Weil Representation, Howe Duality, and the Theta correspondence

Weil Representation, Howe Duality, and the ThetacorrespondenceDipendra PrasadTata Institute of Fundamental ResearchColaba, Bombay, IndiaJanuary 23, 2007These are some expository lectures given in Montreal on the Weil representation.The aim of these lectures was to introduce the audience to this importantrepresentation and to show how it has been useful in constructing representationsof p-adic groups and automorphic forms via dual reductive pairs. These lecturescontain no serious proofs, for given the limitations of time, the inclusion of all thetechnical details would only have served to detract from the global structure. Werefer the reader to the nice book of Moeglin, Vigneras, Waldspurger [MVW] for allthe proofs in the local non-archimedean case, and the basic papers of Howe [H2]and [H3] for the real theory.I would like to thank M. Ram Murty for inviting me to CRM to give theselectures, and for his interest in these lectures and his hospitality in Montreal.Thanks are also due to S. Gelbart for encouragement and to B. Sury for a carefulreading of the manuscript.1 Heisenberg GroupIn this section k will denote a non-archimedean local field which will never be ofcharacteristic 2. All the representations will be over complex numbers and smoothin the sense that every vector in the representation space is fixed by a compactopen subgroup. For any finite dimensional vector space V over k, S(V ) will denotethe space of locally constant compactly supported functions on V .1

Let W be a finite dimensional vector space over k with a non-degenerate alternatingform . Such a vector space will be called a symplectic space. Itsdimension is even and we denote it by 2n. The Heisenberg group H(W ) associatedto the symplectic space W is a non-trivial central extension of W by k and isdefined to be the group of pairswith the law of multiplication{(w, t)|w ∈ W, t ∈ k},(w 1 , t 1 )(w 2 , t 2 ) = (w 1 + w 2 , t 1 + t 2 + 1 2 < w 1, w 2 >)The Heisenberg group H(W) clearly sits in the exact sequence0 → k → H(W ) → W → 0.The commutator subgroup of H(W ) is k and all the one dimensional representationsof H(W ) factor through W . We construct below an infinite dimensionalsmooth representation of H(W ). For this fix a decomposition of W as W = W 1 ⊕W 2where W 1 and W 2 are maximal totally isotropic subspaces of W (ie, subspaces onwhich the alternating form is identically zero). Such a decomposition will be calleda complete polarisation of W .The representation depends on the choice of anadditive character ψ of k which will be fixed in all these lectures.Define the representation ρ ψ of the Heisenberg group H(W ) on S(W 1 ) as follows:ρ ψ (w 1 )f(x) = f(x + w 1 ) for all x, w 1 ∈ W 1 ,ρ ψ (w 2 )f(x) = ψ(< x, w 2 >)f(x) for all x ∈ W 1 , w 2 ∈ W 2ρ ψ (t)f(x) = ψ(t)f(x) for all t ∈ k, x ∈ W 1 .It can be easily checked that this gives a smooth representation of H(W ). Thisrepresentation of H(W ) is called the Schrödinger representation.Remark 1.1: If ¯ψ denotes the character ¯ψ(x) = ψ(−x), then ρ ¯ψ is the smoothdual of ρ ψ . To see this observe that∫(f 1 , f 2 ) → f 1 f 2 dw, f 1 , f 2 ∈ S(W 1 )W 1is a non-degenerate invariant form.There is another model of this representation, called the lattice model whichis also quite useful. In fact we define a representation which is more general thanboth of these.2

Let A ⊂ W be a closed subgroup and define A k = A × k ⊂ H(W ). It is asubgroup of H(W ). Assume now that A = A ⊥ whereA ⊥ = {y ∈ W |ψ(< x, y >) = 1, ∀x ∈ A}In this case the character ψ of k can be extended to A k . Fix such an extensionand call it ψ A .Let S A be the space of functions on H(W ) such that(i) f(ah) = ψ A (a)f(h) for all a ∈ A k , h ∈ H(W ).(ii) f(hl) = f(h) for all l ∈ L, a lattice in W , and h ∈ H(W ).By a simple calculation one can show that functions in S A are compactly supportedmodulo A k . Clearly the Heisenberg group operates on S A via right translationsand this representation is smooth because of (ii).Since k operates via the character ψ on functions in S A , one can realise thisrepresentation also on a certain space of functions on H(W )/k. To do this, letf be a function on H(W ) such that f(ht) = ψ(t)f(h) for t ∈ k, and define afunction ˜f on W by ˜f(w) = f(w, t)ψ(t) −1 for any t ∈ k, w ∈ W . Conversely given˜f, the same equation can be used to define the function f on H(W ). Under thisisomorphism, the space S A gets identified to the space of functions ˜f in S(W ) suchthat ˜f(a + w) = ψ( 2) ˜f(w) for all a ∈ A, and b ∈ W ⊂ H(W ) acts on thisspace of functions ˜f by, (b · ˜f)(w) = ψ( 2) ˜f(b + w).Proposition 1.2: Each S A is an irreducible representation of H(W ).We now give some examples of closed subgroups A with A ⊥ = A(1) A = W 1 , a maximal totally isotropic subspace of W .(2) A = L = ∑ ni=1 O k e i ⊕ ∑ ni=1 O k f i where {e i , f i } is a symplectic basis of Wi.e. < e i , e j >= 0, < f i , f j >= 0, < e i , f j >= δ ij , and O k is the ring of integers of kwith uniformising parameter π k . Then if the conductor of ψ is O k i.e., ψ(O k ) = 1but ψ(π −1k O k) ≢ 1, L ⊥ = L.It is easy to see that in case (1), we get the Schrödinger model of the representationof the Heisenberg group discussed earlier, and the second case gives whatare called lattice models.We have now constructed several smooth irreducible representations of theHeisenberg group on which the centre acts by the character ψ.All these representationsare isomorphic. In fact one has the following basic theorem.3

Theorem 1.3(Stone, von Neuman): The Heisenberg group H(W ) has aunique irreducible smooth representation on which k operates via the character ψ.(ρ ψ , S)We will denote this unique irreducible smooth representation of H(W ) by2 Metaplectic Group and the Weil RepresentationThe main theme of these lectures is not the representation (ρ ψ , S) of the Heisenberggroup constructed in the last section but rather a (projective)-representation ofthe symplectic group which is constructed using intertwining operators of thisrepresentation of the Heisenberg group. To define this, observe that the symplecticgroup Sp(W ) of W (i.e. the automorphisms of W preserving the alternating form) operates on H(W ) by g · (w, t) = (gw, t). Clearly this action is trivial onthe centre of H(W ) and therefore by the uniqueness theorem of Stone and vonNeumann, there is an operator ω ψ (g) (unique up to scaling) on S such thatNow defineThenρ ψ (gw, t) · ω ψ (g) = ω ψ (g) · ρ ψ (w, t), for all (w, t) ∈ H(W ) (∗).˜ Sp ψ (W ) = {(g, ω ψ (g)) such that (∗) holds}.˜ Sp ψ (W ) is a group under pointwise multiplication, called the metaplecticgroup, and fits in the following exact sequence:0 → C ∗ → ˜ Sp ψ (W ) p → Sp(W ) → 0.The metaplectic group comes equipped with a natural representation obtained byprojection on the second factor (g, ω ψ (g)) → ω ψ (g) ∈ Aut(S). This representationof the metaplectic group is called the Weil representation or the metaplecticrepresentation or the oscillator representation.Theorem 2.1: The map p restricted to the commutator subgroup Sp ˆ ψ (W ) =[ Sp ˜ ψ (W ), Sp ˜ ψ (W )] of Spψ ˜ (W ) is a surjection onto Sp(W ) with a kernel of order 2.In particular,Sp ˜ ψ (W ) = Sp ˆ ψ (W ) × Z/2 C ∗ .4

Moreover, the two sheeted covering ˆ Sp ψ (W ) of Sp(W ) is independent of the additivecharacter ψ but the Weil representation restricted to ˆ Sp ψ (W ) does depend onψ.We now give an explicit model, called the Schrödinger model, of the metaplecticrepresentation over a local field. For this, let W = W 1 ⊕ W 2 be a complete polarisationof W . We will write elements of Sp(W ) as matrices with respect to thebasis {e 1 , e 2 , · · · , e n , f 1 , · · · , f n } where e i ∈ W 1 and f i ∈ W 2 and < e i , f j >= δ i,j .One can easily check that( )A 00 t A −1belongs to Sp(W ) if A ∈ GL(W 1 ),and the action of this matrix in the metaplectic representation on S(W 1 ) is givenby( )A 0ω ψ0 t A −1 f(x) = | det A| 1 2 f( t Ax),i.e., the operator so defined has the property (∗) above. (Observe that since g →ω ψ (g) is a {( homomorphism ) on such matrices, } the metaplectic covering splits on theA 0subgroup0 t A −1 |A ∈ GL(W 1 ) of Sp(W )). The purpose of the scalingfactor | det A| 1 2in the above action is to make the action of GL(W 1 ) unitary forthe hermitian structure on S(W 1 ): (f 1 , f 2 ) = ∫ W 1f 1 ¯f2 dw.We have (1 B0 1and the action of this matrix is given by)∈ Sp(W ) if and only if B = t B,( )1 Bt xBxω ψ f(x) = ψ(0 12 )f(x)Finally, we haveω ψ(0 1−1 0)f(x) = γ ˆf(x),where γ is an 8-th root of unity to be described below and ˆf is the Fourier transform,∫n∑ˆf(x) = f(y)ψ( x i y i )dy,k n i=1the Haar measure dy being chosen so that ˆˆf(x) = f(−x).5

We define the γ-factor γ(Q, ψ) more generally for any quadratic form Q on k nto be γ(Q, ψ) = g(Q, ψ)/|g(G, ψ)| where g(Q, ψ) is defined as follows. (Our γ willbe γ(Q, ψ) for Q = ∑ ni=1 x 2 i .)∫∫g(Q, ψ) = P.V. ψ(Q(x))dx def = lim ψ(Q(x))dx,k n m→∞ π −m Oknwhere dx is now chosen so that Fourier inversion holds for B(x, y) = Q(x+y)−Q(x)−Q(y) .2Remark 2.2 : One might wonder why this care about the 8-th root of unityfactor γ when the intertwining operator ω ψ(0 1−1 0)is defined only up to scalaranyway. This has to do with the fact that the metaplectic group ˜ Sp ψ (W ) containsˆ Sp(W ) which is a two sheeted covering of Sp(W ), and therefore the scaling factorcan be normalised up to an ambiguity of sign.Remark 2.3 : It can be proved that for the function ψ Q (x) = ψ(Q(x)) on k n ,ψˆQ = γψQ−1 in the sense of distributions.Remark 2.4 : The construction of the Heisenberg group and the uniquenesstheorem of Stone and von Neumann is also true in the case of finite fields. Thereforeas above, one can construct a projective representation of dimension q n ofSp(n, F q ).This projective representation infact lifts to an ordinary representationof Sp(n, F q ) (in a unique way except if n = 1, q = 3). The representation ofSp(n, F q ) so obtained depends on the choice of the additive chatacter ψ of F q , andthe representation associated to ψ and ψ a (x) = ψ(ax) are isomorphic if and onlyif a is a square in F q .Remark 2.5 : Representation of the Heisenberg group and of the metaplecticgroup can be combined to give a representation of the semi-direct product H(W )ט Sp ψ (W ) where the semi-direct product is via the natural action of Sp(W ) onH(W ).2.6 Lattice Model of the Weil Representation: Let A be a finitelygenerated O k -module of maximal rank in W such that A ⊥ = A. Then in thelattice model of the representation of the Heisenberg group on functions on W ,(g, M[g]) ∈ Sp ˜ ψ (W ) for the operator M[g]:(M[g]f)(w) =∑a∈AgA∩Aψ( < a, w > )f(g −1 (a + w)).26

In particular for K, the stabiliser of A, the above formula simplifies to(M[g]f)(w) = f(g −1 w).Observe now that this K is a maximal compact subgroup of Sp(W ), and sincethe above is clearly a representation of K (instead of a projective representation),the metaplectic covering splits on K.Moreover, from the above action of K it is clear that the characteristic functionof A is invariant under K and it is easy to see that it is the unique vector invariantunder K.Remark 2.7: For W = W 1 ⊕ W 2 , the direct sum of two symplectic spaces,let W 1 = W1 1 ⊕ W2 1 , W 2 = W1 2 ⊕ W2 2 and W = (W1 1 ⊕ W1 2 ) ⊕ (W2 1 ⊕ W2 2 ) be completepolarisations. One has representations (ρ ψ , S(W1 1 )) of H(W 1 ), (ρ ψ , S(W1 2 ))of H(W 2 ), and (ρ ψ , S(W1 1 ⊕W1 2 )) of H(W ). Since S(W1 1 ⊕W1 2 ) = S(W1 1 )⊗S(W1 2 ),it follows that the representation (ρ ψ , S(W1 1 ⊕ W1 2 )) of H(W ) under the mappingH(W 1 ) × H(W 2 ) → H(W ) where (w 1 , t 1 ) × (w 2 , t 2 ) ∈ H(W 1 ) × H(W 2 ) goes to(w 1 + w 2 , t 1 + t 2 ) becomes the tensor product of representations (ρ ψ , S(W1 1 )) ofH(W 1 ) and (ρ ψ , S(W1 2 )) of H(W 2 ). Clearly the intertwining operator correspondingto g ∈ Sp(W 1 ) acting on the first variable is an intertwining operator forg thought of as an element of Sp(W 1 ⊕ W 2 ). This implies that the metaplecticcovering of Sp(W 1 ⊕ W 2 ) restricted to Sp(W 1 ) is the metaplectic covering ofSp(W 1 ), and that the metaplectic representation of Sp(W 1 ⊕ W 2 ) restricted toSp(W 1 ) × Sp(W 2 ) is the tensor product of their metaplectic representations.Remark 2.8: Suppose that W is a symplectic space and V is a non-degeneratequadratic space. Then the product of the bilinear forms gives a symplectic structureon W ⊗V . Suppose that the quadratic form on V is ∑ mi=1 α i x 2 i ; then as a symplecticspaceW ⊗ V =m∑α i W.i=1Therefore, from remark 1, the restriction of the metaplectic representation ofSp(W ⊗ V ) to ∏ mi=1 Sp(α i W ) is the tensor product of the metaplectic representationsof Sp(α i W ). Since the metaplectic representation of Sp(αW ) associated tothe character ψ is isomorphic to the metaplectic representation of Sp(W ) associatedto the character ψ α (x) = ψ(αx), the restriction of the metaplectic representation7

of Sp(W ⊗ V ) to Sp(W ) is isomorphic to the tensor product of the metaplecticrepresentations of Sp(V ) associated to the characters ψ αi .Remark 2.9: For V and W as in the previous remark, O(V ) ⊂ Sp(W ⊗ V )in the obvious way. Let W = W 1 ⊕ W 2 be a complete polarisation of W . ThenW ⊗ V = W 1 ⊗ V ⊕ W 2 ⊗ V is a complete polarisation for W ⊗ V . Recall nowthat the Schrödinger model of the metaplectic representation of Sp(W ˜ ⊗ V ) isobtained on the compactly supported locally constant functions on W 1 ⊗ V . NowO(V ) operates on S(W 1 ⊗ V ) in a natural way, and it is easy to see from theexplicit formulae for the representation of the Heisenberg group that this action isan intertwining operator; therefore we conclude that(i) The metaplectic covering of Sp(W ⊗ V ) splits on the subgroup O(V ).(ii) The metaplectic representation of Sp(W ⊗ V ) restricted to O(V ) is thenatural representation of O(V ) on functions on W 1 ⊗ V (at least, up to a characterof O(V )).Remark 2.10: Similarly if V = V 1 ⊕ V 2 where V 1 and V 2 are maximal totallyisotropic subspaces of the quadratic space V , W ⊗ V = W ⊗ V 1 ⊕ W ⊗ V 2 is acomplete polarisation of W ⊗ V .We conclude as in the previous remark thatthe metaplectic covering of Sp(W ⊗ V ) splits on Sp(W ), and the metaplecticrepresentation of Sp(W ⊗ V ) restricted to Sp(W ) is the natural representation ofSp(W ) on S(W ⊗ V 1 ).Remark 2.11 : By remark 1.1, the dual of the metaplectic representationω ψ of Sp(W ˜ ) associated to the character ψ(x) is the metaplectic representation ofSp(W ˜ ) associated to the character ψ(−x). Therefore by remarks 2.8 and 2.10,ω ψ ⊗ ω ∗ ψ ∼ = S(W ),where Sp(W ) operates on S(W ) in the natural way (observe that ω ψ ⊗ ω ∗ ψ is a representationof Sp(W )). This relation is specially useful for the Weil representationof Sp(n, F q ) where the Weil representation is self-dual if −1 is a square in F q (seeremark 2.4), i.e. for q ≡ 1 mod 4, and the above relation can be used to calculatethe character of the Weil representation up to sign.8

3 Dual Reductive PairsA dual reductive pair is a pair of subgroups (G, G ′ ) in a symplectic group Sp(W )such that(i) G is the centraliser of G ′ in Sp(W ), and G ′ is the centraliser of G in Sp(W ).(ii) The actions of G and G ′ are completely reducible on W (recall that anaction is called completely reducible if every invariant subspace has an invariantcomplement).A dual reductive pair (G, G ′ ) in Sp(W ) will be called irreducible if one can’tdecompose W as the direct sum of two symplectic subspaces each of which isinvariant under both G and G ′ .Remark 3.1: If (G 1 , G ′ 1) ⊂ Sp(W 1 ) and (G 2 , G ′ 2) ⊂ Sp(W 2 ) are dual reductivepairs then (G 1 × G 2 , G ′ 1 × G ′ 2) is a dual reductive pair in Sp(W 1 ⊕ W 2 ).Remark 3.2: Every dual reductive pair is constructed from irreducible onesby repeating the process in remark 3.1.We will give a general method of constructing irreducible dual reductive pairs.For this we begin by defining a hermitian form on a vector space over a divisionalgebra with involution.Let D be a division algebra over k and τ an involution of D over k (i.e. τ(x+y) =τ(x)+τ(y), τ(xy) = τ(y)τ(x), τ 2 (x) = x for all x, y ∈ D, and τ(a) = a for all a ∈ k).Let V be a right vector space over D with a k-bilinear form H : V × V → D suchthat(i) H(v 1 d 1 , v 2 d 2 ) = τ(d 1 )H(v 1 , v 2 )d 2 , for all v 1 , v 2 ∈ V and d 1 , d 2 ∈ D.(ii)τ(H(v 1 , v 2 )) = ɛH(v 2 , v 1 ) for all v 1 , v 2 ∈ V , where ɛ = ±1.A right D-vector space V together with a bilinear form H having the properties(i) and (ii) above is called an ɛ-hermitian space.The group of D-automorphisms of V preserving the ɛ-hermitian form H iscalled the unitary group associated to the ɛ-hermitian space V , and is denoted byU(V, H) (or U(V ) for short).Remark 3.3: If k is a local field then any division algebra over k with aninvolution is at most 4-dimensional over its centre. Moreover, if the involution isnon-trivial on the centre then the division algebra is commutative.Remark 3.4: If D = k, then for ɛ = 1, U(V, H) is the usual orthogonal9

group and for ɛ = −1, U(V, H) is the symplectic group. If D = K = k( √ d) is aquadratic field extension of k, and τ is the non-trivial galois automorphism of K/k,U(V, H) is what is generally called a unitary group. In this case one can multiplyan ɛ-hemitian form by √ d to go from hermitian to skew-hermitian and vice-versawithout changing the unitary group. More generally, multiplying an ɛ-hermitianform by a ∈ D such that τ(a) = −a, takes the ɛ-hermitian form to a −ɛ-hermitianform (for τ ′ = aτa −1 ).Remark 3.5: Let D be a quaternion division algbra over k with the standardinvolution. Then if ɛ = 1, U(V, H) is a form of the symplectic group and if ɛ = −1,the U(V, H) is a form of the orthogonal group.Remark 3.6: The unitary groups U(V, H) for ɛ-hermitian forms H constructedabove, together with general linear groups over division algebras, constitute whatare called the classical groups.Classification Theorem for Irreducible Dual Reductive Pairs :The irreducible dual reductive pairs (G, G ′ ) in Sp(W ) are of two types:Type 1. The action of G · G ′ on W is irreducible.In this case there exists a division algebra D/k with an involution τ, and fori = 1, 2 spaces W i over D and ɛ i -hermitian forms H i on W i with ɛ 1 ɛ 2 = −1 suchthat W ∼ = W 1 ⊗ D W 2 , and the alternating bilinear form on W ∼ = W 1 ⊗ D W 2 is givenby < w 1 ⊗ w 2 , w ′ 1 ⊗ w ′ 2 >= tr D/k (H 1 (w 1 , w ′ 1)H 2 (w 2 , w ′ 2)), and the pair (G, G ′ ) is(U(W 1 ), U(W 2 )).Type 2. The action of G · G ′ on W is reducible.In this case there exists a division algebra D/k and a right D-vector space W 1and a left D-vector space W 2 such that W = W 1 ⊗ D W 2 ⊕ (W 1 ⊗ D W 2 ) ∗ withthe natural symplectic structure, and (G, G ′ ) = (Aut D (W 1 ), Aut D (W 2 )) with thenatural action of Aut D (W 1 ) and Aut D (W 2 ) on W 1 ⊗ D W 2 and its dual, and thereforeon W .4 Howe DualityWe begin with the following general lemma, [MVW] lemma II.5.Lemma 4.1 : If two elements in Sp(W ) commute then their arbitrary lifts inSp ˜ ψ (W ) also do.10

Now for a closed subgroup H ⊂ Sp(W ), let ˜H ⊂ ˜ Sp ψ (W ) be the full inverseimage of H in the metaplectic group. Let R ψ ( ˜H) be the set of smooth irreduciblerepresentations π of ˜H such thatHom ˜H(S, π) ≠ 0,where (ω ψ , S) is the metaplectic representation of ˜ Spψ (W ).Note that for a dual reductive pair (G, G ′ ) in Sp(W ), the subgroups ˜G and ˜G ′ ofSp ˜ ψ (W ) commute by lemma 4.1, and there is a surjective map from ˜G× ˜G ′ to G ˜ · G ′ .Therefore a representation of˜ G · G ′ is a tensor product π ⊗ π ′ of representationsπ of ˜G and π′of ˜G′ . Clearly if π ⊗ π ′ belongs to R ψ ( ˜ G · G ′ ) then π belongs toR ψ ( ˜G) and π ′ to R ψ ( ˜G ′ ). It follows that there is a natural map from R ψ ( ˜ G · G ′ )to R ψ ( ˜G) × R ψ ( ˜G ′ ).Conjecture 1(Howe) : For a dual reductive pair (G, G ′ ), the image of themapping from R ψ ( ˜ G · G ′ ) to R ψ ( ˜G) × R ψ ( ˜G ′ ) constructed above is the graph of abijective correspondence between R ψ ( ˜G) and R ψ ( ˜G ′ ).Actually there is a slightly more refined conjecture which takes into accountthe multiplicities too. To state it, define for an irreducible representation π of ˜GS(π) = ⋂ ker α,where the intersection is taken over all the possible homomorphisms α from theWeil representation (ω ψ , S) onto π, and letS[π] = S/S(π).Note that S[π] is the largest quotient of S which is π-isotypical. Since ˜G ′ commuteswith ˜G, S[π] is also a ˜G ′ module and therefore a ˜G × ˜G ′ -module. We now have thefollowing general lemma.Lemma 4.2: Let τ be a smooth representation of the product G 1 × G 2 of twolocally compact totally disconnected groups, and let τ 1 be a smooth irreducibleadmissible representation of G 1 . If the intersection of the kernels of all the G 1 -homomorphisms of τ onto τ 1 is trivial, then τ ∼ = τ 1 ⊗τ 2 for a smooth representationτ 2 of G 2 .From the above lemma, there is a smooth representation σ 0 (π) of ˜G ′ such thatS[π] ∼ = π ⊗ σ 0 (π) as a ˜G × ˜G ′ -module.11

Conjecture 2 (Howe): The representation σ 0 (π) of ˜G′ has a unique irreduciblequotient denoted by σ(π), and the mapping π → σ(π) is a bijection betweenR ψ ( ˜G) and R ψ ( ˜G ′ ).Remark 4.3: The Howe conjecture is now proved for non-archimedean localfields of residue characteristic not 2. Howe himself proved it in many cases and itwas recently completed in residue characteristic not 2 by Waldspurger [Wa2].The conjecture also makes sense in the archimedean case and was proved byHowe (we take up the archimedean case in the next section).Though the conjecture is technically speaking false for finite fields (as wasobserved by Howe), it should essentially be true there too.It is a theorem of Kudla [Ku1] that σ 0 (π) has finite length and that if π issupercuspidal then σ o (π) is irreducible.The following lemma is very useful in the study of Howe correspondence, i.e.the correspondence given by conjecture 2.Lemma 4.4: For a dual reductive pair (G, G ′ ) in Sp(W ), the metaplecticcovering ˜ Sp ψ (W ) splits on G unless (G, G ′ ) is the pair (Sp(U), O(V )) in Sp(U ⊗V ),with dim V odd.Remark 4.5: The lemma 4.4 is not true for the two sheeted cover of thesymplectic group in place of Spψ ˜ (W ).4.6 Examples of the Howe Correspondence : We will be looking at certainexamples of the Howe correspondence for the dual pair (Sp(W ), O(V )) ⊂ Sp(W ⊗V ), mostly for dim W = 2 in which case Sp(W ) ∼ = SL(2, k).4.6.1 : dim V = 1, q(x) = ax 2 . Therefore O(V ) = {±1} and has two representations.Both of these appear in the Weil representation and the correspondingrepresentation of SL(2, ˆ k) are on the even and odd functions on k. The representationof SL(2, ˆ k) on odd functions is supercuspidal, cf. [Ge1], thm.5.19(c).4.6.2 : dim V = 2, q(x) = a · (norm form of a quadratic field extension K of k).In this case SO(V ) ∼ = K 1 = norm one elements of K, and O(V ) is the semi-directproduct of K 1 with a group of order 2 acting on K 1 by x → ¯x = x −1 . Therefore therepresentations of O(V ) are constructed from the characters of K 1 in the followingway:12

additional problems due to non-compactness, see[R-S].The 3-dimensional representation of O(V ) ∼ = D ∗ k/k ∗ × {±1} can be identifiedto the representation of D ∗ k/k ∗ × {±1} on the Schwartz space of functions on thevector space of trace 0 elements of D k on which D ∗ k acts by conjugation, and {±1}by multiplication.For a quadratic field K/k, let NK ∗ be the normaliser of K ∗ in D ∗ k, and let φ Kbe the character of order 2 of NK ∗ which is trivial on K ∗ . Let H K be the index-2subgroup of NK ∗ × {±1} given by the kernel of the homomorphism φ K × {±1} toC ∗ .The orbits for the action of D ∗ k/k ∗ × {±1} on the trace zero elements of D k are,besides the origin, of the form [D ∗ k/k ∗ × {±1}]/H K for a quadratic field extensionK/k. It follows from the Mackey theory that the only representations of O(V )appearing in S(V ) are those representations of D ∗ k/k ∗ × {±1} which are trivial onH K for some quadratic extension K/k. Therefore to prove that the map taking anirreducible representation of D ∗ k/k ∗ × {±1} appearing in S(V ) to its restriction toD ∗ k/k ∗ is an injection, it suffices to prove thatHom D ∗k[Ind D∗ kNK ∗1, IndD∗ kNL ∗φ L] = 0.We therefore have to prove that there are no non-zero functions f on D ∗ k such thatf(agb) = φ L (b)f(g) ∀a ∈ NK ∗ , g ∈ D ∗ k, b ∈ NL ∗ .To prove this, let j K (resp. j L ) be an element of NK ∗ (resp. NL ∗ ) which is notin K (resp L), and observe that for any g ∈ D ∗ k, j K · K ∩ g(j L · L)g −1 ≠ 0. This isbecause j K · K and g(j L · L)g −1 are 2-dimensional k-vector spaces, contained in the3-dimensional space of trace zero elements of D k . Therefore for all g ∈ D ∗ k, thereexists k ′ ∈ j K · K, l ′ ∈ j L · L such that gl ′ g −1 = k ′ , or g = k ′ gl ′−1 . Thereforef(g) = f(k ′ gl ′−1 ) = φ L (l ′−1 )f(g) = −f(g) = 0.Similarly to prove that the map taking an irreducible representation of D ∗ k/k ∗ ×{±1} appearing in S(V ) to its restriction to D ∗ k/k ∗ is a surjection, it suffices toprove that given any irreducible representation of D ∗ k, there exists a quadratic fieldextension K/k such that the representation has a K ∗ -invariant vector. This is awell-known fact about representations of quaternion division algebras.14

4.6.4 : dim V = 4. Suppose that V does not represent any zeros. In that caseSO(V ) = [D ∗ × D ∗ ] 1 = {(d 1 , d 2 ) ∈ D ∗ × D ∗ |Nd 1 = Nd 2 },where D is the unique quaternion division algebra over k, which can be identifiedto V as a quadratic space over k (with the quadratic form on V getting identifiedto the norm form on D), and the action of (d 1 , d 2 ) ∈ [D ∗ × D ∗ ] 1is by x →d 1 xd −12 . Let GSO(V ) = D ∗ × D ∗ , with the natural action on D. Clearly there areexactly 2 orbits for the action of GSO(V ) = D ∗ × D ∗ on D, one consisting of theorigin only, and the other all of D ∗ . Therefore from the Mackey theory, the onlyrepresentations of D ∗ × D ∗ appearing in its natural action on S(D) are of the typeW ⊗ W ∗ for W an irreducible representation of D ∗ . The representation W ⊗ W ∗ ofGSO(V ) extends in two ways to a representation of GO(V ), the group of orthogonalsimilitudes, and exactly one of these appears in S(D). The representations of O(V )appearing in S(D) are the restrictions of these representations. To conclude, forevery irreducible representation of D ∗ , one can associate a sum of representationsof O(V ) each of which occurs in the Howe correspondence with SL 2 , and converselyevery representation of O(V ) which occurs in the Howe correspondence with SL 2occurs in this way.4.6.5 : The Howe duality conjecture is specially simple to state (though notto prove!) for a dual reductive pair of type 2. Take for example the dual pair(GL(n, k), GL(m, k)). In this case one is looking at the representation of GL(n, k)×GL(m, k) on S(M(n, m; k)), where M(n, m; k) is the space of n × m matrices overk with the action of (A, B) ∈ GL(n, k) × GL(m, k) on X ∈ M(n, m; k) given byAX t B (this is the restriction of the metaplectic representation of Sp(nm, k) toGL(n, k) × GL(m, k) up to twisting by a character). The Howe duality conjecturein this case states that given an irreducible representation π of GL(n, k), thereexists a unique irreducible representation π ′ of GL(m, k) such that π ⊗ π ′ is aquotient of the GL(n, k) × GL(m, k) module S(M(n, m; k)). In this case a veryexplicit form of the Howe duality conjecture is expected, which as far as this authorcould find out, is not known. If n ≤ m, and if σ π and σ π ′are the representationsof the Weil-Deligne group associated by the Langlands correspondence to π and π ′respectively, then one expects that σ π ′= ν m−n2 σ ∗ π ⊕ ν − n 2 , where ν is the character|x| on k ∗ . In particular, if m = n, one expects that the Howe correspondence issimply taking the dual representation.15

Remark 4.7: Howe correspondence for the case when dim V = 3 producesa correspondence between representations of D ∗ /k ∗ , for D a quaternion divisionalgebra, and representations of SL(2, ˆ k), and also between square integrable representationsof P GL(2, k) and representations of SL(2, ˆ k). On the other hand onehas the Jacquet-Langlands correspondence between representations of D ∗ /k ∗ andP GL(2, k) (which does not depend on the additive character ψ). Therefore givena finite dimensional representation of D ∗ /k ∗ one can construct representations ofˆ SL(2, k) in two different ways: either directly using Howe correspondence or firstusing Jacquet-Langlands correspondence and then Howe. It is a basic observationof Waldspurger that these two representations of ˆ SL(2, k) are different (as the firstone does not have a ψ-Whittaker model whereas the second one has).Remark 4.8: We saw in example 4.6.2 above that for dim V = 2, R ψ (O(V ))consists of all the representations of O(V ) except the non-trivial representation ofO(V ) which is trivial on SO(V ). The Howe lift of this representation to Sp(W ) fordim W = 4 is non-zero super-cuspidal representation, and is related to Srinivasan’srepresentation θ 10 of Sp 4 (F q ). This representation occurs as the local componentof a cuspidal automorphic representation on Sp 4 which contradicts Ramanujanconjecture on Sp 4 .Remark 4.9: As in the previous remark, one can more generally ask how doesthe Howe lift vary as we fix the orthogonal group O(V ) but change the symplecticgroup Sp(W ). It is a theorem of Kudla [Ku1] that if we start with a supercuspidalrepresentation of O(V ) then there is a symplectic space W 0 such that the Howe liftto Sp(W 0 ) is non-zero supercuspidal and the Howe lift to all symplectic spaces ofsmaller dimensions are zero. Moreover, for a symplectic space W with dim W >dim W 0 , the Howe lift to Sp(W ) is non-zero and non-supercuspidal and can bedescribed explicitly in terms of parabolic induction from the Howe lift to Sp(W 0 ).Kudla has a similar theorem when one fixes W but instead changes V in the sameWitt class. Both results are local analogues of Rallis’ global theory of towers oftheta series liftings [Ra1].16

5 Howe Conjecture in the Archimedean caseThe Howe conjecture makes sense in the archimedean case too. In this case oneworks with the Harish-Chandra module of the unitary metaplectic representationdefined on square integrable functions on a maximal totally isotropic subspace byformulae exactly similar to the one in the non-archimidean case. We give below theHarish-Chandra module in the Fock model which is more convenient to work with.We will work with the two sheeted covering Sp(n, ˆ R) of Sp(n, R) and the inverseimage Û(n) of U(n) ⊂ Sp(n, R) as the maximal compact subgroup of Sp(n, ˆ R).Let sp be the Lie algebra of Sp(n, ˆ R) and u, the Lie algebra of U(n). One has theCartan decompositionsp = u ⊕ p = sp 1,1 ⊕ sp 2,0 ⊕ sp 0,2 ,with u = sp 1,1 . The Fock model of the Weil representation of (sp, Û(n)) is realisedon the space S of polynomial functions on C n , with representations of u, p 2,0 , p 0,2given as follows.(1) u operates via the differential operatorsz i∂+ 1 ∂z j 2 δ ij, 1 ≤ i, j ≤ n,(2) p 2,0 operates by multiplication by z i z j , 1 ≤ i, j ≤ n,(3) p 0,2 operates via the differential operators∂ 2∂z i ∂z j, 1 ≤ i, j ≤ n.The action of Û(n) is the standard representation of U(n) on polynomials onC n twisted by a character of Û(n) which does not factor through U(n).Let now (G, G ′ ) be a dual reductive pair in Sp(n) with maximal compact subgroupK in Ĝ and K′ in Ĝ′ such that K · K ′ is contained inÛ(n), and let g bethe Lie algebra of G, and g ′ of G ′ . For any irreducible, admissible (g, K)-module(π, V π ), let S(π) and S[π] be defined as in the non-archimedean case but now takinghomomorphisms in the sense of (g, K)-modules. Then Howe proves in [H2] thatjust as in Lemma 4.2,S[π] ∼ = π ⊗ σ 0 (π)17

as (g, K) × (g ′ , K ′ )-modules for a certain finitely generated representation σ 0 (π)of (g ′ , K ′ ) on which the centre of the universal enveloping algebra of g ′ acts by acharacter. Furthermore, the representation σ 0 (π) has a unique irreducible quotient,and the mapping π ↦→ σ 0 (π) is a bijective correspondence between the irreducibleadmissible representation of (g, K) occuring in the metaplectic representation andthe irreducible admissible representation of (g ′ , K ′ ) occuring in the metaplecticrepresentation. In fact Howe proves a much more precise statement which we takeup now; we will follow his fundamental paper on the subject [Ho2] very closely towhich the reader is referred to for all the proofs.Remark 5.1: The simplest example of Howe duality in the real case occursfor the compact pair (U(n), U(m)). In this case the metaplectic representation ofSp(nm) restricted to U(n)×U(m) is essentially the representation of U(n)×U(m)on polynomial functions on M(n, m; C) with the obvious action. If n ≤ m, for anyirreducible representation σ of U(n), there is a unique representation τ of U(m)such that σ ⊗ τ appears in the space of polynomial functions on M(n, m; C).For any irreducible representation µ of a compact subgroup L of Û(n), let deg(µ)denote the smallest integer d such that µ appears in the space S d of polynomialsof degree d under the metaplectic representation defined above.We now discuss Howe duality in the case when one of the groups, say G, in thedual reductive pair (G, G ′ ) is compact. In this case the symmetric space associatedto G ′ is hermitian symmetric, and the Cartan decompositionsp = u ⊕ p = sp 1,1 ⊕ sp 2,0 ⊕ p 0,2restricts to give the Cartan decompositiong ′ = k ′ ⊕ p ′ = k ′ ⊕ g ′2,0 ⊕ g ′0,2 .LetH(G) = {P ∈ S|X · P = 0 for all X ∈ g ′0,2 }.The vector space H(G) is called the space of harmonic polynomials for the compactgroup G. Clearly, H(G) is invariant under G and K ′ . With this notation, we havethe following theorem due to Howe, cf. [H2] and [H3].Theorem 5.2: Let (G, G ′ ) be a dual reductive pair with G compact, and let σbe a finite dimensional irreducible representation of G. Then we have the following.18

(a) The σ-isotypic part H(G) σ of H(G) is precisely the σ-isotypic part in S deg(σ) .(b) The representation H(G) σ of G × K ′ is irreducible.(c) If we write H(G) σ = σ ⊗ τ ′ , for an irreducible representation τ ′ of K ′ ,then σ ↦→ τ ′ is an injective map from the set of irreducible representations of Gappearing in the metaplectic representation to the set of irreducible representationsof K ′ .(d) The σ-isotypic part S σ of S is an irreducible representation of G × G ′ , andis therefore of the form σ ⊗ τ for an irreducible representation τ of G ′ , and one hasS σ = U(g ′ ) · H(G) σ ,where U(g ′ ) is the universal enveloping algebra of g ′ .(e) The representation τ of G ′ contains the representation τ ′ of K ′ on whichg ′0,2 acts trivially. Therefore τ is the unique highest weight module of g ′ containingτ ′ as the highest K ′ -type.This theorem has been used by Howe to prove the general duality for realgroups, to which we come later, using the following structural theorem about dualreductive pairs.Theorem 5.3: Let (G, G ′ ) be a dual reductive pair in Sp(n) with maximalcompact subgroups K and K ′ respectively, which are assumed to be in U(n). Forany subgroup H of Sp(n), let Z(H) denote the centraliser of H in Sp(n), and letH u = H ∩ U(n), which will be assumed to be a maximal compact subgroup of H.Then we have the following.(a) The pair (K, Z(K)) is a dual reductive pair, and similarly (K ′ , Z(K ′ )) is adual reductive pair.(b) The pair (Z(K) u , Z(K ′ ) u ) is a dual reductive pair in which both the groupsZ(K) u , and Z(K ′ ) u are products of compact unitary groups.Example 5.4: For the dual reductive pair (O(p, q), Sp(n)) with pq ≠ 0, themaximal compact subgroup of O(p, q) is O(p)×O(q) whose centraliser in Sp(n(p+q)) is Sp(n) × Sp(n). The maximal compact subgroup of Sp(n) is U(n) whosecentraliser in Sp(n(p + q)) is U(p, q).The Howe duality for real groups has subordinate to it a duality correspondence19

etween the representations of their maximal compact subgroups which appear inthe metaplectic representation. Here is the theorem on the correspondence betweenthe irreducible representations of maximal compact subgroups.Theorem 5.5: For an irreducible representation σ of K, there exists a uniqueirreducible representation σ ′ of K ′ such that H(K) σ ∩ H(K ′ ) σ ′ ≠ 0. Moreover, ifH(K) σ ∩ H(K ′ ) σ ′ ≠ 0, then H(K) σ ∩ H(K ′ ) σ ′∼ = σ ⊗ σ ′ as a K × K ′ -module.Remark 5.6: In the cases such as (O(p, q), Sp(n)) or (U(p, q), U(r, s)), whereK is isomorphic to K 1 × K 2 , and the centraliser of K is isomorphic to G ′ × G ′ ,with (K 1 , G ′ ) and (K 2 , G ′ ) themselves dual reductive pairs, the correspondence oftheorem 5.5 between representations of K and K ′ can be described in terms of thecorrespondence of theorem 5.2(c) as follows. Let σ 1 ⊗ σ 2 be a representation ofK = K 1 × K 2 . If the representations of K ′ associated by theorem 5.2(c) to representationsσ 1 and σ 2 of K 1 and K 2 is τ 1 ′ and τ 2 ′ respectively, then the representationof K ′ associated to the representation σ 1 ⊗σ 2 of K is the irreducible representationin the tensor product τ 1 ′ ⊗ τ 2 ′ whose highest weight is the sum of highest weights ofτ 1 ′ and τ 2.′For an irreducible admissible (g, K)-module π, let deg(π) denote the smallestdegree of any K-type whose image is non-zero under the surjection from S to S[π].Then the following is the more precise version of the Howe correspondence for realgroups.Theorem 5.7: Let π be an irreducible admissible representation of (g, K), andlet µ be a representation of K appearing in π and having degree deg(µ) = deg(π).Then the representation σ(π) of (g ′ , K ′ ) obtained by the Howe correspondence fromπ, contains the representation µ ′ of K ′ which is such that H(K) µ ∩ H(K ′ ) µ ′ ≠ 0.Moreover, the representation σ(π) does not contain any representation of K ′ otherthan µ ′ which appears in H(K) µ .Remark 5.8: For a dual reductive pair (G, G ′ ) in Sp(n, R) with one of G orG ′ compact, the Howe correspondence has been explicitly worked out by Kashiwaraand Vergne in [K-V]. In this case, as already noted in theorem 5.2, all therepresentations are highest weight modules.20

6 The Spherical caseBy an unramified classical group over a non-archimedean local field k, we will meanone of the following groups.(a) The group GL(n, l) where l is an unramified extension of k.(b) The unitary group of an ɛ-hermitian space W over an unramified extensionl of k, with a lattice L in W with L ⊥ψ(O k ) = 1 but ψ(π −1 O k ) ≢ 1).= L (for a character ψ of k such thatThe subgroup GL(n, O l ) of GL(n, l) in case (a), and the stabiliser of the latticeL in case (b) will be called standard maximal compact subgroups.For L = ∑ ni=1 O k e i ⊕ ∑ ni=1 O k f i where {e i , f i } is a symplectic basis of W , i.e.< e i , e j >= 0, < f i , f j >= 0, < e i , f j >= δ ij , it is clear that L ⊥ = L. ThereforeSp(W ) is an unramified classical group, and we let K be the stabiliser of L.Given a dual reductive pair (G, G ′ ) in Sp(W ) consisting of unramified pairs,we can assume that G has a standard maximal compact subgroup K, and G ′ hasa standard maximal compact subgroup K ′ such that K · K ′ is contained in K.From section 2.6, we know that the metaplectic covering of Sp(W ) splits onK. Therefore K can be thought of as a subgroup of ˜G, and K ′ a subgroup of ˜G ′ .Let H ψ −1( ˜G//K) denote the space of K-bi-invariant functions f on ˜G which arecompactly supported modulo C ∗ , such that f(zg) = ψ −1 (z)f(g) for all z ∈ C ∗ .H ψ −1( ˜G//K) is an algebra under convolution (the integral in the convolution isover ˜G/C ∗ ), and operates in a natural way on representations of ˜G with centralcharacter ψ.Theorem 6.1(Howe) : Given a representation π of ˜G with a K fixed vector,either there is no representation π ′ of ˜G ′ such that π ⊗ π ′ is a quotient of the Weilrepresentation (ω ψ , S) of ˜ Sp(W ), or there is exactly one representation π′of ˜G′such that π ⊗π ′ is a quotient of the Weil representation (ω ψ , S) of ˜ Sp(W ), and thisπ ′ has a K ′ fixed vector. Moreover, the algebras H ψ −1( ˜G//K), and H ψ −1( ˜G ′ //K ′ )operate with the same ring of endomorphisms on the space of K · K ′ fixed vectorsof S.7 Seesaw PairsWe introduce in this section the important concept of seesaw pairs due to Kudla.21

Definition 7.1: A pair (G, G ′ ) and (H, H ′ ) of dual reductive pairs in a symplecticgroup Sp(W ) is called a seesaw pair if H ⊂ G and G ′ ⊂ H ′ .It is generally pictorially depicted by the diagramG H ′⊲⊳H G ′where vertical arrows are inclusions and slanted arrows connect members of a dualreductive pair.Examples 7.2:7.2.1 :7.2.2 :7.2.3 :7.2.4 :Sp(W 1 ⊕ W 2 ) O(V ) × O(V )⊲⊳Sp(W 1 ) × Sp(W 2 ) O(V )O(V 1 ⊕ V 2 ) Sp(W ) × Sp(W )⊲⊳O(V 1 ) × O(V 2 ) Sp(W )U(V 1 ⊕ V 2 ) U(W ) × U(W )⊲⊳U(V 1 ) × U(V 2 ) U(W )Sp(W 1 ⊕ W ∗ 1 ) GL(V )⊲⊳GL(W 1 ) O(V )Let (G, G ′ ) and (H, H ′ ) be a pair of dual reductive pairs in Sp(W ) forming aseesaw pair :G H ′⊲⊳H G ′ .Let π H be in R ψ (H) and π G ′ in R ψ (G ′ ), and recall the notation σ 0 (π H ) and σ 0 (π G ′)introduced before Conjecture 2.Lemma 7.3 : With notation as above,Hom H [σ 0 (π G ′), π H ] = Hom G ′[σ 0 (π H ), π G ′].Proof :Let the Weil Representation of Sp(W ) be realised on S. Clearly,Hom H×G ′[S, π H ⊗ π G ′] = Hom H×G ′[σ 0 (π G) ′ ⊗ π G ′, π H ⊗ π G ′]= Hom H [σ 0 (π G ′), π H ].22

Similarly,Hom H×G ′[S, π H ⊗ π G ′] = Hom H×G ′[π H ⊗ σ 0 (π H ), π H ⊗ π G ′]= Hom G ′[σ 0 (π H ), π G ′].This proves the lemma.Remark 7.4: To see how seesaw pairs are used in practice, let us look at thefollowing seesaw pair in the real caseO(p, q) Sp(V ) × Sp(V )⊲⊳O(p) × O(q) Sp(V )Let σ be a representation of Sp(V ), and τ 1 ⊗ τ 2 of O(p) × O(q). As remarkedearlier, since O(p) and O(q) are compact groups, the Howe lifts of τ 1 and τ 2 areknown by [K-V]. By lemma 7.3, σ 0 (σ) contains τ 1 ⊗ τ 2 if and only if σ is a quotientof σ 0 (τ 1 ) ⊗ σ 0 (τ 2 ).This relates the Howe lifting problem to a problem abouttensor product of representations, and information about either one therefore givesinformation about the other one.Here is an example from [HK] who use the following seesaw pair to give thedecomposition of a discrete series representation of Sp(2) restricted to Sp(1) ×Sp(1).Sp(2) O(2, 2) × O(2, 2)⊲⊳Sp(1) × Sp(1) O(2, 2)Towards the analysis by Harris and Kudla on the decomposition of a discreteseries representation of Sp(2) restricted to Sp(1)×Sp(1), we simply note here thatthe Howe correspondence between an orthogonal group and a symplectic group isknown by the work of Adams [Ad1], and one can calculate the tensor product oftwo representations of O(2, 2) by the work of Repka [Re].8 The Theta correspondenceLet k be a number field, and A the adele-ring of k. For G an algebraic group overk, let G(k) denote the group of k-rational points of G, and let G(A) denote theadelic points of G. Let W be a symplectic vector space over k, and H(W ) theassociated Heisenberg group which is an algebraic group over k.23

Let W = W 1 ⊕ W 2 be a complete polarisation of W over k. Let S(W 1 (A))denote the space of Schwartz-Bruhat functions on W 1 (A). Given a character ψ :A/k → C ∗ , one can construct a representation of H(W )(A) on S(W 1 (A)) onwhich the centre of H(W )(A) (which is A) operates via ψ.ρ ψ (w 1 )f(x) = f(x + w 1 ) for all x, w 1 ∈ W 1 (A),ρ ψ (w 2 )f(x) = ψ(< x, w 2 >)f(x) for all x ∈ W 1 (A), w 2 ∈ W 2 (A)ρ ψ (t)f(x) = ψ(t)f(x) for all t ∈ A, x ∈ W 1 (A).Define a linear functional Θ on S(W 1 (A)) byΘ(φ) =∑x∈W 1 (k)The following lemma is easy to prove.Lemma 8.1:φ(x) for φ ∈ S(W 1 (A)).The series defining Θ(φ) converges absolutely, and φ → Θ(φ)defines an H(W )(k)-invariant linear form on S(W 1 (A)).The following basic theorem is due to Weil [We].Theorem 8.2: The space of H(W )(k)-invariant forms on S(W 1 (A)) is generatedby Θ.As in the local case, Sp(W )(A) operates on H(W )(A), and by the uniquenessof the representation of H(W )(A) with central character ψ, we obtain a projectiverepresentation of Sp(W )(A) via intertwining operators. This gives a C ∗ -coveringof Sp(W )(A), denoted by ˜ Sp(W )(A), such that for each place v of k one has thefollowing commutative diagram0 → C ∗ → Sp(W ˜ )(k v ) → Sp(W )(k v ) → 0↓ ↓ ↓0 → C ∗ → Sp(W ˜ )(A) → Sp(W )(A) → 0.We note that ˜ Sp(W )(A) is not the adelic points of an algebraic group over k, sothe notation is not very appropriate but is customary.The global metaplectic group comes equipped with a representation on S(W 1 (A)),called the global metaplectic or Weil representation.For any k-algebraic subgroup H of Sp(W ), let ˜H(A) denote the inverse image ofH(A) in Sp(W ˜ )(A), and let ˜H(k) denote the inverse image of H(k) in Sp(W ˜ )(A).24

Since there is an action of the semi-direct product H(W )(k) × Sp(W ˜ )(k) onS(W 1 (A)), it is clear that for all g ∈ Sp(W ˜ )(k), φ ↦→ Θ(gφ) is also an H(W )(k)-invariant form on S(W 1 (A)). Therefore by the uniqueness theorem 8.2 above,Θ(gφ) = λ g Θ(φ) for some λ g ∈ C ∗ . Clearly g → λ g is a character on Sp(W ˜ )(k),which gives the unique splitting of the exact sequence0 → C ∗ → ˜ Sp(W )(k) → Sp(W )(k) → 0.Because of this splitting, we can think of Sp(W )(k) as a subgroup of ˜ Sp(W )(A),and for any k-algebraic subgroup H of Sp(W ), H(k) can also be thought of as asubgroup of ˜H(A). The reader must be cautioned here that if the global metaplecticcovering of Sp(W )(A) splits on H(A), it is not a priori clear that one canchoose a splitting such that the image of H(A) contains H(k). The problem arisesof course because there may not be a unique splitting of H(A); see [Ge2] for H aunitary group.Given any function φ ∈ S(W 1 (A)), the following formula defines a function θ φon Sp(W )(k)\ Sp(W ˜ (A))θ φ (g) = Θ(g · φ).The function θ φ is known to be an automorphic function on Sp(W (k))\ ˜ Sp(W )(A)(i.e., it is a slowly increasing, K-finite, Z-finite function where Z is the centre ofthe universal enveloping algebra of Sp(W )(R)). It has the property that θ φ (hg) =θ hφ (g).We are now ready to define the so-called theta correspondence which definesa correspondence between automorphic forms on one member of a dual reductivepair to those of the other member of the dual reductive pair.Let (G, G ′ ) be a pair of algebraic groups over k which form a dual reductivepair in Sp(W ). Restricting the function θ φ on ˜ Sp(W )(A) to ˜G(A) × ˜G ′ (A), weget a function θ φ (g, g ′ ) on ˜G(A) × ˜G ′ (A), called the theta-kernel. Now let A bean irreducible space of cusp forms on G(k)\ ˜G(A). For f ∈ A, define a functionθ φ (f)(g ′ ) on G ′ (k)\ ˜G ′ (A) by∫θ φ (f)(g ′ ) =G(k)\ ˜G(A)θ φ (g, g ′ )f(g)dg.The span of θ φ (f) where φ runs over S(W 1 (A)) is a ˜G ′ (A)-invariant space offunctions. From the property θ φ (hg) = θ hφ (g) noted above, it follows that if f 1 and25

f 2 belong to the same irreducible representation of ˜G(A), the span of θ φ (f 1 ) andθ φ (f 2 ) are the same. The span of θ φ (f) as f runs over A, and φ runs over S(W 1 (A))is called the θ-lift of the cuspidal representation A, and will be denoted by Θ(A, ψ).This space is not necessarily irreducible, and any irreducible sub-quotient of it iscalled a representation obtained by theta correspondence.8.3 Examples of the theta Correspondence : We will be looking at certainexamples of the theta correspondence for the dual pair (Sp(W ), O(V )) ⊂ Sp(W ⊗V ), when dim W = 2 in which case Sp(W ) ∼ = SL(2, k).8.3.1 : dim V = 1, q(x) = ax 2 , k = Q. Automorphic forms f S = ∏ f v on O(V )correspond bijectively to finite subsets S of even cardinality of the set of places ofQ such that f v (−1) = −1 if and only if v ∈ S. Theta lift of f S is a modular formof weight 1 or 3 depending on whether ∞ ∉ S, or ∞ ∈ S. If S is the empty set,2 2then the theta lift of f S for an appropriate choice of the function φ is the usualtheta function on the upper half plane: θ(z) = ∑ n∈Z e πin2z . If S is not empty, thetheta lift of f S is a cusp form. See [Ge1] for details on all this.8.3.2 : dim V = 2, q(x) = a · (norm form of a quadratic field extension K ofk). An automorphic form on O(V ) is given by a grossencharacter on the idele classgroup J K /K ∗ , and the theta lift constructs an automorphic form on GL 2 (k). Thisconstruction in the case of k = Q goes back to Hecke and Maass.8.3.3 : dim V = 3, q(x, y, z) = x 2 − yz, SO(V ) ∼ = P GL 2 . The theta lift in thiscase from ˜ SL 2 to P GL 2 is the Shimura lifting which associates a modular form ofweight n − 1 to a modular form of half-integral weight n 2 .8.3.4 : dim V = 4. If V is anisotropic over k, then the theta lifting is relatedto the Jacquet-Langlands correspondence which constructs automorphic forms onGL 2 from automorphic forms on quaternion division algebras.If V has Witt index 1, then there is a unique quadratic extension K/k overwhich V splits. In this case the theta lifting is the base change from GL 2 over kto GL 2 over K. If k = Q, and K is a real quadratic field, it is the Doi-Naganumalift.8.4 Global Howe Conjecture : Let (G, G ′ ) be a dual reductive pair andπ = ⊗π v , a representation of ˜G(A). One knows that at almost all the places vof k, the group G is an unramified classical group and the representation π v hasa fixed vector under a standard maximal compact subgroup of G(k v ). Therefore26

from Theorem 6.1, if the Howe lift σ(π v ) of π v is non-zero, it also has a fixed vectorunder a standard maximal compact subgroup of G ′ (k v ). We can therefore formthe restricted direct product of the representations σ(π v ), to be denoted by σ(π).Howe had conjectured that if π is an automorphic representation of ˜G then σ(π) isalso automorphic. This turned out to be false as was shown by Piatetski-Shapiroin [Ps], following the work of Waldspurger. One hopes that this conjecture, thoughnot entirely correct, is basically correct. Possibly, the problems are related to L-indistiguishability and the counter-example of Piatetski-Shapiro suggests that eventhough σ(π) may not be automorphic, there may be a representation σ ′ (π) whichdiffers from σ(π) in only finitely many places and is automorphic.On the positive side, we have the following (folklore) theorem for which we referto [GRS] for the proof.Theorem 8.5 : Let (G, G ′ ) be a dual reductive pair and π = ⊗π v , a cuspidalautomorphic representation of ˜G such that Θ(π, ψ) consists of cusp forms on ˜G ′ (A).Then(i) Θ(π, ψ) is an irreducible representation π ′ = ⊗π ′ v of ˜G ′ (A).(ii) All the local representations π ′ v are the Howe lifts of π v (for the characterψ v ); in particular the global Howe conjecture is true in this case.π.(iii) The theta lift Θ(π ′ , ψ) of π ′ is non-zero. If it is cuspidal, it is equivalent toRemark 8.6 : To see a situation where Θ(π, ψ) is known to consist only ofcusp forms, we recall a theorem of Rallis [Ra1] (see also [H-Ps]) that for the dualpair (O(V ), Sp(n)), π a cuspidal automorphic representation on O(V ), Θ(π, ψ)consists only of cusp forms if and only if the theta lift of π to Sp(n − 1) is zero forthe dual pair (O(V ), Sp(n − 1)) .The following fundamental theorem of Waldspurger [Wa1] gives criterion forthe non-vanishing of theta lifts in the simplest situation of the dual reductive pair(O(2, 1), ˜ SL2 ). As we saw in 4.6.3, there is no loss of information in going fromO(2, 1) to SO(2, 1) = P GL 2 , and this is what Waldspurger considers: the pair(P GL 2 , SL2 ˜ ).Theorem 8.7:(a) The theta lift of an automorphic cuspidal representation π of P GL 2 to ˜ SL 2is non-zero if and only if L(π, 1 2 ) ≠ 0. 27

(b) The theta lift of an automorphic cuspidal representation π of P GL 2 to SL ˜ 2is orthogonal to all the automorphic forms on SL ˜ 2 arising from the pair (O(1), SL2 ˜ )(and for any choice of the additive character ψ) where O(1) is the orthogonal groupin one variable.(c) Let σ be a cuspidal automorphic representation of SL2 ˜ orthogonal to all theautomorphic forms on SL ˜ 2 arising from the pair (O(1), SL2 ˜ ) (and for any choiceof the additive character ψ), then the theta lift of σ to P GL 2 is an irreduciblecuspidal representation which is non-zero if and only if σ has a Whittaker modelfor ψ −1 . Moreover, the condition that σ has a Whittaker model can be interpretedas a non-vanishing of a certain L-function at 1 2 .We now come to the global analogue of lemma 7.3 about seesaw pairs. Beforewe state it, let us introduce the notation < f 1 , f 2 > E = ∫ C ∗ E(k)\Ẽ(A) f 1 ¯f 2 de where Eis a k-algebraic subgroup of Sp(W ), and f 1 , f 2 are functions on E(k)\Ẽ(A) withG Hthe same unitary central characters. Let ⊲⊳ be a seesaw pair. The proofH ′ G ′of the following analogue of lemma 7.3 is clear and will be omitted.Lemma 8.8: For f 1 a cusp form on H ′ and f 2 a cusp form on G ′ , and anarbitrary φ ∈ S(W 1 (A)), we have< θ φ (f 1 ), f 2 > G ′=< f 1 , ¯θ φ (f 2 ) > H ′,where θ φ (f 1 ) which is a function on ˜H(A) has been restricted to ˜G ′ (A), similarlyθ φ (f 2 ) which is a function on ˜G(A) has been restricted to ˜H ′ (A).This is an identity between inner products of automorphic forms on differentgroups, and is often useful to relate L-functions of automorphic forms to certainperiod integrals involving Eisenstein series (which arise as theta lifts of certainautomorphic forms by the Siegel-Weil theorem). We do not go into the details onthis but refer to [Ku2] for several examples.9 QuestionsHere we want to point out some of the main open problems of the theory that wehave been discussing.28

9.1 Local questionAs the Howe conjecture is now basically proved (except in residue characteristic2 case), the question arises what exactly is the Howe correspondence in termsof familiar parametrisations of representations. For example, one would like tounderstand the Howe correspondence in terms of Langlands’ functoriality. Moreprecisely, let (G, G ′ ) be a dual reductive pair and assume for simplicity that themetaplectic covering of the corresponding symplectic group splits over G × G ′ . Letπ be an irreducible representation of G, and σ(π) the Howe lift of π (which ofcourse depends on the additive character). Then the question is whether thereis a mapping ρ π : L G → L G ′ such that the representation σ(π) of G ′ lies in theL-packet of the lift of the representation π of G under this map. Of course for ageneral pair (G, G ′ ), there may not be any non-trivial maps of L G to L G ′ , so thisnaive question is meaningless but one might hope that there are non-trivial mapsfrom endoscopic subgroups L H of L G to L G ′ , and the representation π comes froman endoscopic lift of a representation of H. So we can modify our naive questionto ask that if σ(π) is non-zero, whether there is an endoscopic subgroup L H π of L G(more precisely an endoscopic data) such that the representation π is an endoscopiclift of a representation of H π with Langlands parameter σ π with values in L H π , anda mapping ρ π : L H π → L G ′ such that σ(π) lies in the L-packet of ρ π (σ π ). Eventhis conjecture seems to be false as two representations of G which are in the sameL-packet might lift to representations of G ′ which are not in the same L-packet.But the question is expected to have an affirmative answer if one replaces L-packetsby the bigger Arthur packets. A related question is to understand how does theHowe correspondence change when one replaces G or G ′ by groups which are innerforms of these. See the work of J. Adams in [Ad2] for precise conjectures andresults in the real case. See also the work of Rallis [Ra2] and the paper of Gelbartin this volume [Ge2] concerning this question. For the dual pair (GL(n), GL(m)),see 4.6.5 for what is expected.9.2 Global questionLet k be a global field, (G, G ′ ) a dual reductive pair over k, and π = ⊗π v acuspidal automorphic representation of ˜G(A). It is clear that if Θ(π, ψ) ≢ 0, then29

for all the local components π v of π, the Howe lift (with respect to ψ v ) is non-zero.The main question concerning the global theory is to find necessary and sufficientconditions for Θ(π, ψ) to be non-zero. If we assume that π = ⊗π v is such thatthe Howe lift of π v is non-zero for all v, then in many cases there is a conditioninvolving global L-function which controls whether or not there is an automorphicrepresentation π ′ = ⊗π v ′ in the same global L-packet as π (i.e., π v and π v ′ belongto the same L-packet for all v and are equal for almost all v) whose theta liftis non-zero. However, in many other cases, there is no such condition involvingglobal L-function and there is an automorphic representation π ′ in the same globalL-packet as π whose theta lift is non-zero as soon as all the local Howe lifts arenon-zero. The situation is not clear at the moment. The work of Waldspurger[Wa1] recalled in theorem 8.7 above, and the subsequent work of Rallis [Ra3] arein this direction. See also the work [GRS] where rather complete results have beenobtained for the theta lifts of global L-packets for the pair (U(2, 1), U(1, 1)).References[Ad1]J.Adams,“Discrete spectrum of the reductive dual pair (O(p, q), Sp(2m))”,Inv. vol 74(1983)449-475.[Ad2] J.Adams,“L-functoriality for Dual Pairs”, Astrisque 171-172, OrbitesUnipotentes et Representations, vol.2(1989), pp. 85-129.[Ge1][Ge2]S.Gelbart,“Weil’s Representation and the Spectrum of the Metaplecticgroup”, LNM 530, Springer-Verlag (1975).S.Gelbart,“On Theta series liftings for Unitary groups”, This volume.[GRS] S.Gelbart, J.Rogawski and D.Soudry,“On periods of cusp forms on U(3)”,to appear.[HK][Ho1]M.Harris and S.Kudla,“Arithmetic Automorphic forms for the nonholomorphicdiscrete series of GSp(2)”, Duke Math J. 66, no 1, (1992) pp. 59-121.R.Howe,“Θ-series and invariant theory”, In: Proc. Symp. Pure Math., vol33, part 1, AMS, 1979, pp. 275-286.30

[Ho2]R.Howe,“Transcending classical invariant theory”, J. of the Amer. Math.Soc. vol.2, no.3 (1989), pp. 535-552.[Ho3] R.Howe,“Remarks on classical invariant theory”, Transactions of AMS 313,vol 2 (1989), pp. 539-570.[H-Ps] R.Howe and I.Piatetski-Shapiro,“Some examples of automorphic forms onSp 4 ”, Duke Math. J.(1983),pp. 55-106.[K-V][Ku1][Ku2]M.Kashiwara and M.Vergne,“On the Segal-Shale-Weil representations andharmonic polynomials”, Inv. math. 44(1978), pp. 1-47.S.Kudla,“On the Local Theta Correspondence, Inv. Math. 83(1986), pp.229-255.S.Kudla,“See-Saw Dual Reductive pairs”, Taniguchi Symposium, Katata,1983, Birkhauser, 1984, pp. 244-268.[MVW] C.Moeglin, M.-F.Vigneras and J.L.Waldspurger, “Correspondence deHowe sur un corps p-adique”, LNM 1291, Springer-Verlag(1987).[Ps][Ra1]I.Piatetski-Shapiro,“Work of Waldspurger”, Lie group representations II,LNM 1041(1984), pp. 280-302.S.Rallis,“On the Howe duality conjecture”, Comp. Math. 51(1984), pp.333-399.[Ra2] S.Rallis,“Langlands Functoriality and the Weil Representation”, Amer. J.Math.104(1982), pp. 469-515.[Ra3] S.Rallis,“L-functions and the Oscillator representation”, LNM 1245,Springer-Verlag, 1988.[R-S]S.Rallis and G.Schiffmann,“On a relation between SL 2 cusp form and cuspform on tube domains associated to orthogonal groups”, Trans. Amer.Math. Soc. 263(1981), pp. 1-58.[Re] J.Repka,“Tensor product of unitary representations of SL 2 (R)”, Amer. J.Math 100(1978), pp. 747-774.31

[Wa1] J.L.Waldspurger,“Correspondence de Shimura”, J. Math. Pures Appl. 59,pp. 1-133(1980).[Wa2]J.L.Waldspurger,“Demonstration d’une conjecture de duality de Howe dansle cas p-adique,p ≠ 2”, In: Festschrift in honor of Piatetski-Shapiro, IsraelMath. Conf. Proc., vol 2, pp. 267-324 (1990).[We] A.Weil,“Sur certains groupes d’operateurs unitaires”, Acta.Math.111(1964), pp. 143-211.32