green forms and the local archimedean ...derivatives with arithmetic intersections of special...

GREEN FORMS AND THE LOCAL ARCHIMEDEANARITHMETIC SIEGEL-WEIL FORMULA

LUIS E. GARCIA, SIDDARTH SANKARAN

Abstract. The goal of this paper is to construct natural Green forms for specialcycles in orthogonal and unitary Shimura varieties, in any codimension, and toshow that the resulting local archimedean height pairings are related to derivativesof Siegel Eisentein series. A conjecture put forward by S. S. Kudla relates thesederivatives with arithmetic intersections of special cycles, and our results essentiallysettle the part of his conjecture involving local archimedean heights, for compactShimura varieties attached to O(p, 2) and U(p, 1) (any p ≥ 1).

Contents

1. Introduction 21.1. Shimura varieties and special cycles 21.2. Kudla’s conjectures 31.3. Main results 41.4. Notation and conventions 62. Green forms on hermitian symmetric domains 72.1. Superconnections and characteristic forms of Koszul complexes 72.2. Hermitian symmetric domains of orthogonal and unitary groups 132.3. Explicit formulas for O(p, 2) and U(p, 1) 162.4. Basic properties of the forms ϕ and ν 172.5. Behaviour under the Weil representation 202.6. Star products 233. Archimedean heights and derivatives of Whittaker functionals 263.1. Degenerate principal series of Mp2r(R) 263.2. Degenerate principal series of U(r, r) 293.3. Whittaker functionals 313.4. The integral of ξ 374. Green forms for special cycles on Shimura varieties 424.1. Orthogonal Shimura varieties 424.2. Unitary Shimura varieties 454.3. Star products on XK 464.4. The degenerate case 475. Archimedean heights and derivatives of Siegel Eisenstein series 535.1. Siegel Eisenstein series and the Siegel-Weil formula 535.2. Archimedean height pairings 56References 58

1

2 LUIS E. GARCIA, SIDDARTH SANKARAN

1. Introduction

The Arakelov theory of Shimura varieties has been intensively studied since, abouttwenty years ago, Kudla [23] launched a program relating their arithmetic Chowgroups with derivatives of Eisenstein series and of Rankin-Selberg L-functions. Letus describe his proposal before stating our main results. For simplicity, in the in-troduction we focus on the case of orthogonal Shimura varieties; in the body of thepaper we treat both the orthogonal and unitary cases.

1.1. Shimura varieties and special cycles. Let F be a totally real number field ofdegree d with real embeddings σ1, . . . , σd : F → R and (V, Q) be a quadratic F -vectorspace such that V = Vσ1 has signature (p, 2) and Vσi is positive definite for i ≥ 2. LetH = GSpin(V) and denote by D the hermitian symmetric domain associated withH(R). For every neat open compact subgroup K ⊂ H(Af ), the Shimura variety

(1.1) XK = H(Q)\D×H(Af )/Kis a quasi-projective complex algebraic variety that carries a natural large family ofalgebraic cycles known as special cycles (see [22]). These exist in any codimensionand are parametrized by pairs (T, ϕf ) consisting of a symmetric r-by-r matrix Twith entries in F and an integer-valued Schwartz function ϕf ∈ S(V(Af )r)K ; thecorresponding cycle has codimension rk(T ) and we denote it by

(1.2) Z(T, ϕf ) ∈ CHrk(T )(XK).

A first step in understanding special cycles is to consider their image [Z(T, ϕf )] ∈H2rk(T )(XK) under the cycle class map. Let L be the Hodge bundle on XK and L∨be its dual. In [24, 25, 26], Kudla and Millson show that the generating series

(1.3)∑T≥0

[Z(T, ϕf )] ∪ (L∨)r−rk(T )qT ∈ H2r(XK)⊗ C[[q]], qT =d∏i=1

e2πitr(σi(T )τi),

is a holomorphic Hilbert-Siegel modular form of genus r and parallel weight 12dim(V ).

For the proof they construct a theta series ΘKM(τ, ϕf ) ∈ A2r(XK) that satisfiesdΘKM(τ, ϕf ) = 0 and whose cohomology class is the generating series (1.3). Themethod shows that the cohomological properties of special cycles can be understoodusing the theta correspondence. For example, the cup product formula

(1.4) [Z(T1, ϕf,1)] ∪ [Z(T2, ϕf,2)] =∑T≥0

T=(T1 ∗∗ T2

)[Z(T, ϕf,1 ⊗ ϕf,2)] ∪ (L∨)r−rk(T )

for T1 and T2 positive definite follows easily. Kudla [22] also shows that, for compactXK , the degrees of special cycles with respect to L are given by

(1.5)∑T≥0

degL(Z(T, ϕf ))qT = Vol(XK ,Ω) · E(τ, λ(ϕf ), s0), s0 = 1

2(p+ 1− r).

Here Ω is the first Chern form of the hermitian line bundle L and E(τ, λ(ϕf ), s) denotesa Siegel Eisenstein series of weight 1

2dim(V ) and level determined by ϕf , normalized

GREEN FORMS AND THE ARITHMETIC SIEGEL-WEIL FORMULA 3

so that the center of the functional equation is at s = 0. The proof follows easilyfrom the Siegel-Weil formula as generalized by Kudla-Rallis [27].

1.2. Kudla’s conjectures. Suppose now thatXK has a regular model X over Spec(Z)

and consider the arithmetic Chow groups CHr(X ) defined in [12], appropriately mod-

ified if XK is not compact [9]. Here and in the following we will ignore many seriousdifficulties regarding the construction of good integral models of Shimura varietiesand their special cycles, places of bad reduction and compactifications, as our focus is

on archimedean aspects. A class Z ∈ CHr(X ) is represented by a pair (Z, gZ), where

Z ∈ CHr(X ) is a Chow cycle whose restriction to X we denote by Z and

(1.6) gZ ∈ Dr−1,r−1(X)/im∂ + im∂

is a Green current for Z (see (2.1)) with appropriate growth at infinity. These groupsadmit an intersection product

(1.7) · : CHr(X )× CH

s(X )→ CH

r+s(X )Q.

If (Z, gZ) ∈ CHr(X ) and (Z ′, gZ′) ∈ CH

s(X ) and Z,Z ′ intersect properly in X, their

product is given by (Z, gZ) · (Z ′, gZ′) = (Z · Z ′, gZ ∗ gZ′), where Z · Z ′ ∈ CHr+s

(X )Qis the usual intersection and gZ ∗ gZ′ is the star product

(1.8) gZ ∗ gZ′ = gZ ∧ δZ′ + [ωZ ] ∧ gZ′ , ωZ := ddcgZ + δZ ∈ A∗(XK).

There is a map CHr(X ) → CHr(XK) given by pullback to XK , and we refer to any

cycle mapping to Z as a lift of Z.Kudla conjectures that, for certain such models X , there is a family of lifts

(1.9) Z(T, y, ϕf ) ∈ CHr(X )

of Z(T, ϕf ) · (L∨)r−rk(T ) ∈ CHr(X), depending on an auxiliary parameter y (a realsymmetric positive definite r-by-r matrix), such that arithmetic analogues of (1.3)-(1.5) hold. More precisely, the generating series

(1.10) θ(r)(τ, ϕf ) :=∑T

Z(T, y1, ϕf )qT , y1 = Im(τ1),

should be a (non-holomorphic) Hilbert-Siegel modular form of genus r and weight12dim(V ), valued in CH

r(X ). We also require that the product formula

(1.11) θ(r)(τ1, ϕf,1) · θ(s)(τ2, ϕf,2) = θ(r+s)((

τ1 00 τ2

), ϕf,1 ⊗ ϕf,2

)holds. Finally, assume that XK is compact. Writing ω ∈ CH

1(X ) for the natural lift

of the Hodge bundle L and deg : CHp+1

(X ) → R for the arithmetic degree map, weask that

(1.12)∑T

deg(Z(T, y1, ϕf ) · ωp+1−r)qT = Vol(XK ,Ω)d

dsE(τ,Φ, s)

∣∣∣∣s=

12

(p+1−r)

for a certain Eisenstein series E(τ,Φ, s) depending on ϕf .


For Shimura curves all this is proved in [30]. There are also analogous conjecturesfor unitary Shimura varieties (see [19, 20]). For recent progress, see [7, 8] and thereferences therein.

1.3. Main results. Let XK be an arbitrary orthogonal or unitary (for a fixed CMextension of F ) Shimura variety as described in Section 4; throughout the paper werefer to the orthogonal group case as case 1 and to the unitary case as case 2. Inthe latter case special cycles on XK are parametrized by pairs (T, ϕf ), where T is ahermitian r-by-r matrix with entries in E and ϕf ∈ S(V(Af )r)K . To treat both casessimultaneously, we write Xr(F ) = Symr(F ) in case 1 and Xr(F ) = Herr(E) in case2.

The sought after arithmetic Chow cycles are of the form

(1.13) Z(T, y, ϕf ) = (Z(T, ϕf ), g(T, y, ϕf )),

and in this paper we focus on the current g(T, y, ϕf ). Assume first that T is non-singular. Then the irreducible components of the special cycle Z(T, ϕf ) admit acomplex uniformization by certain submanifolds Dv in D parametrized by vectorsv = (v1, . . . , vr) ∈ V r such that (Q(vi, vj))i,j = 2T . Using results of Bismut, Gilletand Soule [4, 2, 3] we define, for each such v, a natural Green form

(1.14) g(v) ∈ D∗(D)

for Dv. We give a characterization of g(v) a la Bott-Chern in Theorem 2.1, showingthat it is the unique such form that decreases rapidly with the distance to Dv. Usingthis property we define our Green form

(1.15) g(T, y, ϕf ) ∈ D∗(XK)

as an infinite sum of the forms g(v).Our main results, Theorem 4.4 and Theorem 5.1, show that the currents g(T, y, ϕf )

have the properties required by the conjectural identities (1.11) and (1.12). Concern-ing (1.11) we prove the following result.

Theorem 4.4. Assume that r + s ≤ dim(XK) + 1 and let T1 ∈ Xr(F ) and T2 ∈Xs(F ) be non-singular. If ϕf,i = ⊗pϕp,i (i = 1, 2) and ϕp,1 ⊗ ϕp,2 is supported on theregular locus of V(Fp)

r+s for some finite place p of F , then Z(T1, ϕf,1) and Z(T2, ϕf,2)intersect properly and

g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2) =∑

T=(T1 ∗∗ T2

) g(T, ( y1 y2 ) , ϕf,1 ⊗ ϕf,2)

in D∗(XK)/(im∂ + im∂).

In Section 5, we introduce a Siegel Eisenstein series E(τ, λ(ϕf ), s) of genus r. Fornon-singular T and ϕf = ⊗v-∞ϕv, its Fourier coefficient ET (τ, λ(ϕf ), s) is given by anEuler product ET (τ, λ(ϕf ), s) =

∏vWT,v(τ, λv(ϕv), s), and hence its derivative can be


written as a sum

(1.16)d

dsET (τ, λ(ϕf ), s) =

∑v

E ′T (τ, λ(ϕf ), s)v

over all places of F . Since the arithmetic degree map deg can also be written asa similar sum, the conjectural identity (1.12) suggests comparing the terms at eachplace v. For v = σ1 we obtain the following result.

Theorem 5.1. Assume that XK is compact and attached to a group of the formO(p, 2) or U(p, 1) and that r ≤ p+ 1. Then

(−1)rικ0

2Vol(XK ,Ω)

∫[XK ]

g(T, y1, ϕf ) ∧ Ωp+1−rqT = E ′T (τ, λ(ϕf ), s0)σ1

if T is not totally positive definite and

(−1)rικ0

2Vol(XK ,Ω)

∫[XK ]

g(T, y1, ϕf ) ∧ Ωp+1−rqT

= E ′T (τ, λ(ϕf ), s0)σ1 − ET (τ, λ(ϕf ), s0)ι

2log detσ1(T )

if T is totally positive definite.

(See (3.49) and (5.19) for the values of ι and κ0). This result generalizes those in[29, 10] (the case r = 1), the main result in [23] (r = 2 for Shimura curves) and alsoLiu [33] (unitary Shimura varieties and r = p+ 1).

It seems worthwhile to point out that, in contrast with these works, the proof ofTheorem 5.1 does not proceed by computing both sides explicitly and comparingthem (using often difficult changes of variables). Instead, our proof is conceptual andrelies on understanding: the reducibility and K-types of principal series represen-tations of certain Lie groups; the correspondence of spherical harmonics under thetheta correspondence as described by Howe; results of Shimura giving estimates forWhittaker coefficients.

Now consider the case where T is possibly singular. Assuming again that theunderlying group is O(p, 2) or U(p, 1), in Section 4.4 we define a current g(T, y, ϕf )satisfying

(1.17) ddcg(T, y, ϕf ) + δZ(T,ϕf ) ∧ (−Ω)r−rk(T ) = ω(T, y, ϕf ),

where∑

T ω(T, y, ϕf )qT is the Fourier expansion of ΘKM(τ, ϕf ). The construction

involves regularizing the integral used when T is non-singular (of course, in thatcase this current specializes to the one discussed above). For example, when T isidentically zero, we show that

(1.18) g(0r, y, ϕf ) = − log det y · ϕf (0) · (−Ω)r−1 ∈ D∗(XK)/(im∂ + im∂).


More generally, if T =( τ

0r−r′

)with τ a non-singular r′-by-r′ matrix and ϕf =

ϕ′f ⊗ ϕ′′f , we show that

g(T, y, ϕf ) = ϕ′′f (0)[g(τ, y′, ϕ′f ) ∧ (−Ω)r−r

′

− log(det y′′)δZ(τ,ϕ′f ) ∧ (−Ω)r−r′−1]∈ D∗(XK)/im∂ + im∂.

(1.19)

Thus g(T, y, ϕf ) generalizes the currents considered in [30, §6.4, 6.5] for Shimuracurves. In forthcoming work, we will show that a result similar to Theorem 5.1 holdsfor these currents attached to singular T .

1.4. Notation and conventions. Let k be a field endowed with a (possibly trivial)involution a 7→ a. We write Symr(k) (resp. Herr(k)) for the group of symmetric(resp. hermitian) r-by-r matrices with coefficients in k under matrix addition. Fora ∈ GLr(k) and b ∈ Herr(k), let

(1.20) m(a) =

(a 00 ta−1

), n(b) =

(1r b0 1r

), wr =

(1r

−1r

).

For x = (x1, . . . , xr) ∈ kr, we write

(1.21) d(x) =

x1

. . .xr

.

We fix the following standard choice of additive character ψ = ψF : F → C× whenF is a local field of characteristic zero. If F = R we set ψ(x) = e2πix; if F = Qp

we choose ψ = ψQp so that ψ(p−1) = e−2πi/p; if F is a finite extension of Qv we setψF (x) = ψQv(trF/Qv(x)). If F is a global field, we write A×F for the ideles of F and setψF : F×\A×F → C×, where ψF = ⊗vψFv and the product runs over all places v of F .

We denote the connected component of the identity of a Lie group G by G0.We denote by A∗(X) (resp. D∗(X)) the space of differential forms (resp. currents)

on a smooth manifold X. Given α ∈ A∗(X) = ⊕k≥0Ak(X), we write α[k] for its

component of degree k. If α is closed, we write [α] ∈ H∗(X) for the cohomology classdefined by α.

If X is a complex manifold, we let dc = (4πi)−1(∂− ∂), so that ddc = (−2πi)−1∂∂.We denote by ∗ the operator on ⊕k≥0A

k,k(X) acting by multiplication by (−2πi)−k

on Ak,k(X). The canonical orientation on X induces an inclusion Ap,q(X) ⊂ Dp,q(X)sending a differential form ω to the current given by integration against ω on X. Wewill sometimes denote this current by [ω] (and will be careful to avoid confusion withthe notation for cohomology class described above) but often will still denote it by ω.

If f(s) is a meromorphic function of a complex variable s, we write CTs=0

f(s) for the

constant term of its Laurent expansion at s = 0.


2. Green forms on hermitian symmetric domains

Let X be a complex manifold and Z ⊂ X be a closed irreducible analytic subsetof codimension c. A Green current for Z is a current gZ ∈ Dc−1,c−1(X) such that

(2.1) ddcgZ + δZ = [ωZ ],

where δZ denotes the current of integration on Z and ωZ is a smooth differential formon X. A Green form is a Green current given by a form that is locally integrable onX and smooth on X − Z (cf. [6, §1.1]).

Here we will construct Green forms for certain complex submanifolds of the her-mitian symmetric space D attached to O(p, 2) or U(p, q), where p, q > 0. Throughoutthe paper we will refer to the case involving O(p, 2) as case 1 and to the case involvingU(p, q) as case 2. We would like to emphasize that our methods apply uniformly andall results in both cases are closely parallel.

There is a natural hermitian holomorphic vector bundle E over D, and the sub-manifolds of D that we consider are the zero loci Z(s) of certain natural holomorphicsections s ∈ H0(Er) (r ≥ 1). In this setting, the results in [4, 2, 3] (reviewed in Sec-tion 2.1) can be applied to construct some natural differential forms on D, includinga Green form gZ(s) for Z(s), as we show in Section 2.2. We give explicit examples inSection 2.3, showing that this construction recovers some differential forms consid-ered in previous work on special cycles (cf. [23, 24]), and then (Sections 2.4 and 2.5)we establish the main properties of these forms. The final Section 2.6 considers starproducts and will be used in the proof of Theorem 4.4.

2.1. Superconnections and characteristic forms of Koszul complexes. In thissection we review the construction of some characteristic differential forms attachedto a pair (E, u), where E is a holomorphic hermitian vector bundle and u is a holo-morphic section of its dual. Most of the results in this section are due to Bismut [4]and Bismut-Gillet-Soule [2, 3]. The exceptions are Theorem 2.1, which characterizesthe forms ξ0 that we will use later to construct Green forms for special cycles onShimura varieties, and the results in Section 2.6 leading to the rotation invariance ofstar products proved in Corollary 2.10.

We will use Quillen’s formalism of superconnections and related notions of super-algebra. For more details, we refer the reader to [36, 1]. We briefly recall that a supervector space V is just a complex Z/2Z-graded vector space; we write V = V0⊕V1 andrefer to V0 and V1 as the even and odd part of V respectively. We write τ for the endo-morphism of V determined by τ(v) = (−1)deg(v)v. The supertrace trs : End(V )→ Cis the linear form defined by

(2.2) trs(u) = tr(τu),

where tr denotes the usual trace. Thus if u = ( a bc d ) with a ∈ End(V0), d ∈ End(V1),b ∈ Hom(V1, V0) and c ∈ Hom(V0, V1), then trs(u) = tr(a)− tr(d).

2.1.1. Let E be a holomorphic vector bundle on a complex manifold X and u ∈H0(E∨) be a holomorphic section of its dual E∨. Let K(u) be the Koszul complex


of u: its underlying vector bundle is the exterior algebra ∧E and its differentialu : ∧kE → ∧k−1E is defined by

(2.3) u(e1 ∧ · · · ∧ ek) =∑

1≤i≤k

(−1)i+1u(ei)e1 ∧ · · · ∧ ei ∧ · · · ∧ ek.

The grading on K(u) is given by K(u)−k = ∧kE, so that K(u) is supported innon-positive degrees. We identify K(u) with the corresponding complex of sheaves ofsections of ∧kE, and say that u is a regular section if the cohomology of K(u) vanishesin negative degrees. If u is regular with zero locus Z(u), then K(u) is quasi-isomorphicto OZ(u) (regarded as a complex supported in degree zero).

2.1.2. Assume now that E is endowed with a hermitian metric ‖ · ‖E. This inducesa hermitian metric on ∧E: for any x ∈ X, ∧kEx is orthogonal to ∧jEx if j 6= k,and an orthonormal basis of ∧kEx is given by all elements ei1 ∧ · · · ∧ eik , where1 ≤ i1 < · · · < ik ≤ rkE and e1, . . . , erkE is an orthonormal basis of Ex. Let ∇ bethe corresponding Chern connection on ∧E. We regard ∧E as a super vector bundle,with even part ∧evenE and odd part ∧oddE, and u as an odd endomorphism of ∧E.Let u∗ be the adjoint of u and define the superconnection

(2.4) ∇u = ∇+ i√

2π(u+ u∗)

on ∧E. Then we have Quillen’s Chern form

(2.5) ϕ0(u) = ϕ0(E, ‖ · ‖E, u) = trs(e∇2u) ∈

⊕k≥0

Ak,k(X).

We recall some properties of ϕ0(u) established (in greater generality) by Quillen[36]. The form ϕ0(u) is closed and functorial: given a holomorphic map of complexmanifolds f : X ′ → X, consider the pullback bundle (f ∗E, f ∗‖ · ‖) and the pullbacksection f ∗u ∈ H0(f ∗E∨); then

(2.6) ϕ0(f ∗u) = f ∗ϕ0(u).

Let ∗ be the operator on ⊕k≥0Ak,k(X) acting by multiplication by (−2πi)−k on

Ak,k(X). Writing [ϕ0(u)∗] for the cohomology class of ϕ0(u)∗, we have

(2.7) [ϕ0(u)∗] = ch(∧E) = ch(∧evenE)− ch(∧oddE).

In particular, [ϕ0(u)∗] depends on E, but not on u. Thus the forms ϕ0(tu) for t ∈ R>0

all belong to the same cohomology class, but as t→ +∞ they concentrate on the zerolocus of u. More precisely, recall that ∇2

u is an even element of the (super)algebra

(2.8) A(X,End(∧E)) := A∗(X)⊗C∞(X)Γ(End(∧E)).

Given a relatively compact open subset U ⊂ X whose closure U is disjoint from Z(u)and a non-negative integer k, consider an algebra seminorm ‖·‖U,E,k onA(X,End(∧E))

measuring uniform convergence on U of partial derivatives of order at most k. Let‖ · ‖E∨ be the unique metric on E∨ such that the isomorphism E∨ ' E induced by


‖ · ‖E is an isometry. Quillen [36, §4] shows that there exist positive real numbers C,a such that

(2.9) ‖e∇2tu‖U,E,k ≤ Ce−2πat2 , t ∈ R>0.

Moreover, for a one may take any number strictly less than minx∈U‖u(x)‖E∨. Thusa similar bound holds for ϕ0(tu).

2.1.3. Instead of rescaling u 7→ tu, we may also rescale the metric ‖ · ‖E on E.Replacing ‖ · ‖E with s−1‖ · ‖E preserves the Chern connection ∇ but rescales u∗ tos2u∗ and hence replaces ∇u with ∇+ i

√2π(u+ s2u∗). More generally, given s, t ≥ 1,

consider the superconnection

(2.10) ∇s,tu := ∇+ i√

2π(tu+ ts2u∗).

We will also need an estimate for e∇2s,tu for large values of s, t. To obtain it, write

∇2s,tu = (∇2

s,tu)[0] + Rs,tu, where (∇2s,tu)[0] has form-degree zero and Rs,tu has form-

degree ≥ 1. Note that Rs,tu is nilpotent and that

(2.11) (∇2s,tu)[0] = −2π(tu+ ts2u∗)2 = −2πs2t2‖u‖2 ⊗ id ∈ A0(X)⊗ End(∧E).

In particular, (∇2s,tu)[0] and Rs,tu commute. Hence we have

e∇2s,tu = e(∇2

s,tu)[0]eRs,tu

= e−2πs2t2‖u‖2N∑k=0

1k!Rks,tu

(2.12)

with N ≤ dimRX. Let ‖ · ‖U,E,k be a seminorm as in 2.1.2 and a ∈ R>0 strictlyless than minx∈U‖u(x)‖E∨. Since Rs,tu is a polynomial in s, t, it follows from (2.12)

that we can find n ∈ N such that ‖e∇2s,tu‖U,E,k = O(e−2πas2t2(st)n) for s, t ≥ 1. Since

(st)n = O(eεs2t2) for any ε > 0, we conclude that for any positive a strictly less than

minx∈U‖u(x)‖E∨, there is a positive constant C such that

(2.13) ‖e∇2s,tu‖U,E,k ≤ Ce−2πas2t2 , s, t ≥ 1.

Thus a similar bound holds for ϕ0(E, s−1‖ · ‖E, tu); we say that ϕ0(E, s−1‖ · ‖E, tu)decreases rapidly as st→ +∞ in the C∞ topology of A∗(U).

2.1.4. It follows from (2.7) that the form ddtϕ0(t1/2u) (for t ∈ R>0) is exact, and one

can ask for a construction of a functorial transgression of this form. Bismut, Gilletand Soule [2] construct such a transgression, and the resulting form is key to ourresults. To define it, let N ∈ End(∧E) be the number operator acting on ∧kE bymultiplication by −k and set

(2.14) ν0(u) = trs(Ne∇2u) ∈

⊕k≥0

Ak,k(X).


Then ν0(u) is functorial with respect to holomorphic maps f : X ′ → X and satisfies([2, Thm. 1.15])

(2.15) − 1

t∂∂ν0(t1/2u) =

d

dtϕ0(t1/2u), t > 0.

2.1.5. Assume that the section u has no zeroes on X. Then the Koszul complexK(u) is acyclic and (2.7) shows that ϕ0(u) is exact. In this case one can define acharacteristic form ξ0(u) giving a ∂∂-transgression of ϕ0(u). Namely, let

(2.16) ξ0(u) =

∫ +∞

1

ν0(t1/2u)dt

t.

The bound (2.9) implies that all partial derivatives of ν0(t1/2u) decrease rapidly ast → +∞. In particular, the integral converges and defines a form in ⊕k≥0A

k,k(X);moreover, one can differentiate under the integral sign, so that by (2.15) we have

(2.17) ∂∂ξ0(u) = ϕ0(u).

The following theorem characterizes the forms ξ0(u). Note that its statement andproof are very similar to [39, Thm. IV.2] on Bott-Chern forms. The difference is thatthe form ξ0(u) transgresses Quillen’s Chern form ϕ0(u) instead of the Chern-Weil

form trs(e∇2

), and that instead of requiring that our transgression vanishes for splitcomplexes we impose a rapid decrease condition. For our purposes these conditionsare natural, as we want to define Green forms for cycles on arithmetic quotients Γ\Dby averaging natural forms on D over translates by γ ∈ Γ and the rapid decrease ofξ0(u) and ϕ0(u) as u grows guarantees the convergence of these series.

Given a relatively compact open subset U of X with closure U and a positive integerk, we denote by ‖ · ‖U,k any seminorm on A∗(X) measuring uniform convergence on

U of partial derivatives of order at most k.

Theorem 2.1. For a hermitian vector bundle (E, ‖ ·‖) on a complex manifold X anda nowhere vanishing section u ∈ H0(E∨), define

ξ0(u) = ξ0(E, ‖ · ‖, u) =

∫ +∞

1

ν0(t1/2u)dt

t.

Then ξ0(u) ∈ ⊕k≥0Ak,k(X) and satisfies the following properties:

(i) (Transgression) ∂∂ξ0(u) = trs(e∇2u).

(ii) (Functoriality) For any holomorphic map f : X ′ → X, we have ξ0(f ∗u) =f ∗ξ0(u).

(iii) For any seminorm ‖ · ‖U,k, there are positive constants a, C such that

‖ξ0(E, s−1‖ · ‖E, tu)‖U,k ≤ Ce−2πa(st)2 for all s, t ≥ 1.

Moreover, if (E, ‖ · ‖, u) 7→ ξ0(u) is another assignment as above satisfying (i)-(iii),then

ξ0(u)− ξ0(u) ∈ im∂ + im∂;


more precisely, we can find forms ηi(u) ∈ A∗(X) (i = 1, 2) satisfying

ξ0(u)− ξ0(u) = ∂η1(u) + ∂η2(u),

ηi(f∗u) = f ∗ηi(u) for any holomorphic map f : X ′ → X,

ηi(tu) decrease rapidly as t→ +∞ in the C∞ topology of A∗(X).

Proof. As remarked before the statement, estimate (2.9) shows that one can differen-tiate under the integral sign in (2.16), hence (i) follows from (2.15). Property (ii) isclear.

To prove (iii), let a ∈ R with 0 < a < minx∈U‖u(x)‖E∨. For s, t ≥ 1, estimate(2.13) gives

‖ξ0(E, s−1‖ · ‖E, tu)‖U,k ≤∫ +∞

1

‖ν0(E, s−1‖ · ‖E, t√t′u)‖U,k

dt′

t′

≤ C

∫ +∞

1

e−2πa(st)2t′dt′ = C(2πa(st)2)−1e−2πa(st)2(2.18)

for some positive constant C, and (iii) follows.For the uniqueness statement, the proof follows that of [39, Thm. IV.2]. Namely,

given a pair (E, u) on X and a point x ∈ X, denote the hermitian metric on Ex by‖ · ‖x. Let π : X × C → X be the canonical projection and consider the hermitianvector bundle E on X × C given by E = π∗E, with metric

(2.19) ‖ · ‖(x,z) := (1 + |z|2)−1‖ · ‖x, (x ∈ X, z ∈ C).

Let u = π∗u ∈ H0(E∨). The form ξ0(u) − ξ0(u) ∈ A∗(X × C×) is then ∂∂-closed by(i). Writing the differentials on X ×C as ∂ = ∂X + ∂z and ∂ = ∂X + ∂z, we find that

0 =

∫P1∂∂(ξ0(u)− ξ0(u)) log |z|2

=

∫P1∂z∂z(ξ

0(u)− ξ0(u)) log |z|2

+

∫P1

(∂X∂ − ∂X∂z)(ξ0(u)− ξ0(u)) log |z|2

= −2πi(ξ0(u)− ξ0(u))

+

∫P1

(∂X∂ − ∂X∂z)(ξ0(u)− ξ0(u)) log |z|2

(2.20)

where the last equation follows from the current identity ∂z∂z log |z|2 = −2πi(δ0−δ∞)

on P1 and the rapid decrease of ξ0(E, s−1‖ · ‖E, u) and ξ0(E, s−1‖ · ‖E, u) as s→ +∞.We set

η1(u) = (−2πi)−1

∫P1∂(ξ0(u)− ξ0(u)) log |z|2,

η2(u) = (−2πi)−1

∫P1∂z(ξ

0(u)− ξ0(u)) log |z|2.(2.21)


The integrals converge to smooth forms on X by (iii), which also shows that theforms ηi(tu) decrease rapidly as t→ +∞. We have ηi(f

∗u) = f ∗ηi(u) for i = 1, 2 andany holomorphic map f : X ′ → X, and (2.20) can be rewritten as

(2.22) ξ0(u)− ξ0(u) = ∂η1(u) + ∂η2(u).

This finishes the proof.

We remark that similar forms ϕ0 and ξ0 can be defined for arbitrary acyclic com-plexes of holomorphic hermitian vector bundles, and that the above theorem gener-alizes to this setting; we will not need this more general version.

2.1.6. We now go back to the general setting where we have a hermitian holomorphicvector bundle E on X and a section u ∈ H0(E∨), and now make the assumption thatu is regular (see 2.1.1) and the zero locus Z(u) is smooth; thus K(u) is a resolutionof OZ(u) and the codimension of Z(u) (if non-empty) in X equals rk(E). Define

g(u) = ξ0(u)∗[2rk(E)−2]

=(i

2π

)rk(E)−1∫ +∞

1

ν0(t1/2u)[2rk(E)−2]dt

t∈ Ark(E)−1,rk(E)−1(X).

(2.23)

The following proposition is contained in [3]. It shows that g(u) is a Green form forZ(u).

Proposition 2.2. (1) The integral (2.23) converges to a smooth differential formg(u) on X − Z(u).

(2) The form g(u) is locally integrable on X.(3) As currents on X we have

ddcg(u) + δZ(u) = ϕ0(u)∗[2rk(E)].

Proof. Part (1) has already been discussed in Section 2.1.5.Part (2) is shown to hold in the course of the proof of [3, Thm. 3.3]. Namely, recall

that in that paper one considers an immersion i : M ′ → M of complex manifolds, avector bundle η on M ′ and a complex (ξ, v) of holomorphic hermitian vector bundleson M that gives a resolution of i∗OM ′(η). Assume that Z(u) is non-empty. We setM = X, M ′ = Z(u), η = OZ(u), and for the complex (ξ, v) we take the Koszul complex

(∧E, u); with these choices, the form ν0(t1/2u) agrees with the form αt defined in [3,(3.12)].

Let z0 ∈ Z(u) and chose local coordinates z1, . . . , zN (N = dimX) around z0

such that Z(u) is defined by the equations z1 = · · · = zrk(E) = 0 and let |y| =

(|z1|2 + · · ·+ |zrk(E)|2)1/2. Locally around z0 we can write

(2.24) ν0(u)[2rk(E)−2] =

2rk(E)−2∑p=0

ν0(u)p,

where ν0(u)p corresponds to the part of ν0(u)[2rk(E)−2] whose degree in the variablesdzi, dzi (1 ≤ i ≤ rk(E)) is p. Since the complex normal bundle to Z(u) in X is


canonically identified with E, the equations [3, (3.24), (3.26)] show that, for any p,

(2.25) |y|2rk(E)−1

∫ +∞

1

ν0(t1/2u)pdu

u

is bounded in a neighborhood of z0; part (2) follows since |y|−(2rk(E)−1) is locallyintegrable around z0.

To prove (3), let β be a compactly supported form on X. Then∫X

g(u) ∧ ddcβ = lima→+∞

∫ a

1

∫X

ν0(t1/2u)∗[2rk(E)−2] ∧ ddcβdt

t

= lima→+∞

∫ a

1

∫X

ddcν0(t1/2u)∗[2rk(E)−2] ∧ βdt

t

= lima→+∞

∫ a

1

∫X

(−t ddtϕ0(t1/2u)∗[2rk(E)]) ∧ β

dt

t

=

∫X

ϕ0(u)∗[2rk(E)] ∧ β − lima→+∞

∫X

ϕ0(a1/2u)∗[2rk(E)] ∧ β.

(2.26)

Here the first equality follows from dominated convergence since the integrals

(2.27) |y|2rk(E)−1a

∫1ν0(t1/2u)p

du

u

for a ≥ 1 are uniformly bounded in a neighborhood of z0, as shown in the proofof [3, Thm. 3.3]. The third equality follows from (2.15). This establishes (3) sinceϕ0(a1/2u)∗[2rk(E)] approaches δZ(u) as a→ +∞, as shown by [3, (1.14), (1.19)].

2.2. Hermitian symmetric domains of orthogonal and unitary groups.

2.2.1. Let

(2.28) k =

R case 1C case 2

and

(2.29) σ =

id case 1

complex conjugation case 2.

Let m ≥ 2 be a positive integer and let V be a k-vector space of dimension m endowedwith a non-degenerate σ-Hermitian bilinear form Q of signature (p, q). We assumethat pq 6= 0 and in case 1 we further assume that q = 2. Let U(V ) := Aut(V,Q) bethe isometry group of Q and let

(2.30) G = U(V )0 ∼=

SO(p, 2)0 case 1U(p, q) case 2.

We fix an orthogonal decomposition V = V +⊕V − with dimk V+ = p and dimk V

− =q, and with V + and V − positive and negative definite respectively; in case 1 we alsofix an orientation of V −. Let K be the centralizer in G of the isometry of V acting


as the identity on V + and as −1 on V −. Then K is a maximal compact subgroup ofG, given by

(2.31) K =

SO(V +)× SO(V −) case 1

U(V +)× U(V −) case 2.

Let Gr(q, V ) be the Grassmannian of q-dimensional subspaces of V ; in case 1 weassume furthermore that these subspaces are oriented, so that in this case Gr(q, V )consists of two copies of the usual Grassmannian. Let

(2.32) D = z ∈ Gr(q, V )| Q|z < 0be the open subset of Gr(q, V ) consisting of negative definite subspaces. Then Dhas two connected components in case 1 and is connected in case 2. Let z0 ∈ D bethe point corresponding to V − and D+ be the connected component of D containingz0. Then G acts transitively on D+ and the stabilizer of z0 is K; thus D+ ' G/Kis the symmetric domain associated with G. In case 2 it is clear that D carries aU(V )-invariant complex structure; to see this in case 1, one can use the model

(2.33) D ' [v] ∈ P(V (C))|Q(v, v) = 0, Q(v, v) < 0.The correspondence between both models sends z ∈ D to the line [ez+ie

′z] ∈ P(V (C)),

where ez and e′z form an oriented orthogonal basis of z satisfying Q(ez, ez) = Q(e′z, e′z).

2.2.2. Let E be the tautological bundle on D, whose fiber over z ∈ D is the subspacez ⊂ V . Thus E is a holomorphic line bundle in case 1 (for it corresponds to thepullback of OP(V (C))(−1) under the isomorphism (2.33)), and a holomorphic vectorbundle of rank q in case 2. It carries a natural hermitian metric hE defined by

(2.34) hE(vz) =

−Q(vz, vz) case 1−Q(vz, vz) case 2

for vz ∈ Ez = z. This metric is equivariant for the natural U(V )-equivariant structureon E. We denote by ∇E the corresponding Chern connection on E and by

(2.35) Ω = ΩE := ctop(E,∇E)∗ =(i

2π

)rkEdet(∇2

E) ∈ ArkE,rkE(D)

its Chern-Weil form of top degree.We denote by ΩD the Kahler form

(2.36) ΩD = ∂∂ log kD(z, z),

where kD is the Bergmann kernel function of D. As shown in [42, p. 219], the invariantform − i

2πΩD agrees with the first Chern form c1(Ωtop

X ) of the canonical bundle ΩtopX

on any quotient X = Γ\D+ by a discrete torsion free subgroup Γ ⊂ G. When E hasrank one, the canonical bundle on D is naturally isomorphic to E⊗p in case 1 (as anapplication of the adjunction formula shows) and to E⊗(p+1) in case 2, and so in bothcases − i

2πΩD is a positive integral multiple of ΩE.

An element v ∈ V defines a global holomorphic section sv of E∨: for v′z ∈ Ez, wedefine

(2.37) sv(v′z) = Q(v′z, v).


Let Dv be the zero locus of sv on D and set D+v = Dv ∩ D+. We have

(2.38) Dv = z ∈ D|v ⊥ z

and so Dv is non-empty only if Q(v, v) > 0 or v = 0. Assume that Q(v, v) > 0,so that the orthogonal complement v⊥ of v has signature (p − 1, q). Writing Gv

for the stabilizer of v in G, we find that G0v ' Aut(v⊥, Q)0 acts transitively on D+

v

with stabilizers isomorphic to SO(p − 1) × SO(q) in case 1 and to U(p − 1) × U(q)in case 2. Thus D+

v is the symmetric domain attached to G0v. We conclude that

codimDDv = rkE∨ and hence that sv is a regular section of E∨. In fact, this showsthat sv is regular whenever v 6= 0, since in the remaining case we have Q(v, v) ≤ 0and so sv does not vanish on D.

More generally, given a positive integer r and a vector v = (v1, . . . , vr) ∈ V r, thereis a holomorphic section sv = (sv1 , . . . , svr) of (E∨)r, with zero locus

(2.39) Dv := Z(sv) = ∩1≤i≤rDvi .

Note that Dv depends only on the span 〈v1, . . . , vr〉. It is non-empty if and only if〈v1, . . . , vr〉 is a a positive definite subpace of (V,Q) of positive dimension; in thatcase, its (complex) codimension in D is rk(E)·dimk〈v1, . . . , vr〉. We set D+

v = Dv∩D+.

2.2.3. We will now specialize the constructions in Section 2.1 to the setting ofhermitian symmetric domains. Namely, recall the definitions of (V,Q), D, E andsv ∈ H0(E∨) in 2.2.1 and 2.2.2.

Let r be a positive integer and v = (v1, . . . , vr) be an r-tuple of vectors in V .We write K(v) := K(sv) for the Koszul complex associated with the section sv =(sv1 , . . . , svr) of (Er)∨. On its underlying vector bundle ∧(Er), we consider the su-perconnection

(2.40) ∇v := ∇sv = ∇+ i√

2π(sv + s∗v),

and we define forms

ϕ0(v) := ϕ0(sv)∗ =∑k≥0

(i

2π

)ktrs(e

∇2v)[2k],

ν0(v) := ν0(sv)∗ =∑k≥0

(i

2π

)ktrs(Ne

∇2v)[2k],

(2.41)

where N is the number operator on ∧(Er) acting on ∧k(Er) by multiplication by −k.

Definition 2.3. For v = (v1, . . . , vr) ∈ V r, write Q(v,v) = Q(v1, v1)+ · · ·+Q(vr, vr)and define

ϕ(v) = e−πQ(v,v)ϕ0(v),

ν(v) = e−πQ(v,v)ν0(v).


Thus ϕ(v) and ν(v) belong to ⊕k≥0Ak,k(D). If sv is a regular section of (Er)∨, define

ξ0(v) =

∫ +∞

1

ν0(t1/2v)dt

t∈⊕k≥0

Ak,k(D− Dv),

g(v) = ξ0(v)[2r·rk(E)−2],

ξ(v) = e−πQ(v,v)ξ0(v).

The forms ϕ(v) and ν(v) were already defined and studied in the setting of generalperiod domains in [11]. As to the form g(v), note that the section sv is regular ifdimk〈v1, . . . , vr〉 = r; in this case we say that v is non-degenerate. For non-degeneratev, Proposition 2.2 shows that g(v) is smooth on D−Dv, locally integrable on D and,as a current, satisfies

(2.42) ddcg(v) + δDv = ϕ0(v)[2r·rk(E)],

so that g(v) is a Green form for Dv.

2.3. Explicit formulas for O(p, 2) and U(p, 1). Let us give some explicit formulaswhen the tautological bundle E is a line bundle. Thus V is a either a real vectorspace of signature (p, 2) (case 1) or a complex vector space of signature (p, 1) (case2). There is a unique hermitian metric on E∨ making the isomorphism E ∼= E∨

induced by hE an isometry; we denote this metric by h and its Chern connection by∇E∨ . For v ∈ V we have

ϕ(v)[2] = e−π(Q(v,v)+2h(sv))

(i∂h(sv) ∧ ∂h(sv)

h(sv)− ΩE

),

ϕ(v) = ϕ(v)[2] ∧ Td−1(E∨,∇)∗,

(2.43)

where

(2.44) Td−1(E∨,∇) = det

(1− e−∇2

E∨

∇2E∨

)denotes the inverse Todd form of (E∨,∇E∨). This is a special case of the Mathai-Quillen formula [34, Thm. 8.5]; see also [11, §3] for a proof in our setting, where itis also shown that the form ϕ(v)[2] coincides with the form ϕKM(v) defined by Kudlaand Millson in [24].

Let us now consider the form ν(v) for v ∈ V . Here the Koszul complex K(v) has

just two terms: K(v) = (Esv−→ OD), and the operator N acts by zero on OD and by

−1 on E. For the component of degree zero of ν0(v) we obtain

ν0(v)[0] = trs(Ne∇2v)[0]

= trs(Ne(∇2

v)[0])

= trs(Ne−2πh(sv))

= e−2πh(sv).

(2.45)


Given z ∈ D, let z⊥ be the orthogonal complement of z in V , so that V = z⊕ z⊥; wewrite vz and vz⊥ for the orthogonal projection of v ∈ V to z and z⊥ respectively. LetQz be the (positive definite) Siegel majorant of Q defined by

(2.46) Qz(v, v) = Q(vz⊥ , vz⊥)−Q(vz, vz).

Then we have Qz(v, v) = Q(v, v) + 2hz(sv) and we conclude that ν(v)[0] is just theSiegel gaussian:

(2.47) ν(v)[0] = ϕSG(v) := e−πQz(v,v).

We also compute that, for v 6= 0,

ξ0(v)[0] =

∫ +∞

1

ν0(t1/2v)[0]dt

t

=

∫ +∞

1

e−2πth(sv)dt

t

= −Ei(−2πh(sv)),

(2.48)

where Ei(−z) = −∫∞1 e−zt dtt

denotes the exponential integral. Thus ξ0(v)[0] coincideswith the Green form defined in [23, (11.24)] in case 1 when p = 1.

2.4. Basic properties of the forms ϕ and ν.

2.4.1. We will first give formulas for the restriction of ϕ and ν to special cycles. Letw ∈ V with Q(w,w) > 0 and recall the group Gw and complex submanifold D+

w ⊂ D+

defined in 2.2.2. Then Gw is identified with the isometry group of (w⊥, Q), and wemay identify D+

w with the hermitian symmetric domain attached to G0w. We write

ϕDw(v′) and νDw(v′) for the forms on Dw given in Definition 2.3.

Lemma 2.4. Let v, w ∈ V with Q(w,w) > 0 and write v = v′ + v′′ with v′ ∈ w⊥ andv′′ ∈ 〈w〉. Then

ν(v)|Dw = e−πQ(v′′,v′′)νDw(v′),

ϕ(v)|Dw = e−πQ(v′′,v′′)ϕDw(v′).

Proof. Let Ew be the tautological bundle on Dw whose fiber over z ∈ Dw is z ⊂ w⊥.The restriction of E to Dw is isometric to Ew. For any v ∈ V , this isometry inducesan isomorphism K(v)|Dw ∼= K(v′), where K(v′) denotes the Koszul complex withunderlying vector bundle ∧Ew and differential sv′ . Thus ∇v|Dw = ∇v′ and the lemmafollows.

2.4.2. The proof of the next proposition is a straightforward consequence of generalproperties of Koszul complexes and Chern forms.

Proposition 2.5. Let r ≥ 1 and v = (v1, . . . , vr) ∈ V r. Then:

(a) ϕ(v) = ϕ(v1) ∧ . . . ∧ ϕ(vr).(b) ϕ(v) is closed.(c) ϕ(v)[k] = 0 if k < 2r · rk(E).(d) For every g ∈ U(V ), we have g∗ϕ(gv1, . . . , gvr) = ϕ(v1, . . . , vr).


(e) ϕ(0) = ctop(E∨,∇)∗ ∧ Td−1(E∨,∇)∗ (here we assume r = 1).(f) Let

h ∈

O(r), case 1U(r), case 2.

Then ϕ((v1, . . . , vr) · h) = ϕ(v1, . . . , vr).

Proof. Except for (c), which follows from (a) and the Mathai-Quillen formula [34,Thm. 8.5], all statements are proved in [11, Prop. 2.3 and (2.17)] in case 1, and theproof there extends without modification to case 2.

Consider now the form ν. We write

(2.49) c(F,∇F ) = det(t∇2F + 1rk(F )) = 1 + c1(F,∇F )t+ . . .+ crk(F )(F,∇F )trk(F )

for the total Chern-Weil form of a vector bundle F with connection ∇F .

Proposition 2.6. Let r ≥ 1 and v = (v1, . . . , vr) ∈ V r.

(a) We have ν(v) =∑

1≤i≤r νi(v), where

νi(v) := ν(vi) ∧ ϕ(v1, . . . , vi, . . . , vr).

(b) For t > 0:

ddcν0(t1/2v) = −t ddtϕ0(t1/2v).

(c) ν(v)[k] = 0 if k < 2r · rk(E)− 2.(d) For any g ∈ U(V ), we have g∗ν(gv1, . . . , gvr) = ν(v1, . . . , vr).(e) Assume that r = 1. Then

ν(0)[2rk(E)−2] = crk(E)−1(E∨,∇)∗.

(In particular, ν(0)[0] = 1 when E has rank one.)(f) Let

h ∈

O(r), case 1U(r), case 2.

Then ν((v1, . . . , vr) · h) = ν(v1, . . . , vr).

Proof. Recall that ν(v) = e−πQ(v,v)trs(Ne∇2

v), where N is the number operator onthe Koszul complex K(v) ' ⊗1≤i≤rK(vi). Letting Ni be the number operator onK(vi), we can write N = N1 + · · ·+Nr and ∇2

v = ∇2v1

+ · · ·+∇2vr , where [Ni,∇2

vj] =

[∇2vi,∇2

vj] = 0 for i 6= j; this proves (a).

Part (b) follows from (2.15). By part (a) and Proposition 2.5.(c), it suffices to prove(c) when r = 1. Then (c) is vacuously true if rk(E) = 1, and in general it followsfrom [5, (3.72),(3.35), Thm. 3.10].

For any v and any g ∈ U(V ), the U(V )-equivariant structure on E induces anisomorphism g∗K(gv) ' K(v) preserving the metric; this proves (d).


Part (e) follows from the explicit computation (2.47) when E has rank one. In thegeneral case, by taking u = 0 in [5, (3.35)] we find that

trs(Ne∇2

)∗[2rk(E)−2] = − d

dbdet(i

2π∇2E∨ − b1rk(E)

)∣∣b=0

= (−1)r−1 d

dbdet(i

2π∇2E + b1rk(E)

)∣∣b=0

= (−1)r−1crk(E)−1(E,∇)∗

= crk(E)−1(E∨,∇)∗

(2.50)

(note that the number operator N in op. cit. has sign opposite from ours).To prove (f), note that h induces an isometry

(2.51) K(v1, . . . , vr)i(h)−−→ K((v1, . . . , vr) · h)

that commutes with N and such that ∇(v1,...,vr)·h = i(h)−1∇(v1,...,vr)i(h). Thus (f)follows from the conjugation invariance of trs.

The properties of ϕ(v) and ν(v) established in Propositions 2.5 and 2.6 also holdfor ϕ0(v) and ν0(v). In contrast, the next result, showing that the forms ϕ and ν arerapidly decreasing as functions of v, really requires the additional factor e−πQ(v,v) tohold (note for example that the restriction of ϕ0(tv) to Dv is independent of t ∈ R).

Let S(V r) be the Schwartz space of complex-valued functions on V r all whosederivatives are rapidly decreasing.

Lemma 2.7. For fixed z ∈ D and r ≥ 1, we have

ϕ(·, z), ν(·, z) ∈ S(V r)⊗ ∧T ∗zD.

Proof. By Proposition 2.5.(a) and Proposition 2.6.(a), we may assume that r = 1.Recall that the quadratic form Qz(v) = 1

2Q(v, v) + hz(sv) on V is positive definite.

Write ∇2v(z) = (∇2

v)[0](z) + S(v, z). By Duhamel’s formula ([39, p. 144]) we have

e−πQ(v,v)e∇2v(z) = e−2πQz(v)

+∑k≥1

(−1)k∫

∆k

e−2π(1−tk)Qz(v)S(v, z)e−2π(tk−tk−1)Qz(v) · · ·S(v, z)e−2πt1Qz(v)dt1 · · · dtk.

(2.52)

Here ∆k = (t1, . . . , tk) ∈ Rk|0 ≤ t1 ≤ . . . ≤ tk ≤ 1 is the k-simplex and the sumis finite since S(v, z) has positive degree. Let ‖ · ‖U,k be an algebra seminorm as in2.1.2 and let QU a positive definite quadratic form on V such that QU < Qz for allz ∈ U . Then

(2.53) ‖e−2πQz(v)‖U,k < Ce−2πQU (v), v ∈ V,

for some positive constant C. Since S(v, z) grows linearly with v, (2.52) implies that

(2.54) ‖e−πQ(v,v)e∇2v(z)‖U,k < Ce−2πQU (v), v ∈ V,


(with different C) and hence that the same bound holds for ‖φ(v)‖U,k where φ(v) isany derivative of ϕ or ν in the v variable.

2.5. Behaviour under the Weil representation.

2.5.1. Let r be a positive integer. We write 0 and 1r for the identically zero andidentity r-by-r matrices respectively. Consider the vector space Wr := k

r endowedwith the σ-skew-Hermitian form determined by

(2.55)

(0 1r−1r 0

).

Its isometry group is the symplectic group

(2.56) Sp2r(R) =g ∈ GL2r(R)

∣∣g ( 0 1r−1r 0

)tg =

(0 1r−1r 0

)in case 1 and the quasi-split unitary group

(2.57) U(r, r) =g ∈ GL2r(C)

∣∣g ( 0 1r−1r 0

)tg =

(0 1r−1r 0

)in case 2. We denote by Mp2r(R) the metaplectic double cover of Sp2r(R), and identifyit (as a set) with Sp2r(R)× ±1 as in [37]. Define

(2.58) G′r =

Mp2r(R), case 1U(r, r), case 2.

Let Nr and Mr be the subgroups of G′r given by

Nr = (n(b), 1)|b ∈ Symr(R)(2.59)

Mr = (m(a), ε) |a ∈ GLr(R), ε = ±1(2.60)

in case 1 and by

Nr = n(b)|b ∈ Herr ,Mr = m(a)|a ∈ GLr(C)

(2.61)

in case 2. We also fix a maximal compact subgroup K ′r of G′r as follows. In case 1we let K ′r be the inverse image under the metaplectic cover of the standard maximalcompact subgroup

(2.62)

(a −bb a

)∣∣∣∣ a+ ib ∈ U(r)

∼= U(r)

of Sp2r(R). In case 2 we define K ′r = G′r∩U(2r). Thus in this case K ′r ' U(r)×U(r);an explicit isomorphism is

U(r)× U(r)'−→ K ′r

(k1, k2) 7→ [k1, k2] :=1

2

(k1 + k2 −ik1 + ik2

ik1 − ik2 k1 + k2

).

(2.63)


2.5.2. In case 2, we fix a character χ = χV of C× such that χ|R× = sgn(·)m. ThenG′r × U(V ) acts on S(V r) via the Weil representation

(2.64) ω =

ωψ, case 1ωψ,χ, case 2

(see [14, §1]). Here the action of U(V ) is in both cases given by

(2.65) ω(g)φ(v) = φ(g−1v), g ∈ U(V ), φ ∈ S(V r).

To describe the action of G′r, we write

m(a) =

(m(a), 1), for a ∈ GLr(R) in case 1,

m(a), for a ∈ GLr(C) in case 2,

n(b) =

(n(b), 1), for b ∈ Symr(R) in case 1,

n(b), for b ∈ Herr in case 2,

wr =

(wr, 1), case 1,

wr, case 2.

(2.66)

We write | · |k for the normalized absolute value on k; thus |z|R = |z| and |z|C = zz.Then we have (see [21] and also [16, §4.2] for explicit formulas for χψ and γV r):

ω(m(a))φ(v) = | det a|m/2k

φ(v · a)

·χψ(det a) case 1χ(det a) case 2,

ω(n(b))φ(v) = ψ(tr(bT (v)))φ(v),

ω(wr)φ(v) = γV r φ(v).

(2.67)

Here for v = (v1, . . . , vr) we define T (v) = 12(Q(vi, vj)) and φ(v) denotes the Fourier

transform

(2.68) φ(v) =

∫V rφ(w)ψ(1

2trC/Rtr(Q(v,w)))dw,

where dw is the self-dual Haar measure on V r with respect to ψ.

2.5.3. For our purposes it is crucial to understand the action of K ′r on the Schwartzforms ϕ and ν. This was done for the form ϕ in [24], where it is shown that ϕspans a one-dimensional representation of K ′r. We will show that ν also generates anirreducible representation of K ′r. To describe it we next recall the parametrization ofirreducible representations of K ′r by highest weights; we denote by πλ the (unique upto isomorphism) representation with highest weight λ.

Consider first case 1. Then K ′r is a double cover of U(r) and its irreducible repre-sentations are πλ with

(2.69) λ = (l1, . . . , lr) ∈ Zr ∪ (12

+ Z)r, l1 ≥ · · · ≥ lr.

For an integer k, we write detk/2 for the character of K ′r whose square factors throughU(r) and defines the k-th power of the usual determinant character det : U(r)→ C×.


Kudla and Millson show that

(2.70) ω(k′)ϕ = det(k′)m/2ϕ, k′ ∈ K ′r,

and so ϕ affords the one-dimensional representation πl of K ′r with highest weight

(2.71) l := m2

(1, . . . , 1).

Now consider case 2. Using the isomorphism (2.63) we can write uniquely (upto isomorphism) any irreducible representation of K ′r as πλ1 πλ2 for a unique pairλ = (λ1, λ2) of dominant weights of U(r). Let k(χ) be the unique integer such that

χ(z) = (z/|z|)k(χ), and note that k(χ) and m have the same parity. Kudla and Millsonshow that

(2.72) ω([k1, k2])ϕ = (det k1)(m+k(χ))/2(det k2)(−m+k(χ))/2ϕ, k1, k2 ∈ U(r),

and so ϕ affords the one-dimensional representation πl of K ′r with highest weight

(2.73) l := (m+k(χ)2

(1, . . . , 1), −m+k(χ)2

(1, . . . , 1)).

We will now determine the K ′r representation generated by the Schwartz formν(v)[2r−2]. Recall that by Proposition 2.6 we can write

(2.74) ν(v) =∑

1≤i≤r

νi(v),

where

(2.75) νi(v) = e−πQ(vi,vi)trs(Nie∇2vi ) ∧ ϕ(v1, . . . , vi, . . . , vr).

Let εr = 1r and, for 1 ≤ i < r, set

(2.76) εi =

1i−1

0 11r−i−1

−1 0

∈ SO(r).

Then

(2.77) νi(v) = ω(m(εi))νr(v), 1 ≤ i ≤ r;

thus ν(v)[2r−2] belongs to the K ′r representation generated by νr(v)[2r−2].

Lemma 2.8. Let r ≥ 1 and assume that rk(E) = 1. Under the action of K ′r via theWeil representation, the form ν(v)[2r−2] generates an irreducible representation πλ0with highest weight

λ0 := (m2, . . . , m

2, m

2− 2)

in case 1 and

λ0 := ((m+k(χ)2

, . . . , m+k(χ)2

, m+k(χ)2− 1), (−m+k(χ)

2, . . . , −m+k(χ)

2, −m+k(χ)

2+ 1))

in case 2. In both cases the form νr(v)[2r−2] is a highest weight vector in πλ0.


Proof. Assume first that r = 1. By (2.47), ν(z)[0] is the Siegel gaussian, which hasweight

(2.78) λ0 = p−q2

= m2− 2

in case 1 and

(2.79) λ0 = (p−q+k(χ)2

, q−p+k(χ)2

) = (m+k(χ)2− 1, −m+k(χ)

2+ 1)

in case 2. Now assume that r > 1. By Proposition 2.5.(c), we have

(2.80) νr(v)[2r−2] = ν(vr)[0] · ϕ(v1, . . . , vr−1)[2r−2]

and hence by (2.70), (2.72), (2.78) and (2.79), the form νr(v)[2r−2] has weight λ0.To show that νr(v)[2r − 2] is a highest weight vector, one needs to check thatω(α)νr(v)[2r−2] = 0 for every compact positive root α ∈ ∆+

c (see (3.5) and (3.28)).This can be done by a direct computation using (2.67) and the explicit formulas(2.47) and (2.43) for ν[0] and ϕ, or alternatively as follows. Evaluating at z0, (2.47)and (2.43) show that

(2.81) νr(v, z0) ∈ (S(V r)⊗ ∧p∗)K ;

here S(V r) ⊂ S(V r) is the subspace spanned by functions of the form e−πQz0 (v,v)p(v),where p(v) is a polynomial on V r. For νr(v, z0), the degree of these polynomials(called the Howe degree) is 2r − 2. Now it follows from the formulas in [18, §III.6](see also [13, Prop. 4.2.1] in case 1) that the only K ′r representation containing theweight λ0 realized in S(V r) in Howe degree 2r−2 is πλ0 , and the statement follows.

2.6. Star products. Let k, l be positive integers with k + l ≤ dim(D) + 1. Given anon-degenerate k-tuple v = (v1, . . . , vk) ∈ V k, we have defined a form ξ0(v) such thatg(v) = ξ0(v)[2k−2] is a Green form for the cycle Dv (see Definition 2.3 and (2.42)).

We now consider star products of these currents. Fix tuples v′ = (v′1, . . . , v′k) and

v′′ = (v′′1 , . . . , v′′l ) ∈ V l such that the tuple

(2.82) v = (v1, . . . , vk+l) := (v′1, . . . , v′k, v′′1 , . . . , v

′′l ) ∈ V k+l

is non-degenerate (note that this implies that v′ and v′′ are both non-degenerate).We define

(2.83) g(v′) ∗ g(v′′) = g(v′) ∧ δDv′′+ ϕ0(v′)[2rk(E)k] ∧ g(v′′) ∈ D2(k+l)rk(E)−2(D).

Theorem 2.9. There exist currents α(v′,v′′) ∈ D(k+l)rk(E)−2,(k+l)rk(E)−1(D) and β(v′,v′′) ∈D(k+l)rk(E)−1,(k+l)rk(E)−2(D) such that g∗α(v′,v′′) = α(gv′, gv′′), g∗β(v′,v′′) = β(gv′, gv′′)for g ∈ G and satisfying

g(v′) ∗ g(v′′)− g(v) = ∂α(v′,v′′) + ∂β(v′,v′′).

Proof. We abbreviate r = rk(E). Let π : D × Rk+l>0 → D be the canonical projection

and denote by t1, . . . , tk+l the coordinates on Rk+l>0 . Consider the super vector bundle


π∗∧(Ek+l) on D× Rk+l>0 , endowed with the superconnection ∇t1v1,...,tk+lvk+l . Consider

the form ν0(v) ∈ A∗(D× Rk+l>0 ) defined by

(2.84)

ν0(v) =∑

1≤i≤k+l

ν0(t1/2i vi)[2r−2] ∧ ϕ0(t

1/21 v1, . . . , t

1/2i vi, . . . , t

1/2k+lvk+l)[2(k+l−1)r] ∧

dtiti.

For a piecewise smooth path γ : I → Rk+l>0 (I ⊂ R a closed interval) and α ∈

A∗(D× Rk+l>0 ), let

(2.85)

∫γ

α ∈ A∗−1(D)

be the form obtained by integrating (id × γ)∗α along the fibers of the projection ofD× I → D. Fix a real number M > 1 and consider the paths

γd,M(t) = (t, . . . , t),

γ′k,M(t) = (1, . . . , 1︸︷︷︸k

, t, . . . , t), γ′′k,M(t) = (t, . . . , t︸︷︷︸k

,M, . . . ,M),

γk,M = γ′k,M t γ′′k,M ,

(2.86)

with t ∈ [1,M ]. Let

(2.87) ∆k,M = (t1, . . . , t1︸︷︷︸k

, t2, . . . , t2)|1 ≤ t1 ≤ t2 ≤M ⊂ Rk+l>0 ,

oriented so that ∂∆k,M = γd,M − γk,M . By Proposition 2.6.(a), we have

(2.88)

∫γd,M

ν0(v) =

∫ M

1

ν0(t1/2v)[2(k+l)r−2]dt

t;

this approaches g(v) as M →∞. Next, note that

(id× γ′k,M)∗ν0(v) = ν0(t1/2v′′)[2lr−2] ∧ ϕ0(v′)[2kr] ∧dt

t,

(id× γ′′k,M)∗ν0(v) = ν0(t1/2v′)[2kr−2] ∧ ϕ0(M1/2v′′)[2lr] ∧dt

t.

(2.89)

By [4, Thm. 3.2], as M →∞ we have ϕ0(M1/2v′′)[2lr] → δDv′′as currents, and hence

(2.90) limM→∞

∫γk,M

ν0(v) = g(v′) ∗ g(v′′).

Let d = d1 + d2 be the differential on D × Rk+l>0 , where d1 = ∂ + ∂ is the differential

on D and d2 is the differential on Rp+1>0 . Then we have

(2.91)

∫γd,M

ν0(v)−∫γk,M

ν0(v) =

∫∆k,M

d2ν0(v).


The restriction of ν0(v) to D×∆k,M is

ν0(v)|D×∆k,M= ν0(t

1/21 v′)[2kr−2] ∧ ϕ0(t

1/22 v′′)[2lr] ∧

dt1t1

+ ϕ0(t1/21 v′)[2kr] ∧ ν0(t

1/22 v′′)[2lr−2]

dt2t2.

(2.92)

Applying Proposition 2.6.(b) we obtain

d2ν0(v)|D×∆k,M

=

(t1d

dt1ϕ0(t

1/21 v′)[2kr] ∧ ν0(t

1/22 v′′)[2lr−2]

−ν0(t1/21 v′)[2kr−2] ∧ t2

d

dt2ϕ0(t

1/22 v′′)[2lr]

)∧ dt1dt2

t1t2

= (−2πi)−1(−(∂∂ν0(t

1/21 v′))[2kr] ∧ ν0(t

1/22 v′′)[2lr−2]

+ν0(t1/21 v′)[2kr−2] ∧ (∂∂ν0(t

1/22 v′′))[2lr]

)∧ dt1dt2

t1t2

= ∂α(t1, t2,v′,v′′) + ∂β(t1, t2,v

′,v′′),

(2.93)

with

α(t1, t2,v′,v′′) = (−2πi)−1ν0(t

1/21 v′)[2kr−2] ∧ ∂(ν0(t

1/22 v′′)[2lr−2]) ∧

dt1dt2t1t2

,

β(t1, t2,v′,v′′) = (−2πi)−1∂(ν0(t

1/21 v′)[2kr−2]) ∧ ν0(t

1/22 v′′)[2lr−2] ∧

dt1dt2t1t2

.

(2.94)

We set

α(v′,v′′) =

∫∆k,+∞

α(t1, t2,v′,v′′),

β(v′,v′′) =

∫∆k,+∞

β(t1, t2,v′,v′′).

(2.95)

The estimate (2.9) shows that the integrals converge to smooth forms on D− (Dv′ ∪Dv′′). These forms are locally integrable on D by Proposition 2.2 (note that Dv′ ∩Dv′′ = ∅ since v is non-degenerate) and so we can regard α(v′,v′′) and β(v′,v′′) ascurrents. The statement follows by taking the limit as M → +∞ in (2.91), and theequivariance property under g ∈ G follows from Proposition 2.6.(d).

As a corollary, we obtain the following invariance property of star products.

Corollary 2.10. Let h ∈ O(k+l) (case 1) or h ∈ U(k+l) (case 2) and v = (v′,v′′) ∈V k+l be non-degenerate. Let v′h, v′′h be defined by v · h = (v′h,v

′′h) and set

α(h; v′,v′′) = α(v′,v′′)− α(v′h,v′′h), β(h; v′,v′′) = β(v′,v′′)− β(v′h,v

′′h)

Then

g(v′) ∗ g(v′′)− g(v′h) ∗ g(v′′h) = ∂α(h; v′,v′′) + ∂β(h; v′,v′′) ∈ D2(k+l)rk(E)−2(D).


Proof. This is a consequence of the theorem and the invariance property ξ0(v · h) =ξ0(v), which follows from Proposition 2.6.(f).

When G = SO(1, 2)0, this invariance property is one of the main results in [23],where it is shown to hold by a long explicit computation (see also [33] for a similarproof in arbitrary dimension in case 2 when q = 1 and k + l = p+ 1). Our corollary,and its global counterpart, Corollary 5.2 below, generalizes this to G = SO(p, 2)0

(p ≥ 1) and gives a conceptual proof covering both cases.

3. Archimedean heights and derivatives of Whittaker functionals

Let r ≤ p + 1 and v = (v1, . . . , vr) ∈ V r be non-degenerate (recall that by this wemean that v1, . . . , vr are linearly independent) and denote by StabG〈v1, . . . , vr〉 thepointwise stabilizer of 〈v1, . . . , vr〉 in G. Let ξ(v) be the form given in Definition 2.3.We will next, assuming that rk(E) = 1, compute the integral

(3.1)

∫Γv\D+

ξ(v) ∧ Ωp−r+1,

where Γv is a discrete subgroup of finite covolume in StabG〈v1, . . . , vr〉, under someadditional conditions ensuring that it converges. The result is stated in Theorem 3.10and relates this integral to the derivative of a Whittaker functional defined on adegenerate principal series representation of G′r. Despite the length of this section,its proof is conceptually simple and follows easily from Lemma 2.8 determining theweight of ν(v)[2r−2], together with results in [28, 31] concerning reducibility of theserepresentations and multiplicity one for their K-types (we review these results inSections 3.1 and 3.2) and estimates of Shimura [38] for Whittaker functionals that wereview in Section 3.3.

3.1. Degenerate principal series of Mp2r(R). In this section we fix r ≥ 1 and letG′ = Mp2r(R). We abbreviate N = Nr, M = Mr and K ′ = K ′r (see 2.5.1).

3.1.1. We denote by g′ the complexified Lie algebra of G′. We have the Harish-Chandra decomposition

(3.2) g′ = p+ ⊕ p− ⊕ k′,

with

k′ =

(X1 X2

−X2 X1

)∣∣∣∣X t1 = −X1, X

t2 = X2

,

p+ =

p+(X) =

1

2

(X iXiX −X

)∣∣∣∣X t = X

, p− = p+.

(3.3)

Let K ′ the maximal compact subgroup of G′ with Lie(K ′)C = k′; it is the inverseimage under the metaplectic cover of the standard maximal compact U(r) of Sp2r(R)in (2.62). For x = (x1, . . . , xr) ∈ Cr, we write

(3.4) h(x) =

(−id(x)

id(x)

)


and define ej(h(x)) = xj. Then h′ = h(x)|x ∈ Cr is a Cartan subalgebra of k′, andwe choose the set of positive roots ∆+ = ∆+

c t∆+nc given by

∆+c = ei − ej|1 ≤ i < j ≤ r,

∆+nc = ei + ej|1 ≤ i ≤ j ≤ r,

(3.5)

where ∆+c and ∆+

nc denote the compact and non-compact roots respectively.

3.1.2. The group P = MN is a maximal parabolic subgroup of G′, the inverse imageunder the covering map of the standard Siegel parabolic of Sp2r(R). The group Mhas a character of order four given by

(3.6) χ(m(a), ε) = ε ·i, if det a < 0,1, if det a > 0.

For α ∈ Z/4Z and s ∈ C, consider the character

(3.7) χα| · |s : M → R, (m(a), ε) 7→ χ(m(a), ε)α| det a|s

and the smooth induced representation

(3.8) Iα(s) = IndG′

P χα| · |s

with its C∞ topology, where the induction is normalized so that Iα(s) is unitarywhen Re(s) = 0. In concrete terms, Iα(s) consists of smooth functions Φ : G′ → Csatisfying

(3.9) Φ((m(a), ε)n(b)g′) = χ(m(a), ε)α| det a|s+ρrΦ(g′), ρr := r+12,

with the action of G′ defined by r(g′)Φ(x) = Φ(xg′). Note that, by the Cartandecomposition G′ = PK ′, any such function is determined by its restriction to K ′; inparticular, given Φ(s0) ∈ Iα(s0), there is a unique family (Φ(s) ∈ Iα(s))s∈C such thatΦ(s)|K′ = Φ(s0)K′ for all s. Such a family is called a standard section of Iα(s).

3.1.3. We denote by χα the character of K ′ whose differential restricted to h′ hasweight α

2(1, . . . , 1). We will use some of the results about the K ′-types of Iα(s)

established by Kudla and Rallis. Namely, in [28] they show that the K ′-types thatappear are the irreducible representations πλ of K ′ with highest weight λ = (l1, . . . , lr)(here l1 ≥ · · · ≥ lr) such that πλ ⊗ (χα)−1 descends to an irreducible representationof U(r) and satisfies

(3.10) li ∈α

2+ 2Z, 1 ≤ i ≤ r.

Moreover, these K ′-types appear with multiplicity one in Iα(s) (op. cit., p.31). IfΦλ(·, s) ∈ Iα(s) is a non-zero highest weight vector of weight λ, then Φλ(e, s) 6= 0(op. cit., Prop. 1.1), hence from now on we normalize all such highest weight vectorsso that Φλ(e, s) = 1. Note that the restriction of Φλ to K ′ is independent of s. Forscalar weights λ = l(1, . . . , 1) with l ∈ α

2+ 2Z, we have

(3.11) Φl(k′, s) := Φl(1,...,1)(k′, s) = (det k′)l, k′ ∈ K ′.


3.1.4. Suppose that X ∈ g′ is a highest weight vector for K ′ of weight λ. ThenXΦl(s) is a highest weight vector of weight λ+ l and hence we have

(3.12) XΦl(s) = c(X, l, s)Φλ+l(s)

for some constant c(X, l, s) ∈ C. We will need to determine this constant explicitlyfor certain X and l. Namely, let

(3.13) ei = (0, . . . , 0, 1i−th

, 0, . . . , 0)

and define X−i = p−(d(ei)). Then X−r is a highest weight vector for K ′ of weight−2er. Let ιr : Mp2(R)→ G′ be the embedding defined by

(3.14)

((a bc d

), ε

)7→

1r−1 0r−1

a b0r−1 1r−1

c d

, ε

.

Then, for any Φ ∈ Iα(s), the value of X−r Φ(e, s) only depends on the pullback functionι∗rΦ(s) : Mp2(R)→ C. Recall that every element g′ of Mp2(R) can be written uniquelyin the form

(3.15) g′ = (n(x)m(y1/2), 1)kθ,

where x ∈ R, y ∈ R>0 and θ ∈ R/4πZ, where we define

(3.16) kθ =

(kθ, 1), if − π < θ ≤ π,(kθ,−1), if π < θ ≤ 3π,

kθ =

(cos θ sin θ− sin θ cos θ

).

We think of x, y and θ as coordinates on Mp2(R). In terms of these coordinates, wehave

(3.17) X−r Φ(e) =

(−2iy

d

dτ+i

2

∂

∂θ

)ι∗rΦ(e),

for any Φ ∈ Iα(s), where ddτ

= 12

(∂∂x

+ i ∂∂y

). The following lemma is a straightforward

computation using the explicit expression

(3.18) ι∗rΦl(x, y, θ, s) = y

12

(s+

r+12

)eiθl.

Lemma 3.1. Let l ∈ α2

+ 2Z. Then

X−r Φl(1,...,1)(s) = 12

(s+ r+1

2− l)

Φl(1,...,1)−2er(s).

In other words, this shows that c(X−r , l, s) = 12

(s+ r+1

2− l).


3.1.5. We now return to the quadratic space V over R of signature (p, 2). Letα(V ) = p− 2 and define

I(V, s) = Iα(V )(s),

s0 =p− r + 1

2.

(3.19)

We assume that r ≤ p+ 1, so that

(3.20) s0 ≥ 0.

Consider the Weil representation ω = ωψ of G′×O(V ) on S(V r) and let Rr(V ) beits maximal quotient on which (so(V ),O(V +)×O(V −)) acts trivially. Note that forany φ ∈ S(V r), the function

(3.21) Φ(g′) = ω(g′)φ(0), g′ ∈ G′,belongs to I(V, s0); the resulting map φ 7→ Φ : S(V r) → I(V, s0) is G′-intertwiningand factors through Rr(V ). By [28], it defines an embedding

(3.22) Rr(V ) → I(V, s0).

3.2. Degenerate principal series of U(r, r). In this section we fix r ≥ 1 and letG′ = U(r, r). We abbreviate N = Nr, M = Mr and K ′ = K ′r (see 2.5.1). Our setupfollows [17].

3.2.1. We denote by g′ the complexified Lie algebra of G′ and let g′ss = X ∈g′|trX = 0. We set

(3.23) r =1√2

(1r 1ri1r −i1r

)∈ U(2r)

and define φ(g) = rgr−1 for g ∈ G′; note that φ−1(G′) is the isometry group of theHermitian form determined by

(1r 00 −1r

). Then we have

(3.24) K ′r =

φ

((k1 00 k2

))∣∣∣∣ k1, k2 ∈ U(r)

and the Harish-Chandra decomposition

(3.25) g′ = p+ ⊕ p− ⊕ k′,

with g′ = M2r(C) and

k′ = Lie(K ′r)C =

φ

((X1 00 X2

))∣∣∣∣X1, X2 ∈ Mr(C)

,

p+ =

p+(X) := φ

((0 X0 0

))∣∣∣∣X ∈ Mr(C)

,

p− =

p−(X) := φ

((0 0X 0

))∣∣∣∣X ∈ Mr(C)

.

(3.26)

Let k′ss = k′ ∩ g′ss and

(3.27) h =φ(d(x))|x = (x1, . . . , x2r) ∈ C2r, x1 + . . .+ x2r = 0

.


Then h is a Cartan subalgebra of k′ss. For 1 ≤ i ≤ 2r, the assignment φ(d(x)) 7→ xidefines a functional ei : h→ C. We write ∆ for the set of roots of (gss, h) and fix theset of positive roots ∆+ = ∆+

c t∆+nc given by

∆+c = ei − ej|1 ≤ i < j ≤ r ∪ −ei + ej|r < i < j ≤ 2r,

∆+nc = ei − ej|1 ≤ i ≤ r < j ≤ 2r,

(3.28)

where ∆+c and ∆+

nc denote the compact and non-compact roots respectively.

3.2.2. The group P = MN is the Siegel parabolic of G′. For a character χ of C×and s ∈ C, define a character χ| · |sC : P → C× by

(3.29) χ| · |sC(m(a)n(b)) = χ(det a)| det a|sCand let

(3.30) I(χ, s) = IndG′

P (χ| · |sC)

be the degenerate principal series representation of G′. Thus I(χ, s) is the space ofsmooth functions Φ : G′ → C satisfying

(3.31) Φ(m(a)n(b)g′) = χ(det a)| det a|s+ρrC Φ(g′), ρr := r2,

and the action of G′ is via right translation: r(g′)Φ(x) = Φ(xg′). Here the inductionis normalized so that the I(χ, s) is unitary when χ is a unitary character and s = 0.Note that, by the Cartan decomposition G′ = PK ′, any such function is determinedby its restriction to K ′; in particular, given Φ(s0) ∈ I(χ, s0), there is a unique family(Φ(s) ∈ I(χ, s))s∈C such that Φ(s)|K′ = Φ(s0)K′ for all s. Such a family is called astandard section of I(χ, s).

3.2.3. We will need some results on K ′-types of I(χ, s) that were proved by Lee [31].Namely, I(χ, s) is multiplicity free as a representation of K ′. If Φ(λ1,λ2)(·, s) ∈ I(χ, s)is a highest weight vector of weight (λ1, λ2), then Φ(λ1,λ2)(e, s) 6= 0, hence from nowon we normalize all such highest weight vectors so that Φ(λ1,λ2)(e, s) = 1. Note thatthe restriction of Φ(λ1,λ2) to K ′ is independent of s. For scalar weights

(3.32) l = (l1(1, . . . , 1), l2(1, . . . , 1)), l1, l2 ∈ Z,

we have(3.33)

Φl([k1, k2], s) := Φ(l1(1,...,1),l2(1,...,1))([k1, k2], s) = (det k1)l1(det k2)l2 , k1, k2 ∈ U(r).

3.2.4. Let X−r ∈ p− be as in 3.1.4; that is, X−r = p−(d(er)) with er = (0, . . . , 0, 1) ∈Cr. Then X−r is a highest weight vector for K ′ of weight (−er, er) and so

(3.34) X−r Φl(s) = c(X−r , l, s)Φl+(−er,er)(s)

for some constant c(X−r , l, s) ∈ C. To compute this constant, note that U(r, r) ∩GL2r(R) = Sp2r(R) and X−r ∈ sp2r,C. Moreover, if Φ ∈ I(χ, s), then the restrictionΦ|Sp2r(R) belongs to Iα(s′), where s′ = 2s + (r − 1)/2 and α = 0 if χ|R× is trivial


and α = 2 otherwise (this follows directly from a comparison of (3.9) and (3.31)). Ifl = (l1(1, . . . , 1), l2(1, . . . , 1)) is a scalar weight, then

(3.35) Φl|Sp2r(R)(s) = Φ(l1−l2)·(1,...,1)(s′) ∈ Iα(s′)

and so Lemma 3.1 and the above remarks show that the constant in (3.34) is givenby

(3.36) c(X−r , l, s) = s+ ρr −l1 − l2

2.

3.2.5. We now return to the m-dimensional hermitian space V over C of signature(p, q) with pq 6= 0. From now on we fix a character χ = χV of C× such that χ|R× =sgn(·)m and define

I(V, s) = I(χ, s),

s0 =m− r

2.

(3.37)

We assume that r ≤ p+ 1, so that

(3.38) s0 ≥q − 1

2≥ 0,

with equality only when q = 1 and r = p+ 1.Consider the Weil representation ω = ωψ,χ of G′×U(V ) on S(V r) and let Rr(V ) be

its maximal quotient on which (u(V ), K) acts trivially. Note that for any φ ∈ S(V r),the function

(3.39) Φ(g′) = ω(g′)φ(0), g′ ∈ G′,belongs to I(V, s0); the resulting map ϕ 7→ Φ : S(V r) → I(V, s0) is G′-intertwiningand factors through Rr(V ). By [32, Thm. 4.1], it defines an embedding

(3.40) Rr(V ) → I(V, s0).

3.3. Whittaker functionals.

3.3.1. Let

(3.41) T ∈

Symr(R), case 1Herr, case 2

be a non-singular matrix and let ψT : N → C× the character defined by ψT (n(b)) =ψ(tr(Tb)). Let I(V, s) and s0 be as in (3.19) (resp. (3.37)) in case 1 (resp. in case2). A continuous functional l : I(V, s0) → C is a called a Whittaker functional if itsatisfies

(3.42) l(r(n(b))Φ) = ψT (n(b))l(Φ), n(b) ∈ N, Φ ∈ I(V, s0).

Such a functional can be constructed as follows. Let dn(b) be the Haar measure on Nthat is self-dual with respect to the pairing (b, b′) 7→ ψ(tr(bb′)). Embed Φ ∈ I(V, s0)in a (unique) standard section (Φ(s))s∈C and define

(3.43) WT (Φ, s) =

∫Nr

Φ(w−1r n(b), s)ψT (−n(b))dn(b)


The integral converges for Re(s) 0 and admits holomorphic continuation to all s([41]); its value at s0 defines a Whittaker functional WT (s0). It is shown in op. cit.that WT (s0) spans the space of Whittaker functionals. For g′ ∈ G′r and Φ ∈ I(V, s0),define

(3.44) WT (g′,Φ, s) =

∫Nr

Φ(w−1r n(b)g′, s)ψT (−n(b))dn(b).

and

(3.45) W ′T (g′,Φ, s0) =

d

dsWT (g′,Φ, s)

∣∣∣∣s=s0

.

3.3.2. Let

(3.46) Xr =

Symr(R), case 1

Herr, case 2

and denote by X+r the open subset of Xr consisting of positive definite matrices. Let

Φl(s) be the normalized highest weight vector of I(V, s) of scalar weight

(3.47) l =

m2, case 1(

m+k(χ)2

, −m+k(χ)2

), case 2

(see (3.11) and (3.33)). We will need some results on the value WT (g′,Φl, s0) and itsderivative W ′

T (g′,Φl, s0) established by Shimura in [38]. For s ∈ C, we set

(3.48) (α, β) =

(12(s− s0 +m), 1

2(s− s0)

)case 1,

(s− s0 +m, s− s0) case 2.

Let m(a) ∈Mr and y = ata ∈ X+r . Let

ι = [k : R],(3.49)

κ =

(r + 1)/2, case 1

r, case 2.(3.50)

As in [23, Lemma 9.2] and [33, §4A] one shows that

(3.51) 2−r(κ−1)/2WT (m(a),Φl, s) = | det a|s+ρrk

ξ(y, T ;α, β),

where

(3.52) ξ(y, T ;α, β) =

∫Xr

e−2πitr(Tx) det(x+ iy)−α det(x− iy)−βdx

and dx denotes Lebesgue measure (see [38, (1.11), (1.25)]). The integral in (3.52)converges for Re(α) > κ − 1 and Re(β) > 2κ − 1. As in [38, (1.29)], we can rewritethis as follows. Let Γ0(s) := 1 and

Γr(s) :=

∫X+r

e−tr(x) det(x)s−κdx

= πιr(r−1)/4

r−1∏j=0

Γ (s− ιj/2) .

(3.53)


Then

(3.54) ξ(y, T ;α, β) =(2π)rκ(−i)r(α−β)

2r(κ−1)Γr(α)Γr(β)η(2y, πT ;α, β),

where

(3.55) η(2y, T ;α, β) =

∫Qr(T )

e−2tr(xy) det(x+ T )α−κ det (x− T )β−κ dx

and Qr(T ) = x ∈ Xr|x− T, x+ T ∈ X+r .

Lemma 3.2. Let a ∈ GLr(k) and y = a · ta. Then

WT (m(a),Φl, s0) = 0

if T is not positive definite and

WT (m(a),Φl, s0) =(−2πi)ιrm/2

2r(κ−1)/2Γr(ιm/2)det(T )ιs0| det y|m/4

ke−2πtr(yT )

if T is positive definite.

Proof. Assume that T has r+ positive and r− negative eigenvalues, so that r++r− = r.According to [38, Thm. 4.2], the function

(3.56) Γr−(α− ιr+/2)−1Γr+(β − ιr−/2)−1η(2y, πT ;α, β)

is holomorphic at s = s0. Thus, by (3.51) and (3.54), the order of vanishing ofWT (g′,Φl, s) at s = s0 is bounded below by that of

(3.57)Γr−(α− ιr+/2)Γr+(β − ιr−/2)

Γr(α)Γr(β)=

1

Γr+(α)Γr−(β).

By our hypothesis that s0 ≥ 0, the function Γr+(α) is regular and non-vanishing ats = s0. Moreover, Γr−(β)−1 vanishes at s = s0 if r− > 0; this shows that WT vanishesat s = s0 when T is not positive definite.

Now assume that T is positive definite. By [38, (4.3),(4.6.K),(4.35.K)], setting

(3.58) ω(2y, πT ;α, β) = 2−rαΓr(β)−1 det(2πyT )κ−α det(2y)α+β−κη(2y, πT ;α, β),

we have

(3.59) ω(2y, πT ;α(s0), β(s0)) = ω(2y, πT ; ιm/2, 0) = 2−rκe−2πtr(yT ),

and the expression for WT (s0) in this case follows by a direct computation.

3.3.3. For y > 0, T ∈ Xr invertible and s ∈ C, define

(3.60) fT (y, s) = e2πtr(yT )WT (m(y1/2 · 1r),Φl, s)y−ιrm4 .

Note that by Lemma 3.2, we have

(3.61) fT (y, s0) =(−2πi)ιrm/2

2r(κ−1)/2Γr(ιm/2)det(T )ιs0


if T is positive definite and fT (y, s0) = 0 otherwise. We will need to understand theasymptotic behaviour of the derivative

(3.62) f ′T (y, s0) :=d

dsfT (y, s)|s=s0 = e2πtr(yT )W ′

T (m(y1/2 · 1r),Φl, s0)y−ιrm4

as y → +∞. First we consider the case when T is not positive definite.

Lemma 3.3. Assume that T is not positive definite. Then f ′T (y, s0) = O(e−ay) forsome a > 0 as y → +∞.

Proof. Let r+ and r− be the number of positive and negative eigenvalues of T respec-tively. As in the proof of Lemma 3.2, the function

(3.63)Γr−(α− ιr+/2)Γr+(β − ιr−/2)

Γr(α)Γr(β)=

1

Γr+(α)Γr−(β).

vanishes at s = s0 and the function

(3.64) fT (y, s) = Γr−(α− ιr+/2)−1Γr+(β − ιr−/2)−1η(2y, πT ;α, β)

is regular at s = s0. It follows that W ′T (·,Φl, s0) is a constant multiple of fT (·, s0).

By [38, Thm. 4.2], we have a bound of the form

(3.65) |fT (y, s0)| ≤ Ae−2πyτ(T )yB,

where A and B are positive constants and τ(T ) is the sum of the absolute values ofall eigenvalues of T ; this proves the statement since tr(T )− τ(T ) is negative.

Assume now that T is positive definite. Write T = (T12 ) · t(T 1

2 ), for some matrix

T12 ∈ GLr(k), and let

(3.66) g := 4π t(T12 ) · y · (T

12 );

then using [38, (1.29), (3.3), (3.6)] we may write

(3.67) WT (m(a),Φl, s) = C(β) | det y|m/4k

(detT )β+ιs0 · e−2π tr(Ty) ω(g;α, β)

where

(3.68) C(β) :=(−2πi)ιrm/2

2r(κ−1)/2

πrβ

Γr(β + ιm/2)

is a function independent of T and y, and ω(g;α, β) is an entire function in (α, β) ∈ C2,initially defined by the formula

ω(g;α, β) = Γr(β)−1 det(g)β

×∫X+r

e−tr(gx) det(x+ 1r)α−κ det(x)β−κdx(3.69)

on the region Re(β) > κ− 1, where the integral is absolutely convergent; its analyticcontinuation is proven in [38, Theorem 3.1]. We say that a function F : X+

r → Cvanishes at infinity if F (tg)→ 0 as t→ +∞, for any g ∈ X+

r .

Lemma 3.4. (1) [38, (3.15)] ω(g; ιm/2, 0) = 1.


(2) The special value of the derivative

∂

∂βω(·; β + ιm/2, β)

∣∣β=0

vanishes at infinity.

Proof. We are interested in the value and derivative of ω(g) at the parameters (α, β)in (3.48), which is outside the range of absolute convergence of (3.69). However,Shimura’s proof of analytic continuation involves a useful recurrence: let ∆ denotethe differential operator [38, (3.10.I-II)], which satisfies the identity

(3.70) ∆ e−tr(g) = (−1)r e−tr(g).

Set

(3.71) ε :=

0, if m/2 ∈ Z12, if m/2 /∈ Z

in case 1 and ε = 0 in case 2. Then (3.12) and (3.7) of [38] together imply that

ω(g; β + ιm/2, β) = (−1)r(ιm/2−ε)etr(g) det(g)β ·∆ιm/2−ε (e−tr(g) det(g)−βω(g; β + ε, β))

=(−1)r(ιm/2−ε)etr(g) det(g)β

×∆ιm/2−ε [e−tr(g) det(g)−βω (g;κ− β, κ− β − ε)].(3.72)

The final incarnation of ω in (3.72) is now in the range of convergence of (3.69); forconvenience, rewrite (3.69) for these parameters as

(3.73) ω(g, β) := ω (g;κ− β, κ− β − ε) ,

which may also be expressed as

(3.74) ω(g, β) = Γr (κ− β − ε)−1 det(g)−β ζ(g, β)

where

(3.75) ζ(g, β) := det(g)κ−ε∫X+r

e−tr(gx) det(x+ 1)−β det(x)−β−εdx.

Note that by [38, 1.16],

(3.76)

∫X+r

e−tr(gx) det(x)s−κ dx = Γr(s) det(g)−s, for Re(s) > κ− 1

so

(3.77) ζ(g, 0) = Γr (κ− ε) ,

and is in particular independent of g. This equation implies

(3.78) ω(g, 0) = 1,

and hence ω(g; ιm/2, 0) = 1 as well, via (3.72).


As for the derivative, begin by noting

∂

∂βζ(g, β)

∣∣β=0

= − det(g)κ−ε∫X+r

e−tr(gx) log(det x) det(x)−εdx

− det(g)κ−ε∫X+r

e−tr(gx) log(det(x+ 1)) det(x)−εdx.

(3.79)

For the first integral, writing g = γ · tγ and applying the change of variables x 7→tγ−1 · x · γ−1 yields

− det(g)κ−ε∫X+r

e−tr(gx) log(det x) det(x)−εdx

= −∫X+r

e−tr(x) (log(det x)− log det(g)) det(x)−εdx

= log det g · ζ(g, 0) −∫X+r

e−tr(x) (log(det x)) det(x)−εdx

= log det g · Γr (κ− ε) − Γ′r (κ− ε) .

(3.80)

Turning to the second integral, for convenience write

I(g) := − det(g)κ−ε∫X+r

e−tr(gx) log(det(x+ 1)) det(x)−εdx

=−∫X+r

e−tr(x) log(det(tγ−1 · x · γ−1 + 1

))det(x)−εdx.

(3.81)

A comparison with (3.69) reveals that the integral defining I(g) is absolutely con-vergent for all g. For a fixed g0 = γo ·tγ0 and parameter t ≥ 1, the integrand appearingin I(tg0) can be bounded as follows:

e−tr(x) log(det(t−1[tγ−1

0 · x · γ−10

]+ 1))

det(x)−ε

≤ e−tr(x) log(det(tγ−1

0 · x · γ−10 + 1

))det(x)−ε

(3.82)

As the latter function is absolutely integrable in x, it follows immediately that I(g)vanishes at infinity. By differentiating under the integral, one can similarly show thatall partial derivatives of I(g) also vanish at infinity.

We have shown

(3.83)ζ ′(g, 0)

ζ(g, 0)= log det g − Γ′

Γ+

1

Γ· I(g)

where, for legibility, we have abbreviated Γ = Γr (κ− ε) and Γ′ = Γ′r (κ− ε). Nowtaking the logarithmic derivative in (3.74), and recalling ω(g, 0) = 1, yields

∂

∂βω(g, β)

∣∣β=0

=Γ′

Γ− log det g +

ζ ′(g, 0)

ζ(g, 0)

= Γ−1I(g).

(3.84)


Finally, we obtain part (ii) of the lemma as follows: taking the logarithmic deriva-tive of (3.72) at β = 0 (recall here that ω(g; ιm/2, 0) = 1) gives

∂

∂βω(g; β + ιm/2, β)

∣∣β=0

= log det g + dlog∆ιm/2−ε (e−tr(g) det(g)−βω(g; β)) ∣∣

β=0

= log det g

+ (−1)r(ιm/2−ε)etr(g)∆ιm/2−ε (e−tr(g)− log det g + Γ−1I(g)

).

(3.85)

Note that the partial derivatives of log det g vanish at infinity; since tr(g) is linearin the entries of g, repeated differentiation gives

(3.86) (−1)r(ιm/2−ε)etr(g)∆ιm/2−ε (e−tr(g) − log det g)

= − log det g + J(g)

for some function J(g) vanishing at infinity. Similarly,

(3.87) etr(g)∆ιm/2−ε (e−tr(g)I(g))

can be expressed as a linear combination of partial derivatives of I(g), and hence alsovanishes at infinity. Putting everything together, we find

(3.88)∂

∂βω(g; l + β, β)

∣∣β=0

= f(g)

for some function f(g) vanishing at infinity, as required.

Lemma 3.5. Assume that T is positive definite and let a ∈ GLr(k). Then

limt→∞

W ′T (m(ta),Φl, s0)

WT (m(ta),Φl, s0)=ι

2

(log det(πT )− Γ′r(ιm/2)

Γr(ιm/2)

).

Proof. This follows from (3.67) and (3.68) by taking logarithmic derivatives withrespect to s and applying Lemma 3.4.

Combined with Lemma 3.2, this shows that

(3.89) limy→+∞

f ′T (y, s0) = WT (e,Φl, s0)e2πtr(T ) ι

2


Γr(ιm/2)

)when T is positive definite.

3.4. The integral of ξ. For the rest of Section 3 we assume that rk(E) = 1.

3.4.1. Let r ≥ 1 and εi (1 ≤ i ≤ r) be as in (2.76), so that νi(v) = ω(m(εi))νr(v).Consider the Schwartz functions νi, ν ∈ S(V r) defined by

νi(v, z0) ∧ Ωp−r+1(z0) = νi(v)Ωp(z0), 1 ≤ i ≤ r,

ν := ν1 + . . . ,+νr.(3.90)

Since ϕSG(0) = 1 and ϕ(0) = −Ω, we have νr(0)[2r−2] = (−Ω)r−1 and hence

(3.91) ω(g′)νr(0) = (−1)r−1Φλ0(g′, s0), g′ ∈ G′r,


since both sides define highest weight vectors of weight λ0 (cf. Lemma 2.8) in I(V, s0)that agree for g′ = 1 and the K-types in this representation appear with multiplicityone. Thus, setting

(3.92) Φ(s) =∑

1≤i≤r

r(m(εi))Φλ0(s) ∈ I(V, s),

we have

(3.93) ω(g′)ν(0) = (−1)r−1Φ(g′, s0).

The following lemma relating the Whittaker functional WT (·, s0) evaluated at Φwith the derivative W ′

T (Φl, s0) is the main ingredient in the proof of Theorem 3.10.

Lemma 3.6. Let fT (y, s) and Φ be given by (3.60) and (3.92) respectively. Then,for y > 0,

e2πtr(yT )WT (m(y1/2 · 1r), Φ, s0)y−ιrm4dy

y=

2

ι

d

dyf ′T (y, s0)dy.

Proof. Since fT is real analytic [38, Thm. 5.9], it suffices to show that, for y > 0 ands 6= s0,

(3.94) e2πtr(yT )WT (m(y1/2 · 1r), Φ, s)y−ιrm4dy

y=

2

ι

d

dy

(fT (y, s)− fT (y, s0)

s− s0

)dy.

This is a direct computation using (3.17). By Lemma 3.1 (in case 1) and (3.36) (incase 2), we can write

(3.95) Φ(s) =2

ι(s− s0)−1

∑1≤i≤r

Ad(m(εi))X−r Φl(s).

Let F (g) = WT (g,Φl, s) and F (g) = WT (g, Φ, s). Then we have

ι2−1(s− s0)F (m(y1/2 · 1r) =∑

1≤j≤r

Ad(m(εj))X−r F (m(y1/2 · 1r))

=∑

1≤j≤r

(−2iyjd

dτ j+i

2

d

dθj)F (m(y1/2 · 1r))

=∑

1≤j≤r

(−iyjd

dxj+ yj

d

dyj+i

2

d

dθj)F (m(y1/2 · 1r))

=

(( ∑1≤j≤r

−iyjd

dxj+ yj

d

dyj

)− ιrm

4

)F (m(y1/2 · 1r))

=

(( ∑1≤j≤r

−iyjd

dxj

)+ y

d

dy− ιrm

4

)F (m(y1/2 · 1r))

=

(2πtr(yT ) + y

d

dy− ιrm

4

)F (m(y1/2 · 1r))

(3.96)


and hence

ι2−1(s− s0)e2πtr(yT )WT (m(y1/2 · 1r), Φ, s)y−ιrm4dy

y

= e2πtr(yT )

((2πtr(yT ) + y

d

dy− ιrm

4

)F (m(y1/2 · 1r))

)y−

ιrm4dy

y

= e2πtr(yT )2πtr(yT )F (m(y1/2 · 1r))y−ιrm4dy

y+ e2πtr(yT ) d

dy

(F (m(y1/2 · 1r))y−

ιrm4

)dy

=d

dy

(e2πtr(yT )

)F (m(y1/2 · 1r))y−

ιrm4 dy + e2πtr(yT ) d

dy

(F (m(y1/2 · 1r))y−

ιrm4

)dy

=d

dy

(e2πtr(yT )F (m(y1/2 · 1r))y−

ιrm4

)dy

=d

dy(fT (y, s)− fT (y, s0)) dy,

(3.97)

where the last equality holds since fT (y, s0) is independent of y by (3.61).

3.4.2. For T ∈ Xr with detT 6= 0, let

(3.98) ΩT (V ) = (v1, . . . , vr) ∈ V r|(Q(vi, vj))i,j = 2T.

Thus ΩT (V ) 6= ∅ if and only if (V,Q) represents T , and in this case U(V ) actstransitively on ΩT (V ); assuming this, let dµ(v) be an U(V )-invariant measure onΩT (V ) and consider the functional S(V r)→ C defined by

(3.99) φ 7→∫

ΩT (V )

φ(v)dµ(v).

This functional is obviously U(V )-invariant and non-zero and hence defines a Whit-taker functional on Rr(V ). We denote by dµ(v)SW the unique U(V )-invariant measureon ΩT (V ) such that for any φ ∈ S(V r):

(3.100)

∫ΩT (V )

φ(v)dµ(v)SW = γ−1V r ·WT (e,Φ, s0), where Φ(g) := ω(g)φ(0).

We denote by dg(2) the invariant measure on U(V ) defined as dg(2) = dlpdrk, wheredrk is the unique Haar measure on O(V +)×O(V −) (resp. U(V +)×U(V −)) with totalvolume one in case 1 (resp. case 2), and dlp is the left Haar measure on P inducedby the invariant volume form Ωp on D+ ∼= G/K.

Lemma 3.7. Let λ0 be as in Lemma 2.8, T ∈ Xr with non-zero determinant andv = (v1, . . . , vr) ∈ ΩT (V ). Let Gv ⊂ U(V ) be the pointwise stabilizer of 〈v1, . . . , vr〉and Γv ⊂ G0

v be a torsion free subgroup of finite covolume. Then, for any g′ ∈ G′r,we have ∫

Γv\D+

ω(g′)ν(v) ∧ Ωp−r+1 = − ι2CT,ΓvWT (g′, Φ, s0),


where Φ is as in (3.92) and CT,Γv is the non-zero constant given by

CT,Γv =2

ι(−1)rVol(Γv\Gv, dgv)

dg(2)/dgvdµ(v)SW

γ−1V r .

Here dgv is an arbitrary Haar measure on Gv and dg(2)/dgv denotes the quotientmeasure on ΩT (V ) ∼= U(V )/Gv induced by dg(2) and dgv.

Proof. The hypothesis that detT 6= 0 implies that G0v is reductive, hence the invariant

measure dg(2)/dgv exists. By (3.90) and Proposition 2.6.(d), we have ν(gv) = ν(v)for any g ∈ U(V ) stabilizing V −. We compute

∫Γv\D+

ω(g′)ν(v) ∧ Ωp−r+1 =

∫Γv\U(V )

ω(g′)ν(g−1v)dg(2)

= Vol(Γv\Gv, dgv)

∫Gv\U(V )

ω(g′)ν(g−1v)dg(2)

dgv

= Vol(Γv\Gv, dgv)dg(2)/dgvdµ(v)SW

∫ΩT (V )

ω(g′)ν(v)dµ(v)SW

= (−1)r−1Vol(Γv\Gv, dgv)dg(2)/dgvdµ(v)SW

γ−1V rWT (g′, Φ, s0),

(3.101)

where the last equality follows from (3.100) and (3.93).

3.4.3. We will now apply the results obtained in this section to compute the integral(3.1) under our standing assumption that rk(E) = 1. Rather than aiming for themost general result, we restrict to cocompact arithmetic subgroups Γ and integralvectors v (see below); this suffices for the application to compact Shimura varietiesin Section 5.

Fix a lattice L ⊂ V and a torsion free cocompact arithmetic subgroup Γ ⊂ Gstabilizing L, and let Γv = StabΓ〈v1, . . . , vr〉 and XΓ = Γ\D+.

Lemma 3.8. Assume that v ∈ Lr is non-degenerate and let Z(v)Γ be the image ofthe natural map Γv\D+

v → XΓ.

(a) The sum

g(v)Γ =∑

γ∈Γv\Γ

g(γ−1v)

converges absolutely to a smooth form on XΓ−Z(v)Γ that is locally integrable onXΓ.

(b) As currents on XΓ we have∑γ∈Γv\Γ

∫ M

1

ν0(t1/2γ−1v)[2r−2]dt

t

M→+∞−−−−−→ g(v)Γ.


Proof. Let us first show that g(v)Γ converges as claimed. Let z ∈ D+ and pick arelatively compact neighborhood U of z. Recall that there is a positive definite formQz defined by (2.46) that varies continuously with z. Write Γv = S1 t S2 (disjointunion), with

(3.102) S1 = v′ ∈ Γv|minz∈U

Qz(v′,v′) < 1;

then S1 is finite since Γv ⊂ Lr. We can split the sum defining g(v)Γ accordingly;the sum over S1 converges to a locally integrable form on D+ that is smooth outside∪v′∈S1D+

v′ by Proposition 2.2. The sum over S2 converges to a smooth form on Uby the estimate (2.9) and the standard argument of convergence of theta series. ByProposition 2.6.(d), the current defined by g(v)Γ is invariant under Γ, and this shows(a). For part (b), note that for each v′ ∈ S1 we have ∫M1 ν0(t1/2v′)[2r−2]

dtt→ g(v′) as

M → +∞ (as currents on D+) by dominated convergence, as remarked in the proof ofProposition 2.2. For the sum over S2 we apply the bound (2.54), and (b) follows.

Lemma 3.9. Assume that v ∈ Lr is non-degenerate. Then the integral (3.1) con-verges and

∫Γv\D+

ξ0(v) ∧ Ωp−r+1 = limM→+∞

∫ M

1

∫Γv\D+

ν0(t1/2v) ∧ Ωp−r+1dt

t.

Proof. By Lemma 3.8.(a), the integral (3.1) equals ∫Γ\D+ g(v)Γ ∧ Ωp−r+1 and so itconverges since XΓ is compact. Applying Lemma 3.8.(b) and unfolding the sum andthe integral proves the claim.

Theorem 3.10. Let T ∈ Xr with non-zero determinant and v ∈ ΩT (V ) ∩ Lr. LetCT,Γv be as in Lemma 3.7 and l be the weight in (3.47). Recall that we assume thatrk(E) = 1.

(a) If T is not positive definite (so that Dv = ∅), then∫Γv\D+

ξ(v) ∧ Ωp−r+1 = CT,ΓvW′T (e,Φl, s0).

(b) If T is positive definite (so that Dv 6= ∅), then∫Γv\D+

ξ(v) ∧ Ωp−r+1 = CT,ΓvW′T (e,Φl, s0)

− CT,ΓvWT (e,Φl, s0)ι

2


Γr(ιm/2)

).


Proof. The integral converges by Lemma 3.9. We compute

∫Γv\D+

ξ0(v) ∧ Ωp−r+1 =

∫ ∞1

∫Γv\D+

ν0(y1/2v) ∧ Ωp−r+1dy

y

=

∫ ∞1

e2πtr(yT )

∫Γv\D+

ν(y1/2v) ∧ Ωp−r+1dy

y

=

∫ ∞1

e2πtr(yT )

∫Γv\D+

ω(m(y1/2 · 1r))ν(v) ∧ Ωp−r+1y−ιrm

4dy

y

= − ι2CT,Γv

∫ ∞1

e2πtr(yT )WT (m(y1/2 · 1r), Φ, s0)y−ιrm

4dy

y,

(3.103)

where we have used Lemma 3.9 and Lemma 3.7 for the first and last equality respec-tively. Applying Lemma 3.6, we conclude that

(3.104)

∫Γv\D+

ξ0(v) ∧ Ωp−r+1 = CT,Γv(f ′T (1, s0)− limy→∞

f ′T (y, s0)).

If T is not positive definite, then the limit in the above expression vanishes by part(a) of Lemma 3.3; this proves (a). If T is positive definite, then (b) follows from(3.89).

4. Green forms for special cycles on Shimura varieties

We now shift focus from the hermitian symmetric domain D to its quotients Γ\Dby arithmetic subgroups. Namely, we consider the special cycles Z(T, ϕf ) on orthog-onal and unitary Shimura varieties introduced by Kudla in [22]. We review theirdefinition and apply our results to construct Green forms g(T, y, ϕf ) for those specialcycles Z(T, ϕf ) parametrized by a non-singular matrix T . Section 4.1 deals with theorthogonal case and Section 4.2 with the unitary one. In Section 4.3 we prove ourmain result concerning star products of the Green forms g(T, y, ϕf ).

Motivated by the conjectures in Section 1.2, we would like to define currentsg(T, y, ϕf ) also when T is singular. We will achieve this in Section 4.4 by regularizingthe integral defining ξ0 in Definition 2.3. In Example 4.8 we compute g(T, y, ϕf ) ina special case and show how, specialized to Shimura curves over Q, this constructionrecovers the current in [30, §6.4], introduced there in an ad hoc way to force agreementwith a certain Eisenstein series.

Throughout, we fix a totally real number field F with real embeddings σ1, . . . , σd.

4.1. Orthogonal Shimura varieties.

4.1.1. Let us briefly recall the definition and basic properties of orthogonal Shimuravarieties attached to quadratic spaces over F ; see [22] for more detail.


Let p ≥ 1 and V be a quadratic space over F of dimension p+2, with correspondingbilinear form Q(·, ·). Assume that

(4.1) signature(Vσi) =

(p, 2) if i = 1

(p+ 2, 0) if i 6= 1,

where we abbreviate Vσi := V⊗F,σi R for the real quadratic spaces at each place.Attached to Vσ1 is the symmetric space D(Vσ1) defined in (2.33); we set

(4.2) D(V) = D(Vσ1) = [v] ∈ P(Vσ1(C))|Q(v, v) = 0, Q(v, v)〉 < 0.Here Q(·, ·) is the C−bilinear extension of the bilinear form on Vσ1 . Let

(4.3) H := ResF/Q GSpin(V).

Then

(4.4) H(R) ' GSpin(Vσ1)× · · · ×GSpin(Vσd)

acts transitively on D(V) via the first factor.

Definition 4.1. For a compact open subgroup K ⊂ H(Af ), consider the Shimuravariety

XK := H(Q)\D(V)×H(Af )/KIf K is sufficiently small (neat), then XK is a complex1 quasi-projective algebraicvariety. If V is moreover anisotropic, then XK is projective.2

The space XK may be written in a perhaps more familiar fashion: fix a con-nected component D+ ⊂ D(V) and let H(R)+ denote its stabilizer in H(R)+. SettingH(Q)+ = H(R)+ ∩H(Q), there exist finitely many elements h1, . . . ht ∈ H(Af ) suchthat

(4.5) H(Af ) =∐j

H(Q)+hjK;

then we may write

(4.6) XK '∐j

Γj∖D+ =:

∐j

Xj

as a disjoint union of quotients of D+ by discrete subgroups

(4.7) Γj = H(Q)+ ∩(hjKh

−1j

).

We regard the quotients XK and Γj\D+ as orbifolds. In particular, for a Γj-invariant differential form η of top degree on D+, we define

(4.8)

∫[Γj\D+]

η = [Γ : Γ′]−1

∫Γ′\D+

η,

1In fact, the theory of canonical models implies that XK admits a model defined over F , but wewill not require this fact in the sequel.

2Our assumptions on the signature of V imply that this is always the case when F 6= Q.


where Γ′ ⊂ Γj is a (equivalently, any) neat subgroup of finite index, and set ∫[XK ] =∑j

∫[Γj\D+]

.

4.1.2. Recall that in 2.2.2 we have defined the tautological line bundle E over D(V)and a global section sx of Er for any x ∈ Vrσ1 whose zero locus Z(sx) we denote byDx. Given a rational vector v = (v1, . . . , vr) ∈ Vr, we set

(4.9) Dv = Z(sσ1(v))

and D+v = Dv ∩ D+. Let Hv(Q) be the pointwise stabilizer of 〈v1, . . . , vr〉 in H(Q).

Given a component Xj = Γj\D+ ⊂ XK associated to hj as in (4.6), let Γj(v) =Γj ∩Hv(Q); then the natural map

(4.10) Γj(v)\D+v → Γj\D+

defines an (algebraic) cycle on XK that we denote by c(v, hj, K).We require a final bit of notation: the moment matrix attached to v ∈ Vr is defined

to be

(4.11) T (v) :=(

12Q(vi, vj)

)i,j∈ Symr(F ).

Given T ∈ Symr(F ), we let

(4.12) ΩT (V) = v ∈ Vr|T (v) = T.

Definition 4.2 ([22]). Fix a non-singular matrix T ∈ Symr(F ), and let ϕf ∈S(V(Af )r)K be a K-invariant Schwartz function. Define the weighted special cycleZ(T, ϕf , K) on XK =

∐Xj by

Z(T, ϕf , K) =∑j

∑v∈ΩT (V)mod Γj

ϕf (h−1j v) c(v, hj, K).

Note that this definition is independent of all choices. If K ′ ⊂ K is an opensubgroup of finite index and π : XK′ → XK is the natural covering map, thenπ∗Z(T, ϕf , K) = Z(T, ϕ,K ′); as a consequence, when K is understood we will denotethis cycle simply by Z(T, ϕf ). See [22, §5] for a proof of this and further propertiesof these cycles.

4.1.3. Let T ∈ Symr(F ) with detT 6= 0. Any collection of vectors v = (v1, . . . , vr) ∈Vr, with T (v) = T , is necessarily linearly independent. For such v, we have definedin 2.2.3 a form g(σ1(v)) satisfying

(4.13) ddcg(σ1(v)) + δDv = ϕ0(σ1(v))[2r].

We introduce Green forms that depend on an auxiliary parameter y ∈ Symr(R)>0

as follows: fix some element α ∈ GLn(R) such that y = α · tα and, for a Schwartzfunction ϕf ∈ S(V(Af )r)K , define a form gj(T, y, ϕf ) on D by setting

(4.14) gj(T, y, ϕf ) =∑

v∈ΩT (V)

ϕf (h−1j v) g(σ1(v) · α).


The convergence of this sum on the complement of Z(T, ϕf ) follows from Lemma 3.8.(a)and the fact that the number of orbits of Γj on Supp(ϕf )∩ΩT (V) is finite. Note thatthe definition is independent of the choice of α, by Proposition 2.6.(f). Moreover,the form gj(T, y, ϕf ) is invariant under the action of Γj by Proposition 2.6.(d), andso it descends to a form on the connected component Xj that, abusing notation, wealso denote by gj(T, y, ϕf ).

Finally, let g(T, y, ϕf ) denote the form on XK whose restriction to Xj is gj(T, y, ϕf ).Essentially by construction, it is a Green’s form for the cycle Z(T, ϕf ); more precisely,let

(4.15) ω(T, y, ϕf ) =∑

v∈ΩT (V)

ϕf (h−1v) ϕ0(σ1(v) · α)[2r],

viewed as a differential form on X. Let Hr be the symmetric space of G′r (see (2.58));for τ = x+ iy ∈ Hr, we set

(4.16) g′τ = n(x)m(y1/2) ∈ P ′r.

Then ω(T, y, ϕf ) is the T ’th coefficient of the theta function

ΘKM(τ ;ϕf ) =∑

ω(T, y1, ϕf ) qT ,

where τ = (xi + iyi)1≤i≤d ∈ Hdr , qT =

d∏i=1

e2πitr(τiσi(T )),(4.17)

considered (in much greater generality) by Kudla and Millson [26], who showed thatω(T, y1, ϕf ) is a Poincare dual form to the special cycle Z(T, ϕf ); hence ω(T, y1, ϕf )−δZ(T,ϕf ) is exact.

Applying the identity (4.13) of currents on D, summing over v with T (v) = T anddescending to the Shimura variety XK yields the equation

(4.18) ddcg(T, y, ϕf ) + δZ(T,ϕf ) = ω(T, y, ϕf )

of currents on XK .

4.2. Unitary Shimura varieties. The results in the previous section carry over,essentially verbatim, to the unitary case. To describe the setup, suppose that E isa CM extension of the totally real field F , and that V is a Hermitian space over E,with Hermitian form Q(·, ·).

For each real embedding σi : F → R, the space Vσi = V ⊗σi,F R is a complexHermitian space; we assume

(4.19) signature Vσi =

(p, q), if i = 1

(p+ q, 0), if i = 2, . . . d.

for some integers p, q > 0.Let

D(V) = D(Vσ1) := z ⊂ Vσ1 negative-definite subspace, dimC z = q ,


(cf. (2.32)) be the symmetric space attached to the real points of the unitary group

(4.20) H := ResF/Q GU(V).

Just as in Section 4.1, a sufficiently small compact open subgroup K ⊂ H(Af ) deter-mines a Shimura variety

(4.21) XK := H(Q)\D×H(Af )/K

which is quasi-projective, and projective when V is anisotropic; choosing representa-tives h1, . . . , ht for the double coset space H(Q)\H(Af )/K gives a decomposition

(4.22) X '∐j

Γj\D =:∐j

Xj, where Γj := H(Q) ∩(hjKh

−1j

).

As above we can define E, sσ1(v), Dv and a special cycle c(v, hj, K) for any vectorv ∈ Vr. Given a K-invariant Schwartz function ϕf ∈ S(V(Af )r)K , and a non-singularmatrix T ∈ Herr(E), define the special cycle

(4.23) Z(T, ϕf ) =∑j

∑v∈ΩT (V)mod Γj

ϕf (h−1j v) c(v, hj, K)

exactly as in Definition 4.2, which is either empty or has components of codimensionrq. A Green form g(T, y, ϕf ) for Z(T, ϕf ), depending on an auxiliary parametery ∈ (Herr)>0, can be defined by asking that its restriction to Xj be given by theexpression (4.14), where α ∈ GLr(C) is such that y = α · tα. As before, g(T, y, ϕf ) isindependent of the choice of α and descends to a current on XK satisfying (4.18).

4.3. Star products on XK. We now consider both cases and write

(4.24) Xr(F ) =

Symr(F ), case 1Herr(E), case 2.

Given a place v of F , define the regular locus

(4.25) V(Fv)rreg = v = (v1, . . . , vr) ∈ V(Fv)

r| dim〈v1, . . . , vr〉 = r .

Note that V(Fv)rreg is open in V(Fv)

r.

Lemma 4.3. Let r1, r2 be positive integers such that r = r1 + r2 ≤ dim(XK) + 1.For i = 1, 2, let Ti ∈ Xri(F ) be non-singular and let ϕf,i = ⊗pϕp,i ∈ S(V(Af )ri).Assume that ϕp,1 ⊗ ϕp,2 has support contained in the regular locus V(Fp)

rreg for some

finite place p of F . Then Z(T1, ϕf,1) and Z(T2, ϕf,2) intersect properly and

g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2) = limM→+∞

g(T1, y1, ϕf,1)≤M ∧ ω(T2,My2, ϕf,2)

+ ω(T1, y1, ϕf,1) ∧ g(T2, y2, ϕf,2)≤M .

Proof. Let vi ∈ ΩTi(V) such that ϕf,i(vi) 6= 0. As remarked in the proof of Propo-sition 2.2, we have g(σ1(vi))≤M → g(σ1(vi)) and ϕ0(M1/2σ1(v2))[2r2rk(E)] → δDv2

as


currents on D, as M → +∞. By our hypothesis on ϕf , the cycles Dv1 and Dv2

intersect transversely, hence the product g(σ1(v1)) ∧ δDv2exists and we have

g(σ1(v1)) ∧ δDv2+ ϕ0(σ1(v1))[2r1rk(E)] ∧ g(σ1(v2))

= limM→∞

g(σ1(v1))≤M ∧ ϕ0(M1/2σ1(v2))[2r2rk(E)] + ϕ0(σ1(v1))[2r1rk(E)] ∧ g(σ1(v2))≤M

(4.26)

(see the discussion after equation (2.48) in [5]). The claim now follows by the sameargument as the proof of Lemma 3.8.

Theorem 4.4. With the notation and hypotheses of the Lemma 4.3, we have

g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2) =∑

T∈Xr(F )

T=(T1 ∗∗ T2

)g(T, ( y1 y2 ) , ϕf,1 ⊗ ϕf,2)

in D∗(XK) := D∗(XK)/(im∂ + im∂).

Proof. Lemma 4.3 shows that the left hand side is the limit as M → +∞ of theexpression ∑

v1∈ΩT1 (V)

v2∈ΩT2 (V)

ϕf,1(v1)ϕf,2(v2)·(g(σ1(v1) · α1)≤Mϕ0(M1/2σ1(v2) · α2)[2r2rk(E)]

+ ϕ0(σ1(v1) · α1)[2r1rk(E)]g(σ1(v2) · α2)≤M)

(4.27)

With the notation of the proof of Theorem 2.9, this expression equals

(4.28)∑

v1∈ΩT1 (V)

v2∈ΩT2 (V)

ϕf,1(v1)ϕf,2(v2)

∫γk,M

ν0(σ1(v1) · α1, σ1(v2) · α2)

and so, up to exact forms ∂α and ∂β, it agrees with the same expression with ∫γk,Mreplaced by ∫γd,M , which equals g(T, ( y1 y2 ) , ϕf,1⊗ϕf,2)≤M . This finishes the proof.

4.4. The degenerate case. We now turn to the case where detT = 0. Throughoutthis section we assume that V is anisotropic and that rk(E) = 1.

First, for a tuple v ∈ Vrσ1 and a complex parameter s, let

(4.29) ν0(v; s)[2r−2] :=r∑i=1

ν0(√tivi)[0] ∧ ϕ0(

√t1v1, . . . ,

√trvr︸︷︷︸

i’th term omitted

)[2r−2]dti

t1+si

viewed as a differential form on D×Rr>0, where t1, . . . , tr are the coordinates on Rr>0.Fix a = (a1, . . . , ar) ∈ (R>0)r, consider the path

(4.30) γa : [1,∞) → (R>0)r, γa(t) = (ta1, . . . , tar),

and define

(4.31) g(v, a; s) :=

∫[1,∞)

γ∗a ν0(v; s)[2r−2].


Proposition 4.5. Let v = (v1, . . . , vr) ∈ Vrσ1 and a = (a1, . . . , ar) ∈ Rr>0.

(i) The integral (4.31) converges to a smooth form on D if Re(s) > 0.(ii) As a current, g(v, a; s) extends meromorphically3 to the right half plane Re(s) >−1

2.

(iii) If v is non-degenerate, then the current g(v, a; s) is regular at s = 0 and

g(v, a; s)s=0 = g(v · a12 ) = g(

√a1v1, . . . ,

√arvr).

(iv) Let r′ = dim〈v1, . . . , vr〉. The constant term CTs=0

g(v, a; s) satisfies the current

equation

ddc CTs=0

g(v, a; s) + δDv ∧ (−Ω)r−r′= ϕ0(v · a

12 )[2r].

Proof. Part (i) follows immediately from the expression (2.12), which shows thatν0(√tv) and its partial derivatives stay bounded as t→ +∞.

To show (ii), consider (say) the first term in (4.29). Let v′2 = (√a2v2, . . . ,

√arvr)

and r′ = dim〈v2, . . . , vr〉; then we may rewrite the corresponding contribution in(4.31) as

a−s1

∫ ∞1

ν0(√ta1v1)[0] ∧ ϕ0(

√tv′2)[2r−2]

dt

ts+1

= a−s1

∫ ∞1

ν0(√ta1v1)[0]

(ϕ0(√tv′2)[2r−2] − δDv′2

∧ (−Ω)r−1−r′) dt

ts+1(4.32)

+ a−s1

(∫ ∞1

ν0(√ta1v1)[0]

dt

ts+1

)∧ δDv′2

∧ (−Ω)r−1−r′

Let h ∈ O(r − 1) (case 1) or h ∈ U(r − 1) (case 2) diagonalizing the moment ma-trix T (v′2) and write v′2 · h = (v′′2 , 0r−1−r′), where v′′2 ∈ V r′ is non-degenerate. ByProposition 2.5, we have

(4.33) ϕ0(√tv′2)[2r−2] = ϕ0(

√tv′2 · h)[2r−2] = ϕ0(

√tv′′2)[2r′] ∧ (−Ω)r−1−r′

and hence Bismut’s asymptotic estimate [4, (3.9)] applied to ϕ0(√tv′′2)[2r′] shows that

(4.34) ϕ0(√tv′2)[2r−2] − δDv′2

∧ (−Ω)r−1−r′ = O(t−12 )

as currents on D. By (2.45) we have ν0(√ta1v1)[0] = e−2πta1h(sv1 ) ≤ 1 and we conclude

that the first integral in (4.32) converges to a current when Re(s) > −12.

As to the second integral in (4.32), assume first that v1 /∈ 〈v2, . . . , vn〉. Then thisintegral equals

(4.35) xsΓ(−s, x) ∧ δDv′2∧ (−Ω)r−1−r′ ,

where we have set x := 2πa1h(sv1) and Γ(s, u) = ∫+∞u e−tts−1dt denotes the incomplete

Gamma function. Note that when x > 0 the function xsΓ(−s, x) is entire in s. Thus,

3More precisely, we are asserting that for any compactly supported form η, the expression∫D g(v,a; s) ∧ η admits a meromorphic extension as a function of s, and is continuous in η in the

sense of distributions.


to show meromorphic continuation to Re(s) > −12

of the current (4.35), it sufficesto prove that (2πa1h(sv1))

sΓ(−s, 2πa1h(sv1)) is locally integrable on D for such s.First note that, letting z be a local coordinate for the divisor Dv1 in D, we haveh(sv1) ∼ |z|2 and hence h(sv1)

s is indeed locally integrable when Re(s) > −12. Next,

let γ(s, u) = ∫u0 e−tts−1dt, so that Γ(s, u) + γ(s, u) = Γ(s) and we can write

(4.36) xsΓ(−s, x) = xsΓ(−s)− xsγ(−s, x).

The integrability now follows from the fact that the function zsΓ(−s)−1γ(−s, z) isentire in both s and z (see [35, §8.2]).

Now assume that v1 ∈ 〈v2, . . . , vn〉. Then ν0(√ta1 v1)[0]∧δDv′2

= δDv′2and the second

term becomes

(4.37) a−s1

(∫ ∞1

dt

ts+1

)· δDv′2

∧ (−Ω)r−1−r′ =a−s1

sδDv′2∧ (−Ω)r−1−r′

for Re(s) > 0; the right hand expression defines a meromorphic continuation of thisterm.

The remaining terms in (4.29) can be treated in the same manner, and this proves(ii).

When v is non-degenerate, the proof of (ii) shows that both integrals on the RHSof (4.32) converge when s = 0, and this establishes (iii).

To prove (iv), write g(v, a; s) =∑r

i=1 gi(v, a; s) with

(4.38) gi(v, a; s) =

∫[1,∞)

γ∗a

ν0(√tivi)[0] · ϕ0(

√t1v1, . . . ,

√trvr︸︷︷︸

i’th term omitted

)[2r−2]dti

t1+si

.Consider, say, the first term i = 1, and let v′ = (

√a2v2, . . . ,

√arvr). As in the proof

of (ii), we may write(4.39)

CTs=0

g1(v, a; s) =

∫ ∞1

ν0(√a1tv1)[0]

(ϕ0(√tv′)[2r−2] − δDv′

∧ (−Ω)r−1−r′) dtt

+ µ1,

where

(4.40) µ1 =

− log a1 · δDv′

∧ (−Ω)r−1−r′ , if v1 ∈ 〈v2, . . . , vr〉,g(√a1v1) ∧ δDv′

∧ (−Ω)r−1−r′ , otherwise.

Note that Green’s equation (4.13), together with the fact the Ω is closed, implies that(4.41)

ddcµ1 =

0, if v1 ∈ 〈v2, . . . , vr〉,−δDv ∧ (−Ω)r−1−r′ + ϕ0(

√a1v1)[2] ∧ δDv′

∧ (−Ω)r−1−r′ , otherwise.


On the other hand, as the integral in (4.39) converges uniformly, we may interchangethe integration on t with differentiation, and apply Proposition 2.6.(b), to obtain

ddc

[∫ ∞1

ν0(√a1tv1)[0]

(ϕ0(√tv′)[2r−2] − δDv′

∧ (−Ω)r−1−r′) dtt

]

=

∫ ∞1

ddcν0(√a1tv1)[0] ∧

(ϕ0(√tv′)[2r−2] − δDv′

∧ (−Ω)r−1−r′) dtt

=

∫ ∞1

− d

dtϕ0(√a1tv1)[2] ∧

(ϕ0(√tv′)[2r−2] − δDv′

∧ (−Ω)r−1−r′)dt

=−∫ ∞

1

d

dtϕ0(√a1tv1)[2] ∧ ϕ0(

√tv′)[2r−2]dt

+

∫ ∞1

d

dtϕ0(√a1tv1)[2] ∧ δDv′

∧ (−Ω)r−1−r′dt.

(4.42)

Note that if v1 ∈ 〈v2, . . . , vr〉, then ϕ0(√a1tv1)[2] ∧ δDv′

is independent of t. In anycase,

∫ ∞1

d

dtϕ0(√a1tv1)[2] ∧ δDv′

∧ (−Ω)r−1−r′dt

=[

limt→∞

ϕ0(√a1tv1)[2] ∧ δDv′

∧ (−Ω)r−1−r′ − ϕ0(√a1v1)[2] ∧ δDv′

∧ (−Ω)r−1−r′]

=− ddcµ1,

(4.43)

and so

(4.44) ddc CTs=0

g1(v, a; s) = −∫ ∞

1

d

dtϕ0(√a1tv1)[2] ∧ ϕ0(

√tv′)[2r−2]dt.

Arguing in a similar manner for i = 2, . . . , r, we find that

ddc CTs=0

g(v, a; s) =−∑i

∫ ∞1

d

dtϕ0(√aitvi)[2] ∧ ϕ0(

√a1tv1, . . . ,

√artvr︸︷︷︸

i’th term omitted

)[2r−2]dt

= −∫ ∞

1

d

dtϕ0(√a1tv1, . . . ,

√artvr)[2r]dt

=−[

limt→∞

ϕ0(√tv · a

12 )[2r] − ϕ0(v · a

12 )[2r]

]=− δDv ∧ (−Ω)r−r

′+ ϕ0(v · a

12 )[2r],

(4.45)

establishing (iv).

We can now define a current g(T, y;ϕf ) for possibly singular T , generalizing theconstructions in 4.1 and 4.2. We assume that we are in case 1 (all the results proveduntil the end of this section generalize to case 2 in an obvious way) and begin withthe following invariance property.


Lemma 4.6. Suppose y ∈ Symr(R)>0, and choose elements a = (a1, . . . , ar) ∈ Rr>0

and k ∈ O(r) such that y = kd(a)tk. Then, for any v = (v1, . . . , vr) ∈ Vrσ1, thecurrent g(v · k, a; s) depends only on y, i.e. is independent of the choice of k and a.

Proof. Let

(4.46) 0 < λ1 < · · · < λr′

denote the distinct eigenvalues of y, and let mi denote the corresponding multiplicity.Given two decompositions y = k · a · tk = k′ · (a′) · tk′, the entries of a and a′ are

simply the eigenvalues of y, and so they differ by conjugation by some permutationmatrix σ ∈ Sr. Note that such a matrix acts on D × Rr>0 by permuting the latterfactors, and it is an immediate consequence of the definitions that

(4.47) σ∗ν0(v · σ; s) = ν0(v; s).

Thus, it suffices to assume that a = a′ = (λ1, . . . , λk), with each eigenvalue appearingwith some multiplicity, and so k′ = k · k′′ for some matrix of the form

(4.48) k′′ =

(θ1θr′

)∈ O(m1)× · · · ×O(mr′).

Write

(4.49) v · k = (v′1, . . . , v′m1︸︷︷︸

m1

, . . . , v′r−mr′+1, . . . , v′r︸︷︷︸

mr′

) = (v′1, . . . ,v′r′),

so that

(4.50) v · k′ = (v′1 · θ1, . . . ,v′r′ · θr).

It follows from the rotational invariance of ν0(·) and ϕ0(·), cf. Proposition 2.6.(f) andProposition 2.5.(f), that

γ∗a′ ν0(v · k′; s)

=r′∑i=1

λ−si ν0(λi√tv′i · θi)[2mi−2] ∧ ϕ0(λ1

√tv′1 · θ1, . . . , λr′

√tv′r′ · θr︸︷︷︸

i’th term omitted

)[2r−2mi]dt

ts+1

=r′∑i=1

λ−si ν0(λi√tv′i)[2mi−2] ∧ ϕ0(λ1

√tv′1, . . . , λr′

√tv′r′︸︷︷︸

i’th term omitted

)[2r−2mi]dt

ts+1(4.51)

= γ∗aν0(v · k; s),

which implies the lemma.

Given T ∈ Symr(F ), a Schwartz function ϕf ∈ S(V(Af )r)K , and a parametery ∈ Symr(R)>0, write y = k · d(a) · tk and consider the sum

(4.52) gj(T, y, ϕf ; s) =∑

v∈ΩT (V)

ϕf (h−1j v) g(σ1(v) · k, a; s);

the preceding lemma ensures the construction is independent of the decomposition ofy.


Proposition 4.7. (i) The sum (4.52) converges for Re(s) > 0 to a Γj-invariantcurrent on D, and has meromorphic continuation to Re(s) > −1/2. In particu-lar, the constant term in the Laurent expansion

gj(T, y, ϕf ) := CTs=0

gj(T, y, ϕf ; s)

descends to a current on Xj.(ii) Let g(T, y, ϕf ) be the current on XK whose restriction to Xj is gj(T, y, ϕf ).

Then

ddcg(T, y, ϕf ) + δZ(T,ϕf ) ∧ (−Ω)r−rk(T ) = ω(T, y, ϕf ).

Proof. We proceed as in the proof of Lemma 3.8. Let η ∈ A∗c(D) with compactsupport U . Write ΩT (V) = S1 t S2 with S1 = v ∈ ΩT (V)|minz∈U Qz(v,v) < 1, sothat, by Proposition 4.5, the sum

(4.53)∑v∈S1

ϕf (h−1j v)

∫Dg(σ1(v) · k, a; s)η

converges and has meromorphic continuation since S1 is finite. For the same sumwhere v now runs over S2 we use estimate (2.9), together with the standard argumentfor the convergence of theta series. For (ii), it suffices to prove the given identity,for each component Xj, at the level of Γj-invariant currents on D. The precedingestimates allow us to write

ddcgj(T, y, ϕf ) = ddc

(CTs=0

∑v

ϕ(h−1j v)g(v · k, a; s)

)=∑v

ϕ(h−1j v) ddc CTs=0 g(v · k, a; s)

(4.54)

and hence the result follows from Proposition 4.5.(iv).

Example 4.8. Consider a tuple v = (v′, 0, . . . , 0) ∈ Vrσ1 with v′ = (v1, . . . , vr′) ∈ Vr′σ1

non-degenerate. Then

ν0(v; s)[2r−2] = ν0(v′; s)[2r′−2] ∧ (−Ω)r−r′

+ ϕ0(√

t1v1, . . . ,√tr′vr′

)[2r′]∧ (−Ω)r−r

′−1 ∧

(r∑

i=r′+1

dti

ts+1i

).

(4.55)

Let a = (a1, . . . , ar) ∈ Rr>0 and set α = (a1, . . . , ar′). Then(4.56)

g(v, a; s) = g(v′,α; s)∧ (−Ω)r−r′+

r∑i=r′+1

a−si

(∫ ∞1

ϕ0(√t ·w)[2r′]

dt

ts+1

)∧ (−Ω)r−r

′−1,

where w = v′ · α 12 = (

√a1v1, . . . ,

√ar′vr′). As v′ is non-degenerate, the first term is

holomorphic at s = 0 by Proposition 4.5.(iii), and its value is the current g(v′ ·α 12 )∧


(−Ω)r−r′. For the second term, write

(4.57)

∫ ∞1

ϕ0(√t ·w)[2r′]

dt

ts+1=

∫ ∞1

(ϕ0(√t · v′ ·α

12 )[2r′] − δDv

) dt

ts+1+

1

sδDv .

The first term here converges at s = 0, using the estimate (4.34), and so

CTs=0

g(v, a; s) = g(v′ ·α12 ) ∧ (−Ω)r−r

′

−

(r∑

i=r′+1

log ai

)· δDv ∧ (−Ω)r−r

′−1 + γ(v′ ·α12 ),

(4.58)

where

(4.59) γ(v′ ·α12 ) := (r − r′)

∫ ∞1

(ϕ0(√t · v′ ·α

12 )[2r′] − δDv

) dtt.

Now consider a matrix of the form

(4.60) T =

(τ

0r−r′

)with τ ∈ Symr′(F ) non-degenerate.

Recall that we assume that V is anisotropic; in particular, any v ∈ Vr with T (v) = Tis of the form v = (v′, 0, . . . , 0) with T (v′) = τ .

Suppose further that ϕf = ϕ′f ⊗ ϕ′′f with ϕ′f ∈ S(V(Af )r′) and ϕ′′f ∈ S(V(Af )r−r

′),

and y ∈ Symr(R)>0 is of the form

(4.61) y =

(y′

y′′

), where y′ ∈ Symt(R)>0 and y′′ ∈ Symr−t(R)>0.

It follows from the above discussion that, after descending to the Shimura varietyXK , we have the equation of currents

g(T, y, ϕf ) = ϕ′′f (0)·[g(τ, y′, ϕ′f ) ∧ (−Ω)r−r

′ − log(det y′′)δZ(τ,ϕ′f ) ∧ (−Ω)r−r′−1

+

∫ ∞1

[ω(τ, ty′, ϕ′f )− δZ(τ,ϕ′f )

] dtt∧ (−Ω)r−r

′−1].

(4.62)

Since ω(τ, ty′, ϕ′f ) and δZ(τ,ϕ′f ) are cohomologous, the term appearing on the second

line above is exact, so that

g(T, y, ϕf ) =ϕ′′f (0)[g(τ, y′, ϕ′f ) ∧ (−Ω)r−r

′

− log(det y′′)δZ(τ,ϕ′f ) ∧ (−Ω)r−r′−1]∈ D∗(XK)/im∂ + im∂.

(4.63)

Finally, we note that for T = 0r, the above computation gives the pleasant expres-sion

(4.64) g(0r, y, ϕf ) = − log(det y)ϕf (0) · (−Ω)r−1.

5. Archimedean heights and derivatives of Siegel Eisenstein series

Here we prove Theorem 5.1, which now follows easily from the Siegel-Weil formula.

5.1. Siegel Eisenstein series and the Siegel-Weil formula.


5.1.1. Recall that we have fixed a totally real number field F of degree d and aCM extension E; we denote by η : F×\A×F → ±1 the corresponding quadraticcharacter. We set

(5.1) k =

F, case 1,E, case 2

and denote by σ1, . . . , σd the archimedean places of k. We fix an m-dimensionalhermitian k-vector space (V, Q) such that Vσi is positive definite when i > 1 and

(5.2) sig Vσ1 =

(p, 2), case 1,(p, q), case 2.

with pq 6= 0. From now on we assume that V is anisotropic. We also fix a unitarycharacter χ : E×\A×E → C× such that χ|A×F = ηm.

We let Mp2r(AF ) be the metaplectic double cover of Sp2r(AF ) and for a positiveinteger r we set

(5.3) G′r(A) =

Mp2r(AF ), case 1,

U(r, r)(AF ), case 2.

We denote by Pr(A) the standard Siegel parabolic ofG′r(A); we have Pr(A) = Mr(A)nNr(A), with

Mr(A) = (m(a), ε)|a ∈ GLr(AF ), ε = ±1,Nr(A) = (n(b), 1)|b ∈ Symr(AF )

(5.4)

in case 1 and

(5.5) Mr(A) = m(a)|a ∈ GLr(AE), Nr(A) = n(b)|b ∈ Herr(AE)

in case 2. Define a character χV of Mr(A) by(5.6)

χ ((m(a), ε)) =(det a, (−1)m(m−1)/2 detV

)A ·

ε · γA(det a, ψ)−1, if m is odd,

1, if m is even,

(here γA denotes the Weil index, see [21]) in case 1 and by χ(m(a)) = χ(det a) incase 2. We extend it to a character of Pr(A) whose restriction to Nr(A) is trivial and

set I(V, s) = IndG′r(A)Pr(A) (χ| · |s) (smooth induction), where the induction is normalized

so that s = 0 belongs to the unitary axis. We say that a section Φ(s) ∈ I(V, s) isstandard if its restriction to the standard maximal compact K ′r,A of G′r(A) is K ′r,A-finite and independent of s.

5.1.2. Given a standard section Φ(s) ∈ I(V, s) and g′ ∈ G′r(A), the sum

(5.7) E(g′,Φ, s) =∑

γ∈Pr(F )\G′r(F )

Φ(γg′, s)


converges for Re(s) 0 and admits meromorphic continuation to s ∈ C; the functionE(g′,Φ, s) is the Siegel Eisenstein series. It admits a Fourier expansion

(5.8) E(g′,Φ, s) =∑

T∈Xr(F )

ET (g′,Φ, s)

with Xr(F ) as in (4.24), and

(5.9) ET (g′,Φ, s) =

∫Nr(Q)\Nr(A)

E(n(b)g′,Φ, s)ψ(−tr(Tb))dn(b),

with dn(b) the Haar measure on Nr(AF ) that is self-dual with respect to the pairing(b, b′) 7→ ψ(tr(bb′)).

If detT 6= 0 and Φ =∏

v Φv, then ET (g′,Φ, s) =∏

vWT,v(g′v,Φv, s), where

(5.10) WT,v(g′v,Φv, s) =

∫Xr(Fv)

Φv(w−1r n(b)g′, s)ψ(−tr(Tn(b)))dn(b), Re(s) 0.

For nonsingular T , each WT,v(g′v,Φv, s) admits analytic continuation to s ∈ C, and

we can write

(5.11)d

dsET (g′,Φ, s)

∣∣∣∣s=s0

=∑v

ET (g′,Φ, s0)v,

where the sum runs over places of F and we set

(5.12) E ′T (g′,Φ, s0)v =∏w 6=v

WT,w(g′w,Φw, s0) · ddsWT,v(g

′v,Φv, s0)

∣∣∣∣s=s0

Let K ′r be the standard maximal compact subgroup of G′r(Fv) (v archimedean)described in 2.5.1 and l be the weight of K ′r in (3.47). If Φ∞ = Φl ⊗ · · · ⊗ Φl ∈I(V(R), s) is the normalized vector of parallel weight l, we set

(5.13) E(τ,Φf , s) :=c(s)

c(s0)(det y1 · · · yd)−ιm/4E(g′τ ,Φf ⊗ Φl

∞, s)

in case 1 and

(5.14) E(τ,Φf , s) :=c(s)

c(s0)

∏1≤i≤d

χσi(yi)−1(det y1 · · · yd)−ιm/4E(g′τ ,Φf ⊗ Φl

∞, s)

in case 2, where

(5.15) c(s) = π−ιrs/2Γr(ι(s− s0 +m)/2).

Note that the normalization factor c(s) is natural from the point of view of thefunctional equation of Eisenstein series (cf. [27, Lemma 4.6]). We define similarlyET (τ,Φf , s) and E ′T (τ,Φf , s)v.


5.1.3. Let ω = ωψ,χ be the Weil representation of G′r(AF )×U(V(A)) on S(V(AF )r).For φ ∈ S(V(AF )r), g′ ∈ G′r(AF ) and h ∈ U(V(A)), define the theta series

(5.16) Θ(g′, h;φ) =∑

v∈V(F )r

ω(g′, h)φ(v).

For any φ ∈ S(V(AF )r), the function Φ(g′) = ω(g′)φ(0) belongs to I(V, s0), wheres0 is given by (3.19) and (3.37); these maps define a G′r(A)-intertwining map λ :S(V(AF )r)→ I(V, s0). Since V is anisotropic, E(g′, λ(φ), s) is regular at s = s0 and

(5.17) E(g′, λ(φ), s0) = κ0

∫U(V(F ))\U(V(AF ))

Θ(g′, h;φ)dh,

where

(5.18) κ0 =

1, if s0 > 0,2, if s0 = 0,

and the Haar measure dh is normalized so that Vol(U(V(F ))\U(V(AF )) = 1; this isthe Siegel-Weil formula (see [15, 17, 27, 40]). Comparing the constant terms on bothsides shows that

(5.19) E0(g′, λ(φ), s0) = κ0 · ω(g′)φ(0).

5.2. Archimedean height pairings. In this section we assume that rk(E) = 1. Fora real number M ≥ 1 and a non-degenerate vector v ∈ Vr (r ≤ p+ 1), let

(5.20) g(σ1(v))≤M =

∫ M

1

ν0(t1/2σ1(v))[2r−2]dt

t

and g(T, y, ϕf )≤M be the differential form onXK defined in the same way as g(T, y, ϕf ),but replacing g(σ1(v)) with g(σ1(v))≤M . By Lemma 3.8.(b), we have

(5.21) g(T, y, ϕf )≤MM→+∞−−−−−→ g(T, y, ϕf )

as currents on XK .Let

(5.22) Vol(XK ,Ω) =

∫[XK ]

Ωp.

Note that, for any open compact subgroup K ⊂ H(Af ), the line bundle E onD descends to a positive line bundle L on XK with first Chern form Ω, and soVol(XK ,Ω) = degL(XK) is a positive rational number (a positive integer if K isneat).

Theorem 5.1. Assume that r ≤ p+ 1 and rk(E) = 1 and that detT 6= 0. If T is nottotally positive definite, then

(−1)rικ0

2Vol(XK ,Ω)

∫[XK ]

g(T, y1, ϕf ) ∧ Ωp+1−rqT = E ′T (τ, λ(ϕf ), s0)σ1 .


If T is totally positive definite, then

(−1)rικ0

2Vol(XK ,Ω)

∫[XK ]

g(T, y1, ϕf ) ∧ Ωp+1−rqT

= E ′T (τ, λ(ϕf ), s0)σ1 − ET (τ, λ(ϕf ), s0)ι

2log detσ1(T )

1

Proof. By (5.21), in both cases the left hand side equals the limit of ∫[XK ] g(T, y1, ϕf )≤M∧Ωp+1−r as M →∞. For t ≥ 1, define the Schwartz function

(5.23) φt = ω(m(t1/2 · 1r))ν ⊗ ϕ+ ⊗ · · · ⊗ ϕ+ ∈ S(Vr ⊗F R) = ⊗i≤i≤dS(Vrσi),

where ϕ+ is the Gaussian for Vrσi if i > 1 and ν ∈ S(Vrσ1) is given by (3.90). Considerthe theta series Θ(g′τ , ·;ϕf⊗φt) in (5.16) and its T -th Fourier coefficient ΘT (g′τ , ·;ϕf⊗φt). Note that, for z = hz0 ∈ D+ (h ∈ U(Vσ1)), we have

g(T, y1, ϕf )≤M(z) ∧ Ωp+1−r(z)qT

=

∫ M

1

e2πtr((t−1)σ1(y1T ))ΘT (g′τ , h;ϕf ⊗ φt)α(ty1, . . . , yd)dt

t∧ Ωp(z),

(5.24)

where we write

(5.25) α(ty1, . . . , yd) = (det ty1 · · · yd)−ιm/4 ·

1, case 1,∏1≤i≤d χσi(yi)

−1, case 2.

Applying the Siegel-Weil formula (5.17) shows that∫[XK ]

ΘT (g′τ , h;ϕf ⊗ φt)Ωpdt

t= cET (g′τ , λ(ϕf ⊗ φt), s0)(5.26)

for a constant c. To determine this constant we consider the term corresponding toT = 0: we have

(5.27)

∫[XK ]

Θ0(g′, h;ϕf ⊗ φt)Ωp = ω(g′)(ϕf ⊗ φt)(0)

∫[XK ]

Ωp;

by (5.19), this equals cE0(g′, λ(ϕf ⊗ φt), s0) = cκ0ω(g′)(ϕf ⊗ φt)(0), and we concludethat c = κ−1

0 Vol(XK ,Ω). Using (3.43) and writing ϕf⊗φt = ω(m(t1/2 ·1r))ν⊗v 6=σ1 φv,we find that ∫ M

1

e2πtr((t−1)σ1(y1T ))ET (g′, λ(ϕf ⊗ φt), s)α(ty1, . . . , yd)dt

t

= α(y1, . . . , yd)∏v 6=σ1

WT,v(g′v, λ(φv), s)

·∫ M

1

e2πtr((t−1)σ1(y1T ))WT,σ1(gτ1 , λ(φt), s)t−ιrm/4dt

t.

(5.28)

For i ≥ 2 we have λ(φσi) = λ(ϕ+) = Φl(s0). By (3.93) we have λ(φt) = (−1)r−1r(m(t1/2·1r))Φ(s0). The limit of the integral as M → ∞ can be computed using Lemma 3.6and (3.89).


Note that the sum in Theorem 4.4 runs over non-singular matrices T by the hy-pothesis on ϕf,1⊗ϕf,2. In particular, if r1 + r2 = p+ 1, then all such T are indefinite.Thus Theorem 5.1 has the following corollary generalizing the main result of [23] toarbitrary compact orthogonal Shimura varieties (for U(p, 1) this corollary is due toLiu [33]).

Corollary 5.2. Assume that rk(E) = 1 and r1 + r2 = p + 1. Under the hypothesesof Theorem 4.4, we have

(−1)p+1ι

Vol(XK ,Ω)〈g(T1, y1, ϕf,1) ∗ g(T2, y2, ϕf,2), 1〉qT11 q

T22

=∑

T∈Xr(F )

T=(T1 ∗∗ T2

)E ′T (( τ1 τ2 ) , λ(ϕf,1 ⊗ ϕf,2), 0)σ1 .

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