aa r X i v : . [ m a t h . N T ] A p r STURM’S OPERATOR ACTING ON VECTOR VALUED K -TYPES KATHRIN MAURISCHAT
Abstract.
We define Sturm’s operator for vector valued Siegel modularforms obtaining an explicit description of their holomorphic projection incase of large absolute weight. However, for small absolute weight, Sturm’soperator produces phantom terms in addition. This confirms our earlierresults for scalar Siegel modular forms.
Contents
1. Introduction 12. Poincar´e series 42.1. Definition and convergency 42.2. Lie algebra action 63. Functions on the Siegel upper halfspace 83.1. Petterson scalar product 93.2. Unfolding the Poincar´e series 103.3. Sturm’s operator 104. Phantom terms by Sturm’s operator 115. Gamma integrals 145.1. Alternating powers 145.2. Rank two 155.3. Weyl’s character formula 21References 241.
Introduction
Let G be the symplectic group of rank m . Sturm’s operator St κ is definedon (non-holomorphic) symplectic modular forms f of weight κ for a discretesubgroup Γ ⊂ G by an integral operator on the coefficients of the Fourierexpansion f ( Z ) = P T = T ′ a ( T, Y ) e πi tr( T Y ) for positive definite Ta ( T, Y ) b ( T ) = c ( κ ) − Z Y > a ( T, Y ) det(
T Y ) κ − m +12 e − π tr( T Y ) dY inv It is well-defined for scalar weight κ > m −
1. Here c ( κ ) is a constant dependingonly on weight and rank. The Fourier series St κ ( f )( Z ) = P T > b ( T ) e πi tr( T Z ) Date : 30 th Apr, 2019, 01:31. allows an interpretation as holomorphic cusp form St κ ( f ) ∈ [Γ , κ ] , and indeedis the holomorphic projection pr hol ( f ) of f in case the weight κ is large, i.e.greater than twice the rank of the symplectic group. This result by Sturm [9],[10], and Panchishkin [1] relies on a generating system of Poincar´e series p T ∈ [Γ , κ ] for which the coefficients b ( T ) are essentially given by the scalar product h p T , f i = b ( T ). The same result holds true for weight κ = 2 m in case m ≤ κ = 3 and rank m = 2 we showed jointly withR. Weissauer ([8]) that Sturm’s operator produces, along with the holomorphicprojection, a second term ph ( f ) ∈ [Γ , κ ] St κ ( f ) = pr hol ( f ) + ph ( f ) . This phantom term ph ( f ) = St κ (∆ [ m ]+ ( h )) arises as the non-holomorphic Maassshift of a holomorphic form h ∈ [Γ , κ −
2] of weight one (see section 4 for theexact definition of ∆ [ m ]+ . Later ([7]) we generalized this result to general rank m > κ = m + 1. However, the phenomenon of arising phantom terms incase of small weight is rather non-understood.Therefore, here we study the case of vector valued Siegel modular forms withvalues in the space V ρ of an irreducible rational representation ρ of GL( m, C ).These modular forms for example play an important role for singular weights [3].Consider the operator valued Poincar´e series on the Siegel upper halfspace H (1) p T ( Z ) = X γ ∈ Γ ∞ \ Γ ρ ( J ( γ, Z )) − e πi tr( T γ · Z ) . Here for a matrix g = ( ∗ ∗ C D ) ∈ G and Z ∈ H we use the J -factor J ( g, Z ) = CZ + D . We may evaluate each single summand of these Poincar´e series atspecial vectors v ∈ V ρ to get vector valued series. Candidates for v are thehighest weight vector v ρ or (if it exists) the spherical vector v K . Because of thecocycle relation J (˜ γγ, Z ) = J (˜ γ, γZ ) J ( γ, Z ) valid for all γ, ˜ γ ∈ G , the series P has the transformation property ρ ( J (˜ γ, Z )) − p T (˜ γZ ) = p T ( Z ) . Assuming good convergency properties by proposition 3.1, p T ( Z ) v ∈ [Γ , ρ ] isa vector valued holomorphic cusp form with values in End ( V ρ ). Notice that itdoesn’t transform by Ad ρ , which would be more natural, but is not compatiblewith its interpretation as an operator on V ρ .For a valued non-holomorphic modular form of weight ρ with Fourier expansion f ( Z ) = X T = T ′ ρ ( T ) a ( T, T Y T ) · e πi tr( T X ) we define Sturm’s operator by St ρ ( f )( Z ) = X T > ρ ( T ) b ( T ) e πi tr( T Z ) , TURM VECTOR VALUED 3 where the coefficients b ( T ) are defined by the integral b ( T ) ′ = det( T ) − m +12 Z Y > a ( T, Y ) ′ ρ ( T ) C ( ρ ) − ρ ( Y ) ρ ( T − ) e − π tr( Y ) dY inv det( Y ) m +12 . Here C ( ρ ) is an operator such that on holomorphic cuspforms f Sturm’s opera-tor is the identity. In contrast to the constant c ( κ ) the scalar valued case, C ( ρ )must be placed carefully into the integral. In general it is known that the vectorvalued Γ-integrals converge in case the absolute weight of ρ is large enough ([4]).But it is not clear a priori that the operators are surjective outside a discreteset of zeros and poles. Theoretically, the integrals are computable by usingthe Littlewood-Richardson rule once the Γ-function for all tensor powers st ⊗ n of the standard representation is known. But the latter involves non-trivialcombinatorics. We devote the second part of the paper to obtain some partialresults. We determine the Γ-integrals for alternating powers of the standardrepresentation in section 5.1. Further we obtain all Γ-functions for algebraicrepresentations of GL(2 , C ) by section 5.2. We include some remarks on Weyl’scharacter formula for Γ-functions in section 5.3.We say an irreducible representation ρ of GL( m, C ) with dominant highestweight l = ( l , . . . , l n ), where l ≥ l ≥ · · · ≥ l n , has absolute weight κ = l n .Like in the scalar weight case, for large absolute weight we obtain holomorphicprojection by Sturm’s operator: Theorem 1.1.
Let ρ be an irreducible representation of GL( m, C ) of largeabsolute weight κ > m . Assume C ( ρ ) is an isomorphism. Then Sturm’soperator realizes the holomorphic projection operator. Whereas, again for small absolute weight κ = m + 1 this is no longer true, aswe see by the following special case. Theorem 1.2.
For rank m = 2 let τ be the irreducible representation of GL(2 , C ) of highest weight ( k + 1 , k ) with k ≥ . Let h ∈ [Γ , τ ] be a non-zero vector valued holomorphic cusp form of weight τ . Then the image of itsMaass shift ∆ [ m ]+ ( h ) under Sturm’s operator St τ ⊗ det (cid:0) ∆ [ m ]+ ( h ) (cid:1) is non-zero if and only if k = 1 . In particular, in case of highest weight (4 , Sturm’s operator St ρ does not realize holomorphic projection but produces phan-tom terms. Our results obtained so far are limited by the explicit computability of phantomterms. Nevertheless, by [8], [7], and the above, the following interpretationis at hand. A holomorphic cusp form of weight ρ generates a holomorphicrepresentation of the symplectic group G of minimal K -type ρ . In case ofabsolute weight κ ≥ m + 1 this is a (limit of) discrete series representation.Within the root lattice of sp m and for the consistent choice of positive roots e − e , . . . , e m − − e m , e m , those belong to the cone given by the δ -translate KATHRIN MAURISCHAT of the positive Weyl chamber. More precisely, a representation of minimal K -type of highest weight ( l , . . . , l m ) is situated by its Harish-Chandra parameter( l − , l − , . . . , l m − m ). Here δ = ( m, m − , . . . ,
1) is half the sum of positiveroots. Whereas there are some holomorphic representations outside this cone,for example those generated by h ∈ [Γ , . The wall orthogonal to all shortsimple roots is given by ( r − , r − , . . . , r − m ) for r ≥ m + 1. Here, [7] suggeststhat Sturm’s operator realizes the holomorphic projection operator as long as r > m + 1, i.e. apart from the the apex δ of the cone belonging to the minimal K -type ( m + 1 , . . . , m + 1). In the case of rank two theorem 1.2 shows thatSturm’s operator fails on the wall of the cone perpendicular to the long root.This suggests the following expectation in general. Conjecture 1.3.
Sturm’s operator produces phantom terms on all the facetsof the cone not perpendicular to each of the short simple roots. The phantomterms arise as Maass shifts of holomorphic cusp forms of small absolute weight.
The paper is organized as follows. In section 2 we study non-holomorphicPoincar´e series as functions on the symplectic group. This is the natural pointof view with respect to the Lie algebra action. Section 3 is devoted to theinterplay of functions on group level and on the Siegel half space. We definethe vector valued version of Sturm’s operator, and prove its coincidence withthe holomorphic projection in case of large weight. In section 4 we show theoccurrence of phantom terms. In section 5 we determine the vector valuedgamma functions Γ( ρ ) as described above.2. Poincar´e series
Definition and convergency.
For the irreducible algebraic representa-tion ( ρ, V ρ ) we assume V ρ = C N , ρ : GL( m, C ) → GL( N, C ), to have theproperties ρ ( x ) ′ = ρ ( x ′ ) and ρ (¯ x ) = ρ ( x ) for all x ∈ GL( m, C ). This deter-mines ρ uniquely. Here x ′ denotes the transpose of the matrix x . Then v ′ · ¯ w defines the intrinsic scalar product on V ρ which is ρ ( U ( m ))-invariant. Proposition 2.1.
Let ρ be the irreducible rational representation of GL( m, C ) of dominant highest weight ( l , l , . . . , l m ) . Let κ = l m be its absolute weight.Define the non-holomorphic Poincar´e series P T ( g, s , s ) = X γ ∈ Γ ∞ \ Γ ρ ( J ( γg, i )) − tr( T Im( γg · i )) s det(Im( γg · i )) s e πi tr( T γg · i ) . Applied to any vector v ∈ V ρ the Poincar´e series converge absolutely and uni-formly on compact sets in the sense that this holds for || P T ( g, s , s ) v || in thedomain (cid:26) ( s , s ) ∈ C | Re s > m − κ and Re( ms + s ) > m − P j l j (cid:27) . For fixed such ( s , s ) the function || P T ( g, s , s ) v || is bounded and belongs to L (Γ \ G ) . In particular, in case the absolute weight κ > m is large, at thecritical point ( s , s ) = (0 , the Poincar´e series converge absolutely. TURM VECTOR VALUED 5
The most natural definition of Poincare series on G would be one in m complexvariables, P T ( g, s , . . . , s m ) = X γ ∈ Γ ∞ \ Γ ρ ( J ( γg, i )) − m Y j =1 tr(( T Y ) [ j ] ) s j · e πi tr( T γg · i ) . Here Y [ j ] denotes the j -th alternating power of Y , i.e. a matrix of size (cid:0) mj (cid:1) with entries the ( j × j )-minors of Y . The convergence of these series in( s , . . . , s m ) follows from that of the above in (˜ s , ˜ s ) = ( P j The series S T ( g, k , k ) = X γ ∈ Γ ∞ \ Γ exp(2 πi tr( T γg · i )) tr( T Im( γg · i )) k det(Im( γg · i )) k converges absolutely and uniformly on compact sets in the cone (cid:26) ( k , k ) ∈ C | Re k > m and Re( k + k m ) > m (cid:27) . For ( k , k ) fixed, it is absolutely bounded by a constant independent of τ andbelongs to L (Γ \ G ) .Proof of proposition 2.1. For g ∈ G let Z = X + iY = g · i ∈ H . There exists g Z = Y U Y − ! ∈ G , where Y is the symmetric positive definite square root of Y , such that g Z · i = Z and such that and g = g Z k for some k in the maximal compact subgroup K of G . Further, there exists k ∈ SO( m ) such that D = k Y k ′ is diagonal, D = diag ( d , . . . , d m ) for positive eigenvalues d j of Y . We compute ρ ( J ( g, i )) − = ρ ( J ( g Z , i ) J ( k, i )) − = ρ ( J ( k, i ) − ) ρ ( Y ) = ρ ( J ( k, i ) − k ′ ) ρ ( D ) ρ ( k ) . For computing the norm || ρ ( J ( g, i )) − v || for a vector v ∈ V ρ , unitary factors ρ ( k ) for k ∈ U ( m ) don’t fall into account, so || ρ ( J ( g, i )) − v || ≤ || ρ ( D ) || · || v || . We seize the operator norm || ρ ( D ) || . The action of the diagonal matrix D on V ρ is determined by the weights λ = ( λ , . . . , λ m ) of ρ . For the absolute weight KATHRIN MAURISCHAT κ ≥ ρ we have λ j − κ ≥ j = 1 , . . . , m , and there is j such that λ j = κ .If v is a normalized weight vector for λ , then || ρ ( D ) v || = m Y j =1 d λ j j = m Y j =1 d λ j − κj · det( D ) κ ≤ tr( Y ) P j ( λ j − κ ) · det( Y ) κ . For dominant weights λ we have λ ≥ λ ≥ · · · ≥ λ m = κ ≥ 0, and λ = l − P α i n i α i for some integers n i ≥ α i of gl m . Any otherweight is a conjugate of a dominant one under the Weyl group, which consistsof permutations of the coordinates. So for all weights λ of ρ we have0 ≤ m X j =1 ( λ j − κ ) ≤ m X j =1 ( l j − κ ) . Accordingly, the operator norm is seized by || ρ ( D ) || ≤ tr( Y ) P j ( l j − κ ) · det( Y ) κ . So the absolute series of S T ( g, s + P j ( l j − κ ) , s + κ ) in Theorem 2.2 dominates || P T ( g, s , s ) · v || , and the claim follows from Theorem 2.2. (cid:3) Lie algebra action. We make sure that the Poincar´e series transformadequately under the action of the Lie algebra g C = sp m, C . Following [6] wechoose the following basis of g C = p + ⊕ p − ⊕ k C , where k C is the Lie algebra of K given by the matrices satisfying (cid:18) A S − S A (cid:19) , A ′ = − A , S ′ = S , and p ± = (cid:26)(cid:18) X ± iX ± iX − X (cid:19) , X ′ = X (cid:27) . Let e kl ∈ M m,m ( C ) be the elementary matrix having entries ( e kl ) ij = δ ik δ jl andlet X ( kl ) = ( e kl + e lk ). The elements( E ± ) kl = ( E ± ) lk of p ± are defined to be those corresponding to X = X ( kl ) , 1 ≤ k, l ≤ m . Then( E ± ) kl , 1 ≤ k ≤ l ≤ m form a basis of p ± . A basis of k C is given by B kl , for1 ≤ k, l ≤ m , where B kl corresponds to A kl = 12 ( e kl − e lk ) and S kl = i e kl + e lk ) . For abbreviation, let E ± be the matrix having entries ( E ± ) kl . Similarly, let B = ( B kl ) kl be the matrix with entries B kl and let B ∗ be its transpose havingentries B ∗ kl = B lk .Let us recall some facts on derivatives. In order to compute the action of g C on ( ρ, V ρ )-valued functions, we must evaluate the total differential Dρ atvarious places A . For A in GL( m, C ) let us denote by m A the multiplicationin GL( m, C ) by A from the left, m A ( g ) = Ag , respectively m ρ ( A ) ( G ) = ρ ( A ) G TURM VECTOR VALUED 7 in GL( V ρ ). Then we can compute the differential of ρ ◦ m A = m ρ ( A ) ◦ ρ in m = id GL( m, C ) in two different ways. D ( ρ ◦ m A ) | m = Dρ | m A ( m ) ◦ Dm A | m = Dρ | A ◦ m A , respectively, D ( m ρ ( A ) ◦ ρ ) | m = D ( m ρ ( A ) ) | m ◦ Dρ | m = m ρ ( A ) ◦ dρ , where dρ is the differential of ρ at the identity, i.e. the corresponding Liealgebra representation. It follows that Dρ ( A ) = Dρ | A = m ρ ( A ) ◦ dρ ◦ m − A . Accordingly, for a GL( m, C )-valued C ∞ -function A ( t ) we have ddt ρ ( A ( t )) | t =0 = dρ (cid:18) A (0) − ddt A ( t ) | t =0 (cid:19) ◦ ρ ( A (0)) . We are specially interested in the actions Xρ ( j ( g, i ) − ) for Lie algebra elements X . For elements X of the real Lie algebra g = g R , this action is given by Xρ ( J ( g, i ) − ) = ddt ρ ( J ( g exp( tX ) , i ) − ) | t =0 . For elements of the complex Lie algebra we obtain the action by putting to-gether the actions of the real and the imaginary part. Recalling that the dif-ferential of the inverse mapping f ( g ) = g − is given by Df ( g ) = − g − , wefind Xρ ( J ( g, i ) − ) = − dρ (cid:0) J ( g, i ) − · XJ ( g, i ) (cid:1) ◦ ρ ( J ( g, i ) − ) . We often use the abbreviation J = J ( g, i ). Recalling the actions of the basiselements, B ab J ( g, i ) = J ( g, i ) e ab , ( E − ) ab J ( g, i ) = 0 , ( E + ) ab J ( g, i ) = − J − ¯ J X ( ab ) , we obtain B ab ρ ( J ( g, i ) − ) = − dρ ( e ab ) ◦ ρ ( J ( g, i ) − ) , (2) ( E − ) ab ρ ( J ( g, i ) − ) = 0 , (3) ( E + ) ab ρ ( J ( g, i ) − ) = +2 dρ ( J − ¯ J X ( ab ) ) ◦ ρ ( J ( g, i ) − ) . (4)Here k = J (˜ k, i ) ∈ U ( m ) is the image of the K -component ˜ k of g with respectto the decomposition g = ˜ g z · ˜ k , where˜ g Z = (cid:18) S U S − T (cid:19) with a lower triangular matrix S such that g · i = ˜ g Z · i = Z , i.e. SS ′ = Y =Im( g · i ).Now we give the action of the Lie algebra basis on the summands H T ( g, s , s ) = ρ ( J ( g, i ) − ) h T ( g · i, s , s ) KATHRIN MAURISCHAT of the ρ -valued Poincar´e series. Here we abbreviate h T ( Z, s , s ) = tr( T Y )) s det( Y )) s e πi tr( T Z ) . Recalling the results of [6, Lemma 7.1], we obtain B ab H T ( g, s , s ) = − dρ ( e ab ) ◦ ρ ( J ( g, i ) − ) · h T ( g · i, s , s ) , ( E − ) ab H T ( g, s , s ) = ρ ( J ( g, i ) − ) · s ( k ′ S ′ T Sk ) ab · h T ( g · i, s − , s )+ ρ ( J ( g, i ) − ) · s ( k ′ k ) kl · h T ( g · i, s , s ) , and( E + ) ab H T ( g, s , s ) = + dρ ( J − ¯ J X ( ab ) ) ◦ ρ ( J − ) · h T ( g · i, s , s )+ ρ ( J − ) · (2 s ( J − ¯ J ) ab − π ( ¯ J Y T Y ¯ J ) ab ) · h T ( g · i, s , s )+ ρ ( J − ) · s ( ¯ J Y T Y ¯ J ) ab · h T ( g · i, s − , s ) (cid:1) . Notice that each component of ¯ J Y T Y ¯ J can be sized by tr( T Y ), and that termsin k ∈ U ( m ) only vary in compact sets. Also, dρ ( e ab ) and dρ ( X ( ab ) ) are lineartransformations of V ρ . So the norm of each single term of the above can besized up to a global constant by the norm of H T ( g, s , s ). We conclude thatthe Poincar´e series allow termwise differentiations: Proposition 2.3. The derivatives XP T ( g, s , s ) = X γ ∈ Γ ∞ \ Γ XH T ( γg, s , s ) by elements X of the enveloping Lie algebra U ( g C ) have the same convergencyproperties as the Poincar´e series themselves.In particular, in the case of large weight κ > m , the Poincar´e series convergein ( s , s ) = (0 , , and vanish under the action of E − . Functions on the Siegel upper halfspace Let G = Sp( m, R ) be the symplectic group of genus m . We identify the maximalcompact subgroup K (stabilizer of i ) with the unitary group U ( m ) by k = (cid:18) C S − S C (cid:19) J ( k, i ) = C − iS . For abbreviation, let J ( g ) = J ( g, i ) for g ∈ G . Let C ∞ ( H , V ρ ) be the space of C ∞ -functions on H with values in the space V ρ , and let C ∞ ( G, V ρ ) = C ∞ ( G ) ⊗ V ρ . There is a monomorphism C ∞ ( H , V ρ ) → C ∞ ( G, V ρ ) τ ,f ( Z ) F ( G ) = ρ − ( J ( g )) F ( gKi ) . The images have the following transformation property under KF ( gk ) = ρ − ( J ( gk )) f ( gkKi ) = ρ − ( J ( k )) F ( g ) , TURM VECTOR VALUED 9 so they belong to C ∞ ( G, V ρ ) τ , the subspace of functions in C ∞ ( G, V ρ ) on whichthe action of K by right translations is given by τ = ρ − ◦ J , and the map aboveimplies an isomorphism φ : C ∞ ( H , V ρ ) ˜ −→ C ∞ ( G, V ρ ) τ . In particular, we have F ( g Z ) = ρ ( Y / ) f ( Z ). Under φ the action of the anti-holomorphic differential operator ∂ ¯ Z transforms to the action of E − . Proposition 3.1. Let ρ be an irreducible representation of GL( m, C ) of highestweight l and absolute weight κ > m . The Poincar´e series p T ( Z ) = X γ ∈ Γ ∞ \ Γ ρ ( J ( γ, Z )) − e πi tr( T γZ ) converge absolutely and locally uniformly. They are square-integrable and holo-morphic. In particular, they belong to the space [Γ , ρ ] of holomorphic cusp-forms.Proof of proposition 3.1. Because p T ( s , s ) = φ − ( P T ( s , s )), this is a directconsequence of proposition 2.1 along with proposition 2.3. (cid:3) Petterson scalar product. For f, h ∈ [Γ , ρ ] we define the Pettersonscalar product h f, h i := Z F f ( Z ) ′ ρ (Im Z ) h ( Z ) dV inv , where dV inv = dX det( Y ) m +12 dY det( Y ) m +12 is the invariant measure on H . Here dX = Q i ≤ j dx ij , and likewise dY . Wealso fix the invariant measure dY inv = dY det( Y ) m +12 on the space of positive definite matrices. Using the isomorphism φ , the Pet-terson scalar product equals the L -scalar product on group level if one usesthe normalization dV inv dk = dg for the Haar measures involved. h f, h i = Z F f ( Z ) ′ ρ (Im Z ) h ( Z ) dV inv = Z F F ( g ) ′ ρ ( J ( g )) ′ ρ (Im Z ) ρ ( J ( g )) H ( g ) dV inv = Z Γ \ G F ( g ) ′ H ( g ) dg = hh F, H ii L (Γ \ G ) . Here we used Z = g · i and the formula Im M Z = ( CZ + D ) ′− Im( Z )( CZ + D ) − . Unfolding the Poincar´e series. Let f be a (non-homomorphic) mod-ular form of weight ρ . We have h f, P T v i = Z F f ( Z ) ′ ρ (Im Z ) X γ ∈ Γ ∞ \ Γ ρ − ( J ( γ, z )) e − πi tr( T γ ¯ Z ) v dV inv = Z F X γ ∈ Γ ∞ \ Γ f ( γZ ) ′ ρ − ( J ( γ, Z )) ′ ρ (Im Z ) ρ − ( J ( γ, Z )) e − πi tr( T γ ¯ Z ) v dV inv = Z Γ \H X γ ∈ Γ ∞ \ Γ f ( γZ ) ′ ρ (Im( γZ )) e − πi tr( T γ ¯ Z ) v dV inv = Z Γ ∞ \H f ( Z ) ′ ρ (Im Z ) e − πi tr( T ¯ Z ) v dV inv . More correctly, we must restrict to the case of forms of moderate growth, whichmeans that the above integral exists. Assuming f to have Fourier expansion f ( Z ) = X ˜ T ρ ( ˜ T ) a ( ˜ T , ˜ T Y ˜ T ) e πi tr( ˜ T X ) , (notice that the vector valued coefficients are well-defined because ρ ( ˜ T ) be-longs to GL( V ρ ) and a ( ˜ T , ˜ T Y ˜ T ) belongs to V ρ ) we calculate further h f, P T v i = Z Y > a ( T, T Y T ) ′ ρ ( T ) ρ ( Y ) ve − π tr( T Y ) dY inv det( Y ) m +12 = det( T ) m +12 Z Y > a ( T, Y ) ′ ρ ( Y ) ρ ( T − ) ve − π tr( Y ) dY inv det( Y ) m +12 . If f is assumed to be holomorphic, we may write for its Fourier expansion f ( Z ) = X ˜ T ρ ( ˜ T / ) a ( ˜ T ) e πi tr( ˜ T Z ) , where a ( ˜ T ) = a ( ˜ T , ˜ T Y ˜ T ) · e π tr( ˜ T Y ) is independent of Y . Then we obtain h f, P T v i = det( T ) m +12 a ( T ) ′ Z Y > ρ ( Y ) ρ ( T − ) ve − π tr( Y ) dY inv det( Y ) m +12 . Sturm’s operator. For Sturm’s operator to reproduce holomorphic cusp-forms we must normalize it such that this last expression is a ( T ) ′ · v . So we arein due to calculate the integralsΓ( ρ ) = Z Y > ρ ( Y ) e − π tr( Y ) dY inv for varying ρ . For ρ of large enough absolute weight, this Gamma integralis convergent and belongs to End ( V ρ ) ([4]). It allows analytic continuationto smaller weights. We expect Γ( ρ ) to be invertible in general apart from a TURM VECTOR VALUED 11 discrete set of zeros and poles and prove this for a class of representations insection 5.For all ρ such that the following is well-defined as an element of GL( V ρ ) let C ( ρ ) = Z Y > ρ ( Y ) e − π tr( Y ) dY inv det( Y ) m +12 = (4 π ) ( m +1) − P j l j · Γ( ρ ⊗ det − m +12 ) . Then define the normalized Sturm operator by St ρ ( f ) = X T > ρ ( T ) b ( T ) e πi tr( T Z ) , where b ( T ) is defined by b ( T ) ′ = det( T ) − m +12 Z Y > a ( T, Y ) ′ ρ ( T ) C ( ρ ) − ρ ( Y ) ρ ( T − ) e − π tr( Y ) dY inv det( Y ) m +12 . Then, for holomorphic input f as above and v ∈ V ρ we obtain b ( T ) = a ( T ).The unfolding process above proves theorem 1.1. The assumption that Γ( ρ ⊗ det − m +12 ) is an automorphism is satisfied for example for alternating powers ρ = st [ q ] det κ , κ > m − Phantom terms by Sturm’s operator We will prove theorem 1.2. So fix rank m = 2. We test Sturm’s operatorin case of ρ being the representation of minimal K -type ( κ + 1 , κ ). We showthat in analogy to the case of scalar weight κ the Maass shift of cusp forms h ∈ [Γ , ( κ − , κ − produce phantom terms if and only if κ = 3.we have C ( ρ ) = (4 π ) − (2 κ +1) ( κ − 32 )Γ ( κ − 32 ) . Let c ( ρ ) be the scalar such that C ( ρ ) = c ( ρ ) . Let k = κ − h ∈ [Γ , τ ] be a holomorphic cuspform for τ = ( k + 1 , k ) with Fourier expansion h ( Z ) = X T > τ ( T ) a ( T ) e πi tr( T Z ) . Maass’ shift operator is given by (see [8, 5.1])∆ [2]+ h ( Z ) = (2 i ) ( τ ⊗ det − )( Y − ) · det( ∂ Z ) (cid:16) ( τ ⊗ det − )( Y ) h ( Z ) (cid:17) . The image of h under ∆ [2]+ is a non-holomorphic form of weight τ ⊗ det , i.e.( k + 3 , k + 2) = ( κ + 1 , κ ). Hence (see [8]), its holomorphic projection iszero pr hol (∆ [2]+ ( h )) = 0. We show that Sturm’s operator St τ ⊗ det (∆ [2]+ ( h ))is non-zero if and only if k = 1. For to apply Maass’ operator to h it isenough to apply it to e πi tr( T Z ) . Here ( τ ⊗ det − )( Y ) = det( Y ) k − Y . Let f ( Z ) = det( Y ) k − e πi tr( T Z ) and g ( Z ) = Y . By [2, p. 211] we havedet( ∂ Z )( f · g ) = det( ∂ Z )( f ) · g + 2( ∂ Z ( f ) ⊓ ∂ Z ( g )) + f · det( ∂ Z )( g ) . Here the last term is zero, because det( ∂ Z ) is a differential operator of homoge-neous degree two and g ( Z ) = Y is of degree one. For the first term we obtainfollowing [8, 5.2]det( ∂ Z )( f ( Z )) · Y = − k ( k − 12 ) det( Y ) k − e πi tr( T Z ) · Y − i k − 12 )(2 πi ) tr( T Y ) det( Y ) k − e πi tr( T Z ) · Y +(2 πi ) det( Y ) k − det( T ) e πi tr( T Z ) · Y . For the second term we find ∂ Z ( f ( Z )) = − i k − 12 ) det( Y ) k − e πi tr( T Z ) · Y − + 2 πi det( Y ) k − e πi tr( T Z ) · T , and ∂ Z ( Y jk ) = − i X ( jk ) . Here X ( jk ) = ( e jk + e kj ). So the second term2( ∂ Z ( f ( Z )) ⊓ ∂ Z ( g ( Z )) equals X j,k (cid:18) − 14 ( k − 12 ) det( Y ) k − e πi tr( T Z ) Y − ⊓ X ( jk ) ) · X ( jk ) + π det( Y ) k − e πi tr( T Z ) · T ⊓ X ( jk ) ) · X ( jk ) (cid:17) , which by definition of ⊓ -multiplication ([2, p. 207]) is − 14 ( k − 12 ) det( Y ) k − e πi tr( T Z ) · Y + π det( Y ) k − e πi tr( T Z ) det( T ) · T − . Altogether we obtain∆ [2]+ ( e πi tr( T Z ) ) = ( k + 1)( k − 12 ) 1det( Y ) e πi tr( T Z ) · − π ( k − 12 ) tr( T Y )det( Y ) e πi tr( T Z ) · + (4 π ) det( T ) e πi tr( T Z ) · − π det( T ) e πi tr( T Z ) · ( T Y ) − . The Fourier coefficients of˜ h ( Z ) = ∆ [2]+ ( h ) = X T > ρ ( T ) a ( T, T Y T ) e πi tr( T X ) are given by ρ ( T ) a ( T, T Y T ), which equal e − π tr( T Y ) · (cid:16) ( k − )( k + 1)det( Y ) − π ( k − 12 ) tr( T Y )det( Y ) + (4 π ) det( T ) (cid:17) · τ ( T ) a ( T ) − e − π tr( T Y ) · π det( T ) · Y − T − τ ( T ) a ( T ) , TURM VECTOR VALUED 13 respectively a ( T, Y ) given by e − π tr( Y ) · ( k − )( k + 1)det( Y ) − π ( k − 12 ) tr( Y )det( Y ) + (4 π ) ! · ρ ( T − ) a ( T ) − e − π tr( Y ) · π · Y − a ( T ) . Accordingly, for to compute Sturm’s operator we evaluate the sum of the fol-lowing terms up to the factor det( T ) − c ( ρ ) − · a ( T ) ′ . First,( k − 12 )( k + 1) Z Y > Y det( Y ) k +2+ s − e − π tr( Y ) dY inv · which by Proposition 5.2 equals(5) ( k − 12 )( k + 1)(4 π ) − s + k ) ( s + k − 12 )Γ ( s + k − 12 ) . Second, − π ( k − 12 ) Z Y > Y tr( Y ) det( Y ) k +2+ s − e − π tr( Y ) dY inv which by Lemma 5.5 equals(6) − k − 12 )(4 π ) − s + k ) ( s + k − 12 )( s + k )Γ ( s + k − 12 ) . Third, (4 π ) Z Y > Y det( Y ) k +2+ s − e − π tr( Y ) dY inv which by Proposition 5.2 equals(7) (4 π ) − s + k ) ( s + k + 12 )Γ ( s + k + 12 ) . And(8) 4 π · Z Y > det( Y ) k +2+ s − e − π tr( Y ) dY inv = (4 π ) − s + k ) Γ ( s + k + 12 ) . According to (5)–(8), Sturm’s operator applied to ∆ [2]+ h ( Z ) is given in terms ofcoefficients by the limit lim s → b ( T, s ), where b ( T, s ) = det( T ) − c ( ρ ) − (4 π ) − s + k ) s − s ( s + k − 1) Γ ( s + k + 12 ) a ( T ) . Here we used the identity ( s + k − )Γ ( s + k − ) = ( s + k − − Γ ( s + k + ).The limit lim s → b ( T, s ) = (4 π ) det( T ) − · lim s → s − s s + k − · a ( T )is zero in all cases k > 1, and equals b ( T ) = − (4 π ) T ) − · a ( T ) in case k = 1. So Sturm’s operator applied to ∆ [2]+ h ( Z ) is non-zero exactly incase ρ = ( κ + 1 , κ ) with κ = 3, which is the minimal K -type of the holomorphicdiscrete series representation of Harish-Chandra parameter (4 , − (1 , 2) =(3 , Gamma integrals For an irreducible finite dimensional representation ρ of GL( m, C ) of absoluteweight κ we are interested in the End ( V ρ )-valued integralΓ( ρ ) = Z Y > ρ ( Y ) e − tr( Y ) dY inv . Introducing a factor det( Y ) s the integralΓ( ρ ⊗ det s ) = Z Y > ρ ( Y ) det( Y ) s e − tr( Y ) dY inv exists for Re s + κ > m − ([4]). We denote by Γ( ρ ) its meromorphic continuationto s = 0. Let Γ m ( s ) = π m ( m − m − Y ν =0 Γ( s − ν m which for Re s > m − is givenby the integral Γ m ( s ) = Z Y > det( Y ) s e − tr( Y ) dY inv . In particular we have Γ(det s ) = Γ m ( s ). An important property of the operatorintegrals is their SO( m )-equivariance. Lemma 5.1. The integral Γ( ρ ) is invariant under orthogonal transformations Γ( ρ ) = ρ ( k ′ )Γ( ρ ) ρ ( k ) , for all k ∈ SO( m, R ) ⊂ U ( m ) .Proof of Lemma 5.1. For k ∈ SO( m ) we haveΓ( ρ ⊗ det s ) = Z Y > ρ ( k ′ Y k ) det s ( k ′ Y k ) e − tr( k ′ Y k ) dY inv = ρ ( k ′ )Γ m ( ρ ⊗ det s ) ρ ( k ) . By the uniqueness of meromorphic continuation, this also holds for Γ( ρ ). (cid:3) Alternating powers.Proposition 5.2. For q = 1 , . . . , m , let st [ q ] be the q -th alternating power ofthe standard representation of GL( m, C ) , i.e. the irreducible representation ofhighest weight (1 , . . . , , , . . . , , where the number of ones is q . Define thepolynomial C [ q ] ( x ) = x ( x + ) · · · ( x + q − ) . The automorphism-valued function Γ( st [ q ] ⊗ det s ) = Z Y > Y [ q ] det( Y ) s e − tr( Y ) dY inv = ( − q C [ q ] ( − s )Γ m ( s ) · id st [ q ] TURM VECTOR VALUED 15 is holomorphic on Re s > m − and has meromorphic continuation to the com-plex plane, the pole behavior being that of the scalar function C [ q ] ( − s )Γ m ( s ) .Proof of Proposition 5.2. For a symmetric positive definite matrix T it holds(9) Z Y > det( Y ) s e − tr( T Y ) dY inv = det( T ) − s Γ m ( s ) . Differentiating both sides by ∂ [ q ] T we obtain ([2, p. 210, p. 213])( − q Z Y > Y [ q ] det( Y ) s e − tr( T Y ) dY inv = C [ q ] ( − s ) T − [ q ] det( T ) − s Γ m ( s ) . Evaluating at T = m yields Z Y > Y [ q ] det( Y ) s e − tr( Y ) dY inv = ( − q C [ q ] ( − s )Γ m ( s ) · id st [ q ] . (cid:3) By substitution s ′ = s − κ , Proposition 5.2 determines Γ( ρ ) for the represen-tations ρ = st [ q ] ⊗ det κ . The computation for general ρ may be obtained bychasing Young tableaux, but for rank m > m = 2 by the involved triangle numbers a n,m defined in Proposition 5.3.5.2. Rank two. For a general formula for Γ( ρ ⊗ det − s ) for the irreducible rep-resentations ρ of GL(2 , C ) of highest weights ( r, Proposition 5.3. Define the following triangle numbers a n,m for n, m ∈ N .Let a n, = 1 for all n ∈ N , and let a ,m = 0 for all m > . For n, m > defineby recursion a n,m = (cid:0) n − m − (cid:1) · a n − ,m − + a n − ,m . The triangle numbers have the following properties. (i) a n, = n ( n + 1) . (ii) a n, = n ( n + 1)( n − n − . (iii) a n,m = 0 for all m > ⌊ n +12 ⌋ (Gauss brackets). (iv) a ν − ,ν = a ν − ,ν − . We will be specially interested in the numbers a ν − ,ν , for which we give anexplicit formula in Proposition 5.7. Proof of proposition 5.3. Obviously, a , = a , + a , = 1. Assuming a n − , = n ( n − n by induction and the recursion formula a n, = n · a n − , + a n − , = 12 n ( n + 1) . Property (iii) holds for n = 0 by definition, and by induction the right handside of the recursion formula is zero for all m > ⌊ n ⌋ + 1. So the single case leftto check is that of even n = 2 k and m = k + 1. But here the recursion yields a k,k +1 = (2 k − k ) a k − ,k + a k − ,k +1 = 0. Property (ii) is also obtained by induction using (i) and (iii). For property (iv) notice that by (iii) a n − , n +12 = 0for odd n , so the recursion formula yields a n, n +12 = a n − , n − . (cid:3) Lemma 5.4. Let T be a symmetric two-by-two matrix variable and denote by ∂ ij = δ ij ∂ T ij the normalized partial derivatives. For all n > the derivativesof the function det( T ) − s are given by ∂ ( n ) jj (det( T ) − s ) = ( − n T nii det( T ) − ( s + n ) n − Y l =0 ( s + l ) , for { i, j } = { , } , and ∂ ( n )12 (det( T ) − s ) = n − X k =0 − k a n − ,k det( T ) − ( s + n − k ) T n − k · n − k − Y l =0 ( s + l ) , where the numbers a n,m are defined in Proposition 5.3. Further, ∂ ( n )11 ∂ ( n )22 (det( T ) − s ) = min { n ,n } X k =0 k ! (cid:18) n k (cid:19)(cid:18) n k (cid:19) ( − n + n + k T n − k T n − k ×× det( T ) − ( s + n + n − k ) · n + n − k − Y l =0 ( s + l ) . Proof of lemma 5.4. Iterating ∂ jj (det( T ) − s ) = − sT ii det( T ) − ( s +1) we obtain ∂ ( n ) jj (det( T ) − s ) = ( − n T nii det( T ) − ( s + n ) n − Y l =0 ( s + l ) . Then for ∂ ( n )11 ∂ ( n )22 (det( T ) − s ) we obtain ∂ ( n )11 ( − n T n det( T ) − ( s + n ) n − Y l =0 ( s + l ) ! = ( − n n − Y l =0 ( s + l ) n X k =0 (cid:18) n k (cid:19) ∂ ( k )11 ( T n ) · ∂ ( n − k )11 (det( T ) − ( s + n ) )= min { n ,n } X k =0 (cid:18) n k (cid:19) n !( n − k )! ( − n + n + k T n − k T n − k ×× det( T ) − ( s + n + n − k ) n + n − k − Y l =0 ( s + l ) . Further, ∂ (det( T ) − s ) = sT det( T ) − ( s +1) as well as ∂ (2)12 (det( T ) − s ) = s ( s + 1) T det( T ) − ( s +2) + 12 s det( T ) − ( s +1)TURM VECTOR VALUED 17 satisfy the claimed formula. Then ∂ ( n +1)12 is given by induction ∂ n − X k =0 − k a n − ,k det( T ) − ( s + n − k ) T n − k n − k − Y l =0 ( s + l ) ! = n − X k =0 − k a n − ,k · ( s + n − k ) · det( T ) − ( s + n +1 − k ) T n +1 − k n − k − Y l =0 ( s + l )+ n − X k =0 − k a n − ,k · 12 ( n − k ) · det( T ) − ( s + n − k ) T n +1 − k +1)12 n − k − Y l =0 ( s + l )= n X k =0 − k a n,k det( T ) − ( s + n +1 − k ) T n +1 − k n +1 − k − Y l =0 ( s + l ) , where we have used the product rule and the recursion formula defining thenumbers a n,k (see proposition 5.3) as well as the fact a n,n = 0 for n ≥ (cid:3) Lemma 5.5. Let n , n , n ≥ be integers. The integral Z Y > Y n Y n Y n det( Y ) s e − tr( Y ) dY inv is a holomorphic function on Re s > . For odd n it is zero, while for even n it is given by Γ ( s ) · − n a n − , n · min { n ,n } X k =0 ( − k (cid:18) n k (cid:19)(cid:18) n k (cid:19) k ! · n + n + n − k − Y l =0 ( s + l ) . Here we put a − , = 1 . In particular, the integral has meromorphic continuationto the complex plane, the poles being at most simple and included in those of Γ ( s ) .Proof of lemma 5.5. Starting with the identity Z Y > det( Y ) s e − tr( T Y ) dY inv = det( T ) − s Γ ( s )for Re s > , which holds for all positive definite T , we differentiate both sidesby ∂ ( n )11 ∂ ( n )22 ∂ ( n )12 to determine R Y > Y n Y n Y n det( Y ) s e − tr( T Y ) dY inv byΓ ( s ) · ( − n + n + n ∂ ( n )11 ∂ ( n )22 ∂ ( n )12 (det( T ) − s ) . Evaluating at T = , we obtain a formula for the integral in question byΓ ( s ) · ( − n + n + n ∂ ( n )11 ∂ ( n )22 ∂ ( n )12 (det( T ) − s ) | T = . Lemma 5.4 determines the derivative ∂ ( n )11 ∂ ( n )22 ∂ ( n )12 (det( T ) − s ) = min { n ,n } X k =0 n − X k =0 ( − n + n − k − k (cid:18) n k (cid:19)(cid:18) n k (cid:19) k ! ×× T n − k T n − k T n − k · det( T ) − ( s + n + n + n − k − k ) ×× n + n + n − k − k − Y l =0 ( s + l ) . Evaluating at T = , the factor T n − k is zero apart from the case n = 2 k .In this case the formula reduces to the claimed one, whereas it is zero forodd n . (cid:3) Consider the explicit realization of the representation ρ = ρ r of GL ( C ) ofhighest weight ( r, 0) on the space P r of homogeneous polynomials of degree r in the variable z = ( z , z ), ρ r ( g ) (cid:0) P ( z ) (cid:1) = P ( z · g )for P ∈ P r . We determine Γ ( ρ r ⊗ det s ) by its action on the K = SO(2)-weightspaces. For k = 0 , , . . . , r the polynomial V k ( z ) = ( z − iz ) r − k ( z + iz ) k is a K -eigenfunction of weight − r + 2 k . We find V k ( z ) = r X ν =0 z r − ν z ν i ν min { r − k,ν } X j =0 ( − j (cid:18) r − kj (cid:19)(cid:18) kν − j (cid:19) , whereas ρ r ( Y ) V k ( z ) = (cid:2) ( Y − iY ) z + ( Y + iY )( − iz ) (cid:3) r − k (cid:2) ( Y + iY ) z + ( Y − iY ) iz (cid:3) k = r X ν =0 z r − ν z ν i ν · P k ( ν, Y ) , with P k ( ν, Y ) = min { r − k,ν } X j =0 ( − j (cid:18) r − kj (cid:19)(cid:18) kν − j (cid:19) ( Y − iY ) r − k − j × ( Y + iY ) k + j − ν ( Y + iY ) j ( Y − iY ) ν − j . By lemma 5.1, Γ ( ρ r ⊗ det s ) commutes with K , so acts by scalars on the 1-dimensional K -eigenspaces. Defining c k ( ν ) = min { r − k,ν } X j =0 (cid:18) r − kj (cid:19)(cid:18) kν − j (cid:19) ( − j TURM VECTOR VALUED 19 the integral(10) Γ( r, k, s ) = 1 c k ( ν ) Z Y P k ( ν, Y ) det( Y ) s e − tr( Y ) dY inv is the Γ( ρ r ⊗ det s )-eigenvalue of V k ( z ), which in particular is independent of ν . Proposition 5.6. For k = 0 , , . . . , r we have the functional equation Γ( r, k, s ) = Γ( r, r − k, s ) . For k = 0 , , . . . , ⌊ r ⌋ the function Γ( r, k, s ) is explicitly given by Γ( r, k, s ) = Γ ( s ) ⌊ r ⌋ X µ =0 a µ − ,µ µ k X j =0 (cid:18) kj (cid:19)(cid:18) r − k µ − j ) (cid:19) ( − µ − j r − µ − Y l =0 ( s + l ) . With respect to the SO(2) -weight decomposition, the operator Γ( ρ r ⊗ det s ) isgiven by the diagonal matrix Γ( ρ r ⊗ det s ) = diag (cid:0) Γ( r, , s ) , Γ( r, , s ) , . . . , Γ( r, ⌊ r ⌋ , s ) , . . . , Γ( r, , s ) , Γ( r, , s ) (cid:1) . In particular, Γ( ρ r ⊗ det s ) is divisible by Γ ( s ) Q ⌊ r ⌋− l =0 ( s + l ) . Apart from itsfinite set of zeros and its set of poles which is contained in that of Γ ( s ) , theoperator Γ( ρ r ⊗ det s ) is invertible for Re s > .Proof of proposition 5.6. We determine Γ( r, k, s ) by choosing ν = 0 in (10).For integers a, b ≥ Y + Y ) a ( Y ± iY ) b = a X j =0 b X l =0 (cid:18) aj (cid:19)(cid:18) bl (cid:19) ( ± i ) l Y a − j )+ b − l Y j + l , so by lemma 5.1 only the summands with even Y -exponents contribute to theintegral Z Y ( Y + Y ) a ( Y ± iY ) b det( Y ) s e − tr( Y ) dY inv = Γ ( s ) a X j =0 ⌊ b ⌋ X l =0 (cid:18) aj (cid:19)(cid:18) b l (cid:19) ( − l a j + l ) − ,j + l j + l a + b − ( j + l ) − Y µ =0 ( s + µ ) . Notice that the integral is independent of the sign in ( Y ± iY ) b . AccordinglyΓ( r, k, s ) = Γ( r, r − k, s ), and we may restrict to the case k ≤ r − k , and applythe above formula with a = k and b = r − k . (cid:3) In particular, in case k = 0(11) Γ( r, , s ) = Γ ( s ) ⌊ r ⌋ X µ =0 (cid:18) r µ (cid:19) ( − µ a µ − ,µ µ r − µ − Y l =0 ( s + l ) . On the other hand, we recall the formula valid for all ν Γ( r, , s ) = Z Y ( Y − iY ) r − ν ( Y + iY ) ν det( Y ) s e − tr( Y ) dY inv . Because ( Y + Y ) r = r X ν =0 (cid:18) rj (cid:19) ( Y − iY ) r − ν ( Y + iY ) ν , we obtain Z Y ( Y + Y ) r det( Y ) s e − tr( Y ) dY inv = 2 r · Γ( r, , s ) , which impliesΓ( r, , s ) = Γ ( s )2 r r X j =0 (cid:18) rj (cid:19) min { j,r − j } X µ =0 (cid:18) r − jµ (cid:19)(cid:18) jµ (cid:19) ( − µ µ ! r − µ − Y l =0 ( s + l ) , or equivalently(12) Γ( r, , s ) = Γ ( s ) ⌊ r ⌋ X µ =0 ( − µ µ !2 r r − µ X j = µ (cid:18) rj (cid:19)(cid:18) r − jµ (cid:19)(cid:18) jµ (cid:19) r − µ − Y l =0 ( s + l ) . Noticing that the polynomials Q r − µ − l =0 ( s + l ) for µ = 0 , . . . , ⌊ r ⌋ are linearlyindependent, we obtain by comparing the coefficients of (11) and (12) µ !2 r r − µ X j = µ (cid:18) rj (cid:19)(cid:18) r − jµ (cid:19)(cid:18) jµ (cid:19) = (cid:18) r µ (cid:19) a µ − ,µ µ , which is easily simplified to the identity of proposition 5.7 (a) below. Proposition 5.7. The triangle numbers defined in Proposition 5.3 take thefollowing special values. (a) For all µ = 0 , , , . . . , a µ − ,µ = (2 µ )!2 µ µ ! = (2 µ − . (b) For all µ = 1 , , , . . . , a µ − ,µ − = µ · (2 µ − . (c) For all µ = 1 , , , . . . , a µ,µ − = µ µ + 1)!! . Proof of Proposition 5.7. By the defining recursion formula we obtain a µ,µ = 2 a µ − ,µ − + a µ − ,µ . TURM VECTOR VALUED 21 Because part (a) has already been verified for all ν , we obtain part (b) by usingproposition 5.3 (iv) a µ − ,µ − = 12 (cid:0) (2 µ + 1)!! − (2 µ − (cid:1) = µ · (2 µ − . By recursion a µ +1 ,µ = 3 a µ,µ − + a µ,µ , and applying (a) and (b), we obtainpart (c) a µ,µ − = 13 (cid:0) ( µ + 1)(2 µ + 1)!! − (2 µ + 1)!! (cid:1) = µ µ + 1)!! . (cid:3) Example 5.8 (Symmetric representation) . For r = 2 the representation ρ isisomorphic to the symmetric representation. In terms of the basis of eigenvec-tors V k ( z ), k = − , , 2, for SO(2), the Γ-integral is given by the matrixΓ( ρ ⊗ det s ) = s Γ ( s ) s + s + s + . Equivalently, on the space of symmetric matrices X = (cid:16) X X X X (cid:17) ,Γ(( Sym ⊗ det s )( X )) = Z Y > Y XY det( Y ) s e − tr( Y ) dY inv = s ( s + 1)Γ ( s ) · X + s ( s ) · ˜ X , where ˜ X = (cid:16) X − X − X X (cid:17) is the adjunct matrix for X . In particular, thisexample shows that Γ( ρ ) is not a scalar operator in general.5.3. Weyl’s character formula. Lemma 5.1 suggests the following integraltransformation. For the diagonal torus T of GL( m, R ) let T > = { t = diag ( t , . . . , t m ) ∈ T | t > t > · · · > t m } . Denote by P m the set of positive definite ( m, m )-matrices, P m ⊂ Sym ( R m ).Let K = SO( m ) with unit element E . There is an injective map T > × K → P m , ( t, k ) ktk ′ = ktk − = Y , which has open and dense image. For the pullback φ ∗ we find φ ∗ ( dY ) ( t, E ) = ( dX ′ · t + t · dX ) + dt , where dX = dx . . . dx m − dx . . . ...... . . . . . . . So − dX · t + t · dX equals t − t ) dx . . . ( t − t m ) dx m ( t − t ) dx . . . . . . ...... . . . ( t m − − t m ) dx m − ,m . Accordingly, the pullback φ ∗ (det( Y ) − m +12 Q i ≤ j dY ij ) at ( t, E ) of the invariantmeasure dY inv on P m is given by ± det( t ) − m +12 · Y i 12 ) . This must equal up to a constant depending on the normalization of measuresand their orientation Z t >t > ( t t ) k − ( t − t ) e − t − t dt dt , which equals Z ∞ t k − e − t dt · Z ∞ t k − e − t dt − Z ∞ t k − e − t Z ∞ t t k − e − t dt dt . For the last integral we first notice that by partial integration Z ∞ t t k − e − t dt = t k − e − t + ( k − 12 ) Z ∞ t t k − e − t dt . Let φ ( t ) be an antiderivative of − t k − e − t , in particular φ ( t ) = Z ∞ t t k − e − t dt . Accordingly, Z ∞ φ ′ ( t ) φ ( t ) dt = Z ∞ t k − e − t Z ∞ t t k − e − t dt dt = φ ( t ) | ∞ − Z ∞ φ ( t ) φ ′ ( t ) dt , i.e. − Z ∞ t k − e − t Z ∞ t t k − e − t dt dt = − Γ( k − 12 ) . So we obtain R t >t > ( t t ) k − ( t − t ) e − t − t dt dt to equalΓ( k + 12 )Γ( k − 12 ) − k − Γ(2 k − − ( k − 12 )Γ( k − 12 ) , which simplifies to − (cid:0) (cid:1) k − Γ(2 k − √ π z − Γ( z ) = Γ( z z + 12 ) TURM VECTOR VALUED 23 we conclude Z t >t > ( t t ) k − ( t − t ) e − t − t dt dt = − √ π · Γ( k )Γ( k − 12 ) . Thus indeed,Γ (det k ) = ( − · vol( K ) · Z t >t > ( t t ) k − ( t − t ) e − t − t dt dt with the volume of K normalized by vol( K ) = π .We use (13) to compute the tracetr(Γ( ρ )) = ± vol( K ) Z T > det( t ) − m +12 Y i Non-archimedean L-functions and arithmetical Siegel mod-ular forms, Lecture Notes in Mathematics 1471, second augmented edition, Springer(2004), Heidelberg u.a.[2] E. Freitag: Siegelsche Modulformen, Grundlehren der mathematischen Wissenschaften254 (1983), Springer[3] E. Freitag: Singular modular forms and theta relations, Lecture notes in mathematics1487, Springer (1991).[4] R. Godement: Fonctions holomorphes de carre´e sommable dans le demi-plan de Siegel, Sem. H. Cartan 6, E. N. S. (1957/58), 1-22.[5] B. H. Gross, D. B. Zagier: Heegner points and derivatives of L-series, Invent. Math. 84(1986), no. 2, 225-320.[6] K. Maurischat: On holomorphic projection for symplectic groups, J. Number Theory,Vol. 182 (2018), 131-178.[7] K. Maurischat: Sturm’s operator for scalar weight in arbitrary genus, Int. J. NumberTheory, Vol. 13, No. 10 (2017), pp. 2677-2686.[8] K. Maurischat, R. Weissauer: Phantom holomorphic projections arising from Sturm’sformula, The Ramanujan J., 47(1) (2018), 21-46.[9] Sturm, J.: Projections of C ∞ automorphic forms, Bull. Amer. Math. Soc. 2 (1980),435-439.[10] Sturm, J.: The critical values of Zeta-functions associated to the symplcetic group, DukeMath. J. 48 (1981), 327-350. Kathrin Maurischat , Mathematisches Institut, Universit¨at Heidelberg, Im NeuenheimerFeld 205, 69120 Heidelberg, Germany E-mail address ::