The special values of the standard L-functions for \mathrm{GSp}_{2n} \times \mathrm{GL}_1
aa r X i v : . [ m a t h . N T ] F e b The special values of the standard L -functions for GSp n × GL Shuji Horinaga, Ameya Pitale, Abhishek Saha, Ralf Schmidt
Abstract
We prove the expected algebraicity property for the critical L -values of character twistsof the standard L -function associated to vector-valued holomorphic Siegel cusp forms ofarchimedean type ( k , k , . . . , k n ), where k n ≥ n + 2 and all k i are of the same parity. Forthe proof, we use an explicit integral representation to reduce to arithmetic properties ofdifferential operators on vector-valued nearly holomorphic Siegel cusp forms. We establishthese properties via a representation-theoretic approach. Contents n ( R ) . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Generators for the center of the universal enveloping algebra . . . . . . . . . . . 72.3 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 The relationship between elements of U ( g C ) and differential operators . . . . . . 92.5 Scalar and vector-valued automorphic forms . . . . . . . . . . . . . . . . . . . . . 10 p − )-finite automorphic forms . . . . . . 153.3 Action of Aut( C ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Arithmeticity of certain differential operators on nearly holomorphic cusp forms . 24 L -functions 28 L -values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Bibliography 34
The arithmetic of special values of L -functions is of great interest in modern number theory. Acentral problem here is to prove the algebraicity and Aut( C )-equivariance (up to suitable periods)of critical L -values. For classical cusp forms on the upper-half plane, Shimura [28, 29, 30] andManin [20] were the first to study the arithmetic of their critical L -values in the 1970s. In thispaper, we focus on twists of standard L -functions of vector-valued Siegel cusp forms of degree INTRODUCTION n ; these correspond to L -functions L ( s, Π ⊠ χ ) on GSp n ( A Q ) × GL ( A Q ) such that Π ∞ is aholomorphic discrete series representation. The first results in this case were obtained overforty years ago with the works of Harris [13] and Sturm [33] who (independently) proved theexpected algebraicity results for Π corresponding to a scalar-valued Siegel cusp form of full leveland χ = 1. Subsequent works on the critical L -values of Siegel cusp forms by various authors[3, 5, 7, 17, 22, 32] have strengthened and extended these results in various directions.Nonetheless, a proof of algebraicity of critical L -values for holomorphic forms on GSp n × GL in full generality has not yet been achieved. The case n = 2 has now been largely resolved by arecent result [25, Theorem 1.1]. For general n , one can parameterize the possible archimedeantypes Π ∞ by n -tuples of positive integers ( k , k , . . . , k n ), where k ≥ k ≥ . . . ≥ k n . If Π isassociated to a scalar-valued Siegel cusp form, we have k = k = . . . = k n ; this case is nowessentially solved by the succession of works cited above. The major stumbling block is thegeneral vector-valued case, where the various k i may not be equal. It was proved by Kozima [17]that the expected algebraicity property holds for the critical values of the (untwisted) standard L -function of a vector-valued Siegel cusp form of full level and archimedean type ( k, ℓ, ℓ, . . . , ℓ )with k, ℓ even and k ≥ ℓ ≥ n + 2. Here we prove the conjectured algebraicity property of critical L -values for more general archimedean types, general ramifications, and twists by characters. Let Π be a cuspidal automorphic representation of GSp n ( A Q ) whose archimedean component Π ∞ is the holomorphic discrete series representation with highestweight ( k , k , . . . , k n ) , where the k i are integers of the same parity and k ≥ k ≥ . . . ≥ k n ≥ n + 2 . Let S be a finite set of places of Q including ∞ such that Π p is unramified for p / ∈ S . Let F be a nearly holomorphic Siegel cusp form of scalar weight k with Fourier coefficients lying ina CM field such that (the adelization of) F generates an automorphic representation Π F whoselocal component Π F,p is twist-equivalent to Π p for all p / ∈ S . For each Dirichlet character χ suchthat χ ∞ = sgn k and each integer r such that n + 2 ≤ r ≤ k n − n , r ≡ k n − n (mod 2) , define D (Π , χ, r ; F ) = L S ( r, Π ⊠ χ ) i k π nk + nr + r G ( χ ) n +1 h F, F i . Then σ ( D (Π , χ, r ; F )) = D ( σ Π , σ χ, r ; σ F ) for σ ∈ Aut( C ) ; in particular D (Π , χ, r ; F ) lies in aCM field. Above, G ( χ ) denotes the Gauss sum and the notation L S indicates that we omit the local L -factors corresponding to places in S . The set of critical points for L ( s, Π ⊠ χ ) in the right-half plane are given byintegers r such that { ≤ r ≤ k n − n : r ≡ k n − n (mod 2) } . (1) Thus, Theorem 4.5 fails to include the critical points close to 1 given by the set { ≤ r ≤ n + 1 : r ≡ k n − n (mod 2) } . The reason for this omission is technical and relates to inadequate in-formation about the arithmetic properties of nearly holomorphic (non-cuspidal) modular forms;see Remark 4.4. We also note that the critical points in the left half-plane are related to thosein the right-half plane via the global functional equation (which is known by Theorem 62 of [8]). We show in Section 4.1 that an F satisfying these properties always exists. INTRODUCTION V -valued holo-morphic Siegel cusp form G on H n such that G ( γZ ) = ρ ( J ( γ, Z )) G ( Z ) for all γ in some principalcongruence subgroup of Sp n ( Z ), where ( ρ, V ) is the finite-dimensional representation of GL n ( C )with highest weight ( k , k , . . . , k n ). (We refer to such G as vector-valued holomorphic Siegelcusp forms of archimedean type ( k , k , . . . , k n ).) The vector-valued holomorphic form G isrelated to the scalar valued nearly holomorphic form F via certain differential operators.Theorem 1.1 is an instance of a reciprocity result for the critical L -values in the spirit of afamous conjecture due to Deligne [9]. However, Deligne’s conjecture is in the motivic world andit is a non-trivial problem to relate Deligne’s motivic period to our period h F, F i appearing inTheorem 1.1. One way to observe the compatibility of our result with Deligne’s conjecture isvia ratios of L -values (which eliminates the periods involved). In that direction, Theorem 1.1implies the following consequence of Deligne’s conjecture. Let k ≥ k ≥ . . . ≥ k n ≥ n + 2 be integers, where all k i have the sameparity. Let Π and Π be irreducible cuspidal automorphic representations of GSp n ( A Q ) such that Π , ∞ ≃ Π , ∞ is the holomorphic discrete series representation with highest weight ( k , k , . . . , k n ) . Suppose that for almost all primes p , the representations Π ,p and Π ,p aretwist-equivalent, i.e., there exists a character ψ p of Q × p satisfying Π ,p ≃ Π ,p ⊗ ( ψ p ◦ µ n ) , where µ n is the multiplier homomorphism from GSp n ( Q p ) → Q × p . Let S be a finite set of places includ-ing ∞ such that Π ,p and Π ,p are unramified for p / ∈ S . Then, for primitive Dirichlet characters χ , χ such that χ , ∞ = χ , ∞ = sgn k , and integers r , r such that n + 2 ≤ r , r ≤ k n − n , r ≡ r ≡ k n − n (mod 2) , L S ( r , Π ⊠ χ ) π ( n +1)( r − r ) L S ( r , Π ⊠ χ ) lies in a CM field . (2)In the rest of this introduction we explain briefly the key ideas in our proof of Theorem 1.1.The starting point here is an explicit integral representation (or pullback formula) proved in [25,Theorem 6.4]. This formula roughly says that D E (cid:16)(cid:2) − Z (cid:3) , r + n − k (cid:17) , F E ≈ L S ( r, Π ⊠ χ ) F ( Z ) (3)where E ( Z, s ) is a certain Eisenstein series on H n of weight k , the function F on H n correspondsto a smooth modular form of weight k associated to a particular choice of archimedean vectorinside Π ∞ , the element h Z Z i of H n is obtained from the diagonal embedding of H n × H n ,the Petersson inner product h , i is taken with respect to the Z variable, and the symbol ≈ indicates that the two sides are equal up to some (well-understood) explicit factors.Using the shorthand k = ( k , k , . . . , k n ), we prove in Section 3 that F = D k ( G ) where G isthe vector-valued holomorphic Siegel cusp form on H n of archimedean type k mentioned earlier,and D k is a certain differential operator obtained from a Lie algebra element. Let p χ k denote theprojection operator from the space of nearly holomorphic forms of weight k onto the isotypicsubspace corresponding to the holomorphic discrete series representations of archimedean type k . Then we may rewrite (3) as D p χ k (cid:16) E ( (cid:2) − Z (cid:3) , r + n − k ) (cid:17) , D k ( G ) E ≈ L S ( r, Π ⊠ χ ) F ( Z ) . This essentially means that the two representations are equivalent when restricted to Sp n ( Q p ). INTRODUCTION r ≥ n + 2 in Theorem 1.1 is because in this range thefunction E ( h Z Z i , r + n − k ) is cuspidal in each variable; see also Remark 4.4.)Via a standard linear algebra argument, the proof of Theorem 1.1 can now be reduced tothe heart of this paper, which is to show the Aut( C )-equivariance of D k and p χ k when viewedas operators on spaces of nearly holomorphic cusp forms. This equivariance is a priori not clear,as these operators are defined abstractly. To achieve this, we re-interpret the space of nearlyholomorphic modular forms of degree n in the representation theoretic language, generalizingour work in the n = 2 case done in [26]. After laying the necessary Lie-algebraic foundationsin Section 2, we show in Section 3.2 that the space of (vector-valued) nearly holomorphic mod-ular forms can be identified with the space of p − -finite automorphic forms, where p − is thespan of the non-compact positive roots of Sp n ( R ). We go on to define the operators p χ k and D k representation-theoretically by choosing suitable Lie algebra elements; the crucial Aut( C )-equivariance properties of these operators (Propositions 3.16 and 3.20) are proved via a carefularithmetic analysis of the Lie algebra. A novel aspect of our methodology is that we expand thedomain of functions under consideration from nearly holomorphic modular forms to functionswhich have a Fourier expansion involving polynomials in Im( Z ) ± (see Section 3.3). Along theway, we prove several new results concerning nearly holomorphic Siegel cusp forms and p − -finiteautomorphic forms, including a structure theorem (Proposition 3.2) and a finiteness result forthe dimension of the space of all nearly holomorphic cusp forms of given level and archimedeantype (Proposition 3.4). These results shed new light on nearly holomorphic forms from therepresentation-theoretic point of view and should be of independent interest. Another usefulintermediate result we prove here is that the centre of the universal enveloping algebra of theLie algebra of Sp n ( R ) is generated by elements which act on the space of nearly holomorphicmodular forms diagonalizably, with integral eigenvalues; see Lemma 3.14.Our approach differs from previous works in the direction of Theorem 1.1 in the vector-valued setup such as [17]. There one works directly with holomorphic vector-valued Siegel cuspforms and invokes vector-valued Eisenstein series, relying on arithmetic properties of differentialoperators on tensor products of vector-valued functions [4, 16]. The relevant pullback formulafor that approach has been recently worked out in more generality by Liu [19, 18]. In contrast,our pullback formula (3) involves only scalar-valued (nearly holomorphic) forms and our differ-ential operators correspond to elements of the universal enveloping algebra of the Lie algebra ofSp n ( R ). Thus, our proofs of the algebraicity theorems hinge on understanding the arithmeticproperties of nearly holomorphic cusp forms and of the relevant elements in the Lie algebra. Abyproduct of our method is an explicit formula for the scalar-valued function E ( h Z Z i , r + n − k )— whose p -integrality properties were recently proved by us in [24] — as a bilinear sum overnearly holomorphic Siegel cusp forms, with the coefficients equal to critical values of L -functions(see (64), (66)). We hope to pursue further arithmetic applications of this formula elsewhere. For any ring R , we let M n ( R ) denote the ring of n -by- n matrices over R . We let M sym n ( R )denote the submodule of symmetric matrices. For a commutative ring R , we letGSp n ( R ) = { g ∈ GL n ( R ) | t gJ n g = µ n ( g ) J n , µ n ( g ) ∈ R × } , J n = (cid:2) I n − I n (cid:3) . LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS n ( R ) consists of those elements g ∈ GSp n ( R ) for which the multiplier µ n ( g ) is 1. We let Γ n ( N ) ⊂ Sp n ( Z ) be the preimage of the identity element in Sp n ( Z /N Z )under the natural surjection Sp n ( Z ) → Sp n ( Z /N Z ).The Siegel upper half space of degree n is defined by H n = { Z ∈ M n ( C ) | Z = t Z, i ( Z − Z ) is positive definite } . For g = (cid:2) A BC D (cid:3) ∈ Sp n ( R ), Z ∈ H n , let J ( g, Z ) = CZ + D. For a finite dimensional representation( ρ, V ) of GL n ( C ), a function f ∈ C ∞ ( H n , V ), and g ∈ Sp n ( R ), we define the function f | ρ g ∈ C ∞ ( H n , V ) by ( f | ρ g )( Z ) = ρ ( J ( g, Z )) − f ( gZ ).We let g n = sp n ( R ) be the Lie algebra of Sp n ( R ) and g n, C = sp n ( C ) the complexified Liealgebra. We let U ( g n, C ) denote the universal enveloping algebra and let Z n be its center. For allsmooth functions f : Sp n ( R ) → V where V is a complex vector space, and X ∈ g n , we define( Xf )( g ) = ddt (cid:12)(cid:12) f ( g exp( tX )). This action is extended C -linearly to g n, C . Further, it is extendedto all elements X ∈ U ( g n, C ) in the usual manner. Unless there is a possibility of confusion, wewill omit the subscript n and freely use the symbols g , g C , U ( g C ), Z , etc.We let A = A Q denote the ring of adeles of Q . The symbol f denotes the set of finite places(i.e., primes) of Q and the symbol A f denotes the finite adeles. We define automorphic forms andrepresentations as in [6]. All our automorphic representations are over A and all our L -functionsare normalized so that the expected functional equation takes s − s . All automorphic repre-sentations are assumed to be irreducible. Cuspidal automorphic representations are assumed tobe unitary. Each cuspidal representation π of G ( A ) is isomorphic to a restricted tensor product ⊗ π v , where π v is an irreducible, admissible, unitary representation of G ( Q v ). Given an auto-morphic representation π and a set of places S of Q , we let L S ( s, π ) = Q v / ∈ S L ( s, π v ) be theassociated global L -function where the local factors coming from the places in S are omitted.For a positive integer N , we denote L N ( s, π ) := L S N ( s, π ) where S N consists of the primesdividing N and the place ∞ .We say that a character χ = Q χ p of Q × \ A × is a primitive Dirichlet character if χ is offinite order, or equivalently, if χ ∞ is trivial on R > . We let cond( χ ) denote the conductor ofsuch a χ , and we identify cond( χ ) with a positive integer. A primitive Dirichlet character χ as defined above gives rise to a homomorphism ˜ χ : ( Z / cond( χ ) Z ) × → C × , via the formula˜ χ ( a ) = Q p | cond( χ ) χ − p ( a ), and the association χ ˜ χ is a bijection between primitive Dirichletcharacters in our sense and in the classical sense. We define the Gauss sum G ( χ ) by G ( χ ) = P n ∈ ( Z / cond( χ ) Z ) × ˜ χ ( n ) e πin/ cond( χ ) . For a complex representation π of some group H and an automorphism σ of C , there is acomplex representation σ π of H defined as follows. Let V be the space of π and let V ′ be anyvector space such that t : V → V ′ is a σ -linear isomorphism (that is, t ( v + v ) = t ( v ) + t ( v )and t ( λv ) = σ ( λ ) t ( v )). We define the representation ( σ π, V ′ ) by σ π ( g ) = t ◦ π ( g ) ◦ t − . It canbe shown easily that the representation σ π does not depend on the choice of V ′ or t . We define Q ( π ) to be the fixed field of the set of all automorphisms σ such that σ π ≃ π . Throughout this section, n will be a fixed positive integer. In this section, we will obtainsome information on the action of the Lie algebra of Sp n ( R ) and the corresponding differential LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS n ( R ). Sp n ( R )Recall that g is the Lie algebra of Sp n ( R ). Let K ∞ denote the maximal compact subgroup ofSp n ( R ) consisting of matrices of the form (cid:2) A B − B A (cid:3) . We identify K ∞ with U ( n ) via (cid:2) A B − B A (cid:3) A + iB . Let k be the Lie algebra of K ∞ and k C denote the complexification of k . Then we havethe Cartan decomposition g C = k C ⊕ p C , for some subspace p C of g C . The complex structure of Sp n ( R ) /K ∞ determines the decomposi-tion p C = p + ⊕ p − satisfying (see [31, p. 245])[ k C , p ± ] = p ± , [ p + , p + ] = [ p − , p − ] = 0 , [ p + , p − ] = k C . (4)The explicit description of p ± is given in [31, p. 260] as follows. Set T := M sym n ( C ). As in [31,p. 260], we define the C -linear isomorphisms ι ± from T to p ± by ι ± ( u ) = (cid:2) ∓ iu uu ± iu (cid:3) ∈ p ± for u ∈ T. (5)Let e i,j be the n × n matrix whose ( i, j ) th entry is 1 and all others are 0. For 1 ≤ i, j ≤ n , wedefine, as in [21], B i,j = (cid:20) ( e i,j − e j,i ) − i ( e i,j + e j,i ) i ( e i,j + e j,i ) ( e i,j − e j,i ) (cid:21) , E ± ,i,j = E ± ,j,i = (cid:20) ( e i,j + e j,i ) ± i ( e i,j + e j,i ) ± i ( e i,j + e j,i ) − ( e i,j + e j,i ) (cid:21) . (6)Then { B i,j : 1 ≤ i, j ≤ n } is a basis for k C and { E ± ,i,j : 1 ≤ i ≤ j ≤ n } is a basis for p ± .A Cartan subalgebra h C of k C (and of g C ) is spanned by the n elements B i,i , 1 ≤ i ≤ n . If λ is in the dual space h ∗ C , we identify λ with the element ( λ ( B , ) , λ ( B , ) , . . . , λ ( B n,n )) of C n .In this way we identify h ∗ C with C n . If k C acts on a space V , and v ∈ V satisfies B i,i v = λ i v for λ = ( λ , λ , . . . λ n ) ∈ C n , then we say that v has weight λ .We let Λ = Z n ⊂ h ∗ C be the weight lattice, consisting of the integral weights. Let V be a finite-dimensional k C -module. Then this representation of k C can be integrated to a representationof K ∞ if and only if all occurring weights lie in Λ. The isomorphism classes of irreducible k C -modules, or the corresponding irreducible representations of K ∞ , are called K ∞ -types .We take a system of positive roots to beΦ + = { e i − e j | ≤ i < j ≤ n } ∪ { e i + e j | ≤ i ≤ j ≤ n } , where e i is the element of C n with 1 in the i ’th position and 0 everywhere else. (Concretely, e i ( B j,j ) = δ i,j ). The positive compact roots are { e i − e j | ≤ i < j ≤ n } , and the correspondingroot vectors are { B i,j | ≤ i < j ≤ n } . Let V be a K ∞ -type, which we think of as a k C -module.A non-zero vector v ∈ V is called a highest weight vector if B i,j v = 0 for all 1 ≤ i < j ≤ n. Such a vector v is unique up to scalars. Let k = ( k , k , . . . , k n ) be its weight. Then the k i areintegers and k ≥ k ≥ . . . ≥ k n ; we say that k is the highest weight of V . We let Λ + ⊂ Λ bethe set of tuples k = ( k , k , . . . , k n ) such that each k i ∈ Z and k ≥ k ≥ . . . ≥ k n . Then theelements of Λ + parametrize the irreducible representations of K ∞ via their highest weights, asabove. LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS For notational convenience we define matrices B , E + , E − with matrix valued entries as follows: B = ( B k,ℓ ) k,ℓ ∈ M n ( M n ( C )) , E ± = ( E ± ,k,ℓ ) k,ℓ ∈ M sym n ( M n ( C )) . Let B ∗ = ( B ℓ,k ) k,ℓ be the transpose of B . We consider words in the “letters” B , B ∗ , E + and E − satisfying the following five conditions .1. E + is followed by E − or B ∗ .2. E − is followed by E + or B .3. B is followed by E + or B .4. B ∗ is followed by E − or B ∗ .5. E + and E − occur with the same multiplicity.For a word w = X · · · X m , we denote by Tr( w ) the trace as the M n ( C )-valued matrix, whichwe identify with an element of U ( g C ). Let L ( w ) equal the sum of the number of times E − B and BE + occur isolatedly in w counted cyclicly. By isolatedly, we mean that the E − B and BE + concerned must not intersect each other, so, for e.g., L ( E − BE + B ∗ ) = 1 but L ( E − BBE + ) = 2.And cyclically means that we have to take into account that the trace is cyclically invariant, so,e.g., L ( E + E − BB ) = L ( E − BBE + ) = 2. The following theorem is the main result of [21]. For r ∈ Z > , put D r = X w ( − L ( w ) Tr( w ) , where the sum is over all words w of length r satisfying the conditions (1) to (5) above. Thenthe center Z of U ( g C ) is generated by the n elements D , . . . , D n as an algebra over C . For our purposes, we will only need the following corollaries of the theorem above.
Let ≤ r ≤ n , and let D r be as in Theorem 2.1. Then there exists anexpression of the form D r = t X i =1 c i H ( i )1 · · · H ( i ) r i X ( i )1 · · · X ( i ) p i Y ( i )1 · · · Y ( i ) q i E ( i )+ , · · · E ( i )+ ,s i E ( i ) − , · · · E ( i ) − ,s i . Above, t ∈ Z > , and for each i , we have c i ∈ Z , p i , q i , r i , s i ∈ Z ≥ . Moreover, each H ( i ) k is equalto B a,a for some a , each X ( i ) k is equal to B a,b for some a < b , each Y ( i ) k is equal to B a,b for some a > b , each E ( i )+ ,k is equal to some E + ,a,b , and each E ( i ) − ,k is equal to some E − ,a,b . Conditions 1-4 do not apply to the last letter of a word.
LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS Proof.
Using Theorem 2.1, we can write D r as a Z -linear combination of words in B ∗ , ∗ , E + , ∗ , ∗ ,and E − , ∗ , ∗ where there are an equal number of E + , ∗ , ∗ , and E − , ∗ , ∗ in each word. We now use theLie bracket relations (see Section 1 of [21]):[ E + ,i,j , E + ,k,ℓ ] = 0 , [ E − ,i,j , E − ,k,ℓ ] = 0 , [ E + ,i,j , E − ,k,ℓ ] = δ i,k B j,ℓ + δ j,ℓ B i,k + δ i,ℓ B j,k + δ j,k B i,ℓ [ B i,j , E + ,k,ℓ ] = δ j,k E + ,i,ℓ + δ j,ℓ E + ,i,k [ B i,j , E − ,k,ℓ ] = − δ i,k E − ,j,ℓ − δ i,ℓ E − ,j,k [ B i,j , B k,ℓ ] = δ j,k B i,ℓ − δ i,ℓ B k,j to make the following moves on each word. First we move all the E − , ∗ , ∗ elements one by one tothe right so that they make a block at the end. Then we move all the E + , ∗ , ∗ to the right so thatthey end up just before the E − , ∗ , ∗ part. Note that these two steps do not affect the equality ofthe number of E + , ∗ , ∗ , and E − , ∗ , ∗ elements. Then we move all the B a,b , a > b elements to theright of the other B ∗ , ∗ elements so that they end up just before the E + , ∗ , ∗ part. (Note by therelations above that no new E + , ∗ , ∗ or E − , ∗ , ∗ elements are generated by this step.) Finally wemove the B a,b , a < b elements to the right of the B a,a elements. This brings each word to thedesired form. There exists an expression of the form D r = t X i =1 c i E ( i )+ , · · · E ( i )+ ,s i E ( i ) − , · · · E ( i ) − ,s i H ( i )1 · · · H ( i ) r i X ( i )1 · · · X ( i ) p i Y ( i )1 · · · Y ( i ) q i , where c i ∈ Z , and H ( i ) k , X ( i ) k , B a,b , Y ( i ) k , E ( i )+ ,k and E ( i ) − ,k are elements of g with the same meaningsas in Corollary 2.2. Proof.
This follows from the expression obtained in Corollary 2.2, by moving the B a,b elementsto the very right, using the Lie bracket relations. Recall that H n is the Siegel upper half space of degree n . Let ( ρ, V ) be a finite dimensionalrepresentation of GL n ( C ). Recall that T = { u ∈ M n ( C ) | t u = u } . Let { ǫ ν } ν be any R -rationalbasis for T . For u ∈ T , write u = P ν u ν ǫ ν with u ν ∈ C , and for z ∈ H n write z = P ν z ν ǫ ν with z ν ∈ C .For a non-negative integer e , let S e ( T, V ) denote the vector space of all homogeneous poly-nomial maps T → V of degree e . Note that S ( T, V ) = V . We can identify S e ( T, V ) with thesymmetric elements in the space Ml e ( T, V ) of e -multilinear maps from T e to V (see Lemma 12.4of [32]). We define two representations of GL n ( C ) denoted by ρ ⊗ τ e and ρ ⊗ σ e on Ml e ( T, V )as follows. For h ∈ Ml e ( T, V ) , ( u , · · · , u e ) ∈ T e and a ∈ GL n ( C ), define[( ρ ⊗ τ e )( a ) h ]( u , · · · , u e ) := ρ ( a ) h ( t au a, · · · , t au e a ) , [( ρ ⊗ σ e )( a ) h ]( u , · · · , u e ) := ρ ( a ) h ( a − u t a − , · · · , a − u et a − ) . LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS ρ ⊗ τ e and ρ ⊗ σ e to S e ( T, V ) also by the same notation.Given f ∈ C ∞ ( H n , V ) we can define the functions Df, ¯ Df, Cf, Ef in C ∞ ( H n , S ( T, V )) asfollows. (cid:0) ( Df )( z ) (cid:1) ( u ) := X ν u ν ∂f∂z ν ( z ) , (cid:0) ( ¯ Df )( z ) (cid:1) ( u ) := X ν u ν ∂f∂ ¯ z ν ( z ) , (7) (cid:0) ( Cf )( z ) (cid:1) ( u ) := 4 (cid:0) ( Df )( z ) (cid:1) ( yuy ) , (cid:0) ( Ef )( z ) (cid:1) ( u ) := 4 (cid:0) ( ¯ Df )( z ) (cid:1) ( yuy ) . (8)Here, u = ( u ν ) ∈ T, z = ( z ν ) ∈ H n and y = Im( z ). These are exactly the formulas defined in [32,p. 92] with ξ ( z ) = η ( z ) = 2 y in our case. For 0 ≤ e ∈ Z , we can define D e f , ¯ D e f , C e f and E e f recursively, and these take values in Ml e ( T, V ). For f ∈ C ∞ ( H n , V ), let ρ (Ξ) f ∈ C ∞ ( H n , V )be the function defined by z ρ (2 y ) f ( z ). More generally, for f ∈ C ∞ ( H n , Ml e ( T, V )), let( ρ ⊗ τ e )(Ξ) f ∈ C ∞ ( H n , Ml e ( T, V )) be the function defined by z ( ρ ⊗ τ e )(2 y )( f ( z )). We thendefine D eρ f ∈ C ∞ ( H n , S e ( T, V )) by D eρ f = ( ρ ⊗ τ e )(Ξ) − ( C e ( ρ (Ξ) f )) . (9)The following is Proposition 12.10 of [32].
1. We have D e +1 ρ = D ρ ⊗ τ e D eρ = D eρ ⊗ τ D ρ .2. For f ∈ C ∞ ( H n , V ) and α ∈ Sp n ( R ) , define the slash action by ( f | ρ α )( z ) := ρ ( J ( α, z )) − f ( αz ) . Then, for all α ∈ Sp n ( R ) , we have D eρ ( f | ρ α ) = ( D eρ f ) | ρ ⊗ τ e α, E e ( f | ρ α ) = ( E e f ) | ρ ⊗ σ e α. U ( g C ) and differential operators Let ( ρ, V ) be a finite-dimensional representation of GL n ( C ) and let f ∈ C ∞ ( H n , V ). Define f ρ ∈ C ∞ (Sp n ( R ) , V ) by f ρ ( g ) = ( f | ρ g )( iI n ) . (10)Note that if f is a modular form with respect to a discrete subgroup Γ of Sp n ( Q ), i.e., if f | ρ γ = f for all γ ∈ Γ, then f ρ is left Γ-invariant. Recall the maps ι ± defined in (5) which map u ∈ T to ι ± ( u ) ∈ p ± . The elements of p ± act on functions on Sp n ( R ) in the usual way. Givenany collection { ε , . . . , ε e } of symbols ± , we get a map from Sp n ( R ) to Ml e ( T, V ) as follows T e ∋ ( u , . . . , u e ) (cid:0) ι ε ( u ) . . . ι ε r ( u e ) f ρ (cid:1) ( g ) , g ∈ Sp n ( R ) . The following proposition gives the relation between the above action of p ± on V -valued functionson the group and the differential operators D ρ and E acting on V -valued functions on the Siegelupper half space. Let ( ρ, V ) be a finite dimensional representation of GL n ( C ) . Let f be in C ∞ ( H n , V ) and let f ρ ∈ C ∞ (Sp n ( R ) , V ) be the corresponding function defined in (10). Let u , . . . , u e ∈ T , and g ∈ Sp n ( R ) . LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS
1. We have ( ι + ( u ) . . . ι + ( u e ) f ρ )( g ) = (( D eρ f ) ρ ⊗ τ e ( g ))( u , . . . , u e ) , ( ι − ( u ) . . . ι − ( u e ) f ρ )( g ) = (( E e f ) ρ ⊗ σ e ( g ))( u , . . . , u e ) .
2. Set D + ρ = D ρ , D − ρ = E , µ + = τ and µ − = σ . Let ε , . . . , ε e ∈ {±} . Then ( ι ε ( u ) . . . ι ε e ( u e ) f ρ )( g )= ( D ε ρ ⊗ µ ε ⊗···⊗ µ εe D ε ρ ⊗ µ ε ⊗···⊗ σ µe · · · D ε e ρ f ) ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u , . . . , u e ) . Proof.
Part 1 follows from [31, Proposition 7.3], part 1 of Proposition 2.4, and E e +1 = EE e .For part 2, we proceed inductively in e . If e = 1, it is already proven in part 1. Put F = D ε ρ ⊗ µ ε ⊗···⊗ µ εe · · · D ε e ρ f. By the induction hypothesis, F ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u , . . . , u e ) = ι ε ( u ) · · · ι ε e ( u e ) f ρ ( g ) , and by the e = 1 case, ι ε ( u ) F ρ ⊗ µ ε ⊗···⊗ µ εe ( g ) = ( D ε ρ ⊗ µ ε ⊗···⊗ µ εe F ) ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u ) . (11)Note that F ρ ⊗ µ ε ⊗···⊗ µ εe is a Ml e − ( T, V )-valued function on Sp n ( R ). By substituting the tuple( u , . . . , u e ) ∈ T e − into (11), the right hand side equals(( D ε ρ ⊗ µ ε ⊗···⊗ µ εe F ) ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u ))( u , . . . , u e )= ( D ε ρ ⊗ µ ε ⊗···⊗ µ εe F ) ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u , u , . . . , u e ) . This equals ( D ε ρ ⊗ µ ε ⊗···⊗ µ εe D ε ρ ⊗ µ ε ⊗···⊗ µ εe · · · D ε e ρ f ) ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u , . . . , u e ) . The left hand side of (11) becomes( ι ε ( u ) F ρ ⊗ µ ε ⊗···⊗ µ εe )( g )( u , . . . , u e ) = ι ε ( u )( F ρ ⊗ µ ε ⊗···⊗ µ εe ( g )( u , . . . , u e ))= ι ε ( u ) ι ε ( u ) · · · ι ε e ( u e ) f ρ ( g ) . This concludes the proof.
Let Γ be a congruence subgroup of Sp n ( Q ) and let V be a finite dimensional complex vectorspace. We let A (Γ; V ) denote the space of V -valued automorphic forms on Γ \ Sp n ( R ), i.e.,the space of smooth V -valued functions on Sp n ( R ) that are left Γ-invariant, Z -finite, K ∞ -finite and slowly increasing. Let A (Γ; V ) ◦ ⊂ A (Γ; V ) be the subspace of V -valued cusp forms. LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS A (Γ) := A (Γ; C ) and A (Γ) ◦ := A (Γ; C ) ◦ denote the usual spaces of (complex valued)automorphic forms and cusp forms on Γ \ Sp n ( R ).Let ( ρ, V ) be a rational, finite dimensional representation of GL n ( C ). We will abuse notationand use ρ to denote its restriction to U ( n ), as well as use ρ to denote the correspondingrepresentation of K ∞ via the identification K ∞ ∼ → U ( n ) given by (cid:2) A B − B A (cid:3) A + iB , as definedearlier. Concretely, for g ∈ K ∞ , the element J ( g, iI n ) belongs to U ( n ) and our identificationmeans that ρ (¯ g ) = ρ − ( ι ( g )) = ρ ( J ( g, iI n )) for g = (cid:2) A B − B A (cid:3) ∈ K ∞ , where the involution ¯ g on K ∞ corresponds to complex conjugation on U ( n ) and the anti-involution ι on K ∞ correspondsto the transpose on U ( n ), i.e., ¯ g = (cid:2) A − BB A (cid:3) , ι ( g ) = h t A t B − t B t A i . The derived map, also denotedby ι , is an anti-involution of gl n ( C ) and of k C , and it extends naturally to an anti-involution oftheir respective universal enveloping algebras.Given f ∈ C ∞ ( H n , V ), the function f ρ defined in (10) satisfies f ρ ( gk ) = ρ ( ι ( k )) f ρ ( g ) (12)for all g ∈ Sp n ( R ), k ∈ K ∞ . We will abuse notation and also use ρ to denote the derivedrepresentation of gl n ( C ) ≃ k C , i.e., for X ∈ gl n ( C ), we denote ρ ( X ) v := ddt (cid:12)(cid:12) ρ (exp( tX )) v. (13)This action extends to U ( gl n ( C )) ≃ U ( k C ). It can be checked that if f ρ ∈ C ∞ (Sp n ( R ) , V )satisfies (12) then it also satisfies ( Xf ρ )( g ) = ρ ( ι ( X ))( f ρ ( g )) (14)for X ∈ U ( k C ) and g ∈ Sp n ( R ).We fix once and for all a rational structure on ρ , i.e., we fix a Q -vector space V Q such that V Q ⊗ C = V and such that the representation ρ restricts to a homomorphism from GL n ( Q ) toGL Q ( V Q ).We define A ρ (Γ) = { f ∈ A (Γ; V ) | f ( gk ) = ρ ( ι ( k )) f ( g ) for all k ∈ K ∞ } , A ρ (Γ) ◦ = A ρ (Γ) ∩ A (Γ; V ) ◦ . (15)Recall here that ρ ( ι ( k )) = ρ − ( J ( k, iI n )).We consider the following general result. Let G be a group, K a subgroup, ( σ, V ) a finite-dimensional, irreducible repre-sentation of K , and let f : G → V be a function satisfying f ( gk ) = σ ( k ) − f ( g ) for all g ∈ G, k ∈ K. Let (ˆ σ, ˆ V ) be the contragredient of ( σ, V ) .1. The K -module spanned by the right K -translates of f is isomorphic to dim( V ) copies of ˆ σ . Recall that the restriction functor is an equivalence between the categories of rational, irreducible, finitedimensional representations of GL n ( C ) and U ( n ). LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS
2. Let L be any fixed non-zero linear map from V to C , and define the C -valued function f on G by f := L ◦ f . Then the K -module spanned by the right K -translates of f isisomorphic to ˆ σ . Proof.
For part 1, let W be the space spanned by all right K -translates of f . For L ∈ ˆ V , wedefine a map α L : W → ˆ V by α L (cid:16) X i c i f ( · k i ) (cid:17) = X i c i ˆ σ ( k i ) L ( c i ∈ C , k i ∈ K ) . It is easy to verify that α L is well-defined, linear, non-zero if L = 0, and satisfies α L ( k.w ) = ˆ σ ( k ) α L ( w ) for all w ∈ W, k ∈ K, where k.w means right translation. Hence, if L = 0, then α L is a surjection W → ˆ V thatintertwines the right translation action on W with ˆ σ . We claim that the linear mapˆ V −→ Hom K ( W, ˆ V ) , L α L , (16)is an isomorphism. Clearly, it is injective. To see the surjectivity, let β : W → ˆ V be anintertwining operator. Set L := β ( f ). Then it is easy to see that β = α L .We proved that W contains n := dim ˆ V = dim V copies of ˆ σ . To see that W contains noother representations, let L , . . . , L n be a basis of ˆ V , and consider the map W −→ ˆ V × . . . × ˆ V ( n copies) , (17) w ( α L ( w ) , . . . , α L n ( w )) , It is straightforward to verify that this map is injective. Considering dimensions, we see thatthe map (17) is in fact an isomorphism, and that W consists of n = dim V copies of ˆ V as a K -module.For part 2, we change notation and let W be the space spanned by the right K -translates of f . Then, with similar arguments as above, we see that the map α : W −→ ˆ V , X i c i f ( · k i ) X i c i ˆ σ ( k i ) L ( c i ∈ C , k i ∈ K )is well-defined, injective, and commutes with the K -action, so that W ∼ = ˆ V as K -modules. Assume that the representation ( ρ, V ) of GL n ( C ) is irreducible.1. For each f ∈ A ρ (Γ) , the K ∞ -module generated by the K ∞ -translates of f is isomorphicto dim( V ) ρ .2. Let L be any fixed non-zero linear map from V to C , and define the C -valued function f on Sp n ( R ) by f := L ◦ f . Then the K ∞ -module generated by the K ∞ -translates of f isisomorphic to ρ . LIE ALGEBRA ACTION AND DIFFERENTIAL OPERATORS Proof.
Let σ be the representation of K ∞ (on the same representation space V as of ρ ) defined by σ ( k ) := ρ ( ι ( k ) − ). Then it is well-known that the contragredient representation ˆ σ is isomorphicto ρ . Recall that the function f satisfies f ( gk ) = σ ( k ) − f ( g ) for all g ∈ G, k ∈ K . Hence ourassertions follow from Lemma 2.6Recall that the set of (isomorphism classes of) irreducible rational representations ρ ofGL n ( C ) is parameterized via their highest weights by integers k ≥ k ≥ . . . ≥ k n , i.e., ele-ments of Λ + . More precisely, we say that such a ρ has highest weight ( k , k , . . . , k n ) if thereexists v ρ ∈ V that, under the action of the Lie algebra k C , satisfies ρ ( B i,j ) v ρ = 0 for i < j and ρ ( B i,i ) v ρ = k i v ρ . As recalled already, such a v ρ ∈ V is known as a highest weight vector in V ρ and is unique up to multiples. For k = ( k , k , . . . , k n ) ∈ Λ + , we use ρ k to denote the irreduciblerational representation of GL n ( C ) (as well as of U ( n ) and of K ∞ ) with highest weight k . Weremark that the highest weight ( k, k, . . . , k ) corresponds to the character det k of GL n ( C ) andthe character det( J ( g, iI n )) − k of K ∞ .It is well-known that an irreducible rational representation ρ of GL n ( C ) with highest weight( k , k , . . . , k n ) with k n ≥ homogeneous degree k ( ρ ) = k + k + . . . + k n . More generally, given any finite-dimensional (not necessarily irreducible) representation ρ of GL n ( C ), we say that ρ is polynomial of homogeneous degree k ( ρ ) if ρ is a direct sum ofirreducible polynomial representations and ρ ( tg ) = t k ( ρ ) ρ ( g ) for all t ∈ C × , g ∈ GL n ( C ) . (18)Given an irreducible rational representation ( ρ, V ), we fix a highest weight vector v ρ that isrational with respect to the given rational structure on ρ . Let h , i be the unique U ( n )-invariantinner product on V normalized such that h v ρ , v ρ i = 1. Let L ρ : V → C be the orthogonalprojection operator from V to C v ρ followed by the isomorphism C v ρ ∼ = C . Equivalently, we maydefine L ρ ( w ) := h w, v ρ i .Given ρ as above, let A (Γ; ρ ) ⊂ A (Γ) denote the set of all C -valued automorphic forms f in A (Γ) with the following properties: a) Either f equals 0, or the K ∞ -module generated bythe K ∞ -translates of f is isomorphic to ρ , b) f is a highest weight vector in the above module.Using the fact that the highest weight vector in ρ is unique up to multiples, we see that A (Γ; ρ )is a vector-space . One may equivalently define A (Γ; ρ ) as the space of all the automorphic forms f satisfying B i,j f = 0 for 1 ≤ i < j ≤ n and B i,i f = k i f where ( k , . . . , k n ) is the highest weightof ρ . Let A (Γ; ρ ) ◦ = A (Γ; ρ ) ∩ A (Γ) ◦ . We have the following lemma.
Let ( ρ, V ) be an irreducible, rational, finite dimensional representation of GL n ( C ) .The map f L ρ ◦ f gives an isomorphism of vector spaces A ρ (Γ) → A (Γ; ρ ) and A ρ (Γ) ◦ →A (Γ; ρ ) ◦ . Proof.
Let V be the space of ρ and let f ∈ A ρ (Γ). By Lemma 2.7, the K ∞ -translates of thefunction L ρ ◦ f generate a module W isomorphic to ρ . A calculation shows that( X ( L ρ ◦ f ))( g ) = h f ( g ) , ρ ( X ) v ρ i (19)for g ∈ Sp n ( R ) and X ∈ k C . It follows that L ρ ◦ f is a highest weight vector in W . Therefore, L ρ ◦ f ∈ A (Γ; ρ ) . Next, we show that the map f L ρ ◦ f is injective. If not, there exists somenon-zero f ∈ A ρ (Γ) such that L ρ ( f ( g )) = 0 for all g ; this contradicts the fact that the vectors f ( g ), as G runs through Sp n ( R ), span all of V . NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS f L ρ ◦ f is surjective. Supposethat h ∈ A (Γ; ρ ) . Let V h be the space generated by the right-translates R ( k ) h with k ∈ K ∞ .By assumption, there exists an isomorphism τ : V h → V as K ∞ -modules. Consider the element˜ h ∈ A (Γ; V ) given by ˜ h ( g ) = Z K ∞ h ( gι ( k − )) τ ( R ( k ) h ) dk, where dk is a Haar measure on K ∞ . A calculation shows that ˜ h ∈ A ρ (Γ). Using the Schurorthogonality relations, one can further show that L ρ ◦ ˜ h ( g ) = h ˜ h ( g ) , v ρ i is a constant multipleof h ( g ). By renormalizing if needed, we obtain that L ρ ◦ ˜ h = h . This completes the proof that f L ρ ◦ f gives an isomorphism A ρ (Γ) → A (Γ; ρ ). Finally, it is clear that f is cuspidal if andonly if L ρ ◦ f is. In this section we define nearly holomorphic Siegel modular forms and reframe them in therepresentation-theoretic language, construct some key differential operators linking holomorphicand nearly holomorphic forms, and prove the arithmetic properties of these operators that willbe crucial for our main theorem on L -values. Let ( ρ, V ) be a finite dimensional representation of GL n ( C ). Write z ∈ H n as z = x + iy . For aninteger e ≥
0, define N eρ ( H n ) to be the space of all functions f : H n → V of the form f = P i f i v i where v i ranges over some fixed basis of V and each f i : H n → C is a polynomial of degree ≤ e in the entries of y − with holomorphic functions H n → C as coefficients. Note that the definitiondoes not involve the representation ρ but merely the representation space V . The space N ρ ( H n ) = [ e ≥ N eρ ( H n )is called the space of V -valued nearly holomorphic functions. By [32, Lemma 13.3 (3) andequation (13.10)] we see that for f ∈ C ∞ ( H n , V ), we have f ∈ N eρ ( H n ) if and only if E e +1 f = 0 , (20)where E is defined as in (8). Now, by Proposition 2.5, part 1, we conclude that f ∈ N ρ ( H n ) if and only if U ( p − ) f ρ is finite-dimensional , (21) f ∈ N ρ ( H n ) if and only if f ρ is annihilated by p − . (22)We say that a ( g , K ∞ )-module V ′ is locally ( p − )-finite, if U ( p − ) v is finite-dimensional for all v ∈ V ′ . It follows from the above and (4) that if f ∈ N ρ ( H n ), then U ( g C ) f ρ is a locally ( p − )-finite( g , K ∞ )-module. This definition does not depend on the choice of basis.
NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS n ( Q ), let N ρ (Γ) be the space of nearly holomorphicmodular forms of weight ρ with respect to Γ. Precisely, N ρ (Γ) consists of the space of functions f ∈ N ρ ( H n ) such that f | ρ γ = f for all γ ∈ Γ; if n = 1, we also require that the Fourierexpansion of f | ρ γ is supported on the non-negative rationals for all γ ∈ Sp n ( Z ). Let N eρ (Γ) = N ρ (Γ) ∩ N eρ ( H n ). Any f ∈ N eρ (Γ) has a Fourier expansion of the form f ( z ) = X h ∈ M sym n ( Q ) c h (( πy ) − ) exp(2 πi tr( hz )) (23)where c h ∈ S ≤ p ≤ e S p ( T, V ). We use N ρ (Γ) ◦ to denote the space of cusp forms in N ρ (Γ), whichconsists of the forms f for which the Fourier expansion of f | ρ γ is supported on positive definitematrices for all γ ∈ Sp n ( Z ).For k = ( k , k , . . . , k n ), we denote N k (Γ) = N ρ k (Γ) , N e k (Γ) = N eρ k (Γ) , N k (Γ) ◦ = N ρ k (Γ) ◦ , N e k (Γ) ◦ = N eρ k (Γ) ◦ . Furthermore, for a non-negative integer k , we denote N k (Γ) = N det k (Γ), N k (Γ) ◦ = N det k (Γ) ◦ . Note that N k (Γ) = N k,k,...,k (Γ). Given forms f , f in N k (Γ) ◦ , we define the Petersson innerproduct h f , f i by h f , f i = vol(Γ \ H n ) − Z Γ \ H n det( y ) k f ( Z ) f ( z ) dz. (24)Above, dz is any Sp n ( R )-invariant measure on H n (the definition of the inner product does notdepend on the choice of dz ).Finally we denote the set of holomorphic cusp forms as follows: S ρ (Γ) = N ρ (Γ) ◦ , S k (Γ) = N k (Γ) ◦ . ( p − ) -finite automorphic forms We let A (Γ; V ) p − -fin denote the subspace of A (Γ; V ) consisting of all f ∈ A (Γ; V ) such that U ( p − ) f is finite-dimensional. If V = C we denote this space by A (Γ) p − -fin . Let A (Γ; V ) ◦ p − -fin and A (Γ) ◦ p − -fin denote the corresponding spaces of cusp forms. It is easy to see that A (Γ; V ) p − -fin , A (Γ; V ) ◦ p − -fin , A (Γ) p − -fin and A (Γ) ◦ p − -fin are all locally ( p − )-finite ( g , K ∞ )-modules.Define A ρ (Γ) p − -fin = A (Γ; V ) p − -fin ∩ A ρ (Γ) , A ρ (Γ) ◦ p − -fin = A (Γ; V ) p − -fin ∩ A ρ (Γ) ◦ . A (Γ; ρ ) p − -fin = A (Γ) p − -fin ∩ A (Γ; ρ ) , A (Γ; ρ ) ◦ p − -fin = A (Γ) p − -fin ∩ A (Γ; ρ ) ◦ . Recall that A ρ (Γ) was defined in (15) and A (Γ; ρ ) was defined before Lemma 2.8. The followingcrucial proposition, which generalizes Proposition 4.5 of [26], gives the relation between nearlyholomorphic modular forms and ( p − )-finite automorphic forms. For a function f on H n , recallthe associated function f ρ on Sp n ( R ) defined in (10).
1. The map f f ρ gives isomorphisms of vector spaces N ρ (Γ) ∼ −→A ρ (Γ) p − - fin and N ρ (Γ) ◦ ∼ −→ A ρ (Γ) ◦ p − - fin . NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS
2. Assume that ρ is irreducible. The map f L ρ ◦ f ρ gives isomorphisms of vector spaces N ρ (Γ) ∼ −→ A (Γ; ρ ) p − - fin and N ρ (Γ) ◦ ∼ −→ A (Γ; ρ ) ◦ p − - fin . Proof.
In view of Lemma 2.8 and the fact that L ρ preserves the ( p − )-finiteness property, itsuffices to prove the first part of the proposition.Let us first show that for each f in N ρ (Γ), the function f ρ lies in A ρ (Γ) p − -fin . Clearly, f ρ isleft Γ-invariant and K ∞ -finite. Let π = U ( g C ) f ρ . Since f is a nearly holomorphic modular form, π is a ( g , K ∞ )-module, and is locally p − -finite; see (21). The action of the Cartan subalgebra h is semisimple, hence π is a weight module. This implies that π lies in Category O in the senseof [15]. By [15, Thm 1.1 (e)] we have that f ρ is Z -finite. Finally, f ρ has the moderate growthcondition from Theorem 1.1 of [23]. This completes the proof that f ρ lies in A ρ (Γ) p − -fin .To show that the map f f ρ is an isomorphism, we construct an inverse map. Let f ′ ∈A ρ (Γ) p − -fin , z ∈ H n and g ∈ Sp n ( R ) such that g ( iI n ) = z . Then define f ( z ) := ρ ( J ( g, iI n )) f ′ ( g ).Then f ∈ N ρ (Γ) by the left Γ-invariance of f ′ and (21) and it can be easily checked that themap f ′ f defined above is the inverse of the map f f ρ .Finally, the argument of Proposition 4.5 of [26] shows that f ρ is cuspidal if and only if f is.For the rest of this subsection, we assume that ρ is irreducible. The above result impliesthat for any f ∈ N ρ (Γ) ◦ the function L ρ ◦ f ρ is a cuspidal automorphic form on Sp n ( R ) andthe ( g , K ∞ )-module generated by L ρ ◦ f ρ decomposes into a finite direct sum of irreducible,admissible, unitary ( g , K ∞ )-modules. We now make this observation more precise.For each k ∈ Λ + , let F k be any model for ρ k , and consider F k as a module for k C + p − by letting p − act trivially. Let N ( k ) := U ( g C ) ⊗ U ( k C + p − ) F k . Then N ( k ) is a locally p − -finite( g , K ∞ ) module, and by the general theory of category O p (see Section 9.4 of [15]), it admitsa unique irreducible quotient, which we denote by L ( k ). The ( g , K ∞ ) module L ( k ) is locally p − -finite and contains the K ∞ -type ρ k with multiplicity one.From the theory developed in Chapter 9 of [15], any irreducible, locally p − -finite ( g , K ∞ )-module is isomorphic to L ( k ) for some k ∈ Λ + . More precisely, given an irreducible, locally p − -finite ( g , K ∞ )-module V , there exists a k ∈ Λ + and a v ∈ V such that L ( k ) ∼ = U ( g C ) v andthe following properties hold: a) v is a highest weight vector of weight k generating the K ∞ -type ρ k , b) v is annihilated by p − . These two properties identify v ∈ V uniquely up to multiples, andensure that V ≃ L ( k ).For any k ∈ Λ + , we define A (Γ; L ( k )) ◦ to be the subspace of A (Γ) ◦ spanned by the forms f such that U ( g C ) f is isomorphic to a sum of copies of L ( k ) as a ( g , K ∞ )-module. Since L ( k ) islocally p − -finite, it follows that A (Γ; L ( k )) ◦ ⊆ A (Γ) ◦ p − -fin . Hence A (Γ; L ( k )) ◦ is the L ( k )-isotypical component of A (Γ) ◦ p − -fin . As ( g , K ∞ ) -modules, we have A (Γ) ◦ p − - fin = M k ∈ Λ + A (Γ; L ( k )) ◦ where A (Γ; L ( k )) ◦ ∼ = dim( S k (Γ)) L ( k ) . The highest weight vectors of weight k in A (Γ; L ( k )) ◦ correspond to elements of S k (Γ) via themap from Proposition 3.1. The direct sum decomposition above is orthogonal with respect tothe Petersson inner product. NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS Proof.
The existence of the Petersson inner product implies that the ( g , K ∞ )-module A (Γ) ◦ p − -fin decomposes as a direct sum of irreducible, locally p − -finite ( g , K ∞ )-modules. As noted earlier,any such module is isomorphic to L ( k ) for some k ∈ Λ + .Next we show that the ( g , K ∞ )-module L ( k ) occurs in A (Γ) ◦ p − -fin with multiplicity equal todim( S k (Γ)). For this, let W ′ be the subspace of A (Γ; L ( k )) ◦ spanned by all the highest weightvectors of weight k . In each copy of L ( k ), the highest weight vector of weight k is unique up tomultiples, and W ′ is spanned by these highest weight vectors. Therefore the dimension of W ′ equals the multiplicity of L ( k ) in A (Γ) ◦ p − -fin . It suffices to then show that the map S k (Γ) → W ′ given by f L ρ k ◦ f ρ k is an isomorphism. Note here that W ′ is the subspace of A (Γ; ρ k ) ◦ p − -fin that is annihilated by p − . Now the required result follows from (22) and Proposition 3.1.As is well-known, for k = ( k , k , . . . , k n ) ∈ Λ + , we have S k (Γ) = { } if k n ≤ L ( k ) cannot occur in A (Γ) ◦ p − -fin when k n ≤ Let k ′ , k ∈ Λ + with k ′ = ( k ′ , k ′ , . . . , k ′ n ) and k = ( k , k , . . . , k n ) . Suppose that L ( k ′ ) contains the K ∞ -type ρ k . Then k i ≥ k ′ i for ≤ i ≤ n . Proof.
It suffices to prove the analogous statement for the representation N ( k ′ ). Let B i,j bethe elements defined in (6), and set k + = h B i,j : 1 ≤ i < j ≤ n i (the space spanned by the rootvectors for the positive compact roots), k − = h B i,j : 1 ≤ j < i ≤ n i . Then k C = h C + k + + k − .Let v be a highest weight vector in F k ′ , which we recall is a model for ρ k ′ . Then, as vectorspaces, N ( k ′ ) = U ( g C ) ⊗ U ( k C + p − ) F k ′ = U ( p + ) F k ′ = U ( p + ) U ( k − ) v = U ( k − ) U ( p + ) v . The last equality follows from [ k C , p + ] ⊂ p + . It is clear that all the highest weight vectors of the K ∞ -types occurring in N ( k ′ ) must be contained in U ( p + ) v . Hence the weight of such a highestweight vector is of the form ( k , k , . . . , k n ) with k i ≥ k ′ i for 1 ≤ i ≤ n .We can now prove the following fact. Let ρ be a rational, irreducible, finite dimensional representation of GL n ( C ) ,and let Γ be a congruence subgroup of Sp n ( Q ) . Then the space N ρ (Γ) ◦ ≃ A (Γ; ρ ) ◦ p − - fin is finitedimensional. A well-known result of Shimura [32, Lemma 14.3] asserts that N eρ (Γ) is finite-dimensional for each non-negative integer e . However, Proposition 3.4 goes much further andasserts that the space S ∞ e =1 N eρ (Γ) is itself finite-dimensional, i.e., N eρ (Γ) = N e +1 ρ (Γ) for allsufficiently large e . Proof.
Let k = ( k , k , . . . , k n ) be such that ρ = ρ k . Recall that each element of A (Γ; ρ ) ◦ p − -fin generates a K ∞ -module isomorphic to ρ . Since a given irreducible ( g , K ∞ )-module contains the K ∞ -type ρ with finite multiplicity, it suffices to show that the ( g , K ∞ )-module U ( g C ) A (Γ; ρ ) ◦ p − -fin decomposes into a direct sum of finitely many irreducible ( g , K ∞ )-modules.Using Proposition 3.2 and the remarks following it, let U ( g C ) A (Γ; ρ ) ◦ p − -fin = ⊕ λ ∈ Λ + d λ L ( λ ) , (25) NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS d λ ≤ dim( S λ (Γ)) and d λ = 0 if λ = ( λ , λ , . . . , λ n ) satisfies λ n ≤
0. We now claim thatif d λ = 0 then λ i ≤ k i for all i . This will complete the proof, since there are only finitely manyelements ( λ , λ , . . . , λ n ) ∈ Λ + satisfying λ n > λ i ≤ k i for 1 ≤ i ≤ n .To prove the aforementioned claim, we appeal to Lemma 3.3. Suppose that d λ = 0. Thenthere exists an element f in A (Γ; ρ ) ◦ p − -fin whose component f λ in (25) along d λ L ( λ ) is non-zero.Clearly, f λ generates the K ∞ -type ρ = ρ k , and hence ρ k occurs in L ( λ ). Now, Lemma 3.3implies that λ i ≤ k i for 1 ≤ i ≤ n .Next recall the space A (Γ; L ( k )) ◦ defined before Proposition 3.2, and define A (Γ; ρ, L ( k )) ◦ = A (Γ; ρ ) ◦ ∩ A (Γ; L ( k )) ◦ . If ρ = ρ λ then by the results above we have A (Γ; ρ, L ( k )) ◦ = { } ⇒ ≤ k i ≤ λ i for 1 ≤ i ≤ n. (26)Let N ρ (Γ; L ( k )) ◦ ⊂ N ρ (Γ) ◦ and A ρ (Γ; L ( k )) ◦ ⊂ A ρ (Γ) ◦ p − -fin be the isomorphic images of A (Γ; ρ, L ( k )) ◦ under the isomorphisms N ρ (Γ) ◦ ∼ −→ A ρ (Γ) ◦ p − -fin ∼ −→ A (Γ; ρ ) ◦ p − -fin given by Propo-sition 3.1. Then we have orthogonal (with respect to the Petersson inner product) direct sumdecompositions of finite-dimensional vector spaces A (Γ; ρ ) ◦ p − -fin = M k ∈ Λ ++ A (Γ; ρ, L ( k )) ◦ , N ρ (Γ) ◦ = M k ∈ Λ ++ N ρ (Γ; L ( k )) ◦ , (27)where Λ ++ consists of those k = ( k , . . . , k n ) ∈ Λ + with k n ≥
1. Above, if λ = ( λ , . . . , λ n )is the highest weight of ρ , then the sum in (27) can be taken over the (finitely many) k =( k , . . . , k n ) ∈ Λ ++ such that k i ≤ λ i for all i , since the summands are zero otherwise. The decomposition (27) together with Proposition 3.2 may be viewed as a struc-ture theorem for the space of nearly holomorphic cusp forms from the representation-theoreticpoint of view.
Aut( C )Recall that T denotes the space of symmetric n × n complex matrices. Let V be a vector spacewith a rational structure V Q , i.e., V Q ⊂ V is a vector-space over Q such that V Q ⊗ C = V . Then V Q gives rise to a rational structure on Ml e ( T, V ), as follows. Let { v , . . . , v d } be a basis of V Q .For i , . . . , i e , j , . . . , j e ∈ { , . . . , n } , define the function f ( i ,j ) ,..., ( i e ,j e ) ∈ Ml e ( T, C ) by f ( i ,j ) ,..., ( i e ,j e ) ( y (1) , . . . , y ( e ) ) := y (1) i ,j · · · y ( e ) i e ,j e . (28)Then the elements f ( i ,j ) ,..., ( i e ,j e ) v i , where i , . . . , i e , j , . . . , j e run through { , . . . , n } and i runsthrough { , . . . , d } , define a rational structure on Ml e ( T, V ), which is independent of the choiceof basis of V Q . We also obtain a rational structure on S e ( T, V ), which we recall can be identifiedwith the symmetric elements of Ml e ( T, V ).Let t i,j , 1 ≤ i ≤ j ≤ n be indeterminates. For i, j as above, define t j,i = t i,j and let T denote the symmetric n × n matrix of indeterminates whose ( i, j ) entry equals t i,j . For a field F , let F [ T ] denote the algebra of polynomials in the indeterminates t i,j ; clearly F [ T ] can be NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS F in n + n variables. We let F ( T ) denote the field offractions of F [ T ]. We let T − denote the (formal) inverse of T according to Cramer’s rule, i.e., T − = T adj( T ) where adj denote the adjugate. We define the algebra F [ T ± ] ⊂ F ( T ) asfollows. F [ T ± ] := { polynomials over F in the entries of T and T − } . (29)It is an easy exercise that F [ T ± ] equals the extension of the polynomial ring F [ T ] by T .Equivalently, F [ T ± ] is the localization of F [ T ] in the multiplicative set consisting of the non-negative powers of det( T ). Given c ∈ F [ T ± ] we obtain a well-defined function from the set of invertible symmetric matrices over F to F . As a special case of this which will be relevant forus, recall that a real, positive-definite matrix y has a unique positive definite square-root y / and so for any c ∈ C [ T ± ], the complex number c ( y / ) makes sense.We say that a non-zero c ∈ C [ T ± ] is homogeneous of degree m if c ( r T ) = r m c ( T ). Thismakes the space C [ T ± ] into a graded algebra, graded by degree m ∈ Z . We say that an element c ∈ C [ T ± ] is rational if c ∈ Q [ T ± ]. Let V [ T ± ] := C [ T ± ] ⊗ C V. (30)We define an action of Aut( C ) on the space V [ T ± ] as follows. Given c ∈ V [ T ± ], write itas c ( y ) = P i c i ( y ) v i where v i ranges over a fixed rational basis of V Q and c i ∈ C [ T ± ]. For σ ∈ Aut( C ), we define σ c ( y ) = P i σ c i ( y ) v i where σ c i is obtained by letting σ act on the coefficients of c i . This gives a well defined actionof Aut( C ) on the space V [ T ± ] (that does not depend on the choice of the basis { v i } of V Q ). Anelement c ∈ V [ T ± ] is defined to be rational if the components c i all belong to Q [ T ± ]. We let V Q [ T ± ] denote the Q -vector space of rational elements in V [ T ± ] so that V Q [ T ± ] = Q [ T ± ] ⊗ Q V Q . It is clear that an element c in V [ T ± ] belongs to V Q [ T ± ] if and only if σ c = c for all σ ∈ Aut( C ).We let N S ( H n , V ) ⊂ C ∞ ( H n , V ) consist of all functions f ∈ C ∞ ( H n , V ) with the propertythat there exists an integer N (depending on f ) such that f has an absolutely and uniformly(on compact subsets) convergent expansion f ( z ) = f ( x + iy ) = X h ∈ N M sym n ( Z ) q h (( πy ) / ) exp(2 πi tr( hz )) , (31)where each q h = q h,f is an element of V [ T ± ]. Observe that for any finite dimensional rationalrepresentation ρ of GL n ( C ) on V we have N ρ (Γ) ⊆ N S ( H n , V ). Indeed, if f ∈ N ρ (Γ), thecorresponding q h in (31) actually belong to the subalgebra C [ T − ] ⊗ C V of V [ T ± ] (cf. (23)).Given a function f ∈ N S ( H n , V ), we define σ f ∈ N S ( H n , V ) by σ f ( z ) = X h σ q h (( πy ) / ) exp(2 πi tr( hz )) . (32)A very special case of all this is the (well-known) action of Aut( C ) on N ρ (Γ). Suppose that ρ is a representation of GL n ( C ) on V that respects the rational structure on V (meaning that ρ NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS n ( Q ) to GL Q ( V Q )). The action of Aut( C ) on elements of S e ( T, V ) leads to an action of Aut( C ) on N ρ (Γ), via σ (cid:18) X h c h (( πy ) − ) exp(2 πi tr( hz )) (cid:19) = X h σ c h (( πy ) − ) exp(2 πi tr( hz )) , (33)where c h ∈ S e S e ( T, V ). This is a special case of the definition (32). From Theorem 14.12 (2)of [32], we get the following result in this special setup.For f ∈ N eρ (Γ) and σ ∈ Aut( C ) , we have σ f ∈ N eρ (Γ) . (34)We will need the following result, which is Theorem 14.12 (3) of [32]. Let f ∈ N ρ (Γ) . For a positive integer p , let D pρ and E p be defined as in (9) .Then we have σ (( πi ) − p D pρ f ) = ( πi ) − p D pρ ( σ f ) , σ (( πi ) p E p f ) = ( πi ) p E p ( σ f ) . We will now prove some general results on Aut( C )-equivariance of operators, culminating inProposition 3.12 below, which will be crucial for the results of the next subsection. To beginwith, we prove a lemma that will allow us to move from Ml b ( T, V )-valued functions to V -valuedfunctions, and will also clarify why we cannot restrict ourselves to the space N ρ (Γ) and insteadneed to consider the larger space N S ( H n , V ). Let e , b be positive integers, and for ≤ i ≤ b , let u i ∈ M sym n ( Q ) and ǫ i ∈ Z .1. Let T > denote the set of real, symmetric, positive-definite, n × n matrices. Let q ∈ S a ≤ e S a ( T, Ml b ( T, V )) . Then there exists r q ∈ V [ T ± ] such that for all y ∈ T > , r q ( y / ) = q ( y − )( y ǫ u y ǫ , . . . , y ǫb u b y ǫb ) and for any σ ∈ Aut( C ) we have σ ( r q ) = r σ q .
2. Let ρ be a finite dimensional, rational representation of GL n ( C ) on Ml b ( T, V ) that re-spects the rational structure on Ml b ( T, V ) . For each f ∈ N ρ (Γ) , define the function θf ∈ C ∞ ( H n , V ) by ( θf )( z ) = ( f ( z ))(( πy ) ǫ u ( πy ) ǫ , . . . , ( πy ) ǫb u b ( πy ) ǫb ) . Then θf ∈ N S ( H n , V ) and for all σ ∈ Aut( C ) we have θ ( σ f ) = σ ( θf ) . Proof.
We first prove part 1 of the lemma. Recall the definition of the functions f ( i ,j ) ,..., ( i b ,j b ) in (28), and the rational structure of Ml b ( T, V ) given by the elements f ( i ,j ) ,..., ( i b ,j b ) v i . Fromour definitions, we have that the function from T > to V given by y f ( i ,j ) ,..., ( i b ,j b ) ( y ǫ u y ǫ , . . . , y ǫb u b y ǫb ) v i NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS V Q [ T ± ] at y / , i.e., there is g ( i ,j ) ,..., ( i b ,j b ); i ∈ V Q [ T ± ] such that g ( i ,j ) ,..., ( i b ,j b ); i ( y / ) = f ( i ,j ) ,..., ( i b ,j b ) ( y ǫ u y ǫ , . . . , y ǫb u b y ǫb ) v i . Now write q = P c q ;( i ,j ) ,... ( i b ,j b ); i f ( i ,j ) ,..., ( i b ,j b ) v i where c q ;( i ,j ) ,..., ( i b ,j b ); i ∈ C [ T ] are of degree ≤ e and the sum runs over all i , . . . , i b , j , . . . , j b in { , . . . , n } and v i runs through a fixedrational basis of V Q . By definition, σ c q ;( i ,j ) ,..., ( i b ,j b ); i = c σ q ;( i ,j ) ,..., ( i b ,j b ); i and (35) q ( y − )( y ǫ u y ǫ , . . . , y ǫb u b y ǫb ) = X c q ;( i ,j ) ,..., ( i b ,j b ); i ( y − ) g ( i ,j ) ,..., ( i b ,j b ); i ( y / ) . (36)Now define r q ( T ) = P c q ;( i ,j ) ,..., ( i b ,j b ); i ( T − ) g ( i ,j ) ,..., ( i b ,j b ); i ( T ) . Then r q ∈ V [ T ± ] satisfies therequired properties by (35), (36).Finally, part 2 is an immediate consequence of part 1. Let f in N S ( H n , V ) and let ρ be a finite-dimensional, rational representation of GL n ( C ) on V of homogeneous degree k such that ρ respects the rational structure on V . For u = ± , define the function R uρ ( f ) on H n by ( R uρ ( f ))( z ) = ρ ( y u ) f ( z ) . Then R uρ ( f ) ∈ N S ( H n , V ) ,and for each σ ∈ Aut( C ) we have π uk R uρ ( σ f ) = σ ( π uk R uρ ( f )) . Proof.
By assumption, we can write f ( z ) = X h X a q h,a (( πy ) / ) v a ! exp(2 πi tr( hz )) , where q h,a ∈ C [ T ± ] and the v a form a rational basis of V . Furthermore, from the assumptionson ρ we have ρ ( y u/ ) v a = P b r u,a,b ( y / ) v b where the r u,a,b ∈ Q [ T ± ] have homogeneous degree uk . In particular r u,a,b (( πy ) / ) = π uk/ r u,a,b ( y / ). This gives us π uk/ R uρ ( f )( z ) = X h X a,b q h,a (( πy ) / ) r u,a,b (( πy ) / ) v b ! exp(2 πi tr( hz )) , from which the assertion of the lemma is clear.We define U ( k Q ) to be the Q -subalgebra of U ( k C ) generated by the various B a ,b (see (6)).The elements of U ( k Q ) are sums of the form c + P ni =1 c i B a i ,b i · · · B a iui ,b iui where n ≥ c i ∈ Q for 0 ≤ i ≤ n , u i ≥ ≤ i ≤ n , and 1 ≤ a ik , b ik ≤ n for 1 ≤ k ≤ u i .Recall that for each representation ( ρ, V ) of GL n ( C ), the derived representation of U ( k C ) on V is also denoted by ρ . For a general X ∈ U ( k C ), ρ ( X ) ∈ End( V ) may not be invertible (i.e., ρ ( X ) may not be in GL( V )). However the following lemma is immediate. Let X ∈ U ( k Q ) and let ρ be a finite-dimensional representation of GL n ( C ) on V . Then for i = 1 , , there exists X i ∈ U ( k Q ) such that X = X + X and ρ ( ι ( X i )) is invertible. NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS Proof.
We choose a sufficiently large c ∈ Q such that ρ ( ι ( X )) + cI and ρ ( ι ( X )) − cI areboth invertible, where I denotes the identity map on V . Now the result follows from taking X = ( X + c ), X = ( X − c ). Let ρ be an irreducible rational representation of GL n ( C ) on V such that ρ respects the rational structure on V . Let X ∈ U ( k Q ) and for each f in N S ( H n , V ) , define e Xf ∈ C ∞ ( H n , V ) by ( e Xf )( z ) = ρ ( ι ( X )) f ( z ) . Then e Xf ∈ N S ( H n , V ) and for each σ ∈ Aut( C ) ,we have σ ( e Xf ) = e X ( σ f ) . Furthermore, suppose that ρ ( ι ( X )) is invertible and let the representation ρ ′ of GL n ( C ) on V be given by ρ ′ ( h ) = ρ ( ι ( X )) ρ ( h ) ρ ( ι ( X )) − . Then the following hold:1. If f ∈ N ρ (Γ) , then e Xf ∈ N ρ ′ (Γ) ,2. ( e Xf ) ρ ′ = Xf ρ ,3. ρ ′ respects the rational structure on V . Proof.
First of all, we note that ρ ( ι ( X )) ∈ End Q ( V Q ) (37)This follows from the definition of the rational structure and the fact that the ρ ( B a i ,b i ) preserve V Q .We now show that e Xf ∈ N S ( H n , V ) and that σ ( e Xf ) = e X ( σ f ) for each σ ∈ Aut( C ). Since f ∈ N S ( H n , V ), we can write f ( z ) = X h X a q h,a (( πy ) / ) v a ! exp(2 πi tr( hz )) , where q h,a ∈ C [ T ± ] and the v a form a rational basis of V . Using (37) we can write ρ ( ι ( X )) v a = P b r a,b v b where the r a,b ∈ Q . This gives us( e Xf )( z ) = X h X a,b q h,b (( πy ) / ) r a,b v b ! exp(2 πi tr( hz )) , from which the required facts follow immediately.Next, for each z = x + iy ∈ H n , put g z = h y / xy − / y − / i . Using (10) and (14), we see that( e Xf ) ρ ′ ( g z ) = ρ ′ ( y / )( e Xf )( z ) = ρ ( ι ( X )) ρ ( y / ) f ( z ) = ρ ( ι ( X )) f ρ ( g z ) = Xf ρ ( g z ) . On the other hand, using (12), (14) we observe that for each g ∈ Sp n ( R ), k ∈ K ∞ , ( e Xf ) ρ ′ ( gk ) = ρ ′ ( ι ( k ))( e X f ) ρ ′ ( g ), Xf ρ ( gk ) = ρ ′ ( ι ( k )) Xf ρ ( g ). Since each element of Sp n ( R ) can be written asa product of an element of the form g z and an element of K ∞ , it follows that ( e Xf ) ρ ′ = Xf ρ .Next, suppose that f ∈ N ρ (Γ). We will show then that e Xf ∈ N ρ ′ (Γ). It is clear that e Xf is nearly holomorphic in this case, so we only need to show that ( e Xf ) | ρ ′ γ = e Xf for all γ ∈ Γ.This follows from the calculation( e Xf )( γz ) = ρ ( ι ( X )) f ( γz ) = ρ ( ι ( X )) ρ ( J ( γ, z )) f ( z ) = ρ ′ ( J ( γ, z )) e X f ( z ) . NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS ρ ′ respects the rational structure on V follows from (37) and the fact that ρ respects the rational structure on V . Let ρ and ρ be finite-dimensional rational representations of GL n ( C ) on V of homogeneneous degrees k ( ρ ) and k ( ρ ) respectively and admitting a common rationalstructure on V ; assume also that ρ is irreducible. Let R ∈ U ( g C ) be of the form R = X i c i E + ,m ,n · · · E + ,m si ,n si E − ,p ,q · · · E − ,p ti ,q ti B a ,b · · · B a ui ,b ui , where the c i are rational numbers and s i , t i , u i are non-negative integers. Let Γ be a congruencesubgroup of Sp n ( Q ) . For each f in N ρ (Γ) , define the function e Rf ∈ C ∞ ( H n , V ) by ( e Rf )( z ) = ρ ( y − / ) (cid:16) ( Rf ρ ) h y / xy − / y − / i(cid:17) . (38) Then e Rf ∈ N S ( H n , V ) and σ (cid:16) π k ( ρ − k ( ρ e Rf (cid:17) = π k ( ρ − k ( ρ e R ( σ f ) (39) for all σ ∈ Aut( C ) . Proof.
By linearity and using Lemma 3.10, it suffices to consider the case R = E + ,m ,n · · · E + ,m s ,n s E − ,p ,q · · · E − ,p t ,q t X where X ∈ U ( k Q ) and ρ ( ι ( X )) is invertible. Put E = E + ,m ,n · · · E + ,m s ,n s E − ,p ,q · · · E − ,p t ,q t , g f = ρ ( ι ( X )) f , and ρ ′ ( h ) = ρ ( ι ( X )) ρ ( h ) ρ ( ι ( X )) − . For each z = x + iy ∈ H n , put g z = h y / xy − / y − / i . By Lemma 3.11,( e Rf )( z ) = ρ ( y − / ) (cid:16) ( Eg ρ ′ f )( g z ) (cid:17) (40)and g f ∈ N ρ ′ (Γ) , σ ( g f ) = g ( σ f ) . (41)For 1 ≤ a ≤ s , 1 ≤ b ≤ t , let u a and v b be elements such that ι + ( u a ) = E + ,m a ,n a , ι − ( v b ) = E − ,p b ,q b . By (5) and (6), we see that iu a , iv b ∈ M sym n ( Q ). Using part 2 of Proposition 2.5 andProposition 3.7, we see that there exists an operator D E : C ∞ ( H n , V ) → C ∞ ( H n , Ml s + t ( T, V ))such that σ (( πi ) t − s D E g f ) = ( πi ) t − s D E ( σ g f ) = ( πi ) t − s D E ( g σ f ) (42)and such that the following identities hold( Eg ρ ′ f )( g z ) = ( ι + ( u ) . . . ι + ( u s ) ι − ( v ) . . . ι − ( v t ) g ρ ′ f )( g z )= (cid:16) ( D E g f ) ρ ′ ⊗ τ s ⊗ σ t ( g z ) (cid:17) ( u , . . . , u s , v , . . . , v t )= (cid:16) ( ρ ′ ⊗ τ s ⊗ σ t )( y / )( D E g f )( z ) (cid:17) ( u , . . . , u s , v , . . . , v t ) NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS ρ ′ ( y ) (cid:16) ( D E g f )( z )( y u y , . . . y u s y , y − v y − , . . . , y − v t y − ) (cid:17) . Combining this with (40) and using multilinearity we obtain( e Rf )( z ) = ρ ( y − ) ρ ′ ( y ) (cid:16) (( πi ) t − s D E g f )( z ) (cid:16) ( πy ) ( iu )( πy ) , . . . , ( πy ) − ( − iv t )( πy ) − (cid:17)(cid:17) . (43)Now, combining (43) with Lemma 3.8, Lemma 3.9, and (42), we obtain the desired result. For each k ∈ Λ + , let χ k be the character by which Z acts on L ( k ). (Note that this notationdiffers from what we used in [26]; χ k was denoted χ k + ̺ there.) Let k = ( k , . . . , k n ) and k ′ = ( k ′ , . . . , k ′ n ) be elements of Λ + such that either k n , k ′ n ≥ n or k n + k ′ n > n . Then χ k ′ = χ k if and only if k ′ = k . Proof.
Suppose that χ k = χ k ′ . Then k ′ = w · k for a Weyl group element w , where · denotesthe dot action of the Weyl group (see Sections 1.8 to 1.10 of [14]). The element w decomposesas w = τ I σ with σ ∈ S n , the symmetric group for { , . . . , n } , and I a subset of { , . . . , n } . Theaction τ I σ · k is given by τ I σ · k = (cid:0) ǫ ( k σ (1) − σ (1)) , . . . , ǫ n ( k σ ( n ) − σ ( n )) (cid:1) + (1 , . . . , n ) , where ǫ i = 1 if i / ∈ I and ǫ i = − i ∈ I . (Note that the dot action is given by w · k = w ( k + ̺ ) − ̺ ,where ̺ is half the sum of the positive roots. In order to be in the setting of [14], the positiveroots have to be e i − e j for 1 ≤ i < j ≤ n and − ( e i + e j ) for 1 ≤ i ≤ j ≤ n , resulting in ̺ = − (1 , . . . , n )). Hence ǫ j ( k σ ( j ) − σ ( j )) = k ′ j − j for j ∈ { , . . . , n } . (44)Let S = { j ∈ { , . . . , n } | σ ( j ) = j } . Then σ induces a permutation of S without fixed points.The condition (44), together with k j + k ′ j ≥ n , forces ǫ j = 1 for all j ∈ S . Hence x σ ( j ) = x ′ j for j ∈ S , where x i := k i − i and x ′ i = k ′ i − i . Since the x i and the x ′ i are strictly ordered fromlargest to smallest, this is only possible if S = ∅ .We proved that σ is the identity, so that ǫ j ( k j − j ) = k ′ j − j for all j ∈ { , . . . , n } . If ǫ j = 1,then k j = k ′ j . If ǫ j = −
1, then our hypothesis implies j = n and k j = k ′ j = n . This proves k = k ′ .For 1 ≤ i ≤ n , let D i be the generators of Z from Theorem 2.1. Note that Z , andhence each D i , acts on the space A (Γ; ρ ) ◦ p − -fin . By the isomorphism N ρ (Γ) ∼ −→ A (Γ; ρ ) p − -fin ofProposition 3.1, it follows that D i gives rise to an operator Ω i on the space N ρ (Γ). Precisely,for each f ∈ N ρ (Γ), (Ω i f ) ρ = D i ( f ρ ) , L ρ ((Ω i f ) ρ ) = D i ( L ρ ( f ρ )) . More concretely, for z = x + iy ∈ H n and f ∈ N ρ (Γ), the above definition is equivalent to(Ω i f )( z ) = ρ ( y − / ) (cid:16) ( D i f ρ ) h y / xy − / y − / i(cid:17) . (45) NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS χ of Z , we define A (Γ; ρ, χ ) ◦ p − -fin ⊆ A (Γ; ρ ) ◦ p − -fin to be the subspaceconsisting of all the elements on which Z acts via the character χ . We let N ρ (Γ; χ ) ◦ be thecorresponding subspace of N ρ (Γ) ◦ . In the notation introduced in Sect. 3.2, it is clear that A (Γ; ρ, L ( k )) ◦ ⊆ A (Γ; ρ, χ k ) ◦ p − -fin , N ρ (Γ; L ( k )) ◦ ⊆ N ρ (Γ; χ k ) ◦ . (46)More precisely, M k ′ ∈ Λ ++ χ k ′ = χ k A (Γ; ρ, L ( k ′ )) ◦ = A (Γ; ρ, χ k ) ◦ p − -fin , M k ′ ∈ Λ ++ χ k ′ = χ k N ρ (Γ; L ( k ′ )) ◦ = N ρ (Γ; χ k ) ◦ . (47)It follows from Lemma 3.13 that N ρ (Γ; L ( k )) ◦ = N ρ (Γ; χ k ) ◦ if k = ( k , . . . , k n ) ∈ Λ ++ with k n ≥ n, (48)and similarly for A (Γ; ρ, χ k ) ◦ p − -fin . Let k ∈ Λ + . Then for all ≤ i ≤ n , χ k ( D i ) ∈ Z . Proof.
Since D i v = χ k ( D i ) v for any v in the space of L ( k ), it is enough to prove that D i v k is an integral multiple of v k , where we fix v k to be a highest weight vector of weight k in L ( k ). For this we appeal to Corollary 2.2. We know that v k is annihilated by all the E − , ∗ , ∗ elements, as well as all the B a,b elements for a > b , and the number of E − , ∗ , ∗ elements is thesame as the number of E + , ∗ , ∗ elements in the expression of D i . Hence, we may write D i v k = P ti =1 c i H ( i )1 · · · H ( i ) r i X ( i )1 · · · X ( i ) p i v k where the c i are integers, r i , p i ≥ H ( i ) k is equal to B a,a for some a , and each X ( i ) k is equal to B a,b for some a < b . We break up the expression aboveinto two parts corresponding to whether p i = 0 or p i >
0, and obtain D i v k = w + w where w = P ui =1 c i H ( i )1 · · · H ( i ) r i v k and w = P ti = u +1 c i H ( i )1 · · · H ( i ) m i X ( i )1 · · · X ( i ) n i v k with each n i > w = 0. Since B a,a v k = k a v k , it follows that w is an integralmultiple of v k , completing the proof. Let σ ∈ Aut( C ) . Then for any f ∈ N ρ (Γ) σ (Ω i f ) = Ω i ( σ f ) . Proof.
This follows from Corollary 2.3, Proposition 3.12 and (45).For each k ∈ Λ ++ , let p χ k denote the orthogonal projection map from N ρ (Γ) ◦ to its subspace N ρ (Γ; χ k ) ◦ . (We omit ρ from the notation of p χ k for brevity.) If k = ( k , . . . , k n ) ∈ Λ + suchthat k n ≥ n , then (48) implies that p χ k is precisely the orthogonal projection map from N ρ (Γ) ◦ to its subspace N ρ (Γ; L ( k )) ◦ . For any σ ∈ Aut( C ) , f ∈ N ρ (Γ) ◦ , and k ∈ Λ ++ , we have σ ( p χ k ( f )) = p χ k ( σ f ) . NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS Proof.
We can give an explicit formula for p χ k as follows, p χ k = C − k Y k ′ ∈ Λ ++ k ′ = k n X i =1 sgn (cid:0) χ k ( D i ) − χ k ′ ( D i ) (cid:1)(cid:0) Ω i − χ k ′ ( D i ) (cid:1) , (49)where C k = Y k ′ ∈ Λ ++ k ′ = k n X i =1 | χ k ( D i ) − χ k ′ ( D i ) | . (50)In both (49) and (50), the product extends only over those finitely many k ′ for which the space N ρ (Γ; χ k ) ◦ is non-zero. Now the proposition follows from Lemmas 3.14 and 3.15.In the following we denote by k Q the Lie algebra over Q spanned by the elements B ij definedin (6). Then k C = k Q ⊗ Q C . For k ∈ Λ + , let F k be a model for ρ k , and let v k be a highest weight vectorin F k . Let F k , Q := U ( k Q ) v k . Then F k , Q is an irreducible k Q -module, and F k , Q ⊗ Q C ∼ = F k as k C -modules. Proof.
By Weyl’s theorem [34, Theorem 7.8.11], F k , Q is a direct sum U ⊕ . . . ⊕ U m of irreducible k Q -modules. Evidently, v k must have a non-zero component v i in each of the U i . Then each v i has weight k . Since, up to multiples, v k is the only vector of weight k in F k , there exist c i ∈ C with v i = c i v k . We can get from v k to any other vector in F k , Q with an appropriate element of U ( k Q ). In particular, there exists X i ∈ U ( k Q ) with X i v k = v i . Looking at weights, we see that X i must be a constant, so that in fact X i = c i . It follows that all the c i are rational. On theother hand, the v i are Q -linearly independent. This is only possible if m = 1.We proved the irreducibility assertion. It follows that F k , Q ⊗ Q C is an irreducible k C -module.The obvious map from F k , Q ⊗ Q C to F k is then an isomorphism.Let p ± , Q be the Q -span of the elements E ± ,i,j defined in (6); then p ± , Q is a Lie algebra over Q . Set g Q = p + , Q ⊕ k Q ⊕ p − , Q . Then g Q is a rational form of g C . For k ∈ Λ + , recall the g C -module N ( k ) = U ( g C ) ⊗ U ( k C + p − ) F k . Consider the g Q -module N ( k ) Q := U ( g Q ) ⊗ U ( k Q + p − , Q ) F k , Q . Then N ( k ) ∼ = N ( k ) Q ⊗ Q C as g C -modules. Thereexists a direct sum decomposition of N ( k ) Q into irreducibles U j such that the U j ⊗ Q C are the K ∞ -types of N ( k ) . Proof.
Inside U ( p + ), let U ( p + ) ( m ) be the C -linear span of the elements E + ,i ,j . . . E + ,i m ,j m , andlet U ( p + ) ( m ) Q be the Q -linear span of the same elements. Using the PBW theorem, we have N ( k ) = U ( g C ) ⊗ U ( k C + p − ) F k ∼ = U ( p + ) ⊗ C F k ∼ = ∞ M m =0 (cid:16) U ( p + ) ( m ) ⊗ C F k (cid:17) (51)as complex vector spaces. It follows from the PBW theorem and Lemma 3.17 that U ( p + ) ( m ) ⊗ C F k ∼ = ( U ( p + , Q ) ( m ) ⊗ Q F k , Q ) ⊗ Q C (52) NEARLY HOLOMORPHIC SIEGEL MODULAR FORMS C -dimension). Hence N ( k ) ∼ = (cid:18) ∞ M m =0 (cid:16) U ( p + , Q ) ( m ) ⊗ Q F k , Q (cid:17)(cid:19) ⊗ Q C ∼ = (cid:16) U ( p + , Q ) ⊗ Q F k , Q (cid:17) ⊗ Q C ∼ = (cid:16) U ( g Q ) ⊗ U ( k Q + p − , Q ) F k , Q (cid:17) ⊗ Q C . (53)This proves N ( k ) ∼ = N ( k ) Q ⊗ Q C as C -vector spaces. It is easy to see that the isomorphism iscompatible with the action of g C ∼ = g Q ⊗ Q C .It follows from [ k Q , p + , Q ] = p + , Q that U ( p + , Q ) ( m ) ⊗ Q F k , Q is a k Q -module. By Weyl’s theorem,we may decompose it into irreducibles U j . Then the U j ⊗ Q C are irreducible under the actionof k C ∼ = k Q ⊗ C , i.e., they are the K ∞ -types of U ( p + ) ( m ) ⊗ C F k . For k ∈ Λ + , let v k be a highest weight vector in the K ∞ -type ρ k of L ( k ) . Thenfor any K ∞ -type ρ of L ( k ) there exists a non-negative integer m such that ρ admits a C -basisconsisting of vectors of the form Y i v k , where Y i ∈ U ( p + , Q ) ( m ) U ( k Q ) . Proof.
Let U j ⊗ Q C be one of the K ∞ -types from Lemma 3.18 such that it maps onto ρ under theprojection N ( k ) → L ( k ). By construction there exists an m such that U j has a basis consistingof vectors of the form Y i v with Y i ∈ U ( p + , Q ) ( m ) U ( k Q ); here v is a highest weight vector in F k , Q .The vectors Y i v ⊗ Y i ( v ⊗
1) are a C -basis of U j ⊗ Q C . Our assertion follows since, afterproper normalization, v ⊗ v k .For the rest of this subsection let k = ( k , k , . . . , k n ) where k ≥ . . . ≥ k n > n are integerswith the same parity . In this case, L ( k ) is a holomorphic discrete series representation thatcontains the K ∞ -type ρ k ,k ,...,k with multiplicity one (see Lemma 5.3 of [25]). We use thenotation N k (Γ; L ( k )) ◦ := N det k (Γ; L ( k )) ◦ . We will construct a differential operator that maps S k (Γ) isomorphically onto N k (Γ; L ( k )) ◦ . The idea is to find a Q -rational element of U ( g C )that maps a highest-weight vector of weight k inside L ( k ) onto a vector in the one-dimensional,multiplicity one K ∞ -type ρ k ,k ,...,k in L ( k ). This can be done thanks to Lemma 3.19. Let k = ( k , k , . . . , k n ) where k ≥ . . . ≥ k n > n are integers with the sameparity. Then there exists an injective linear map D k from S k (Γ) to N k (Γ) ◦ with the followingproperties.1. The image D k ( S k (Γ)) is equal to N k (Γ; L ( k )) ◦ .2. For any σ ∈ Aut( C ) and f ∈ S k (Γ) we have D k ( σ f ) = σ ( D k f ) . Proof.
For brevity, we write ρ = ρ k . Let f ∈ S k (Γ), and put f ′ = L ρ ◦ f ρ . By Proposition 3.2, f ′ is a highest weight vector of weight k inside V = U ( g C )( f ′ ) ≃ L ( k ).By Lemma 3.19, there exists an element Y ∈ U ( p + , Q ) ( m ) U ( k Q ) such that when Y is viewed asan operator on V , then Y v k is a non-zero vector in the one-dimensional K ∞ -type ρ := ρ k ,k ,...,k . The K ∞ -type ρ k ,k ,...,k corresponds to the character det( J ( k ∞ , iI n )) − k of K ∞ , or equivalently, the char-acter det k of U ( n ). SPECIAL VALUES OF L -FUNCTIONS e Y = π k ... + kn − nk Y and define the map D k from S k (Γ) to C ∞ ( H n ) by( D k f )( z ) := det( y ) − k / e Y ( L ρ ( f ρ ))( g z )= L ρ (cid:16) det( y ) − k / ( e Y ( f ρ ))( g z ) (cid:17) . (54)From the construction and the aforementioned property of Y , it is clear that ( D k f ) ρ = L ρ ( e Y f ρ ) = e Y ( L ρ f ρ ). This, together with the fact that ρ is one-dimensional and occurs withmultiplicity one in L ( k ) (and using Propositions 3.1 and 3.2), it follows that D k maps S k (Γ)surjectively onto N k (Γ; L ( k )) ◦ . This proves part 1. Using Proposition 3.12 and the expression(54) (and the fact that the projection map L ρ is rational) we obtain part 2. In the case n = 2 , we may take D k = π − k − k U k − k using the notation of [26](see Propositions 3.15 and 5.6 of [26]). L -functions Throughout this section, we fix an integer k and an n -tuple k = ( k , k , . . . , k n ) of positiveintegers such that k = k ≥ . . . ≥ k n ≥ n + 2 and all k i have the same parity. We recall that L ( k ) is the holomorphic discrete series representation of Sp n ( R ) with highestweight ( k , k , . . . , k n ). Let GSp n ( R ) + be the index two subgroup of GSp n ( R ) consisting ofelements with positive multiplier. We may extend L ( k ) in a trivial way to GSp n ( R ) + ∼ =Sp n ( R ) × R > . This extension induces irreducibly to GSp n ( R ). We denote the resultingrepresentation of GSp n ( R ) by the same symbol L ( k ).Let p be a prime and let σ be an irreducible spherical representation of GSp n ( Q p ). Let b , b , . . . b n be the Satake parameters associated to σ (see, e.g., [1, § χ of Q × p , put α χ = χ ( ̟ ) for any uniformizer ̟ of Q p if χ is unramified, and put α χ = 0 if χ isramified. Define the local GSp n × GL standard L -function L ( s, σ ⊠ χ, ̺ n +1 ) := (1 − α χ p − s ) − n Y i =1 (cid:0) (1 − b i α χ p − s )(1 − b − i α χ p − s ) (cid:1) − . Given another irreducible spherical representation σ ′ of GSp n ( Q p ) with Satake parameters b ′ , b ′ , . . . b ′ n we say that σ ∼ σ ′ if ( b , . . . , b n ) and ( b ′ , . . . , b ′ n ) represent the same tuple under theaction of the Weyl group. Using Lemma 2.4 of [12] for G = GSp n ( Q p ), H = Z ( Q p )Sp n ( Q p ),we see that the following conditions are equivalent:1. σ ∼ σ ′ ,2. there is an unramified character χ of Q × p such that σ ′ ≃ σ ⊗ ( χ ◦ µ n ),3. there exists an irreducible admissible representation of Sp n ( Q p ) that occurs as a directsummand inside both σ | Sp n ( Q p ) and σ ′ | Sp n ( Q p ) , SPECIAL VALUES OF L -FUNCTIONS σ | Sp n ( Q p ) ≃ σ ′ | Sp n ( Q p ) .Given any character χ of Q × p , and irreducible spherical representations σ , σ ′ of GSp n ( Q p )we have σ ∼ σ ′ ⇒ L ( s, σ ⊠ χ, ̺ n +1 ) = L ( s, σ ′ ⊠ χ, ̺ n +1 ) . (55)We let R k ( N ) denote the set of irreducible cuspidal automorphic representations Π ≃ ⊗ Π v ofGSp n ( A ) such that Π ∞ ≃ L ( k ) and such that Π has a vector right invariant under the principalcongruence subgroup K n ( N ) of GSp n (ˆ Z ). Note that if Π ∈ R k ( N ) then Π p is spherical for all p ∤ N . For Π , Π ∈ R k ( N ), we define an equivalence relation Π ∼ Π if Π ,p ∼ Π ,p for all p ∤ N . We define e R k ( N ) = R k ( N ) / ∼ . For Π ∈ R k ( N ), let [Π] denote its equivalence class in e R k ( N ). For Π , Π ∈ R k ( N ), and σ ∈ Aut( C ), it follows from the definition that if Π ∼ Π then σ Π ∼ σ Π ; therefore forany class [Π] ∈ e R k ( N ), the element σ [Π] = [ σ Π] ∈ e R k ( N ) is well-defined. For [Π] ∈ e R k ( N ),define L N ( s, [Π] ⊠ χ, ̺ n +1 ) = Q p ∤ N L ( s, Π p ⊠ χ p , ̺ n +1 ) (this is well-defined by (55)). Given[Π] ∈ e R k ( N ), we let Q ([Π]) be the fixed field of the set of σ ∈ Aut( C ) satisfying σ [Π] = [Π].Since Q ([Π]) is contained in the field of rationality Q (Π) of any automorphic representation inits equivalence class, it follows that Q ([Π]) is a totally real or CM field [2, Theorems 3.2.1 and4.4.1]. By the strong approximation theorem, we haveGSp n ( A ) = G ≤ a For i = 1 , 2, let σ i be the representation of GSp n ( A ) generated by φ i . By assumption, φ | Sp n ( A ) = φ | Sp n ( A ) . It follows that the restrictions to Sp n ( A ) of σ and σ have a commonnon-zero quotient. So for some irreducible constituents Π , Π of σ , σ respectively, there existsan irreducible cuspidal automorphic representation of Sp n ( A ) that occurs as an automorphicrestriction (in the sense of [11, § and Π . Hence using [11, Lemma 5.1.1] we seethat Π ∼ Π .Let K ′ n ( N ) be the subgroup of elements g ∈ GSp n (ˆ Z ) such that g ≡ (cid:2) I n aI n (cid:3) (mod N ) forsome a ∈ ˆ Z . Given any F in V N, k , let the adelization Φ F be the function on GSp n ( A ) definedas Φ F ( g ) = det( J ( g ∞ , iI n )) − k F ( g ∞ ( iI n )) , where we write any element g ∈ GSp n ( A ) as g = λg Q g ∞ k f , g Q ∈ GSp n ( Q ) , g ∞ ∈ Sp n ( R ) , k f ∈ K ′ n ( N ) , λ ∈ Z ( R ) + . SPECIAL VALUES OF L -FUNCTIONS φ F ∈ A GSp n ( A ) ( K ′ n ( N ); ρ k , L ( k )) ◦ . One has the following commutativediagram. A GSp n ( A ) ( K n ( N ); ρ k , L ( k )) ◦ ≃ −−−−−−−→ φ ( F ( a ) φ ) a L ≤ a ≤ N ( a,N )=1 N k (Γ n ( N ); L ( k )) ι x x F ( F,F,...,F ) A GSp n ( A ) ( K ′ n ( N ); ρ k , L ( k )) ◦ φ F F ≃ −−−−→ φ F φ N k (Γ n ( N ); L ( k )) = V N, k (59)In the above diagram, the top row coincides with the bottom row of (58), and the map ι is the inclusion. Prima facie , the bottom isomorphism in (59) appears to give a cleaner wayto go back and forth between classical and adelic forms than the top one; however there isone major disadvantage. Namely, every automorphic representation Π ∈ R k ( N ) is generatedby some element in A GSp n ( A ) ( K n ( N ); ρ k , L ( k )) ◦ but the same is not known to be true for A GSp n ( A ) ( K ′ n ( N ); ρ k , L ( k )) ◦ (unless n ≤ R k ( N ) is generated via the adelization process. Nonetheless, we will soon observe (see nextlemma) that for each Π ∈ R k ( N ), some representative Π ′ ∈ [Π] is generated by an element of A GSp n ( A ) ( K ′ n ( N ); ρ k , L ( k )) ◦ , and hence obtained via adelization.For each F ∈ V N, k , let Π F denote the representation of GSp n ( A ) generated by φ F . ClearlyΠ F is a finite direct sum of irreducible cuspidal automorphic representations. We now make thefollowing definition. Given a class [Π] ∈ e R k ( N ), we let V N, k ([Π]) be the space generated by all F ∈ V N, k with the property that each irreducible constituent of Π F belongs to [Π]. We define S k (Γ n ( N ); [Π]) similarly. It is clear that V N, k ([Π]) = D k ( S k (Γ n ( N ); [Π])) . We have direct sumdecompositions V N, k = M [Π] ∈ e R k ( N ) V N, k ([Π]) (60) S k (Γ n ( N )) = M [Π] ∈ e R k ( N ) S k (Γ n ( N ); [Π])into a sum of orthogonal subspaces with respect to the Petersson inner product. Let Π ∈ R k ( N ) .1. There exists F ∈ V N, k such that Π F is irreducible and satisfies Π F ∼ Π . In particular, V N, k ([Π]) = { } .2. The space V N, k ([Π]) has a basis consisting of forms F as above. Proof. Let φ ∈ A GSp n ( A ) ( K n ( N ); ρ k , L ( k )) ◦ be such that φ generates Π. Let F ′ = F (1) φ . UsingLemma 4.1, we see that some irreducible constituent of Π F ′ lies inside [Π]. Letting F be theprojection of F ′ onto this constituent, we see that Π F ∼ Π. This proves the first assertion. Thesecond assertion is immediate from the definitions.Let [Π] ∈ e R k ( N ). By Theorem 4.2.3 of [2] (see also the proof of Theorem 3.13 of [27]), itfollows that F ∈ S k (Γ n ( N ); [Π]) = ⇒ σ F ∈ S k (Γ n ( N ); σ [Π]) . SPECIAL VALUES OF L -FUNCTIONS C )-equivariance of the D k map given by Proposition 3.20 (see also (58)) we obtain F ∈ V N, k ([Π]) = ⇒ σ F ∈ V N, k ( σ [Π]) . (61)In particular, the space V N, k ([Π]) is preserved under the action of the group Aut( C / Q ([Π])).Using Lemma 3.17 of [27], it follows that the space V N, k ([Π]) has a basis consisting of forms whoseFourier coefficients are in Q ([Π]). We note also that given any irreducible cuspidal automorphicrepresentation Π of GSp n ( A ) such that Π ∞ ≃ L ( k ), there exists N such that Π ∈ R k ( N ), andconsequently, using Lemma 4.2 we have V N, k ([Π]) = { } . Given a positive integer N = Q p p m p , a τ ∈ ˆ Z × , and a primitive Dirichlet character χ satisfyingcond( χ ) | N and χ ∞ = sgn k , we define the Eisenstein series E χk,N ( Z, s ; Q τ ) for each s ∈ C as in[25, (117)] (see also Section 2.3 of [24]). For an integer 0 ≤ m ≤ k − n − 1, by Theorem 1.1 of[24] we have that E χk,N ( h Z Z i , − m ; Q τ ) ∈ N k (Γ n ( N )) ◦ ⊗ N k (Γ n ( N )) ◦ ; (62)furthermore, for m as above and σ ∈ Aut( C ) we have by Proposition 6.8 of [25], σ (cid:16) π − m n E χk,N ( Z, − m ; Q ) (cid:17) = π − m n E σ χk,N ( Z, − m ; Q τ ) , (63)where τ ∈ ˆ Z × is the element corresponding to σ via the natural map Aut( C ) → Gal( Q ab / Q ) ≃ ˆ Z × . L -values4.3 Proposition. Let Π ∈ R k ( N ) , χ a primitive Dirichlet character such that cond( χ ) | N and χ ∞ = sgn k , and F ∈ V N, k ([Π]) be such that the Fourier coefficients of F lie in a CM field. Thenfor any σ ∈ Aut( C ) and an integer r such that n + 2 ≤ r ≤ k n − n , r ≡ k n − n (mod 2) , we have σ (cid:18) G ( χ ) n C N ([Π] , χ, r ) h F, F i (cid:19) = G ( σ χ ) n C N ( σ [Π] , σ χ, r ) h σ F, σ F i , where C N ([Π] , χ, r ) := i nk π n ( r + n − k ) π n ( n +1) / L N ( r, Π ⊠ χ, ̺ n +1 ) A k ( r − Q p | N vol(Γ n ( p m p ))vol(Sp n ( Z ) \ Sp n ( R )) L N ( r + n, χ ) Q nj =1 L N (2 r + 2 j − , χ ) and the rational number A k ( r − is defined in [25, (106)]. Proof. First of all, we note that C N ([Π] , χ, r ) depends only on the class [Π]. For each [Π ′ ] in e R k ( N ), we pick an orthogonal basis B [Π ′ ] of V N, k ([Π ′ ]) such that B [Π] includes F and B σ [Π] includes σ F . As usual, the Eisenstein series is given by an absolutely convergent series for Re( s ) sufficiently large, and byanalytic continuation outside that region. SPECIAL VALUES OF L -FUNCTIONS n + 2 ≤ r ≤ k n − n , r + n − k ≡ τ ∈ ˆ Z , define G χk,N ( Z , Z , r ; Q τ ) := π n ( r + n − k ) × ( p χ k ⊗ p χ k )( E χk,N ( h Z Z i , n − k + r ; Q τ ) , (64)where p k is defined as in Section 3.4. Combining Proposition 3.16, (62) and (63), we have that σ G χk,N ( Z , Z , r ; Q ) = G σ χk,N ( Z , Z , r ; Q τ ) . (65)On the other hand, using Corollary 6.5 of [25] and part 2 of Lemma 4.2, we deduce the followingidentity: G χk,N ( Z , Z , r ; Q τ ) = χ ( τ ) − n X [Π ′ ] ∈ e R k ( N ) C N ([Π ′ ] , χ, r ) X G ∈ B [Π ′ ] G ( Z ) ¯ G ( Z ) h G, G i . (66)Using (61), (65), (66), and comparing the V N ( σ [Π]) components (in the Z variable) for thefunction σ G χk,N ( Z , Z , r ; Q ) , we see that σ ( C N ([Π] , χ, r )) X G ∈ B [Π] σ G ( Z ) σ ¯ G ( Z ) σ ( h G, G i ) = σ χ − n ( τ ) C N ( σ [Π] , σ χ, r ) X G ∈ B σ [Π] G ( Z ) ¯ G ( Z ) h G, G i . Taking inner products of each side with σ F ( Z ) (which eliminates the Z variable) and thencomparing the coefficients of σ ¯ F ( Z ) = σ F ( Z ) on each side of the resulting equation (note that σ F = σ ¯ F by our hypothesis on the Fourier coefficients of F being in a CM field) together withthe well known fact σ ( G ( χ ) n ) = σ χ ( τ n ) G ( σ χ ) n , we obtain the desired equality. The reason that we restrict ourselves to the range n + 2 ≤ r ≤ k n − n is that inthis range E χk,N ( h Z Z i , n − k + r ; Q τ ) is cuspidal in each of the variables Z , Z , and so we canapply the projection operator p χ k ⊗ p χ k on it and use the arithmetic properties of this operatorproved in Proposition 3.16. If we knew that the “cuspidal projection” operator from the spaceof nearly holomorphic modular forms to the space of nearly holomorphic cusp forms is Aut( C ) -equivariant, it would be possible to extend Proposition 4.3 to the larger range ≤ r ≤ k n − n by first applying this projection operator followed by p χ k on each variable. In the following theorem, we will obtain an algebraicity result for the special values L N ( r, Π ⊠ χ, ̺ n +1 ) at all the critical points r in the right half plane which lie in the range r ≥ n + 2. Let k = k ≥ k ≥ . . . ≥ k n ≥ n + 2 be integers where all k i have the sameparity. Let Π be an irreducible cuspidal automorphic representation of GSp n ( A ) such that Π ∞ isthe holomorphic discrete series representation with highest weight ( k , k , . . . , k n ) . Let N be aninteger such that Π p is unramified for all primes p ∤ N . Let F be a nearly holomorphic cusp formof scalar weight k (with respect to some congruence subgroup) whose Fourier coefficients lie in aCM field, such that any irreducible constituent Π F = ⊗ v Π F,v of the automorphic representationgenerated by (the adelization) Φ F satisfies Π F, ∞ ≃ Π ∞ and Π F,p ∼ Π p for p ∤ N . Then,for a Dirichlet character χ such that χ ∞ = sgn k , a σ ∈ Aut( C ) , and an integer r such that n + 2 ≤ r ≤ k n − n , r ≡ k − n (mod 2) , we have σ (cid:18) L N ( r, Π ⊠ χ, ̺ n +1 ) i k π nk + nr + r G ( χ ) n +1 h F, F i (cid:19) = L N ( r, σ Π ⊠ σ χ, ̺ n +1 ) i k π nk + nr + r G ( σ χ ) n +1 h σ F, σ F i . (67) EFERENCES Proof. By enlarging N if necessary, we can assume that cond( χ ) | N and F ∈ V N, k ([Π]). (Weexploit here the fact that if p is a prime such that Π p is unramified and χ p is ramified, then L ( s, Π p ⊠ χ p ) = 1). By Proposition 4.3, σ i nk G ( χ ) n L N ( r, Π ⊠ χ, ̺ n +1 ) π n ( k − r − n ) L N ( r + n, χ ) Q nj =1 L N (2 r + 2 j − , χ ) h F, F i ! = i nk G ( σ χ ) n L N ( r, σ Π ⊠ σ χ, ̺ n +1 ) π n ( k − r − n ) L N ( r + n, σ χ ) Q nj =1 L N (2 r + 2 j − , σ χ ) h σ F, σ F i . (68)For a Dirichlet character ψ and a positive integer t satisfying ψ ∞ = sgn t , we have by [29,Lemma 5]) σ (cid:18) L N ( t, ψ )( πi ) t G ( ψ ) (cid:19) = L N ( t, σ ψ )( πi ) t G ( σ ψ ) . (69)Plugging (69) (for ψ = χ and ψ = χ ) into (68), we obtain (67). One can choose F in Theorem 4.5 such that its Fourier coefficients lie in Q ([Π]) (see the last paragraph of Section 4.2). For such an F , Theorem 4.5 implies that L N ( r, Π ⊠ χ, ̺ n +1 ) i k π nk + nr + r G ( χ ) n +1 h F, F i ∈ Q ([Π]) Q ( χ ) . Proof of Corollary 1.3. This follows immediately from Theorem 4.5, and the fact that theGauss sums G ( χ ) are algebraic numbers. 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