is scalar action. (cid:3) Proposition 13.3.
Let H = PGSp . As a GL × H -module, r Q (Π) (the normalized Jacquetfunctor) has a filtration with three successive sub quotients (top to bottom): (1) δ − / Q · Π U = Π A · | det | / ⊕ Π A · | det | . (2) Ind GL × H ¯ B × P ( δ · C c (GL )) . (3) Ind HP W . - In (1), the center of L ∼ = GL acts trivially on both Π A and Π A , the minimal and aprincipal series representation of the two factors of L . - In (2) δ = | · | / × | · | is a character of the group ¯ B of lower triangular matrices in GL .Proof. This proposition is entirely similar to Proposition 6.8 in [GS99], which treated thecase of non-split form of H , except the character δ was not determined there. This is doneas follows. For a generic character χ of GL , representations I Q ( χ ) and I ( χ ⊗ χ ) are bothirreducible and I Q ( χ ) ⊗ I ( χ ⊗ χ ) is a quotient of Π, this follows from the bottom factor (3)of the filtration. Hence r Q ( I Q ( χ )) ⊗ I ( χ ⊗ χ ) is a quotient of r P (Π). Now determining δ isan easy exercise using r Q ( I Q ( χ )). (cid:3) Non-tempered representations.
We enumerate the nontempered irreducible repre-sentations of G using the discussion from Section 3. Let P = M N and Q = LU be thetwo maximal parabolic subgroups in G as before. Their Levi groups are isomorphic to GL .Let τ be a representation of GL , and let I P ( τ ) and I Q ( τ ) be the corresponding normalizedinduced representations of G . Irreducible, non-tempered representations of G are describedas follows, where τ is irreducible, and ω τ is the central character of τ . (a) Unique irreducible quotient of I Q ( τ ) where τ is an unramified twist of a temperedrepresentation such that | ω τ | = | · | s for some s > I P ( τ ) where τ is an unramified twist of a temperedrepresentation such that | ω τ | = | · | s for some s > I P ( τ ) where τ is the unique quotient of a representation inducedfrom an ordered pair of characters χ , χ such that | χ | = | · | s , | χ | = | · | s where s > s > I Q ( τ ) and I P ( τ ) are standard modules, while in (c), I P ( τ ) is a quotient ofa standard module associated to the minimal parabolic P ∩ Q . In any case, each of theseinduced representations has a unique irreducible quotient which we denote by J Q ( τ ) in (a)and by J P ( τ ) in (b) and (c). These representations J Q ( τ ) and J P ( τ ) exhaust the irreduciblenontempered representations of G .We also enumerate some relevant nontempered representations of PGSp . Let P i = M i N i , i = 1 , , . Let I i ( σ ) denote the representa-tion of PGSp obtained by normalized parabolic induction from P i , and let I jk ( σ ) denote therepresentation of PGSp obtained by normalized parabolic induction from P j ∩ P k . We shallconsider the following non tempered representations of PGSp , corresponding to the cases(a), (b) and (c) above:(a’) If τ is an irreducible representation of L = GL satisfying the conditions of (a) above,let σ = τ ⊗ τ be a representation of M ∼ = GL × GL / GL ∇ ∼ = GSO . Then I ( σ ) isa standard module, with unique irreducible quotient J ( σ ) = J ( τ ⊗ τ ).(b’) If τ is an irreducible representation of M = GL , satisfying the conditions of (b)above, let σ = τ ⊗ M ∩ M ∼ = GL × GL . Then I ( σ ) is astandard module with unique irreducible quotient J ( σ ) = J ( τ ⊗ τ is an irreducible representation of M = GL , satisfying the conditions of (c)above, let σ = τ ⊗ M ∩ M ∼ = GL × GL . Then I ( σ ) isa quotient of a standard module associated to the Borel subgroup, Hence, it has aunique irreducible quotient which we denote by J ( σ ) = J ( τ ⊗ Theta lifts from G . Now the following lemma attempts to compute the theta liftsof the above non tempered representations of G to PGSp . Lemma 13.4.
Let π ∈ Irr( G ) be non-tempered. • If π ⊂ I Q ( τ ∨ ) where τ is as in (the first bullet (a) above, then Θ( π ) is a quotient of I ( τ ⊗ τ ) and hence has finite length. Moreover, Θ( J ( τ ⊗ τ )) = 0 where J ( τ ⊗ τ ) isthe unique irreducible quotient of I ( τ ⊗ τ ) . • If π ⊂ I P ( τ ∨ ) where τ is as in (the second bullet) (b) and (c) above, then Θ( π ) is aquotient of I ( τ ⊗ and hence has finite length. Moreover, Θ( J ( τ ⊗ = 0 where J ( τ ⊗ is the unique irreducible quotient of I ( τ ⊗ .Proof. Let Π be the minimal representation, and π ∈ Irr( G ). We shall use the fact thatΘ( π ) ∗ ∼ = Hom G (Π , π ) OWE DUALITY AND DICHOTOMY 43 as non-smooth H = PGSp -modules, where former is the linear dual of Θ( π ). Assume that π ⊂ I Q ( τ ∨ ). ThenΘ( π ) ∗ = Hom G (Π , π ) ⊂ Hom G (Π , I Q ( τ ∨ )) ∼ = Hom L ( r Q (Π) , τ ∨ ) . Now we shall use the filtration of r Q (Π) from Proposition 13.3.Let Π , Π and Π denote the three sub quotients in the same order. Observe thatExt iL (Π , τ ∨ ) are trivial from the central character considerations, since the central characterof τ ∨ is a negative power of | z | . Hence we have a long exact sequence0 → Hom L (Π , τ ∨ ) → Hom L ( r Q (Π) , τ ∨ ) → Hom L (Π , τ ∨ ) → Ext L (Π , τ ∨ )Since Π is induced from ¯ B , by the second adjointness,Ext iL (Π , τ ∨ ) ∼ = Ext iT (Ind HP ( δ · C c (GL )) , r B ( τ ∨ ))where T = GL × GL , the maximal torus in B . Observe that the action of the second GL on Ind HP ( δ · C c (GL )) is | · | , and this is different from the action on r B ( τ ∨ ) by our assumptionon τ . Hence Ext iL (Π , τ ∨ ) = 0 for all i , and we can conclude thatHom L ( r Q (Π) , τ ∨ ) ∼ = Hom L (Π , τ ∨ ) ∼ = Hom L (Ind HP W, τ ∨ ) , where, for the second isomorphism, we have simply substituted the explicit expression forΠ given in Proposition 13.3. By [GG06, Lemma 9.4], the maximal τ ∨ isotypic quotientof Ind HP W is (Ind HP Θ( τ ∨ )) ⊗ τ ∨ where Θ( τ ∨ ) is the big theta lift for the similitude thetacorrespondence on W . Since τ is generic, Lemma 13.2 shows that Θ( τ ∨ ) = τ ⊗ τ and itfollows that Hom L (Π , τ ∨ ) ∼ = I ( τ ⊗ τ ) ∗ . Hence Θ( π ) ∗ ⊂ I ( τ ⊗ τ ) ∗ , and Θ( π ) ∨ ⊂ I ( τ ⊗ τ ) ∨ by taking smooth vectors. Thus Θ( π )is a quotient of I ( τ ⊗ τ ). Observe that we have proved in the process that I ( τ ⊗ τ ) is aquotient of Π, so that Θ( J ( τ ⊗ τ ) = 0. This proves the first bullet. The proof of the secondis completely analogous. (cid:3) Jacquet functors for
PGSp . Recall that in PGSp , we have fixed three standardmaximal parabolic subgroups P , P and P . They correspond to Z -gradings of the Liealgebra of PGSp given by three fundamental co-characters. The action of each of thesethree co-characters gives a Z -grading of the Lie algebra of E , and these gradings definethree parabolic subgroups P , P and P of E . To recognize these parabolic subgroups,perhaps it is easiest to proceed as follows. Observe that the E Dynkin diagram contains aunique D subdiagram. We embed G into D . The centralizer of G in the split, adjoint E is PGSp . Let P be the parabolic subgroup of E , whose Levi factor has the type D . Thisparabolic is contained in precisely three maximal parabolic subgroups denoted by P , P and P , whose Levi factor types are, D , A × D and E , respectively. The intersection of P i and PGSp is P i , for each i . We write P i = M i N i and P i = M i N i the Levi decompositionsfor these parabolic subgroups.Case P : This is treated in [MS, § P and P are abelian, M ∼ = GL and the modular character is δ P ( m ) = | det( m ) | . Let O denote the space of trace 0 elements in the octonion algebra O . On the space O , wehave the natural diagonal action ( x, y, z ) ( gx, gy, gz ) of g ∈ G and the row-vector action( x, y, z ) ( gx, gy, gz ) m − of m ∈ GL . Let Ω ⊂ O be the set of all ( x, y, z ) such that thelinear subspace h x, y, z i ⊂ O is a null-space for octonion multiplication i.e. the product ofany two elements in the space is 0. Such non-zero null-spaces in O are of dimension 1 or 2.We have an exact sequence of G × GL -modules0 → C c (Ω) → Π N → Π N → g, m ) ∈ G × GL acts on f ∈ C c (Ω) by(( g, m ) · f )( x, y, z ) = | det( m ) | · f (( g − x, g − y, g − z ) m ) . The group G × GL acts on Ω with two orbits Ω and Ω where Ω i is the subset of triples( x, y, z ) such that h x, y, z i has dimension i . Thus C c (Ω) has a filtration with C c (Ω ) asubmodule and C c (Ω ) a quotient. Each of these can be explicitly described as G × GL -modules.In order to state the result, let Q and Q be the maximal parabolic subgroups of GL stabilizing subspaces consisting of row vectors ( ∗ , ,
0) and ( ∗ , ∗ , × GL andGL × GL , respectively. Their modular characters are δ Q ( g , g ) = | g | − · | det( g ) | and δ Q ( g , g ) = | det( g ) | − · | g | . Recall that r P (Π) = δ − / P · Π N is the normalized Jacquet module. Then: Proposition 13.5.
As a G × GL -module, r P (Π) has a filtration with three successivesubquotients (from top to bottom): (1) δ − / P · Π N = Π E ⊕ Π ∅ · | det | . (2) Ind G × GL Q × Q ( δ · C c (GL )) . (3) Ind G × GL P × Q ( C c (GL )) .Here, note that: - In (1), the center of M ∼ = GL acts trivially on both Π E and Π ∅ , the minimal andthe trivial representation of the Levi M . - In (2), δ ( g , g ) = | g | − / × | det( g ) | / is a character of Q . - For i = 1 , , Q i acts on C c (GL i ) by right translations via the factor GL i as describedabove in § Case P : This case is not in the literature; however, it is similar to the computation of theJacquet module of the minimal representation of E with respect to a maximal parabolicsubgroup of F in [SWo, § P and P are Heisenberggroups with M ∼ = GSp . Let ν be the similitude character of GSp . The modulus characterof M is δ P ( m ) = | ν ( m ) | . Recall that O is the space of trace 0 octonions. On O , we have the row-vector action( x, y, x ′ , y ′ ) ( x, y, x ′ , y ′ ) m − of m ∈ GSp preserving the form O → ∧ O given by( x, y, x ′ , y ′ ) x ∧ x ′ + y ∧ y ′ . Let Ω ⊂ O be the set of all nonzero ( x, y, x ′ , y ′ ) such that the linear subspace h x, y, x ′ , y ′ i ⊂ O is a null-space for octonion multiplication and x ∧ x ′ + y ∧ y ′ = 0. We have an exactsequence of G × GSp -modules0 → C c (Ω) → Π N → Π N → g, m ) ∈ G × GSp acts on f ∈ C c (Ω) by(( g, m ) · f )( x, y, x ′ , y ′ )) = | ν ( m ) | · f (( g − x, g − y, g − x ′ , g − y ′ ) m ) . Now the group G × GSp acts on Ω with two orbits Ω and Ω , where Ω i is the subsetof quadruples ( x, y, x ′ , y ′ ) such that h x, y, x ′ , y ′ i has dimension i . Thus C c (Ω) has a filtrationwith C c (Ω ) as a submodule and C c (Ω ) as a quotient. Each of these can be explicitlydescribed as G × GSp -modules.In order to state the result, let Q and Q be the maximal parabolic subgroups of GSp stabilizing subspaces consisting of row vectors ( ∗ , , ,
0) and ( ∗ , ∗ , , L ∼ = GL × GL be the Levi subgroup of Q such that ( g , g ) ∈ GL × GL acts on thequadruples, after rearranging the order, by( x, x ′ , y, y ′ ) ( xg − , x ′ g det( g ) − , ( y, y ′ ) g − ) . Let L ∼ = GL × GL be the Levi subgroup of Q such that ( g , g ) ∈ GL × GL acts on thequadruples by ( x, y, x ′ , y ′ ) (( x, y ) g − , ( x ′ , y ′ ) g − g ⊤ ) . The similitude character ν , restricted to L and L , is given by ν ( g , g ) = det g and ν ( g , g ) = g respectively, and the modulus characters are δ Q ( g , g ) = | g | − · | det( g ) | and δ Q ( g , g ) = | det( g ) | − · | g | . Recalling that r P (Π) = δ − / P · Π N is a normalized Jacquet module, we have: Proposition 13.6.
As a G × GSp -module, r P (Π) has a filtration with three successivesubquotients (from top to bottom): (1) δ − / P · Π N = Π D · | ν | / ⊕ Π ∅ · | ν | / . (2) Ind G × GSp Q × Q ( δ · C c (GL )) . (3) Ind G × GSp P × Q ( C c (GL )) .Here, note that - In (1), the center of M ∼ = GSp acts trivially on both Π D and Π ∅ , the minimal andthe trivial representation of the Levi M . - In (2) δ ( g , g ) = | g | − / × | det( g ) | / , a character of Q . - For i = 1 , , Q i acts on C c (GL i ) by right translations via the factor GL i as describedabove in § Case P : A variant of this case can be found in [GS99] for the non-split form of PGSp .However, for the split case considered in this paper, the Jacquet module filtration containsan additional “middle” term.The unipotent radical subgroups N ⊂ P and N ⊂ P are two-step nilpotent subgroups.Let Z ⊂ N be the center of N . We now explain how the kernel of the natural projectionΠ Z → Π N contributes to Π N . We have0 → C c ( ω ) → Π Z → Π N → ω is the M -highest weight orbit in¯ N / ¯ Z ∼ = O ⊗ M ( F ) = M ( O )where ¯ N is the unipotent group opposite to N , and M ( F ) is the set of two-by-two matrices.(In the non-split case M ( F ) is replaced by a division algebra, so Ω is empty; see the discussionon [GS99, Pg. 137].) Recall that the type of M is D × A and ¯ N / ¯ Z ∼ = F ⊗ F where F is a spin-module of D . In the above isomorphism we assume that A acts from theright on M ( O ), and columns are vectors in the spin-module. Thus ω is the set of non-zeromatrices (cid:18) x x ′ y y ′ (cid:19) where the two columns are linearly dependent over F and each column (if non-zero) is ahighest weight vector in the spin-module. Let Ω be the subset of ω such that x, x ′ , y, y ′ aretraceless octonions. We have an exact sequence of G × M -modules0 → C c (Ω) → (Π Z ) N → Π N → g, ( α, β )) ∈ G × (GL × GL ) / GL ∇ acts on f ∈ C c (Ω) by(( g, ( α, β )) · f ) (cid:18)(cid:18) x x ′ y y ′ (cid:19)(cid:19) = δ ′ ( αβ ) · f (cid:18) ¯ α (cid:18) g − x g − x ′ g − y g − y ′ (cid:19) β (cid:19) for some (unknown) character δ ′ . The highest weight orbit in the 16-dimensional spin moduleis described in [MS]. That result, applied to each column of M ( O ), implies that x, x ′ , y, y ′ (of an element in Ω) generate a nil-subalgebra. The group G × M acts on Ω with two orbitsΩ and Ω , where Ω i consists of elements such that h x, x ′ , y, y ′ i has dimension i . Thus C c (Ω),as a G × M -module, has C c (Ω ) as a submodule and C c (Ω ) as quotient. Proposition 13.7.
As a G × (GL × GL ) / GL ∇ -module, r P (Π) has a filtration with foursuccessive sub quotients: (1) δ − / P · Π N = Π D · | det | / ⊕ Π A · | det | / . (2) Ind G × (GL × GL ) / GL ∇ Q × ( ¯ B × ¯ B ) / GL ∇ ( δ · C c (GL )) . (3) Ind G × GL P × ¯ B ( C c (GL )) . (4) Ind G Q W .Here, note that: - In (1), the second SL ⊂ M acts trivially on the first summand, and the first SL ⊂ M acts trivially on the second summand. The center of M acts trivially on both OWE DUALITY AND DICHOTOMY 47 Π D and Π A , the minimal and a principal series representation of the two factors of M . - In (2) δ = | · | / × | · | on each ¯ B . - In (3), ¯ B is the subgroup of the second factor GL of M . It acts on C c (GL ) by righttranslation by the scalar given by the (1 , matrix entry. The first factor GL of M acts by right translations on C c (GL ) . - In (4), W is the Weil representation of GL × (GL × GL ) / GL ∇ ∼ = GL × GSO . This proposition is a combination of [GS99, Proposition 8.1], which accounts for the bottompiece of the filtration (4), and the above discussion. The pieces (2) and (3) are the spacesof functions C c (Ω ) and C c (Ω ), respectively. This also assumes that we have explicated thecharacter δ ′ appearing in the action on C c (Ω). To that end, observe that (3) (or any unknowntwist) gives a correspondence of generic principal series representations of G and PGSp thathas to be compatible with the one in Lemma 13.4, and this determines δ ′ uniquely.13.5. Theta lifts from
PGSp . Using Propositions 13.5, 13.6 and 13.7, we can now provethe following analog of Lemma 13.4.
Lemma 13.8.
Let σ ∈ Irr(PGSp ) be non-tempered. Then Θ( σ ) = 0 unless σ is as describedin Lemma 13.4. More precisely, • If σ ⊂ I ( τ ∨ ⊗ τ ∨ ) , then Θ( σ ) is a quotient of I Q ( τ ) and hence has finite length.Moreover, Θ( J Q ( τ )) = 0 where J Q ( τ ) is the unique irreducible quotient of I Q ( τ ) . • If σ ⊂ I ( τ ∨ ⊗ , then Θ( σ ) is a quotient of I P ( τ ) , and hence has finite length.Moreover, Θ( J P ( τ )) = 0 where J P ( τ ) is the unique irreducible quotient of I P ( τ ) .Proof. We set H = PGSp . Assume that σ is a Langlands quotient of a standard modulefor the maximal parabolic P . Then σ ⊆ I ( τ ∨ ⊗ τ ∨ ) where τ and τ have the same centralcharacter and are both tempered representations of GL twisted by a positive power of | det | .Then Hom H (Π , σ ) ⊆ Hom H (Π , I ( τ ∨ ⊗ τ ∨ ) ∼ = Hom M ( r P (Π) , τ ∨ ⊗ τ ∨ ) . Let Π i , i = 1 , , , r P (Π) as in Proposition 13.7, in the same order.We clam that Hom M ( r P (Π) , τ ∨ ⊗ τ ∨ ) ∼ = Hom M (Π , τ ∨ ⊗ τ ∨ ). Assume this claim for amoment. ThenHom M ( r P (Π) , τ ∨ ⊗ τ ∨ ) ∼ = Hom M (Π , τ ∨ ⊗ τ ∨ ) ∼ = Hom M (Ind G Q W, τ ∨ ⊗ τ ∨ )where W is the Weil representation of GL . This implies that Θ( σ ) = 0 unless τ ∼ = τ , and ifwe denote this representation as τ , then Θ( σ ) is a non-zero quotient of the standard module I Q ( τ ). In order to prove the claim, we need to show that Ext nM (Π i , τ ∨ ⊗ τ ∨ ) = 0 for all n and i <
4. Consider i = 3. Then, using the (second) Frobenius reciprocity for induction from¯ B to GL , the first factor of M , we haveExt iM (Ind G × GL P × ¯ B ( C c (GL )) , τ ∨ ⊗ τ ∨ ) ∼ = Ext iT × GL (Ind G P ( C c (GL )) , r B ( τ ∨ ) ⊗ τ ∨ )where T ∼ = GL × GL is the torus of diagonal matrices in GL . Now recall that the secondGL acts trivially on Ind G P ( C c (GL )). On the other hand, since τ is a tempered with apositive twist of | det | , the second GL acts on r B ( τ ∨ ) with characters χ such that | χ | is anegative power of absolute value. This prove the vanishing for i = 3. The other two cases arejust as easy or even easier: for i = 1 vanishing follows from central character considerations, and for i = 2 using Frobenius reciprocity where it suffices that either τ or τ is twist of atempered representation by a positive power of | det | .Now assume that σ is a Langlands quotient of a standard module for the parabolic P = P ∩ P . Then, by induction in stages, we get that σ ⊆ I ( τ ∨ ⊗ τ ∨ ) where τ is a twistof a tempered representation by a positive power of | det | . This is enough to show thatHom M (Π i , τ ∨ ⊗ τ ∨ ) = 0 for i <
4. Thus, if Θ( σ ) = 0 then Hom M (Π , τ ∨ ⊗ τ ∨ ) = 0. Thisimplies that τ ∼ = τ , contradicting that σ is a Langlands quotient of a standard module forthe parabolic P . Hence Θ( σ ) = 0.If σ is a Langlands quotient of a standard module for the parabolic P = P ∩ P then,by induction in stages, we get that σ ⊆ I ( τ ∨ ⊗ τ ∨ ) where now τ is a twist of a temperedrepresentation by a positive power of | det | . In this case Hom M (Π i , τ ∨ ⊗ τ ∨ ) = 0 for i = 3by repeating the above arguments. For i = 3 we haveHom M (Ind G × GL P × ¯ B ( C c (GL )) , τ ∨ ⊗ τ ∨ ) ∼ = Hom T × GL (Ind G P ( C c (GL )) , r B ( τ ∨ ) ⊗ τ ∨ )and the last space is isomorphic toHom T × GL (Ind G P ( τ ) ⊗ τ ∨ , r B ( τ ∨ ) ⊗ τ ∨ ) . Recall that T = GL × GL and the second GL acts trivially on Ind G P ( C c (GL )) and henceon its quotient Ind G P ( τ ) ⊗ τ ∨ . The first GL acts on this space by the central characterof τ , which is equal to the central character of τ , hence it is a nontrivial character, say χ .Hence the above Hom space, if non-zero, is non-trivial if and only if χ ⊗ τ , and then it is isomorphic toHom GL (Ind G P ( τ ) ⊗ τ ∨ , τ ∨ ) ∼ = Hom(Ind G P ( τ ) , C ) = I P ( τ ) ∗ . Summarizing, Θ( σ ) = 0 implies that Θ( σ ) is a quotient of I P ( τ ). It follows that J P ( τ ) ⊗ σ is a quotient of Π, where J P ( τ ) is the unique irreducible quotient of I P ( τ ). But, by Lemma13.4, J P ( τ ) does not lift to σ . This is a contradiction, hence Θ( σ ) = 0.The remaining non-tempered representations of H (associated to standard modules inducedfrom P , P , P or P ) are easily dealt with using r P (Π) and r P (Π). We leave details tothe reader. (cid:3) Consequences of Jacquet Module Computations
We can now draw some definitive consequences of the Jacquet module computations ofthe previous section. In particular, we shall determine the theta lift of nontempered repre-sentations explicitly, and also complete the proofs of Lemmas 6.2 and 6.3 for the dual pair G × PGSp .14.1. Lift of nontempered representations.
Taken together, Lemmas 13.4 and 13.8 allowus to determine the theta lift of nontempered representations explicitly:
Theorem 14.1.
We have: (a) Θ( J Q ( τ )) is a nonzero quotient of I ( τ ⊗ τ ) and hence has finite length with uniqueirreducible quotient J ( τ ⊗ τ ) . Likewise, Θ( J ( τ ⊗ τ )) is a nonzero quotient of I Q ( τ ) and hence has finite length with unique irreducible quotient J Q ( τ ) . OWE DUALITY AND DICHOTOMY 49 (b) Θ( J P ( τ )) is a nonzero quotient of I ( τ ⊗ and hence has finite length with uniqueirreducible quotient J ( τ ⊗ . Likewise, Θ( J ( τ ⊗ is a nonzero quotient of I P ( τ ) and hence has finite length with unique irreducible quotient J P ( τ ) . (c) For all other nontempered σ ∈ Irr(PGSp ) different from those in (a) and (b), Θ( σ ) =0 .In particular, if π ⊗ σ ∈ Irr( G × PGSp ) is such that π ⊗ τ is a quotient of the minimalrepresentation Π , then π nontempered ⇐⇒ σ nontempered . Hence, we have shown Lemma 6.2 for nontempered representations and also Lemma 6.3.
Finiteness of Θ( π ) nc . To complete the proof of Lemma 6.2, we need to show that fortempered π ∈ Irr( G ) and σ ∈ Irr(PGSp ), the noncuspidal components Θ( π ) nc and Θ( σ ) nc are of finite length.To show that Θ( π ) nc has finite length, it suffices to show that for each maximal parabolicsubgroup P i = M i N i (with 1 ≤ i ≤
3) of PGSp , the Jacquet module J P i (Θ( π )) has finitelength as an M i -module. In other words, we need to show that the multiplicity space of themaximal π -isotypic quotient of r P i (Π) has finite length as an M i -module.We have described in Propositions 13.5, 13.6 and 13.7 an equivariant filtration of r P i (Π)as an G × M i -module and described the successive quotients. It suffices to show that, foreach of these successive quotients Σ, the multiplicity space of the π -isotypic quotient of Σhas finite length. We shall explain how this can be shown, depending on whether Σ is atop piece of the filtration or not. The difference lies in the fact that the top piece of thefiltration involves a minimal representation of a smaller group M i and hence one needs toconsider theta correspondence in lower rank situations. When Σ is not the top piece of thefiltration, the finite length of the multiplicity space of the maximal π -isotypic quotient of Σas an M i -module follows readily from the explicit description of Σ. We give two examples asillustration: • Consider the case of P = GL · N . The bottom piece of the filtration in Proposition13.5 is Σ = Ind G × GL P × Q C c (GL ) . Then for π ∈ Irr( G ),Θ Σ ( π ) ∗ := Hom G (Σ , π ) ∼ = Hom M (cid:16) Ind M × GL M × Q C c (GL ) , r ¯ P ( π ) (cid:17) where M ∼ = GL . Now r ¯ P ( π ) is a finite length M -module and for any of its irreduciblesubquotient σ , Hom M (cid:16) Ind M × GL M × Q C c (GL ) , σ (cid:17) ∼ = (cid:16) Ind GL Q σ ∨ (cid:17) ∗ using the fact that the maximal σ -isotypic quotient of the regular representation C c (GL ) is of the form σ ∨ ⊗ σ . On taking smooth vectors (which is a left exactfunctoir), we see that Θ Σ ( π ) ∨ has a finite filtration whose successive quotients aresubmodules of Ind GL Q σ ∨ for some irreducible σ . In particular, Θ Σ ( π ) has finite length. • Consider the case of P = M · N with M = (GL × GL ) / GL ∇ ∼ = GSO . Thebottom piece of the filtration in Proposition 13.7 isΣ = Ind G × M Q × M W where W is the Weil representation for GL × GSO . Then for π ∈ Irr( G ),Θ Σ ( π ) ∗ := Hom G (Σ , π ) ∼ = Hom L × M ( W, r Q ( π ))Now r Q ( π ) has finite length as L -module (where L ∼ = GL ) and if σ is an irreduciblesubquotient, Hom L × M ( W, σ ) = Θ W ( σ ) ∗ where Θ W ( σ ) is the big theta lift of σ ∈ Irr(GL ) to GSO , which has finite length by the Howe duality theorem for classical(similitude) theta correspondence. From this, one deduces as above that Θ Σ ( π ) hasfinite length as an M -module.Now let’s consider the case when Σ is the top piece of the filtration. From Propositions13.5, 13.6 and 13.7, we see that we need to consider the following theta correspondences inlower rank: • G × PGL in E : for this case, the finite length of the big theta lift has been verifiedin Theorem 8.5. • G × SO ⊂ SO or G × SO ⊂ SO ; we shall now treat these two cases togetherin the following proposition. Proposition 14.2.
Let Π n be the minimal representation of SO(2 n ) for n = 5 or . Thenfor tempered π ∈ Irr( G ) , Θ n ( π ) is a finite length H n -module where H n = SO n − .Proof. We shall use the fact that the minimal representation of SO n ( n = 5 or 6) is the bigtheta lift of the trivial representation of SL (see [Y, Prop. 8.4] for the irreducibility of thisbig theta lift) and then appeal to the seesaw identity arising from the seesaw diagram:˜ SL × ˜ SL PPPPPPPPPPPPP SO n SL ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ G × SO n − From this, we see that(14.3) Θ n ( π ) ∗ = Hom G (Π n , π ) = Hom SL (Ω n − ,ψ ⊗ ˜Θ ¯ ψ ( π ) , C ) ∼ = Hom ˜ SL (Ω n − ,ψ , ˜Θ ψ ( π ))as H n -modules, where • Ω n − ,ψ is the Weil representation of ˜SL × SO n − (with respect to a nontrivialadditive character ψ of F ); • ˜Θ ψ ( π ) denotes the big ψ -theta lift of π to ˜SL , with respect to the Weil representationΩ ψ of ˜ SL × SO ⊃ ˜SL × G .We see in particular that if Θ n ( π ) is nonzero, then π has nonzero ψ -theta lift to ˜SL . More-over, it remains now to show that ˜Θ ψ ( π ) has finite length as an ˜SL -module; the desiredresult would then follow from this and the Howe duality theorem for ˜SL × H n . OWE DUALITY AND DICHOTOMY 51
Now the theta correspondence for ˜SL × G has been completely determined in [GG06],though the finiteness of ˜Θ ψ ( π ) was not formally stated there. Let us see how this finitenesscan be deduced from [GG06].As before, let us write ˜Θ ψ ( π ) = ˜Θ( π ) c ⊕ ˜Θ( π ) nc as a sum of its cuspidal and noncuspidalcomponent. To show that ˜Θ( π ) nc has finite length as a ˜ SL -module, it suffices to show thatthe Jacquet module of ˜Θ( π ) with respect to a Borel subgroup ˜ B = ˜ T · N of ˜SL has finitelength as a ˜ T -module. Now [GG06, Prop. 8.1] gives a short exact sequence of G -modules:0 −−−−→ Ind G Q C c (GL ) −−−−→ Ω N −−−−→ C −−−−→ L ∼ = GL on GL is via det. The finite length of Θ( π ) N follows from thisvia a similar argument as above, by examining Hom G (Ω N , π ).It remains to show that ˜Θ( π ) c has finite length. In fact, it was shown in [GG06, Thm. 9.1(c)and (d)] that for genuine supercuspidal representations σ ≇ σ of ˜SL , one has ˜ θ ψ ( σ ) ≇ ˜ θ ψ ( σ ). In other words, ˜Θ( π ) c is irreducible or 0. This shows that ˜Θ ψ ( π ) has finite length. (cid:3) Finiteness of Θ( σ ) nc . For tempered σ ∈ Irr(PGSp ), the finite length of Θ( σ ) nc as a G -module is shown in the same way, using Propositions 13.1 and 13.3. We leave the detailsto the reader and only consider the top pieces in the filtration of the two Jacquet modules. • For the maximal parabolic subgroup P , we have to consider the theta correspondencefor PGSp × PGL with respect to the minimal representation Π of PGSO . For thepurpose of showing finiteness, there is no harm in working with Sp × SL . Hence,the theta correspondence in question arises as follows. If V and V denote the 2-dimensional and 6-dimensional symplectic vector spaces, then we are considering themap Sp( V ) × Sp( V ) −→ SO( V ⊗ V )and pulling back the minimal representation Π of SO( V ⊗ V ). As before, we shalluse the fact that this minimal representation is the big theta lift of the trivial repre-sentation of SL . More precisely, let V ′ be another symplectic space of dimension 2,then we have the mapSp( V ′ ) × Sp( V ) × Sp( V ) −→ Sp( V ′ ) ⊗ SO( V ⊗ V ) −→ Sp( V ′ ⊗ V ⊗ V ) . Given the Weil representation Ω of Sp( V ′ ) × SO( V ⊗ V ) and σ ∈ Irr(Sp( V )), wehaveΘ( σ ) ∗ ∼ = Hom Sp( V ) (Π , σ ) ∼ = Hom Sp( V ′ ) × Sp( V ) (Ω , Sp( V ′ ) ⊗ σ ) ∼ = Hom(Θ ′ ( σ ) , Sp( V ′ ) )where Θ ′ ( σ ) is the big theta lift of σ to SO( V ′ ⊗ V ). Note that there is a naturalisogeny Sp( V ′ ) × Sp( V ) −→ SO( V ′ ⊗ V )whose image is of finite index. Hence, by the classical Howe duality theorem, Θ ′ ( σ )is a finite length representation of Sp( V ′ ) × Sp( V ). This implies that Θ( σ ) has finitelength. • For the maximal parabolic subgroup Q , we need to consider the restriction of Π A , aminimal representation of SL to Sp . Note that Π A is a degenerate principal seriesrepresentation induced from a maximal parabolic subgroup which stabilizes a line inthe standard representation. Since Sp acts transitively on such lines, we see that therestriction of Π A to Sp is simply a degenerate principal series representation of Sp .This implies the desired finiteness.We have thus completed the proofs of Lemmas 6.2 and 6.3.15. Howe Duality for G × PGSp : General Case Finally, by combining Theorem 12.4 and Theorem 14.1, we can establish the Howe dualitytheorem for G × PGSp . Theorem 15.1.
Let π ∈ Irr( G ).(i) Θ( π ) is nonzero if and only if π has zero theta lift to P D × .(ii) If Θ( π ) = 0 , then Θ( π ) is a finite length representation of PGSp with a uniqueirreducible quotient θ ( π ) .(iii) For π , π ∈ Irr( G ) , θ ( π ) ∼ = θ ( π ) = 0 = ⇒ π ∼ = π . (iv) If Θ( π ) = 0 , then θ ( π ) is tempered if and only if π is tempered.(v) If π is non-tempered, then θ ( π ) is nonzero and the L-parameter of θ ( π ) is obtainedfrom that of π by composing with the natural inclusion G ( C ) ⊂ Spin ( C ) . Explicit correspondence.
We can in fact determine the theta lift θ ( π ) explicitly if π is tempered and noncuspidal. Indeed, we may also determine θ ( π ) for those tempered π which has nonzero theta lift to PGL . To achieve this, we shall use the following four facts: • If π does not appear in the correspondence with P D × , then θ ( π ) = 0 (Theorem15.1(i)). • If π is tempered and θ ( π ) = 0, then θ ( π ) is irreducible and tempered (Theorem15.1(v)). • If π is nongeneric, then θ ( π ) is nongeneric (Corollary 11.2(ii)). • The cuspidal support of θ ( π ) can be computed (from the Jacquet module computa-tions of § Theorem 15.2.
Let π be an irreducible tempered representation of G . Assume that π is alift of a (necessarily tempered) representation τ of PGL , that is, π = θ B ( τ ǫ ) for some ǫ = ± .Then we have the following:(i) If τ ≇ τ ∨ then θ ( π ) ∼ = I ( τ ) ∼ = I ( τ ∨ ) ∈ Irr(PGSp ) .(ii) If τ ∼ = τ ∨ and the parameter of τ contains a trivial summand, then θ ( π ) ∼ = I ( τ ) . OWE DUALITY AND DICHOTOMY 53 (iii) If τ ∼ = τ ∨ and the parameter of τ does not contain a trivial summand, then π is one ofthe two representations π gen = θ B ( τ + ) and π deg = θ B ( τ − ) . In this case I ( τ ) = I ( τ ) gen ⊕ I ( τ ) deg , θ ( π gen ) = I ( τ ) gen and θ ( π deg ) = I ( τ ) deg . We now deal with the remaining tempered representations of G . Non-supercuspidal rep-resentations are mostly constituents of the principal series I Q ( τ ) where τ is a discrete seriesrepresentation. These representations lift to constituents of the principal series I ( τ ⊗ τ ).More precisely, we have: Theorem 15.3.
Let π be an irreducible tempered representation of G which is not a liftfrom PGL . Then we have the following:(i) Let τ be a unitary discrete series representation of GL . Then I Q ( τ ) is irreducible if andonly if I ( τ ⊗ τ ) is irreducible. We have: • If I Q ( τ ) is irreducible then θ ( I Q ( τ )) ∼ = I ( τ ⊗ τ ) . • If I Q ( τ ) is reducible then θ ( I Q ( τ ) gen ) ∼ = I ( τ ⊗ τ ) gen and θ ( I Q ( τ ) deg ) ∼ = I ( τ ⊗ τ ) deg . (ii) Assume that τ ∼ = τ ∨ is a supercuspidal representation of GL with the trivial central char-acter. Let δ Q ( τ ) and δ P ( τ ) be the square integrable constituents of I Q (1 / , τ ) and I P (1 / , τ ) .Then θ ( δ Q ( τ )) ∼ = δ ( τ ) and θ ( δ P ( τ )) ∼ = δ ( τ ) where δ ( τ ) and δ ( τ ) are the square integrable constituents of I (1 / , τ ⊗ τ ) and I (1 / , τ ⊗ .(iii) Assume that τ ∼ = τ ∨ is a supercuspidal representation of GL whose Langlands parameterhas the image S . Recall that I Q (1 , τ ) has a square integrable constituent denoted by π gen [ τ ] .Then θ ( π gen [ τ ]) ∼ = σ gen [ τ ] where σ gen [ τ ] is the square integrable constituent of I (1 , τ ⊗ τ ) .(iv) Assume that χ = 1 and χ = 1 . Recall that I Q (1 / , st χ ) has a square integrable con-stituent denoted by π gen [ χ ] . Then θ ( π gen [ χ ]) ∼ = σ gen [ χ ] where σ gen [ χ ] is the square integrable constituent of I (1 / , st χ ⊗ st χ ) .(v) Steinberg lifts to Steinberg: θ (St G ) = St PGSp . Finally we need to deal with supercuspidal representations. In view of Theorem 15.1(i)and Theorem 15.2, we only need to consider those supercuspidal representations which do notlift to PGL or P D × . We first introduce a thin family of supercuspidal representatons of G ,namely those which participate in the theta correspondence for ˜SL × G . We have already encountered this theta correspondence in the proof of Proposition 14.2. As mentioned there,this theta correspondence has been studied in detailed in [GG06].We first introduce some notation. For each cuspidal representation ρ of PGL ∼ = SO , let J L ( ρ ) be its Jacquet-Langlands lift to the an isotropic inner form P B × = SO ∗ (where B isthe quaternion division algebra) and let σ ρ be the ψ -theta lift of J L ( τ ) to ˜SL (where ψ isa fixed nontrivial additive character of F ). Then σ ρ is an irreducible supercuspidal genuinerepresentation of ˜SL . Consider now the ψ -theta lift π ρ := θ ( σ ρ ) ∈ Irr( G )of σ ρ from ˜ SL to G . Now we recall some results from [GG06, Thm. 9.1]: Lemma 15.4.
With the above notations, we have:(i) The representation π ρ is nonzero irreducible supercuspidal. Moreover, Θ( π ρ ) = σ ρ underthe theta correspondence for ˜SL × G .(ii) The map ρ π ρ is an injective map from the set Irr sc (PGL ) of supercuspidal represen-tations of PGL to Irr sc ( G ) .(iii) Any π ∈ Irr sc ( G ) which lifts to ˜ SL but not PGL or P D × is of the form π ρ for some ρ ∈ Irr sc (PGL ) . For σ ρ ∈ Irr( ˜SL ) as above, we may also consider its ψ -theta lift from ˜SL to SO and set τ ρ = Θ( σ ρ ) = θ ( σ ρ ) ∈ Irr(SO ) . Then τ ρ is a nongeneric supercuspidal representation of SO belonging to a so-called Saito-Kurokawa A-packet. The representations π ρ and τ ρ are related as follows: Lemma 15.5.
Consider the restriction of the minimal representation of SO to G × SO .Then for ρ ∈ Irr sc (PGL ) , Θ( π ρ ) = τ ρ . Proof.
We shall use the seesaw diagram in the proof of Proposition 14.2. The ensuing seesawidentity (14.3) and Lemma 15.4(i) give:Θ( π ρ ) ∗ ∼ = Hom ˜SL (Ω , σ ρ ) = τ ∗ ρ . Hence Θ( π ρ ) = τ ρ . (cid:3) Now we have:
Proposition 15.6.
Let π be an irreducible supercuspidal representation of G that is not alift from PGL or P D × . Then we have the following two possibilities: • If π = π ρ for some ρ ∈ Irr sc (PGL ) (as in Lemma 15.4), then θ ( π ρ ) = δ ( τ ρ ) where δ ( τ ρ ) is the square integrable subquotient of I (1 / , τ ρ ) given in Proposition10.4. • If π is not of the above form, then θ ( π ) is supercuspidal. OWE DUALITY AND DICHOTOMY 55
Proof.
Let Π be the minimal representation of E . Recall that r P i is the normalized Jacquetfunctor with respect to the maximal parabolic P i in PGSp . Then π ⊗ r P i ( θ ( π )) is a quotientof r P i (Π). By the assumption that π does not lift to PGL , it follows that r P i ( θ ( π )) = 0 for i = 2 ,
3. Thus either r P ( θ ( π )) = 0, in which case θ ( π ) is supercuspidal, or r P ( θ ( π )) is asupercuspidal representation of the Levi factor L = GSp . In fact, from Proposition 13.6, itfollows that r P ( θ ( π )) = τ ⊗ | ν | / where τ is a (possibly reducible) supercuspidal represen-tation of PGSp ∼ = SO such that π ⊗ τ appears as a quotient of the minimal representationof SO . By the seesaw in the proof of Proposition 14.2, we see that π must have nonzerotheta lift to ˜SL and hence is of the form π ρ for some ρ ∈ Irr(PGL ) by Lemma 15.4(iii).Then Lemma 15.5 implies that τ = τ ρ . By Frobenius reciprocity and the fact that θ ( π ρ ) istempered, we see that θ ( π ρ ) = δ ( τ ρ ), as desired. (cid:3) As a consequence of the explicit results in this section, we have:
Corollary 15.7. If π ∈ Irr( G ) is a discrete series representation which does not lift to PGL or P D × , then θ ( π ) is an irreducible discrete series representation of PGSp . As a result, anydiscrete series representation of G lifts to a discrete series of exactly one of P D × , PGL or PGSp . That lift is Whittaker generic iff and only if π is. Acknowledgments:
The authors would like to thank MFI in Oberwolfach for hospitalityduring a conference in October of 2019 when some of the ideas needed to finish this workemerged. Thanks are due to Petar Baki´c, Baiying Liu and Yiannis Sakellaridis for help withsome finer points. W.T. Gan is partially supported by an MOE Tier 1 grant R-146-000-320-114. G. Savin is partially supported by a National Science Foundation grant DMS-1901745.
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W.T.G.: Department of Mathematics, National University of Singapore, 10 Lower KentRidge Road Singapore 119076
Email address : [email protected] G. S.: Department of Mathematics, University of Utah, Salt Lake City, UT
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