Geometric Aspects of the Complex Heisenberg Group

Maurizio Candilera and Valentino Cristante

Dipartimento di Matematica Pura e Applicata Via Belzoni, 7 — 35131 PADOVA (Italy) e-mail: candilera@pdmat1.unipd.it or cristanv@pdmat1.unipd.it

Introduction.

In his article Groupe de Heisenberg et Réalité, P. Deligne relates the Heat Equation with the irreducible representation of the real Heisenberg group ℂ_{1}^{∗} × ℝ^{2}. First he defines a holomorphic vector bundle over the projective line, W, on which the complex Heisenberg group ℂ^{∗} × ℂ^{2} acts. Then, by using W, he proves that the space Γ of solutions of the Heat Equation on an open disc is naturally endowed with a pre-Hilbert space structure and that its completion, H_{1}, is the irreducible representation of ℂ_{1}^{∗} × ℝ^{2}. Moreover he shows that the Metaplectic group acts on Γ, and that its action extends to H_{1}, normalizing the action of ℂ_{1}^{∗} × ℝ^{2}.

The present paper is motivated by the desire to understand the Deligne’s ideas and to see how they apply to the analytic theory of the abelian varieties. In order to reach this goal, most of the results of the original paper are extended to the case of many variables, and the place occupied by the Riemann theta function in this picture is brought out. A pleasant feature of this point of view is that all the representations of the real Heisenberg group, used to obtain the Riemann theta function as a matrix coefficient (cf. [T3]), arise naturally from the regular right representation of the complex Heisenberg group.

Even though not treated here, there is another point which should be remarked: the Lie algebra of the universal vectorial extension of a polarized abelian scheme over the integers of a p-adic field has a non-degenerate alternating form. Thus the method used here for the complex case seems to give good suggestions for an analytic theory in a p-adic context. We hope to return soon on these arguments.

The paper consists of four sections whose contents can be summarized as follows:

1. Starting from a complex vector space V, of dimension 2n, endowed with a non degenerate alternating form ⟨−, −⟩, and taking the associated complex Heisenberg group H, we define a holomorphic vector bundle W over the manifold S of the lagrangian sub-spaces of V. The fibers of W are spaces of functions on H and H acts on these fibers by means of the regular right representation. We then twist W with the line bundle Det^{− \frac{1}{2}} and define a connection on the twisted bundle. This connection commutes with the action of H and coincides locally with the Heat Equations.

2. We consider the family of real Heisenberg groups coming from the maximal real subspaces of V on which ⟨−, −⟩ assumes real values. By restricting the previous representations of H to the groups of this family, in view of the uniqueness of the irreducible representation of a real Heisenberg group, we can define a connection on the projective bundle ℙ(W). This turns out to be the projectivization of the connection defined in section 1. In this context, the Riemann theta function appears as a horizontal section of W.

3. We consider solutions of Heat Equations on (symplectic) open discs, showing how the radius of the disc can be estimated by the growth of the value of the solution in the center of the disc. This allows us to characterize the germs of the horizontal sections of W, and to define a pre-Hilbert pairing on the space Γ of horizontal sections on a disc. The completion of this space, H_{1}, is shown to be an irreducible (Fock) representation of a group of the family introduced in section 2.

4. We show how the action of the real symplectic group on a disc lifts to the action of its double cover, Mp(2n, ℝ), on Γ, inducing the classical action on the irreducible representation of the real Heisenberg group. In this context the Riemann theta function appears as a section invariant, up to a constant, under the action (of the double cover) of Γ_{1,2}.

1. A representation of the complex Heisenberg group.

We start with the following

1.1 Definition. Given a complex vector space V of dimension 2n, endowed with a non-degenerate alternating bilinear form ⟨−, −⟩, the complex Heisenberg Group over V, H = H_{V}, is the group-extension of V by the multiplicative group ℂ^{∗}, whose composition law is

(λ, v)(λ′, v′) = (λλ′ exp(−πi⟨v, v′⟩), v + v′),

for any λ, λ′ ∈ ℂ^{∗}, v, v′ ∈ V.

1.2 Remarks. (a). Let V and ⟨−, −⟩ be as above, we recall that a symplectic basis of V is a basis for which the alternating form ⟨−, −⟩ has matrix ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). If one chooses a symplectic basis, H_{V} is identified with the standard Heisenberg Group H_{ℂ^{2n}}. The group law of H_{ℂ^{2n}} is

(λ, x_{1}, x_{2})(μ, y_{1}, y_{2}) = (λμ exp[πi(^{t}x_{1}y_{2} −^{t}x_{2}y_{1})], x_{1} + y_{1}, x_{2} + y_{2}),

for any x_{1}, x_{2}, y_{1}, y_{2} ∈ ℂ^{n} and λ ∈ ℂ^{∗}. The standard Heisenberg group acts on the space O(ℂ^{n}) of holomorphic functions on ℂ^{n} by the rule

(U_{(λ,x,y)}f)(z) = λ exp πi(2^{t}yz +^{t}yx)f(z + x),

for any (λ, x, y) ∈ H_{ℂ^{2n}} and f ∈ O(ℂ^{n}). This action is called the Schrödinger action. (b). Given a symplectic basis of V, the real subspace L spanned by that basis is naturally endowed with a non-degenerate, real-valued alternating form, so that the subgroup H_{L} = ℂ_{1}^{∗}.L is a real Heisenberg group. (c). In what follows we shall use the same symbol D to denote either a maximal isotropic subspace of V or the corresponding level subgroup { (1, d) | d ∈ D } of H.

1.3 Notations. Let G(n, V) be the Grassmannian variety of the n-dimensional subspaces of V, and denote by S the subvariety of G(n, V) whose points are isotropic subspaces of V. As E varies in S, the open sets U′_{E} = { D ∈ G(n, V) | V = E ⊕ D } form a covering of G(n, V). If U ∈ U′_{E} is fixed, then for any other D in U′_{E} there exists a unique ϕ_{D} ∈ Hom(U, E) such that D = { u + ϕ_{D}(u) | u ∈ U }. The correspondence D ↦ ϕ_{D} is a bijection, so that after a choice of bases for U and E, the entries of the matrix T_{D} of ϕ_{D} become coordinates on U′_{E}. In particular, if the subspaces E and U are both in S, once the basis {e_{1}, …, e_{n}} of E is fixed, we shall choose the dual basis {e_{n+1}, …, e_{2n}} of U, which is determined by the conditions ⟨e_{n+i}, e_{j}⟩ = δ_{ij} for i, j = 1, …, n. After this choice, we get a symplectic identification of V with ℂ^{2n} and we shall write E = ( \begin{matrix}1 \\ 0 \end{matrix} ) and D = ( \begin{matrix}−T_{D} \\ 1 \end{matrix} ). More generally, we associate to the matrix

( \begin{matrix}T_{1} \\ T_{2} \end{matrix} ) the subspace of ℂ^{2n} spanned by its columns. Finally we observe that D = ( \begin{matrix}T \\ 1 \end{matrix} ) belongs to U_{E} := U′_{E} ∩ S if, and only if, T is a symmetric matrix (^{t}T = T).

In what follows we’ll attribute the affine coordinate T to the point ( \begin{matrix}−T \\ 1 \end{matrix} ) ∈ U_{E} .

1.4 Remark. The objects described above have a natural interpretation in a geometric context. Let us give a brief sketch. Let V and ⟨−, −⟩ be as above and consider the n-dimensional subspaces D of V for which:

(i) there exists a free abelian group Λ spanned by a symplectic basis of V, such that its image in the quotient V/D is a lattice Λ_{D};

(ii) the extension by ℝ-linearity of the alternating form induced by ⟨−, −⟩ on Λ_{D} is the imaginary part of a positive definite hermitian form h on V/D. Once a lattice Λ as above is fixed, it’s well known that for any one of the subspaces

D satisfying the above conditions relative to Λ, the quotient B_{D} of the vector space V/D by the lattice Λ_{D} is a principally polarized abelian variety with an isomorphism Λ ≅ H_{1}(B_{D}, ℤ), which identifies the bilinear form ⟨−, −⟩ with the (Chern) class in H^{2}(B_{D}, ℤ) of any line bundle on B_{D} associated to the hermitian form h. Finally, the quotient V/Λ is canonically isomorphic to the maximal non-trivial vectorial extension H_{1}(B_{D}, ℂ)/H_{1}(B_{D}, ℤ) of any torus B_{D} of the family determined by Λ (cf. [MM] Cor. 4.4.2).

Let {e_{1}, …, e_{2n}} be a symplectic basis of Λ; then in the frame given by the subspaces E and U spanned by {e_{1}, …, e_{n}} and {e_{n+1}, …, e_{2n}}, respectively, the subspaces D ∈ U_{E} satisfying (i) and (ii) relative to Λ, are exactly those with affine coordinates Ω, where Ω ∈ 𝔖_{n}, the Siegel upper half-space of degree n (cf. [T1], Ch.II, Lemma 4.1). Finally, by observing that the symplectic group Sp(V) acts transitively on S, we can conclude that any element of S produces a principally polarized abelian variety for a suitable lattice Λ in V.

Our goal now is to construct a vector bundle W, over S, of representations of H. The fiber of W over D ∈ S, is the complex vector space W_{D} of the holomorphic functions ϕ: H → ℂ satisfying the condition ϕ(λdh) = λϕ(h) for any λ ∈ ℂ^{∗}, d ∈ D and h ∈ H; and the action of H on W_{D} is given by the regular right representation. The structure of W is described in the following

1.5 Proposition. Notations as above.

(a) Let E ∈ S. For any D ∈ U_{E}, there exist isomorphisms φ_{D}: H → H_{ℂ^{2n}} and ρ_{D}: W_{D} → O(ℂ^{n}) which fit into the commutative diagram

H × W_{D}  ⟶^{R} W_{D}
↓^{φ_{D}×ρ_{D}}  ↓^{ρ_{D}}
H_{ℂ^{2n}} × O(ℂ^{n}) ⟶^{U} O(ℂ^{n}),

where R denotes the regular right action of H on W_{D} and U denotes the Schrödinger action (cf. Remark 1.2.(a)).

(b) The collection of spaces W_{D} has a structure of holomorphic vector bundle, W, such that W|_{U_{E}} ≅ U_{E} × O(ℂ^{n}) by means of the maps ρ_{D}, as E varies in S. W is a bundle of representations.

Proof. (a). Denote the space of the holomorphic functions on E by O(E) and consider the isomorphism r_{E}: W_{D} → O(E) induced by the restriction to E of the functions on H (cf. Remark 1.2.(c)). Its inverse map can be written explicitly as (r_{E}^{−1}f)(λ, d+e) = (r_{E}^{−1}f)(λ exp(πi⟨d, e⟩)(1, d)(1, e)) = λ exp(πi⟨d, e⟩)f(e), for f ∈ O(E).

Now fix a symplectic basis of V, {e_{1}, …, e_{2n}}, in such a way that {e_{1}, …, e_{n}} spans E and denote the subspace generated by {e_{n+1}, …, e_{2n}} by U. Then for any D ∈ U_{E}, the corresponding map ϕ_{D} ∈ Hom(U, E) produces the basis (d_{i} = e_{n+i} + ϕ_{D}(e_{n+i}))_{i=1,...,n}, dual to {e_{1}, …, e_{n}}. After these choices, the isomorphisms ρ_{D} and φ_{D} are defined by

ρ_{D}(ϕ)(z) = ϕ(1, ∑_{i=1}^{n} z_{i}e_{i}) and φ_{D}(λ, ∑_{i=1}^{n} x_{i}e_{i} + ∑_{i=1}^{n} y_{i}d_{i}) = (λ, x, y).

Now, let e′ = ∑_{i=1}^{n} z_{i}e_{i}, e = ∑_{i=1}^{n} x_{i}e_{i}, d = ∑_{i=1}^{n} y_{i}d_{i} and h = (λ, e + d); then, recalling the definition of the Schrödinger action (cf. Remark 1.2.(a)), by a direct computation one has

\begin{matrix}ρ_{D}(R_{h}ϕ)(z) = & R_{h}ϕ(1, e′) = ϕ((1, e′)(λ, e + d)) = \\ = & ϕ(λ exp[−πi(2⟨e′, d⟩ + ⟨e, d⟩)](1, d)(1, e′ + e)) = \\ = & λ exp πi(2^{t}yz +^{t}yx) ϕ(1, e′ + e) = [U_{φ_{D}(h)}ρ_{D}(ϕ)](z). \end{matrix}

(b) To prove this point we must show that, for any E, E′ ∈ S, the map D ↦ ρ_{D} ∘ρ′_{D}^{−1} defined on U_{E} ∩ U_{E′} with values in Aut(O(E)), is holomorphic (ρ′_{D} is obtained from the couple E′, D in the same way as ρ_{D} was obtained from E, D).

In order to compute ρ_{D} ∘ρ′_{D}^{−1} let’s identify V with ℂ^{2n} by means of {e_{1}, …, e_{2n}}. Let E = ( \begin{matrix}1 \\ 0 \end{matrix} ), U = ( \begin{matrix}0 \\ 1 \end{matrix} ), and assume E′ = ( \begin{matrix}E_{1} \\ E_{2} \end{matrix} ); then if D = ( \begin{matrix}−T \\ 1 \end{matrix} ), its basis

dual to the columns of E′ is given by the columns of the matrix ( \begin{matrix}−T^{t}(E_{1}+T E_{2})^{−1} \\ ^{t}(E_{1}+T E_{2})^{−1} \end{matrix} ) (D ∈ U_{E′} implies that E_{1} + T E_{2} ∈ GL(n, ℂ)). Now let (1, E′z) ∈ E′ ⊂ H, then

E′z = ( \begin{matrix}E_{1} \\ E_{2} \end{matrix} ) z = ( \begin{matrix}−T \\ 1 \end{matrix} ) u + ( \begin{matrix}1 \\ 0 \end{matrix} ) v, where u = E_{2}z and v = (E_{1} + T E_{2})z. From this

decomposition it follows that

ρ′_{D}(ρ_{D}^{−1}(f))(z) = ρ_{D}^{−1}(f)(1, E′z) = exp(πi^{t}z^{t}E_{2}(E_{1} + T E_{2})z) f((E_{1} + T E_{2})z),

and this shows that the relation between f and ρ′_{D}(ρ_{D}^{−1}(f)) is holomorphic with respect to T.

The same holds for the automorphism φ′_{D} ∘ φ_{D}^{−1} of H_{ℂ^{2n}}: in fact by direct calculation, one sees that it induces the identity on ℂ^{∗} and acts on ℂ^{2n} by the symplectic matrix

( \begin{matrix}(E_{1} + T E_{2})^{−1} & 0 \\ −^{t}(E_{1} + T E_{2})E_{2}(E_{1} + T E_{2})^{−1} & ^{t}(E_{1} + T E_{2}) \end{matrix} ) ,

so that also the family φ′_{D} ∘ φ_{D}^{−1} is holomorphic as D varies in U_{E} ∩ U_{E′}. The last claim follows from the commutativity of the above diagram. QED

The structural isomorphisms defining W are linear substitutions in the variables followed by multiplication by a quadratic exponential. Thus, for v ∈ W_{D} the condition

(1.6)  ∃A such that |r_{E}(v)(z)| ≤ O(exp A|z|^{2}),

doesn’t depend on the choice of E or on the choice of the coordinate z on E. So we can state the following

1.7 Corollary. The subspaces W_{D}^{0} of W_{D} consisting of the functions satisfying the above condition (1.6), form a sub-vector bundle W^{0} of W.

Our next task is to prove that the vector bundle W, after a suitable twist, carries a connection ∇, for which the group H acts horizontally. We denote by Det the restriction to S of the n^{th} exterior power of the tautological vector bundle on G(n, V) and by Det^{\frac{1}{2}} one of its square roots, i.e. a line bundle L with a fixed isomorphism L^{⊗2} ≅ Det. The remaining part of this section is devoted to proving the following

1.8 Theorem. Let W(−\frac{1}{2}) := W ⊗ Det^{− \frac{1}{2}} , then there exists a connection

∇: W(−\frac{1}{2}) → W(−\frac{1}{2}) ⊗ Ω_{S}^{1}

for which the group H acts horizontally.

Proof. We shall carry out the construction of ∇ in three steps:

(i) first we fix E and for any D ∈ U_{E} we identify the tangent space T_{D}(S) = Hom_{ℂ}^{sym}(D, V/D) with Hom_{ℂ}^{sym}(D, E); then to each t ∈ Hom_{ℂ}^{sym}(D, E) we associate the Kodaira deformation of W_{D} and the corresponding connection ∇^{E} on W|_{U_{E}};

(ii) we explain how t acts on Det and Det^{− \frac{1}{2}} and so we define a connection ∇^{E} on

W(−\frac{1}{2})|_{U_{E}} ; (iii) finally we show that, given E and E′ in S, the connections ∇^{E} and ∇^{E′} coincide

on W(−\frac{1}{2})|_{U_{E}∩U_{E′}}.

Let’s start with the first step. Given E and D as in (i), we choose a basis {e_{1}, …, e_{n}} of E and denote by {d_{1}, …, d_{n}} the dual basis of D; then we identify t ∈ T_{D}S with the element t_{E} ∈ Hom(D, E), defined by

t_{E}(d_{j}) = ∑_{i=1}^{n}⟨d_{i}, t(d_{j})⟩e_{i} for j = 1, …, n.

The symmetry condition implies that t_{E} coincides with its adjoint, so that the matrix T = (T_{ij}) defined by T_{ij} = ⟨d_{i}, t(d_{j})⟩ is symmetric. Recalling that the tangent space to H at the identity is ℂ × V, we shall denote by δ_{v} the right-invariant vector field on H determined by the tangent vector (0, v); then

σ = \frac{1}{4πi} ∑_{1≤i,j≤n} T_{ij}δ_{e_{i}}δ_{e_{j}}  (1.9)

is a symmetric operator on W_{D}. Observing that [δ_{u}, δ_{v}] = 2πi⟨u, v⟩δ_{(1,0)}, by a direct computation one checks that [−σ, δ_{d_{h}}] = δ_{t_{E}(d_{h})}, for h = 1, …, n, and then, by linearity, one gets

[−σ, δ_{d}] = δ_{t_{E}(d)}, for any d ∈ D.  (1.10)

Now let ϕ ∈ W_{D}; assuming ϵ^{2} = 0, we prove that ϕ − ϵ(σϕ) ∈ W_{D+ϵt}. This, in view of the definition of W_{D}, amounts to verifying that

δ_{(1,0)}(ϕ − ϵ(σϕ)) = (ϕ − ϵ(σϕ)) and δ_{d+ϵt_{E}(d)}(ϕ − ϵ(σϕ)) = 0, for all d ∈ D.

The first check is immediate because [δ_{(1,0)}, δ_{v}] = 0, for any v ∈ V; the second one is a consequence of (1.10); in fact,

δ_{d+ϵt_{E}(d)}(ϕ − ϵ(σϕ)) = ϵδ_{t_{E}(d)}(ϕ) − ϵ[−σ, δ_{d}]ϕ = 0.

The connection ∇^{E} on W|_{U_{E}} is defined by

(∇_{t}^{E}(μ))(D) = (∂_{t}μ)(D) + σ(μ(D)),

for every section μ and every D ∈ U_{E}.

We now turn to (ii). The line bundle Det on S is the inverse image of O(1) under (the restriction to S of) the Plücker embedding p: G(n, V) → ℙ(Λ^{n}V). The O(U_{E})-module of its sections over U_{E} is generated by the section D ↦ (d_{1} ∧ ⋯ ∧ d_{n}); as a consequence, ∇^{E} is uniquely defined on the restriction of Det and Det^{− \frac{1}{2}} to U_{E}, by setting ∇_{t}^{E}(d_{1} ∧ ⋯ ∧ d_{n}) = ∑_{j=1}^{n} d_{1} ∧ ⋯ ∧ t_{E}(d_{j}) ∧ ⋯ ∧ d_{n}.

If μ ⊗ s is a section of W(−\frac{1}{2}) over U_{E} we put

∇_{t}^{E}(μ ⊗ s) = ∇_{t}^{E}μ ⊗ s + μ ⊗ (∇_{t}^{E}s).

Finally we prove (iii). Let E′ ∈ U_{D} and {e′_{1}, …, e′_{n}} be its basis dual to {d_{1}, …, d_{n}}; then there exists α ∈ Hom(E, D) such that e′_{i} = e_{i} + α(e_{i}), i = 1, …, n. Since E and E′ are isotropic, as previously observed, the matrix A = (a_{ij}) of α, with respect to the basis (e_{i}), (d_{i}), is symmetric. Now let’s compute δ_{e′_{i}}δ_{e′_{j}} as an operator on the space W_{D}:

\begin{matrix}δ_{e′_{i}}δ_{e′_{j}} = & (δ_{e_{i}} + ∑_{h=1}^{n} a_{hi}δ_{d_{h}})(δ_{e_{j}} + ∑_{k=1}^{n} a_{kj}δ_{d_{k}}) = \\ = & δ_{e_{i}}δ_{e_{j}} + ∑_{h} a_{hi}[δ_{d_{h}}, δ_{e_{j}}] = \\ = & δ_{e_{i}}δ_{e_{j}} + 2πi ∑_{h} a_{hi}⟨d_{h}, e_{j}⟩ = δ_{e_{i}}δ_{e_{j}} + 2πia_{ji}. \end{matrix}

So, if σ′ is built up from the δ_{e′_{i}}’s in the same way as σ was built from the δ_{e_{i}}’s, by comparing the two operators on W_{D}, we get

(1.11)  σ′ = σ + \frac{1}{2} ∑_{1≤i,j≤n} T_{ij}a_{ji} = σ + \frac{1}{2}tr(t_{E}α).

Now, since t_{E′} = (1 + α) ∘ t_{E}, we get

\begin{aligned}∇_{t}^{E′}(d_{1} ∧ ⋯ ∧ d_{n}) & = ∇_{t}^{E}(d_{1} ∧ ⋯ ∧ d_{n}) + ∑_{j=1}^{n} d_{1} ∧ ⋯ ∧ α(t_{E}(d_{j})) ∧ ⋯ ∧ d_{n} = \\ & = ∇_{t}^{E}(d_{1} ∧ ⋯ ∧ d_{n}) + tr(αt_{E})(d_{1} ∧ ⋯ ∧ d_{n}), \end{aligned}

and more generally, if f is a section of Det over U_{E} ∩ U_{E′},

(1.12)  \begin{aligned}(∇_{t}^{E′}f^{− \frac{1}{2}} ) & = −\frac{1}{2}f^{− \frac{3}{2}} (∇_{t}^{E′}f) = \\ & = (∇_{t}^{E}f^{− \frac{1}{2}} ) − \frac{1}{2}tr(αt_{E})f^{− \frac{1}{2}} . \end{aligned}

By applying (1.11) and (1.12) to a section μ ⊗ s of W(−\frac{1}{2}) over U_{E} ∩ U_{E′}, we conclude that

∇_{t}^{E′}(μ ⊗ s) = (∂_{t}μ + σ′μ) ⊗ s + μ ⊗ (∇_{t}^{E′}s)) =
= (∂_{t}μ) ⊗ s + (σμ) ⊗ s + μ ⊗ (∇_{t}^{E}s)) =
= ∇_{t}^{E}(μ ⊗ s).

Finally, the connection ∇ commutes with the action of the group H because the operators used in its construction are right-invariant. QED

1.13 Remarks. (a). After the identification of W(−\frac{1}{2})|_{U_{E}} with O(U_{E} × ℂ^{n}), described in Proposition 1.5 (b), the invariant vector fields δ_{e_{i}} become the partial derivatives ∂_{z_{i}} of O(ℂ^{n}), so that the sections horizontal with respect to ∇ are the solutions of the system of the heat equations (∂_{t_{ij}} − \frac{1}{4πi}∂_{z_{i}}∂_{z_{j}})g = 0 for 1 ≤ i ≤ j ≤ n (cf. Notations 1.3 for the choice of affine coordinates on U_{E}).

(b). With the notations of Remark 1.4, let A = \{ B_{D} | D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) , Ω ∈ 𝔖_{n} \} be the family of principally polarized abelian varieties related to the lattice Λ spanned by {e_{1}, …, e_{2n}} and denote by ϑ = ϑ(Ω, z) its Riemann theta function. Then the deformation ϑ ↦ ϑ − ϵ(σϑ) induces on A the deformation given by t_{E} in the sense of Kodaira (cf. [KS], §14, (γ) and (δ)).

2. Building horizontal sections.

In this section we consider maximal real Heisenberg subgroup of H. Any such group G, defines an open subset X ⊂ S, with the property that for any D ∈ X, there exists an irreducible representation of G, F_{D} ⊂ W_{D}. Thus, as a consequence of the Stone-Von Neumann Theorem, W_{|X} carries a projective connection. Our goal is to explain the relation between this connection and the connection ∇ on W(−\frac{1}{2}) defined in the previous section.

We start by fixing some notations regarding the real subgroups of H.

2.1 Definition. A real structure on the complex Heisenberg group H is a real subspace L of V spanned by a symplectic basis.

For later use let’s recall some elementary and well known results about real structures.

2.2 Proposition. Let L be a real structure on H. Then one has:

(a) dim_{ℝ} L = 2n and L ∩ iL = {0};

(b) the restriction of ⟨−, −⟩ to L is an alternating form with real values;

(c) the subgroup H_{L} := ℂ_{1}^{∗}.L is a real Heisenberg group;

(d) the decomposition of real vector spaces V = L⊕iL induces a complex conjugation σ_{L}: V → V defined by setting σ_{L}(l_{1} + il_{2}) = l_{1} − il_{2} for any l_{1}, l_{2} ∈ L.

Now we define a subset of S canonically associated to a real structure L on H.

2.3 Definition. The open disc associated to the real structure L is the subset

X_{L} = \{ D ∈ S | \frac{i}{2}⟨D, σ_{L}D⟩ < 0 \},

where \frac{i}{2}⟨D, σ_{L}D⟩ < 0 means that the pairing \frac{i}{2}⟨d, σ_{L}d′⟩, for d, d′ ∈ D, is a negativedefinite hermitian form.

The real structures and the related open discs can be described by means of coordinates. In fact let’s start with a fixed symplectic basis {e_{1}, …, e_{2n}} of V and denote by L the real structure spanned by that basis; then, if Sp(V) denotes the group of symplectic automorphisms of V and Sp(L) its subgroup which stabilizes L, the set of all the real structures on H can be parameterized by Sp(V)/Sp(L). More explicitly, if V is identified with ℂ^{2n} by means of {e_{1}, …, e_{2n}}, then the real structure and its related disc corresponding to (the coset of) a matrix Γ ∈ Sp(2n, ℂ) are:

L_{Γ} = \{ Γx | x ∈ ℝ^{2n} \} ⊂ ℂ^{2n} and X_{L_{Γ}} = \{ Γ ( \begin{matrix}−Ω \\ 1 \end{matrix} ) | Ω ∈ 𝔖_{n} \} .

Finally, the complex conjugation σ_{L_{Γ}} is given by σ_{L_{Γ}}(Γz) = Γz̄, for any z ∈ ℂ^{2n}.

Now it’s clear that the well-known result describing the action of Sp(2n, ℝ) on Siegel upper half-space can be rephrased as follows:

2.4 Proposition. Let L be a real structure on H; then the group Sp(L) acts transitively on the open disc X_{L} and the stabilizer of any point is isomorphic to the group of unitary transformations in ℂ^{n}.

Here is a first result connecting W with the irreducible representation of a real subgroup of H.

2.5 Proposition. Let L be a real structure on H and let D ∈ S be such that D ∩ L = 0. Given E ∈ U_{D}, denote by λ : E → D the isomorphism defined by e + λ(e) ∈ L, for any e ∈ E, and consider the following norm on O(E):

(2.6)  ∥f∥_{E} = ( ∫_{E} |f(e) exp πi⟨λ(e), e⟩|^{2} dx)^{\frac{1}{2}} ,

where dx denotes the Haar measure on E. Then:

(a) The action of the real group H_{L} on O(E) is unitary.

(b) Let r_{E}: W_{D} → O(E) be the usual isomorphism of restriction, and define ∥ϕ∥_{D} = ∥r_{E}ϕ∥_{E}, for any ϕ ∈ W_{D}; then ∥ϕ∥_{D} is independent of the choice of E. In particular, if D ∈ X_{L}, the subspace F_{D} ⊂ W_{D}^{0} of elements with finite norm is

a Hilbert space on which the action of H_{L} is irreducible.

Proof. (a). Denote by U the action of H on O(E) which renders r_{E} an isomorphism of representations. Given f = r_{E}ϕ ∈ O(E) and (τ, l) ∈ H_{L}, we must prove that

∥U_{(τ,l)}f∥_{E}^{2} = ∫_{E} |U_{(τ,l)}f(e) exp πi⟨λ(e), e⟩|^{2} dx = ∥f∥_{E}^{2}.

Let l = λ(e′) + e′ be the decomposition of l ∈ L, relative to the sum D ⊕ E, then one has

\begin{matrix}U_{(τ,l)}f(e) = & [r_{E}^{−1}f]((1, e)(τ, λ(e′) + e′)) = \\ = & [r_{E}^{−1}f](τ exp πi⟨λ(e′), e′ + 2e⟩(1, λ(e′))(1, e + e′)) = \\ = & τ exp πi⟨λ(e′), e′ + 2e⟩ f(e + e′). \end{matrix}

Now by observing that τ ∈ ℂ_{1}^{∗} and ⟨λ(e′), e⟩ − ⟨λ(e), e′⟩ = ⟨λ(e′) + e′, λ(e) + e⟩ ∈ ℝ, the previous integral may be rewritten as follows:

∥U_{(τ,λ(e′)+e′)}f∥_{E}^{2} = ∫_{E} |f(e + e′) exp πi⟨λ(e + e′), e + e′⟩|^{2} dx,

and so it coincides with ∥f∥_{E}^{2} since the measure is invariant by translations. (b). Let f = r_{E}ϕ, in order to check that ∥r_{E}ϕ∥_{E} doesn’t depend on E, it suffices to observe that

f(e) exp πi⟨λ(e), e⟩ = ϕ(1, λ(e) + e),

and actually this shows that ∥r_{E}ϕ∥_{E} is just the L^{2}-norm of the restriction of ϕ to L.

To prove the last claim we make use of coordinates. Let’s fix a symplectic basis of

L so that L = ℝ^{2n}, D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) ∈ X_{L} and E = ( \begin{matrix}1 \\ 0 \end{matrix} ); then λ(x_{1}+Ωx_{2}, 0) = (−Ωx_{2}, x_{2}),

for any (x_{1}, x_{2}) ∈ L, so that

∥f∥^{2} = ∫_{ℝ^{2n}} |f(\underline{x})|^{2} exp πi^{t}x_{2}(Ω − \overline{Ω})x_{2} dx_{1} dx_{2},  (2.7)

where \underline{x} = x_{1} + Ωx_{2}. This expression shows that the subspace of elements of O(E) of finite norm, endowed with the action U of H_{L}, is the Fock Representation of the real group H_{L}, which is well-known to be irreducible (cf. [T3], §.2). Finally the convergence of the last integral implies that f satisfies the condition (1.6) defining the subspace W_{D}^{0}. QED

2.8 Remark. For any D ∈ S denote by W_{D}^{−} the vector space of holomorphic functions ϕ on H, such that ϕ(λdh) = λ^{−1}ϕ(h), for any λ ∈ ℂ^{∗}, d ∈ D and h ∈ H. Assume that H acts on W_{D}^{−} by means of the regular right representation. Then, using the arguments of §.1, we can see that W^{−} := (W_{D}^{−})_{D∈S} has a structure of holomorphic vector bundle of representations over S, and that W(−\frac{1}{2})^{−} := W^{−} ⊗ Det^{− \frac{1}{2}} carries a connection commuting with the action of H.

The conjugation σ_{L} on V gives an antilinear isomorphism between W(−\frac{1}{2})^{−} and W(−\frac{1}{2}) defined by σ_{L}ϕ(h) = \overline{ϕ(σ_{L}h)}, where σ_{L}h denotes the image of h by the automorphism of H defined by σ_{L}(λ, v) = (\overline{λ^{−1}}, σ_{L}v). Observing that R_{g}(σ_{L}ϕ) = σ_{L}(R_{σ_{L}g}ϕ), we conclude that ϕ ↦ σ_{L}ϕ is an isomorphism of representations.

Each W_{D}^{−}, like W_{D}, possess a Fock norm (cf. Proposition 2.5) obtained by taking the L^{2}-norm of the functions restricted to L, and its subspace of elements with finite norm, F_{D}^{−}, is an irreducible representation of H_{L}. By restricting σ_{L} to F_{D} we get a unitary isomorphism σ_{L}: F_{D} → F_{σ_{L}D}^{−}, thus the Hilbert pairing on F_{D} gives a non degenerate bilinear paring (−, −): F_{D} × F_{σ_{L}D}^{−} → ℂ.

By Proposition 2.5, for any D ∈ X_{L} there exists the irreducible representation F_{D} of H_{L}. Since all the irreducible representations of H_{L} are unitarily equivalent (Theorem of Stone-Von Neumann), the restriction of the projective bundle ℙ(W) to X_{L} carries a connection. More explicitly: if H_{L} is identified with H_{ℝ^{2n}} by means of a symplectic basis of L, and L^{2}(ℝ^{n}) is endowed with the Schrödinger action of H_{ℝ^{2n}}, for any fixed D ∈ X_{L}, there exists an isomorphism of representations, ψ_{D}: L^{2}(ℝ^{n}) → F_{D}, unique up to a constant in ℂ_{1}^{∗}. The last part of this section is devoted to showing how the projective connection induced by the maps ψ_{D} is related to the connection ∇ defined in the previous section.

In view of the irreducibility of L^{2}(ℝ^{n}), ψ_{D} is completely determined by its value on a non-zero element; so it is sufficient to consider its restriction to the (dense) subspace L^{2}(ℝ^{n})^{hol} of holomorphic vectors. By holomorphic vector we mean an element f ∈ L^{2}(ℝ^{n}) such that h ↦ U_{h}f extends to a holomorphic map from H to L^{2}(ℝ^{n}) (the definition and the basic properties of holomorphic maps in infinite dimensional vector spaces can be found in [H], Ch. III).

Let ϵ: W_{D} → ℂ be the map defined by ϵ(ϕ) = ϕ(1_{H}); then, by observing that the previous identification H_{L} ≅ H_{ℝ^{2n}} extends to H ≅ H_{ℂ^{2n}}, we have

(2.9)  [ψ_{D}(f)](h) = [R_{h}ψ_{D}(f)](1_{H}) = [ψ_{D}(U_{h}f)](1_{H}) = (ϵ ∘ ψ_{D})(U_{h}f),

for any h ∈ H, f ∈ L^{2}(E_{ℝ})^{hol}. This shows that ψ_{D} is determined by the linear form ω_{D} = ϵ ∘ ψ_{D}. From (2.9) it follows that ω_{D}(U_{h}f) = ω_{D}(U_{dh}f), for any f ∈ L^{2}(ℝ^{n})^{hol} and d ∈ D; so ω_{D} is a left D-invariant form.

In order to compute ω_{D}, let’s describe the identification H_{L} ≅ H_{ℝ^{2n}} more precisely. Let {e_{1}, …, e_{2n}} be a symplectic basis of L, denote by E_{ℝ} the real subspace spanned by {e_{1}, …, e_{n}}, and consider the the corresponding identifications V ≅ ℂ^{2n}, E_{ℝ} ≅ ℝ^{n} Then X_{L} = \{ ( \begin{matrix}−Ω \\ 1 \end{matrix} ) | Ω ∈ 𝔖_{n} \}; moreover, if E = ( \begin{matrix}1 \\ 0 \end{matrix} ) and U = ( \begin{matrix}0 \\ 1 \end{matrix} ), it’s

easy to check that L^{2}(E_{ℝ})^{hol} ⊂ O(E), where the action of H on O(E) is induced by the regular right representation on W_{U} by means of r_{E} : W_{U} → O(E).

Now we can prove the following result (cf. [T3] Thm. 2.2):

2.10 Proposition. Notations as above. Let D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) ∈ X_{L}, and ω_{D} be a left D-

invariant form on L^{2}(E_{ℝ})^{hol}. Then there exists an element e_{Ω} ∈ L^{2}(E_{ℝ})^{hol}, invariant with respect to the level subgroup lifting D′ = ( \begin{matrix}Ω \\ 1 \end{matrix} ), such that

ω_{D}(g) = ∫_{E_{ℝ}} g(x)e_{Ω}(x) dx,

for any g ∈ L^{2}(E_{ℝ})^{hol}.

Proof. By general facts about Fock representations (e.g. cf. [I], Ch.I, §.8), it follows that the restriction to F_{D} of the linear form ϵ is induced by a function; thus the same must happen for ω_{D}. Since ω_{D} is left D-invariant, our task amounts to finding the left D′-invariant functions on L^{2}(E_{ℝ})^{hol}. Let f ∈ L^{2}(E_{ℝ})^{hol} ⊂ O(E), ϕ = r_{E}^{−1}f ∈ W_{U}, and denote by ρ_{d} the left invariant vector field on H determined by d ∈ D′. Then f is left D′-invariant if, and only if, ρ_{d}(ϕ) = 0, for each d ∈ D′. Recalling the relations among right- and left-invariant vector fields, one sees that

ρ_{d}(ϕ)(1, v) = δ_{d}ϕ(1, v) − 2πi⟨v, d⟩ϕ(1, v) = 0.

By using the coordinate z on E given by {e_{1}, …, e_{n}}, the last relation may be rewritten as follows:

∑_{i=1}^{n} t_{ij}∂_{z_{i}}f(z) + 2πiz_{j}f(z) = 0,

for j = 1, …, n and (t_{ij})_{1≤i,j≤n} = Ω. So by a direct computation we get f(z) = c exp −πi^{t}zΩ^{−1}z, where c is a constant. QED

2.11 Remarks. (a) From now on we shall denote ω_{D}(g) by (g, e_{Ω}). Let’s observe that for any g ∈ L^{2}(E_{ℝ}) and h = (λ, e + u) ∈ H_{L}, where λ ∈ ℂ^{∗}, u ∈ U and e ∈ E, the following identity holds

(U_{h}g, e_{Ω}) = ∫_{E_{ℝ}} λ exp πi⟨u, 2x + e⟩g(e + x)e_{Ω}(x) dx =
= ∫_{E_{ℝ}} λ exp πi⟨u, 2y − e⟩g(y)e_{Ω}(y − e) dy = (g, U_{h′}e_{Ω}),

(2.12)

where h′ = (λ, −e + u) ∈ H; the map h ↦ h′ is an anti-isomorphism of H.

(b) Since the restriction of U_{h}e_{Ω} to ℝ^{n} = E_{ℝ}, for any h ∈ H, is a Schwartzfunction (e.g. cf. [I], Ch.I, §.9), the linear forms (−, U_{h}e_{Ω}) extend to the space of tempered distributions on E_{ℝ}.

The following theorem explains how to choose e_{Ω} to get horizontal sections of W(−\frac{1}{2}).

2.13 Theorem. Notations as above. Let (det Ω)^{1/2} be the function on 𝔖_{n} defined by

(−i)^{n/2}(det Ω)^{1/2} = (det −iΩ)^{1/2} = ∫_{ℝ^{n}} exp(−πi^{t}xΩ^{−1}x)dx,

and let

(2.14)  e_{Ω}(z) = (det Ω)^{− \frac{1}{2}} exp(−πi^{t}zΩ^{−1}z).

Then, if f ∈ L^{2}(E_{ℝ}) and g_{Ω} denotes the function on H defined by g_{Ω}(h) = (f, U_{h′}e_{Ω}), the map D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) ↦ g_{Ω} is a horizontal section of W(−\frac{1}{2}) over X_{L}. The above map

L^{2}(E_{ℝ}) → Γ(X_{L}, W(−\frac{1}{2})^{∇=0}) extends naturally to the dual space of L^{2}(E_{ℝ})^{hol}.

Proof. The function g_{Ω} is D-invariant; thus by Remark 1.13.(a), it suffices to prove that g_{Ω}(1, z, 0) lies in the kernel of the operators ∂_{t_{ij}} − \frac{1}{4πi}∂_{z_{i}}∂_{z_{j}}, for 1 ≤ i ≤ j ≤ n, and this can be done directly. QED

Now we shall describe the link between the above construction of horizontal section of W(−\frac{1}{2})_{|X_{L}} and the geometry of the family of principally polarized abelian varieties associated to X_{L} (cf. Remark 1.4). Maintaining the notation given above, denote by Λ the free abelian group spanned by the symplectic basis of L used in the previous identification, and for any λ ∈ Λ write λ = λ_{E} + λ_{U}, where λ_{E} ∈ E = ( \begin{matrix}1 \\ 0 \end{matrix} ),

λ_{U} ∈ U = ( \begin{matrix}0 \\ 1 \end{matrix} ). Let Λ_{E} = \{ λ_{E} | λ ∈ Λ \}, and denote by ω_{Λ} the tempered distribution on E_{ℝ} defined by ω_{Λ} = ∑_{λ∈Λ_{E}} Δ_{λ}, where Δ_{λ} denotes the Dirac measure centered at the point λ. Then we can prove the following

2.15 Proposition. Let Θ be the section of W(−\frac{1}{2})_{|X_{L}}, horizontal with respect to ∇, defined by setting

Θ(D, h) = (ω_{Λ}, U_{h′}e_{Ω}),

for each h ∈ H and D ∈ X_{L} (cf. Remark 2.11). Then r_{E}(Θ(h, D)) = cϑ(z, Ω), where c is a suitable root of 1 and ϑ(z, Ω) is the Riemann theta function.

Proof. From the above definitions it follows that

(2.16)

r_{E}(Θ(D, h)) = (ω_{Λ}, U_{(1,−z,0)}e_{Ω}) = ∑_{ν∈ℤ^{n}} e_{Ω}(ν − z) =

= (det Ω)^{− \frac{1}{2}} ∑_{ν∈ℤ^{n}} exp πi[−^{t}(ν − z)Ω^{−1}(ν − z)].

Now, since

ϑ(Ω^{−1}z, −Ω^{−1}) = ∑_{ν∈ℤ^{n}} exp πi[2^{t}νΩ^{−1}z −^{t}νΩ^{−1}ν],

the conclusion follows from (2.16) on recalling the classical identity

ϑ(Ω^{−1}z, −Ω^{−1}) = (det −iΩ)^{\frac{1}{2}} exp πi[^{t}zΩ^{−1}z]ϑ(z, Ω),

where the branch of the square root has positive value when Ω is purely imaginary (cf. [T1] Ch.II, p. 195). QED

2.17 Remark. Notations as in Proposition 2.15. If one considers the restriction of the function h ↦ (ω_{Λ}, U_{h′}e_{Ω}) to L, one gets a slight variation of the Riemann theta function (cf. [T3], §.2, Corollary 2.4). In fact we have the identity

(ω_{Λ}, U_{(1,−Ex_{1}+Ux_{2})}e_{Ω}) = c exp πi(^{t}x_{2}(x_{1} + Ωx_{2})) ϑ(x_{1} + Ωx_{2}, Ω),

where c is a root of 1.

3. Local characterization of horizontal sections.

Here we shall characterize the germs of horizontal sections of W(−\frac{1}{2}), and show how the growth of their values determines the extent to which they can be continued. To achieve our task we need some notations of symplectic geometry (cf. [S]). Let M(n, ℂ) be the space of the n × n complex matrices, then A > 0 means that A is a positive-definite hermitian matrix. Let r > 0 be a real number and Ω_{0} = ^{t}Ω_{0} ∈ M(n, ℂ), then the open disc D(Ω_{0}, r), with center Ω_{0} and radius r, is the subset of M(n, ℂ) defined as follows:

D(Ω_{0}, r) = \{ Ω = ^{t}Ω | r^{2}1 − (Ω − Ω_{0})\overline{(Ω − Ω_{0})} > 0 \} ,  (3.1)

where 1 denotes the identity matrix.

Since there exists a “linear fractional transformation” which sends the Siegel upper half plane 𝔖_{n} onto the open unit disc D(0, 1), given a real structure L, there exists a symplectic frame on V such that X_{L} = D(Ω_{0}, r) for any Ω_{0}, r as above (cf. the proof of Corollary 3.5).

Let Ω = ^{t}Ω, and let Δ denote the diagonal real matrix having as entries the nonnegative square roots of the characteristic values of ΩΩ̄. Then there exists a unitary matrix V such that (^{t}V ΔV̄)^{2} = ΩΩ̄; the matrix ^{t}V ΔV̄ will be denoted by |Ω| and called the modulus of Ω.

Let’s observe that Ω ∈ D(0, r) if, and only if, Δ ∈ D(0, r), i.e. all the characteristic values of |Ω| are less than r. For later use we recall the following elementary result (cf. [S], II §.7, Lemma 1).

3.2 Lemma. Let Ω and Δ be as above; then there exists a unitary matrix U such that Ω = ^{t}UΔU.

With the notations of the above Lemma, the matrix X = ^{t}UΔ^{1/2} is called a square root of Ω.

Now we are ready to state the main result.

3.3 Theorem. Let T be a positive real number, f ∈ O(ℂ^{n}), and consider the following conditions:

(i)_{T} There exists a holomorphic function g(Ω, z), where z ∈ ℂ^{n} and Ω ∈ D(0, T), which satisfies the Heat Equations (∂_{t_{ij}} − ∂_{z_{i}}∂_{z_{j}})g = 0, for 1 ≤ i ≤ j ≤ n, with initial condition g(0, z) = f(z);

(ii)_{T} |f(z)| ≤ O(exp(\frac{1}{4T} |z|^{2}));

(iii)_{T} ∫_{ℂ^{n}} f(z)\overline{f(z)} exp(−\frac{1}{2T} |z|^{2})|dz ∧ dz̄| < ∞.

Then, if 0 < T < T_{1}, any one of (i)_{T_{1}}, (ii)_{T_{1}}, (iii)_{T_{1}} implies all of (i)_{T}, (ii)_{T}, (iii)_{T}.

Proof. (i)_{T_{1}} ⇒ (ii)_{T} Let μ, α, β ∈ ℕ^{n}, such that μ = 2α + β with 0 ≤ β_{i} ≤ 1 for i = 1, …, n, and denote by α′ = (α′_{ij}) the element of of ℕ^{n^{2}} defined by α′_{ij} = α_{i}δ_{ij}, where δ_{ij} is the Kronecker symbol.

Since the derivatives ∂_{z_{i}} and ∂_{t_{ij}} commute, ∂_{z}^{β}g is again a solution of the Heat Equations, so ∂_{z}^{μ}g = ∂_{t}^{α′}(∂_{z}^{β}g). Now, since (∂_{z}^{β}g)_{z=0} is holomorphic on D(0, T_{1}), by taking its Taylor expansion at Ω = 0, we have

|(∂_{z}^{μ}g(0, z))_{z=0}| = |[∂_{t}^{α′}(∂_{z}^{β}g)_{z=0}]_{Ω=0}| ≤ O(α′!T_{2}^{−|α′|}),

for any 0 < T_{2} < T_{1}, where α′! = α! = α_{1}! ⋯ α_{n}! and |α′| = |α| = α_{1} + ⋯ + α_{n}. As a consequence, the Taylor expansion of f at z = 0 shows the following asymptotic behavior:

|f(z)| ≤ O( ∑_{α∈ℕ^{n}} \frac{α!}{(2α)!}T_{2}^{−|α|}|z_{1}|^{2α_{1}} ⋯ |z_{n}|^{2α_{n}}).

Now let’s recall that \frac{(m!)^{2}}{(2m)!} ≤ \frac{2m}{2^{2m}}, for any positive integer m ∈ ℕ; so, if |α| = k, one

has \frac{(α!)^{2}}{(2α)!} ≤ \frac{(2k)^{n}}{2^{2k}}. This, for 0 < T < T_{2}, implies that

\begin{aligned}∑_{|α|=k} \frac{α!}{(2α)!}T_{2}^{−|α|}|z_{1}|^{2α_{1}} ⋯ |z_{n}|^{2α_{n}} & ≤ \frac{(2k)^{n}}{2^{2k}} \frac{1}{k!} ∑_{|α|=k} \frac{k!}{α!}T_{2}^{−|α|}|z_{1}|^{2α_{1}} ⋯ |z_{n}|^{2α_{n}} = \\ & = (2k)^{n} \frac{1}{k!}(\frac{1}{4T_{2}} ^{t}zz̄)^{k} ≤ O(\frac{1}{k!}(\frac{1}{4T} ^{t}zz̄)^{k}), \end{aligned}

so that |f(z)| ≤ O(exp(\frac{1}{4T} |z|^{2})). (ii)_{T_{1}} ⇒ (i)_{T} Given Ω ∈ D(0, T), let X be one of its square roots (in the sense of Lemma 3.2) and consider the following integral

g(Ω, z) = ∫_{Xℝ^{n}} f(z + x)\frac{1}{det(2\sqrt{π}X)} exp(−\frac{1}{4}^{t}xΩ^{−1}x)dx.

By a substitution of variables, recalling that (det X)^{2} = det Ω, one checks that g(Ω, z) is independent of the choice of X; thus, if we prove that the integral converges, we get a function. By hypothesis |f(z + Xy)| ≤ O(exp(\frac{1}{4T_{1}} ^{t}y^{t}X\overline{Xy})), for any y ∈ ℝ^{n}; so the conclusion follows by observing that ^{t}X\overline{X} has the same characteristic values as |Ω| and therefore lies in D(0, T). In fact we have

\frac{1}{4T_{1}} ^{t}y^{t}X\overline{Xy} < \frac{T}{4T_{1}} ^{t}yy < \frac{1}{4}^{t}yy = \frac{1}{4}^{t}y^{t}XΩ^{−1}Xy ,

which implies the convergence of the above integral for Ω ∈ D(0, T). (ii)_{T_{1}} ⇒ (iii)_{T} is clear. Now we prove that (iii)_{T_{1}} ⇒ (ii)_{T}. Denote by B_{z} the ball in ℂ^{n} with center in z and radius \frac{1}{|z|}, and by V_{z} its volume. Since f is a holomorphic function, f(z) = \frac{1}{V_{z}} ∫_{B_{z}} f(u)|du ∧ dū|; then

|f(z)| ≤ \frac{1}{V_{z}} ∫_{B_{z}} |f(u)||du ∧ dū| = \frac{1}{V_{z}} ∫_{B_{z}} |f(u)| exp(−\frac{1}{4T_{1}} ^{t}uū) exp(\frac{1}{4T_{1}} ^{t}uū)|du ∧ dū|.

Now, by applying the Cauchy-Schwarz inequality for the L^{2}-norm of the functions on B_{z}, we get

\frac{1}{V_{z}} ∫_{B_{z}} |f(u)| exp(−\frac{1}{4T_{1}} ^{t}uū) exp(\frac{1}{4T_{1}} ^{t}uū)|du ∧ dū| ≤

≤ (\frac{1}{V_{z}} ∫_{B_{z}} |f(u)|^{2} exp(−\frac{1}{2T_{1}} ^{t}uū)|du ∧ dū|)^{\frac{1}{2}} (\frac{1}{V_{z}} ∫_{B_{z}} exp(\frac{1}{2T_{1}} ^{t}uū)|du ∧ dū|)^{\frac{1}{2}} ≤

≤ V_{z}^{− \frac{1}{2}} ( ∫_{ℂ^{n}} |f(u)|^{2} exp(−\frac{1}{2T_{1}} ^{t}uū)|du ∧ dū|)^{\frac{1}{2}} (max_{u∈B_{z}} exp(\frac{1}{2T_{1}} ^{t}uū))^{\frac{1}{2}} .

Finally, since T < T_{1} and (max_{u∈B_{z}} exp(\frac{1}{2T_{1}} ^{t}uū))^{\frac{1}{2}} = exp(\frac{1}{4T_{1}} (1 + \frac{1}{|z|})^{2}|z|^{2}), we conclude that |f(z)| ≤ O(exp(\frac{1}{4T} |z|^{2})). QED

Now, we collect a couple of consequences of Theorem 3.3; the first one concerns the sub-vector bundle W^{0} described in Corollary 1.6.

3.4 Corollary. Let D ∈ S. For any element ϕ ∈ W_{D}^{0} there exists a horizontal section s of W(−\frac{1}{2}) such that s(D) = ϕ.

Proof. The condition (1.6) says that an element ϕ ∈ W_{D} belongs to the subspace W_{D}^{0} if, and only if, there exists a constant such that |r_{E}(ϕ)(z)| ≤ O(exp A|z|^{2}) for some, and hence for any, complement E of D. By point (ii) of the previous theorem, this condition implies that ϕ is the value in D of a horizontal section s of W(−\frac{1}{2}) and the constant A gives an estimate of the radius of the disc centered in D on which s is defined. QED

The second fact concerns the spaces of horizontal sections of W(−\frac{1}{2}) on the open discs X_{L} and σ_{L}X_{L} associated to a real structure L on H, so let’s observe first that σ_{L}X_{L} = X_{iL}.

3.5 Corollary. Let X_{L} be the open disc, H_{L} the real group, and σ_{L} the conjugation associated to a real structure L on H; then the following chain of inclusions

Γ(X_{iL}, W(−\frac{1}{2})^{∇=0}) ⊆ H_{1} ⊆ Γ(X_{L}, W(−\frac{1}{2})^{∇=0})

holds, where H_{1}, is an irreducible unitary representation of H_{L}.

The space Γ(X_{iL}, W(−\frac{1}{2})^{∇=0}) is dense in H_{1}, so that it has structure of pre-Hilbert space.

Proof. Let’s start by explaining how H_{1} arises. Let {e_{1}, …, e_{2n}} be a symplectic basis of L, and set e′_{i} = \frac{1}{\sqrt{2}}(e_{i} − ie_{n+i}), e′_{n+i} = \frac{1}{\sqrt{2}}(e_{n+i} − ie_{i}), for i = 1, …, n; then {e′_{1}, …, e′_{2n}} is the symplectic basis we will use to identify V with ℂ^{n}. After this choice, let E = ( \begin{matrix}1 \\ 0 \end{matrix} ) and U = ( \begin{matrix}0 \\ 1 \end{matrix} ), then

H_{L} = \{ (λ, z, iz̄) | λ ∈ ℂ_{1}^{∗}, z ∈ ℂ^{n} \} , and X_{L} = \{ ( \begin{matrix}(Ω−i)(Ω+i)^{−1} \\ 1 \end{matrix} ) | Ω ∈ 𝔖_{n} \} ⊂ U_{E}.

Since { (Ω − i)(Ω + i)^{−1} | Ω ∈ 𝔖_{n} } = D(0, 1) (cf. (3.1) and [S], II §.4), we shall write X_{L} = D(0, 1).

Now consider the fibre W_{U} with the regular right action of H_{L}; then, the action on O(E) induced by r_{E}: W_{U} → O(E) is as follows:

(U_{(λ,z,iz̄)}f)(y) = λ exp[−π(2^{t}yz̄ +^{t}zz̄)] f(y + z).

As one can directly check, the action U is unitary with respect to the (Fock) norm

(3.6)  ∥f∥_{(1)}^{2} = ∫_{ℂ^{n}} |f(y)|^{2} exp(−2π^{t}yȳ) |dy ∧ dȳ|.

Then, (cf. Proposition 2.5) the space of the elements of O(E) having finite norm is an irreducible unitary representation of H_{L}: this is H_{1}.

Now we describe the inclusion of H_{1} into Γ(X_{L}, W(−\frac{1}{2})^{∇=0}). Let f(y) ∈ H_{1}, then the function f̃(z) := f(\frac{z}{2\sqrt{πi}}) satisfies the condition (iii)_{T} of Theorem 3.3 with T = 1,

so there exists a function g̃(Ω, z) ∈ O(D(0, 1)×E) such that f̃(z) = g̃(0, z), and (∂_{t_{ij}} − ∂_{z_{i}}∂_{z_{j}})g̃ = 0, for 1 ≤ i ≤ j ≤ n. As a consequence, g(Ω, y) := g̃(Ω, 2\sqrt{πi} y) satisfies the equations (∂_{t_{ij}} − \frac{1}{4πi}∂_{y_{i}}∂_{y_{j}})g = 0, for 1 ≤ i ≤ j ≤ n, and then g corresponds to a section of W(−\frac{1}{2}) on X_{L}, horizontal with respect to ∇ (cf. Remark 1.13.(a)).

Now there remains only to prove that Γ(X_{iL}, W(−\frac{1}{2})^{∇=0}) ⊆ H_{1}. By a direct

computation one checks that X_{iL} ∩ U_{E} = \{ ( \begin{matrix}T \\ 1 \end{matrix} ) | |T | − 1 > 0 \}. As a consequence,

if r > 0 is a real number and D_{0} = ( \begin{matrix}S \\ 1 \end{matrix} ), with |S| − (r + 2)1 > 0, it can be shown

that D(D_{0}, r) ⊆ X_{iL} ∩ U_{E}^{(∗)}. Let s ∈ Γ(X_{iL}, W(−\frac{1}{2})^{∇=0}) and denote by g(Ω, z) the function it induces on O(D(D_{0}, r) × ℂ^{n}) (cf. Proposition 1.5 (b)). Since (∂_{t_{ij}} − \frac{1}{4πi}∂_{z_{i}}∂_{z_{j}})g = 0, for 1 ≤ i ≤ j ≤ n, the function g̃(Ω, z) defined on O(D(0, r) × ℂ^{n}) by g̃(Ω, z) = g(Ω + S, \frac{z}{2\sqrt{πi}}), satisfies the Heat Equations (∂_{t_{ij}} − ∂_{z_{i}}∂_{z_{j}})g̃ = 0. Now assume r > 1, then Theorem 3.3 tells us that

∫_{ℂ^{n}} |g̃(0, z)|^{2} exp(−\frac{^{t}zz̄}{2}) dz ∧ dz̄ < ∞,

i.e. ∥g̃(0, z)∥_{(1)} < ∞; so s ↦ g̃(0, z) is the required inclusion.

Finally, Γ(X_{iL}, W(−\frac{1}{2})^{∇=0}) is dense in H_{1} because it is a non-zero sub-represen-tation of an irreducible representation. QED

(∗) In fact, from Lemma 3.2 it follows that |S| > s ⟺ |Sz| > s|z| ∀z ∈ ℂ^{n}. Then, if |S| > r + 2 and |T | < 1, one has |(S + T)z| ≥ ||Sz| − |T z|| > (r + 1)|z| for all z ∈ ℂ^{n}.

4. Actions of the Metaplectic Group.

Let L be a real structure on H, X_{L} and H_{L} the corresponding disc and Heisenberg group. In this section we shall lift the action of Sp(L) on X_{L} to an action of its double cover Mp(L) on Γ(X_{L}, W(−\frac{1}{2})^{∇=0}), and then we shall show how this induces the classical action of Mp(L) on the irreducible representation of H_{L}.

First of all let’s explain how the group Sp(V) of symplectic automorphisms of (V, ⟨−, −⟩), acts on the sections of the bundle W. Let γ ∈ Sp(V) and denote by the same name both the automorphism (λ, v) ↦ (λ, γ(v)) of H, and the bijection D ↦ γ(D) of S. Then, for any open set U ⊆ S and any s ∈ Γ(U, W), we get a section s_{γ} ∈ Γ(γ^{−1}(U), W), by setting

(4.1)  [s_{γ}(D)](h) = [s(γ(D))](γ(h)),

for any D ∈ γ^{−1}(U) and h ∈ H.

Now fix a real structure L on H and consider the group Sp(L) acting on the corresponding open disc X_{L} (cf. Definition 2.3). By means of the above definition one gets an action of Sp(L) on Γ(X_{L}, W); in fact it is immediate to check that (s_{γ})_{τ} = s_{γτ} for any γ, τ ∈ Sp(L) and s ∈ Γ(X_{L}, W).

If V is identified with ℂ^{2n} by means of a symplectic basis of L, so that X_{L} =

\{ ( \begin{matrix}−Ω \\ 1 \end{matrix} ) | Ω ∈ 𝔖_{n} \} ⊂ U_{E}, where E = ( \begin{matrix}1 \\ 0 \end{matrix} ), Γ(X_{L}, W) = O(𝔖_{n} × ℂ^{n}) and Sp(L) =

Sp(2n, ℝ), we can write the action of Sp(L) on Γ(X_{L}, W) explicitly. In fact let γ =

( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(2n, ℝ), D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) ∈ X_{L}, and g(Ω, z) ∈ O(𝔖_{n} × ℂ^{n}); then γ(D) =

( \begin{matrix}−γ(Ω) \\ 1 \end{matrix} ), where γ(Ω) = (AΩ − B)(D − CΩ)^{−1}; thus using the decomposition

(1, γ(z, 0)) = exp[−πi^{t}z(D − CΩ)^{−1}Cz] (1, −γ(Ω)Cz, Cz) (1,^{t}(D − CΩ)^{−1}z, 0),

from (4.1) we get the formula

(4.2)  g_{γ}(Ω, z) = exp[−πi^{t}z(D − CΩ)^{−1}Cz] g(γ(Ω),^{t}(D − CΩ)^{−1}z).

The group Sp(L) also acts on the line bundle Det, but its action doesn’t extend to Det^{− \frac{1}{2}} . In order to have an action on W(−\frac{1}{2}) we must use Mp(L). Now let’s describe Mp(L): let s ∈ Γ(X_{L}, Det) be a section such that s(D) ≠ 0 for each D ∈ X_{L}; denote by f_{γ} the element of O^{∗}(X_{L}) defined by s_{γ}(D) = f_{γ}(D)s(γ(D)), for any γ ∈ Sp(L),

and by fγ^{\frac{1}{2}} one of its square roots. Then the Metaplectic group is defined as follows:

Mp(L) = \{ (γ, ±fγ^{\frac{1}{2}} ) | γ ∈ Sp(L) \} ,

with the group law (τ, fτ^{\frac{1}{2}} )(γ, fγ^{\frac{1}{2}} ) = (τγ, fτ^{\frac{1}{2}} (fγ^{\frac{1}{2}} )^{τ}), where (fγ^{\frac{1}{2}} )^{τ} (D) = fγ^{\frac{1}{2}} (τ(D)). Now, using the coordinates and the identification Γ(X_{L}, Det^{− \frac{1}{2}} ) ≅ O(X_{L}) given by s^{− \frac{1}{2}} ↦ 1, we have:

Mp(L) ≅ Mp(2n, ℝ) = \{ (( \begin{matrix}A & B \\ C & D \end{matrix} ) , ±[det(D − CΩ)]^{\frac{1}{2}} ) | ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(2n, ℝ) \} ,

and fγ̃(Ω) = [det(D − CΩ)]^{− \frac{1}{2}} f(γ(Ω)), for γ̃ = (( \begin{matrix}A & B \\ C & D \end{matrix} ) , [det(D − CΩ)]^{\frac{1}{2}} ). Finally,

identifying Γ(X_{L}, W(−\frac{1}{2})) with O(𝔖 × ℂ^{n}) by means of the trivializations of W_{|X_{L}}

and Det_{|X_{L}}^{− \frac{1}{2}} defined above, and combining both the actions of Mp(L), we get

gγ̃(Ω, z) = [det(D − CΩ)]^{− \frac{1}{2}} exp −πi^{t}z(D − CΩ)^{−1}Cz g(γ(Ω),^{t}(D − CΩ)^{−1}z). (4.3)

Now we shall explain how the action of Mp(L) on Γ(X_{L}, W(−\frac{1}{2})^{∇=0}) induces a unitary representation A : Mp(L) → Aut (L^{2}(E_{ℝ})) which normalizes the action of H_{L}. Let γ̃ ∈ Mp(L) and γ ∈ Sp(L) be its projection. Since we are asking that Aγ̃(U_{h}f) = U_{γ(h)}(Aγ̃f), for any h ∈ H_{L} and f ∈ L^{2}(E_{ℝ}), in view of the irreducibility of L^{2}(E_{ℝ}) it’s sufficient to define Aγ̃v for a vector v ≠ 0 in L^{2}(E_{ℝ}): we shall use the holomorphic vector e_{Ω} given by the formula (2.14).

Recall that in Theorem 2.13 we described how to build a horizontal section of W(−\frac{1}{2}) starting with an element of L^{2}(E_{ℝ}): given f ∈ L^{2}(E_{ℝ}), the corresponding

horizontal section s was defined by setting s(D, h) = (f , U_{h′}e_{Ω}), for any D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) ∈ X_{L}, where the map ρ : h ↦ h′ is the involution of H defined in Remark 2.11.(a). Now we are able to define the action A:

let f, s be as above, and denote by γ′ the element of Sp(L) which gives the automorphism ργρ of H; then Aγ̃e_{Ω} is defined by requiring that

sγ̃′(D, 1_{H}) = (f, Aγ̃e_{Ω}),  (4.4)

for any f.

Now we can prove the following

4.5 Proposition. Notations as above. There exists a (unique) unitary representation A : Mp(L) → Aut L^{2}(E_{ℝ}) satisfying (4.4) and normalizing the action of H_{L} on L^{2}(E_{ℝ}).

Let A^{∗} be the action of Mp(L) on the dual space of L^{2}(E_{ℝ})^{hol} defined by setting

(A∗ ; γ̃ω, f) = (ω, Aγ̃′f),

for any f ∈ L^{2}(E_{ℝ})^{hol} and ω in the dual space. Then, the map described in Theorem 2.13, which sends the dual space of L^{2}(E_{ℝ})^{hol} to Γ(X_{L}, W(−\frac{1}{2})^{∇=0}), is Mp(L)- equivariant.

Proof. Let I = { U_{h}e_{Ω} | h ∈ H_{L} } and denote by G the set of generators of Sp(L) given by the following matrices:

( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ) , ( \begin{matrix}1 & 0 \\ C & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ,

where A, C = ^{t}C ∈ GL_{n}(ℝ).

Since the subspace spanned by I is dense L^{2}(E_{ℝ}), there exists a unique A such that

(4.6)  Aγ̃(U_{h}e_{Ω}) = U_{γ(h)}(Aγ̃e_{Ω}),

for any γ ∈ G and h ∈ H, where Aγ̃e_{Ω} is the element defined by (4.4). By construction this A normalizes the action of H_{L}; thus all that remains to check is the unitarity of the restriction to I of the maps Aγ̃, as γ moves in G.

First, let’s compute an explicit expression for Aγ̃e_{Ω}. From (4.1) and (4.3) it follows that sγ̃′(Ω, 1_{H}) = [det(D + CΩ)]^{− \frac{1}{2}} (f, e_{γ′(Ω)}), thus using (2.14) we get

(4.7)  Aγ̃e_{Ω}(y) = [det(AΩ + B)]^{− \frac{1}{2}} exp(−πi^{t}y(D + CΩ)(AΩ + B)^{−1}y),

for any γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(2n, ℝ). In particular:

(i) Aγ̃e_{Ω}(y) = (det A)^{− \frac{1}{2}} e_{Ω}(A^{−1}y), for any γ = ( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ) ∈ G;

(ii) Aγ̃e_{Ω}(y) = e_{Ω}(y) exp(−πi^{t}yCy), for any γ = ( \begin{matrix}1 & 0 \\ C & 1 \end{matrix} ) ∈ G;

(iii) Aγ̃e_{Ω}(y) = exp(πi^{t}yΩy) = (i)^{\frac{n}{2}} ê_{Ω}(y), for γ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), where ê_{Ω}(y) denotes the Fourier transform. Now it’s straightforward to check that for any f ∈ I and γ ∈ G the element

Aγ̃f(y) defined by (4.6) is related to f(y) in the same way as Aγ̃e_{Ω}(y) was to e_{Ω}. Since all the transformations (i), (ii) and (iii) are unitary, the proof of the first claim is complete.

Given an element ω of the dual space, consider the section s ∈ Γ(X_{L}, W(−\frac{1}{2})^{∇=0})

defined by s(D, h) = (ω , U_{h′}e_{Ω}), where D = ( \begin{matrix}−Ω \\ 1 \end{matrix} ) ∈ X_{L}. Then, for any γ =

( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(2n, ℝ), one has

(A∗ ; γ̃ω , U_{h′}e_{Ω}) = (ω , Aγ̃′U_{h′}e_{Ω}) = (ω , U_{γ′(h′)}Aγ̃′e_{Ω}) =
= [det(D − CΩ)]^{− \frac{1}{2}} (ω , U_{γ(h)′}e_{γ(Ω)}) = sγ̃(D, h) .

This concludes the proof. QED

4.8 Remark. Let’s maintain the above notations. Recall that in Proposition 2.15 we obtained the Riemann theta function (up to multiplication by a root of 1) as a horizontal section of W(−\frac{1}{2}) on the open disc X_{L}. Precisely, the horizontal section Θ was the image of the (tempered) distribution ω_{Λ} and ϑ was obtained by restriction: ϑ(Ω, z) = c(Θ(D, (1, z, 0)), where c is a suitable root of 1.

Now observe that the distribution ω_{Λ} is characterized (up to a constant) by its invariance under the action of the subgroup Λ̃ = { (exp πi⟨λ_{E}, λ_{U}⟩ , λ) | λ ∈ Λ } of H, where Λ is the free abelian group generated by a (fixed) symplectic basis of the real structure L and λ_{E} and λ_{U} are the projections of λ ∈ Λ on the subspaces E = ( \begin{matrix}1 \\ 0 \end{matrix} )

and U = ( \begin{matrix}0 \\ 1 \end{matrix} ) (cf. [T3], Theorem 2.3).

By means of Proposition 4.5, one can compute (up to a constant) the classical functional equations for the Riemann theta function. In fact, denote by Γ_{1,2} the subgroup of Sp(L) consisting of the elements γ such that γ(Λ̃) ⊆ Λ̃, and let γ̃ be an element of Mp(L) which projects onto γ ∈ Γ_{1,2}; then the distribution A∗ ; γ̃ω_{Λ} is again invariant under the action of the subgroup Λ̃, therefore there exists a constant aγ̃ such that A∗ ; γ̃ω_{Λ} = aγ̃ω_{Λ}, so that the horizontal sections Θ and Θγ̃ are related by

Θγ̃(D, h) = (A∗ ; γ̃ω_{Λ} , U_{h′}e_{Ω}) = aγ̃(ω_{Λ} , U_{h′}e_{Ω}) = aγ̃Θ(D, h),

for any D ∈ X_{L} and h ∈ H. Now, by means of (4.3), one can write these identities in terms of the Riemann theta function; in particular, if γ ∈ Γ_{1,2} is represented by the matrix ( \begin{matrix}A & −B \\ −C & D \end{matrix} ) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}A & B \\ C & D \end{matrix} ) ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), the above relation assumes the classical form

ϑ((AΩ + B)(CΩ + D)^{−1},^{t}(CΩ + D)^{−1}z) =
= aγ̃[det(CΩ + D)]^{\frac{1}{2}} exp[πi^{t}z(CΩ + D)^{−1}Cz] ϑ(Ω, z).

One can obtain the last equations directly using the quasi-periodicity of the Riemann theta function. In fact, by its construction, the section Θ satisfies the identities

Θ(D, λdhξ) = λ(ω_{Λ} , U_{(dhξ)′}e_{Ω}) = λ(ω_{Λ} , U_{ξ′}U_{h′}U_{d′}e_{Ω}) = λΘ(D, h),  (4.9)

for any D ∈ X_{L}, λ ∈ ℂ, d ∈ D, h ∈ H and ξ ∈ Λ̃. These identities imply the quasi-periodicity of ϑ; in fact, for any n_{1}, n_{2} ∈ ℤ^{n} we have the decomposition

(1, z + n_{1} + Ωn_{2}, 0) =

= exp[−πi(^{t}n_{2}Ωn_{2} + 2^{t}n_{2}z)] (1, Ωn_{2}, −n_{2}) (1, z, 0) (exp πi^{t}n_{2}n_{1} , n_{1}, n_{2}). (4.10)

Thus, by observing that d = (1, Ωn_{2}, −n_{2}) ∈ D and ξ = (exp πi^{t}n_{2}n_{1}, n_{1}, n_{2}) ∈ Λ̃, in view of (4.9) one gets

(4.11) ϑ(Ω, z + n_{1} + Ωn_{2}) = cΘ(D, exp[−πi(^{t}n_{2}Ωn_{2} + 2^{t}n_{2}z)] d(1, z, 0)ξ) = = exp[−πi(^{t}n_{2}Ωn_{2} + 2^{t}n_{2}z)] ϑ(Ω, z).

Given γ̃, which projects on γ ∈ Γ_{1,2}, let ϑγ̃ = r_{E}Θγ̃; then ϑγ̃ and ϑ have the same quasi-periodicity. In fact from (4.9) and (4.10) one deduces

ϑγ̃(Ω, z + n_{1} + Ωn_{2}) = f_{γ}(Ω)^{− \frac{1}{2}} Θ(γ(D), γ(1, z + n_{1} + Ωn_{2}, 0))
= f_{γ}(Ω)^{− \frac{1}{2}} Θ(γ(D), exp[−πi(^{t}n_{2}Ωn_{2} + 2^{t}n_{2}z)] γ(d)γ(1, z, 0)γ(ξ))
= exp[−πi(^{t}n_{2}Ωn_{2} + 2^{t}n_{2}z)] ϑγ̃(Ω, z),

because γ(d) = γ(1, Ωn_{2}, −n_{2}) ∈ γ(D) and, after the hypothesis on γ, γ(ξ) = γ(exp πi^{t}n_{2}n_{1}, n_{1}, n_{2}) ∈ Λ̃. As a consequence there exists a function k on X_{L}, such that ϑγ̃(Ω, z) = k(Ω)ϑ(Ω, z). Finally, since ϑ comes from a horizontal section, the same holds for ϑγ̃, therefore k is a constant.

References

[D] P. Deligne, Groupe de Heisenberg et Réalité, Journal of AMS 4 (1991) p.197–206

[H] M. Hervé, Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces, LNM 198, Springer Verlag 1971

[I] J-I. Igusa, Theta Functions, Grundlehren der Math. Wiss. Band 194, Springer Verlag 1972

[KS] K. Kodaira, D.C. Spencer, On Deformations of complex analytic Structures, II, Annals of Math. 67 (1958) p.403–466

[MM] B. Mazur, W. Messing, Universal Extensions and One Dimensional Crystalline Cohomology, LNM 370, Springer Verlag 1974

[S] C.L. Siegel, Symplectic Geometry, Academic Press 1964

[T1] D. Mumford, Tata Lectures on Theta I, Progr. in Math. 28, Birkhäuser 1983

[T3] D. Mumford, Tata Lectures on Theta III, Progr. in Math. 97, Birkhäuser 1991