Local theta correspondence and the lifting of Duke, Imamoglu and Ikeda
aa r X i v : . [ m a t h . N T ] M a r LOCAL THETA CORRESPONDENCE AND THE LIFTING OFDUKE, IMAMOGLU AND IKEDA
RAINER SCHULZE-PILLOT
Abstract.
We use results on the local theta correspondence to prove that forlarge degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form isnot a linear combination of theta series. Introduction
In the article [7] it is proved as a side remark that a Siegel modular form F of degree2 n and weight n + k obtained by the lifting of Duke, Imamoglu and Ikeda [5] (calledthe DII-lifting in the sequel) from an elliptic modular form of weight 2 k is not alinear combination of theta series of even unimodular positive definite quadraticforms of rank m = 2( n + k ) if n is bigger than k , whereas for n = k ≡ L -functions of elliptic cuspidal Hecke eigenforms. The proof uses B¨ocherer’scharacterization of the cuspidal Siegel eigenforms that lie in the space of thetaseries by special values of their standard L -functions.In this article we use results on the local theta correspondence to give a differentproof of the first result. In fact we prove a more general version including thetaseries attached to arbitrary non degenerate quadratic forms and also theta serieswith spherical harmonics. The underlying local fact has been noticed by severalpeople (including the author) immediately after the preprint version of [5] becameavailable in 1999 but has apparently never been published.In the case n = k we show that the local representations attached to a DII-liftare in the image of the local theta correspondence with the orthogonal group of asuitable local quadratic space of dimension 4 n = 2( n + k ) for all places.I thank Ralf Schmidt for discovering a mistake in an earlier version of this article.2. The case n > k
Let f be an elliptic modular form of weight 2 k for the full modular group SL ( Z ).It was conjectured by Duke and Imamoglu and proven by Ikeda in [5] that for any n ≡ k mod 2 there exists a nonzero Siegel cusp form F = F n ( f ) of weight n + k for the group Sp n ( Z ) ⊆ SL n ( Z ) whose standard L -function is equal to ζ ( s ) n Y i =1 L ( s + k + n − i, f ) , where L ( s, f ) is the usual Hecke L -function of f . We call F n ( f ) the DII-lift ofdegree 2 n of f .In fact, it has been shown in [16, Lemma 1.3.1], that the Satake parameters α ( j ) p of F at the prime p are given by(2.1) α (0) p = β − np , α ( j ) p = β p p − n + j − for 1 ≤ j ≤ n, MSC 2000: Primary 11F46, Secondary 11F27. where we write the p -factor of L ( s, f ) as (1 − β p p − s + k − ) − (1 − β − p p − s + k − ) − ,i. e., we have (by Ramanujan-Petersson) the normalization | β p | = 1.Moreover, Ikeda has recently announced a generalization of this result. In this gen-eralized version f is replaced by an irreducible cuspidal automorphic representation τ = ⊗ τ v of GL ( A E ), where E is a totally real number field and τ v is assumed tobe a principal series representation attached to a character µ v for all finite places v of E and to be discrete series with minimal weight ± κ w for the infinite places w of E .Ikeda then proves the existence of an irreducible cuspidal automorphic represen-tation π = π ( m, τ ) of the metaplectic group g Sp m ( A E ) with local components π v described in terms of τ v as follows: For a real place w the representation π w is thelowest weight representation of lowest K -type det κ w + m/ , for a finite place v therepresentation π v is a degenerate principal series representation which is inducedfrom a character µ ( m ) v on the maximal parabolic f P m derived from µ v If m = 2 n is even one can obtain from π a representation of Sp n ( A E ), also denotedby π = π (2 n, τ ); this representation is induced from the 2 n -tuple of characters µ v | | − n + j − for 1 ≤ j ≤ n . If (for E = Q ) in addition f is as above and τ is the automorphic representation of GL ( A Q ) associated to f this gives therepresentation of Sp n ( A Q ) associated to F n ( f ).We are interested in the question whether the Siegel modular form F n ( f ) can beobtained as a linear combination of theta series, respectively in the more generalsituation whether the representation π (2 n, τ ) is in the image of the theta corre-spondence between Sp n and a suitable orthogonal group.We let V be a vector space over E of even dimension 2 r with a nondegeneratequadratic form q and associated symmetric bilinear form B ( x, y ) = q ( x + y ) − q ( x ) − q ( y ) on it. If E = Q and q is positive definite we consider a homogenouspluriharmonic form P ∈ C [ { X ij | ≤ i ≤ r, ≤ j ≤ m } ] in 2 rm variables ofweight ν (see [3, III, 3.5]) as a function on ( V ⊗ R ) m by identifying the latter spacewith R rm using a basis of V which is orthonormal with respect to the symmetricbilinear form B .The theta series of degree m of L with respect to P is defined for any Z - lattice L of full rank 2 r on V by ϑ ( m ) ( Z, L, P ) = X x ∈ L m P ( x ) exp(2 πi tr( q ( x ) Z )) , where Z ∈ H m is in the Siegel upper half space H m ⊆ M Sym m ( C ) and where q ( x ) = ( B ( x i , x j )) is half the Gram matrix of the m -tuple x = ( x , . . . , x m ). It is a Siegelmodular form of weight r + ν for some congruence subgroup of Sp m ( Z ). One hasa similar definition for arbitrary totally real E and totally positive definite q .If L is even unimodular the theta series is a Siegel modular form for the full modulargroup Sp m ( Z ).More generally one considers for arbitrary E and (non degenerate) q the theta cor-respondence associating to a subset of the set of irreducible automorphic represen-tations of the adelic orthogonal group O ( V,q ) ( A E ) a subset of the set of irreducibleautomorphic representations of the adelic group Sp m ( A E ). There is by now a vastliterature (starting with Weil’s [18] and Howe’s [4]) but no textbook reference onthis correspondence, for definitions and properties of it see [14, 11, 12, 8, 10]. OCAL THETA CORRESPONDENCE AND THE DII LIFTING 3
Theorem 2.1.
Let
E, V, q, τ be as above. (1) If n > r − and there is a finite place v of E for which the completion V v of the quadratic space ( V, q ) is not split (i. e. is not an orthogonal sumof hyperbolic planes) the representation π (2 n, τ ) is not in the image of thetheta correspondence with O ( V,q ) ( A E ) . (2) If n > r the representation π (2 n, τ ) is not in the image of the theta corre-spondence with O ( V,q ) ( A E ) . Corollary 2.2.
Let f be an elliptic modular form of weight k as above, ν ∈ N .Then for n > k − ν the DII-lift F n ( f ) is not a linear combination of theta seriesof positive definite quadratic forms with pluriharmonic forms of degrees ν ′ ≥ ν .In particular for n > k the DII-lift F n ( f ) is not a linear combination of thetaseries attached to positive definite quadratic forms (with or without pluriharmonicforms).Proof of the Theorem. If π (2 n, τ ) is in the image of the global theta correspon-dence with O ( V,q ) ( A E ) its local components π v (2 n, τ ) are in the image of the localtheta correspondence with O ( V,q ) ( E v ) for all places v of E ; we denote by π ′ v thecorresponding representation of this orthogonal group.The local theta correspondence has been described explicitly in terms of the Bern-stein–Zelevinsky data of the representations in [14, 8]. By construction the Bern-stein–Zelevinsky data of π v (2 n, τ ) are µ v | | − n + j − for 1 ≤ j ≤ n . The characters µ v | | − n + j − are never of the type | | i for some integral i ; this is clear if µ v isramified, and follows from | µ v ( ω v ) | < q v (see [6]) if µ v is unramified (where ω v is a prime element at the place v and q v is the norm of ω v ). Since by the resultsof [14, 8] only characters of the type | | i for some integral i can be missing inthe Bernstein–Zelevinsky data of the representation π ′ on the orthogonal side wesee that all the µ v | | − n + j − for 1 ≤ j ≤ n have to appear in these Bernstein–Zelevinsky data. Obviously the rank of the orthogonal group O ( V,q ) ( E v ) has to beat least 2 n for this to be possible. If V v is split this requires r ≥ n , if V v is notsplit it requires r − ≥ n (and even r − ≥ n if the anisotropic kernel of V v isof dimension 4). (cid:3) Proof of the Corollary.
It is well known that one can associate to a cuspidal Siegelmodular form F for Sp n ( Z ) which is an eigenfunction of all Hecke operators anirreducible cuspidal automorphic representation π ( F ) of Sp n ( A Q ) with the sameSatake parameters, see e.g. [2]. Moreover, if F has weight ˜ k and can be writtenas a linear combination of theta series of positive definite quadratic forms withpluriharmonic forms of weights ν ′ ≥ ν , the fact that the space generated by thetaseries with pluriharmonic forms of weight ν ′ for lattices on a fixed quadratic spaceis Hecke invariant guarantees that there is a positive definite quadratic space ( V, q )of dimension 2 r = 2(˜ k − ν ′ ) for some ν ′ ≥ ν such that the irreducible representation π ( F ) is in the image of the theta correspondence with O ( V,q ) ( A Q ).Considering this for the DII lift F n ( f ) and ν, ν ′ , k as in the assertion we have˜ k = k + n and therefore r = k + n − ν ′ ≤ k + n − ν and k < n + ν , hence r < n .By the Theorem π ( F n ( f )) is not in the image of the theta correspondence with O ( V,q ) ( A Q ), so F n ( f ) can not be a linear combination of theta series as describedabove. (cid:3) R. SCHULZE-PILLOT
Remark. (1) If we omit the restriction that the quadratic form is positive definite westill get the same result as long as we restrict attention to the holomorphictheta series of weight r associated to indefinite quadratic forms of rank 2 r that have been constructed by Siegel and Maaß [17, 9]. Although there isno method known to use suitable test functions in the oscillator represen-tation in order to construct holomorphic theta series of weight k + n − ν ′ associated (by the theta correspondence) to indefinite quadratic forms ofrank larger than 2( k + n − ν ′ ), the results of Rallis [15] seem to imply thatsuch a construction is indeed possible. Such a construction would then yield F n ( f ) without contradicting the Theorem.(2) The Corollary could in principle be proved without using representationtheoretic tools by computing the possible Hecke eigenvalues of a linearcombination of theta series with the help of results of Andrianov [1] andYoshida [19] and comparing with the eigenvalues of a DII-lift; we expectthis to be a rather tedious and unpleasant computation.3. The case n = k With the notations of the previous section we assume now E = Q and n = k , sothat the weight k + n of a DII lift is equal to the rank 2 n of the symplectic groupconsidered. Theorem 3.1.
For n = k the local components π p of the representation π ( F n ( f )) of the Duke-Imamoglu-Ikeda lift F n ( f ) are in the image of the local theta corre-spondence with the split quadratic space over Q p of dimension n = 2( k + n ) (whichis the orthogonal sum of n hyperbolic planes) for all (finite) primes p .The component π ∞ at the real place is in the image of the theta correspondence withthe orthogonal group of the positive definite quadratic space over R of dimension n and also in the image of the theta correspondence with the orthogonal group ofthe quadratic space of dimension n and signature (4 n − , over R .Proof. The assertion for the finite primes is again an immediate consequence of theresults of [14, 8]. The assertion at the real place follows from Theorem 15 of [13]since π ∞ is a limit of discrete series. (cid:3) Remark. (1) At the real place it follows from Theorem 15 of [13] that π ∞ occurs also inthe images of the theta correspondence with the orthogonal groups of thequadratic spaces of dimension 4 n + 2 and signatures (4 n + 1 ,
1) and (4 n, R , but not for any other quadratic space of dimension 4 n or 4 n + 2.(2) Since the split quadratic space of dimension 4 n has square discriminant,there is no quadratic space over Q which is split at all finite primes andof signature (4 n − , Q of even dimension 2 r which is split at allfinite primes if and only if r is divisible by 4 and that the same conditionis necessary and sufficient for the existence of a quadratic space over Q ofeven dimension 2 r + 2 and signature (2 r + 1 ,
1) which is split at all finiteprimesWe see that π ( F n ( f )) can (in the case n = k ) be in the image of the globaltheta correspondence with some quadratic space of dimension 4 n = 2( n + k )only if n is even and the quadratic space is the unique positive definite spaceof that dimension which is split at all finite places.In a similar way for even n there is also a unique quadratic space ( V, q )over Q of dimension 4 n + 2 and signature (4 n + 1 ,
1) for which π v is in the OCAL THETA CORRESPONDENCE AND THE DII LIFTING 5 image of the local theta correspondence with the orthogonal group of thecompletion V v for all (finite or infinite) places v of Q ; this space is againsplit at all finite primes.There are also spaces of signature (4 n,
2) for which π v is in the image ofthe local theta correspondence with the orthogonal group of the completion V v for all (finite or infinite) places v of Q ; such a space can be obtainedas the orthogonal sum of an imaginary quadratic field equipped with thenorm form scaled by some negative number and a positive definite spaceof dimension 4 n which is split at all finite places. In all cases our purelylocal methods allow no statement about occurrence in the global thetacorrespondence for the respective space.4. Iterated theta liftings
One might ask whether it is possible to construct the Duke–Imamoglu–Ikeda lift F n ( f ) or the associated automorphic representation by a sequence of theta liftingsbetween groups G i , where for each i the pair G i , G i +1 consists (in either order)of a symplectic or metaplectic group and an orthogonal group, starting with therepresentation associated to f on SL or the representation on the metaplecticgroup g SL associated to the form g which corresponds to f under the Shimuracorrespondence. We have more generally: Proposition 4.1.
Let the notations be as in Sections 1 and 2 and n > .The generalized Duke–Imamoglu–Ikeda lift π (2 n, τ ) can not be constructed by aseries of theta liftings as described above.Proof. If one starts out with the metaplectic group g SL all subsequent groups G i will be orthogonal groups of quadratic spaces in odd dimension or metaplecticgroups g Sp m (with a genuine representation of g Sp m on it) so that we will never arriveat Sp n , with the exception that the initial step may lead to SO (3 ,
2) identifiedwith
P GSp or to SO (2 ,
1) identified with
P GL (in which case we obtain a Saito–Kurokawa lift respectively a Shimura lift). After this initial step there have toappear correspondences which raise the rank of the group, and again by [14, 8] alllocal representations occurring will (at each finite place v of E ) have at most twoterms µ v | | j with j ∈ Z \ Z among their Bernstein–Zelevinsky data, so π v (2 n, τ )can never occur for n > SL the representation π v (2 n, τ ) can not occur if n > (cid:3) References [1] A. N. Andrianov: Quadratic forms and Hecke operators.
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