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Theta functions and arithmetic quotients of loop groups

In this paper we observe that isomorphism classes of certain metrized vector bundles over P^1-{0,infinity} can be parameterized by arithmetic quotients of loop groups. We construct an asymptotic version of theta functions, which are defined on these quotients. Then we prove the convergence and extend the theta functions to loop symplectic groups. We interpret them as sections of line bundles over an infinite dimensional torus, discuss the relations with loop Heisenberg groups, and give an asymptotic multiplication formula.


Introduction
Let E be a vector bundle of rank n on the affine scheme X := P 1 Z − {0, ∞} = SpecR, where R = Z[t, t −1 ]. Then the global section H 0 (E) of E is isomorphic to R n . Fix q ∈ R, q > 1. If we choose a trivialization H 0 (E) ≃ R n , then we may define an inner product on H 0 (E) via the following inner product (, ) 1 on R n : where q −1 S 1 = {t ∈ C : |t| = q −1 }, v(t) denotes the complex conjugate, and the supscript T is the transpose (here we write w(t), v(t) as column vectors). The Lebesgue measure on q −1 S 1 is normalized such that meas(q −1 S 1 ) = 1. For any g ∈ G := GL n (R[t, t −1 ]), we define (1.2) (w, v) g := (g −1 w, g −1 v) 1 .
We extend these inner products to R[t, t −1 ] n = R n ⊗ Z R by the same formulas. Our starting point is to introduce the arithmetic quotient (see Section 2 for precise details) Q n,q = Γ\ G/ K, where Γ, K are certain "discrete" and "maximal compact" subgroup of G respectively, G is a central extension of G by R × , and Γ, K are double covers of Γ, K.
The key observation (Theorem 2.2) is that Q n,q classifies above metrized vector bundles on X with an additional datum called covolume theory. Roughly speaking, a covolume theory, denoted by c, will assign a positive number c(L) to each lattice L of H 0 (E) in such a way that the information of relative covolumes between different lattices will be captured. Via this moduli interpretation of Q n,q , for any g ∈ G where L runs over all lattices L of H 0 (E), and the limit is taken with respect to inclusions of lattices. The interesting point of our construction is the (tautological!) fact that ϑ is defined on Q n,q . In other words, we obtain an automorphic function on G.
The most technical problem is the convergence of the limit defining ϑ( g). Using a variant of the lemmas in [4] and [12], together with some elementary Fourier analysis, in particular the Poisson summation formula, we establish in Section 3 the uniform convergence of ϑ( g) for g varying in certain Siegel subset of G. In the course of the proof, the use of Iwasawa decompositions for loop groups simplify our considerations. Motivated by the work of Y. Zhu [12] on Weil representations, we also extend the theta functions to loop symplectic groups in Section 4. In all the above, the reparametrization t → qt for the variable t, plays an important role for the convergence result. Note that it is also a crucial ingredient in the proof of the convergence of the Eisenstein series (see [1], [3]) on loop groups.
We try to give some interpretations for our theta functions from representationtheoretic and geometric point of view. D. Mumford's idea [5] is to bring together three ways of viewing theta functions: (a) as holomorphic functions in the vector/period matrix, (b) as matrix coefficients of a representation of the Heisenberg/metaplectic groups, (c) as sections of line bundles on abelian varieties/moduli space of abelian varieties.
In Section 5, we interpret the theta functions as global sections of line bundles over an infinite dimensional torus, discuss the action of loop Heisenberg groups, and give an asymptotic multiplication formula, which is analogous to the theta relations given by Mumford in [5].
Inspirations from Kapranov's work [2] are indispensable for the writing of this paper, especially for introducing the notion of covolume theory. We also use the notions of semiinfinite Grassmannian and dimension theory in [2] to write our multiplication formula in Section 5 in a more conceptual way.
It should not be too surprising to get some generalizations of part of this paper to more general framework, say, to the situation of higher local fields and adelic spaces, or the so-called C n -categories (see [6], [7], [8] and [10]). For instance, Parshin [9] considered Heisenberg groups associated with objects from C f.g. 0 , the category of finitely generated abelian groups. The loop Heisenberg groups introduced in Section 5 should be viewed as Heisenberg groups associated with objects from C f.g.
1 , the category of filtered abelian groups with finitely generated quotients.
Notations For an abelian group A, we write A R = A ⊗ Z R. For a finite dimensional real vector space V , write V * = Hom R (V, R). Let (, ) : V × V * → R be the canonical pairing. If L is a lattice of V , denote by L * ⊂ V * the dual lattice with respect to the bi-character e −2πi(,) . Let S(V ) be the space of Schwartz functions on V. If a Haar measure on V has been chosen, then define the Fourier transform The Haar measure on V * is determined by the condition that F is an isometry.
For a positive integer l, Z l stands for Z/lZ (in contrast to the usual meaning of l-adic integers for prime l), U l is the group of l-th roots of unity. The pairing Z l × U l → U l ⊂ S 1 gives the Pontryagin duality. Denote ζ l = e 2πi/l .
For an invertible matrix g write g * = (g −1 ) T .

Moduli interpretations of arithmetic quotients
Our purpose in this section is to give a moduli interpretation of arithmetic quotients of loop groups, which in turn classify the vector bundles with metrics described in the introduction. We start from the loop group GL n (R[t, t −1 ]). It is well-known that this group has a central extension GL n (R[t, t −1 ]) which we shall now define.
A lattice L of a free R[t, t −1 ]-module V of rank n is a free R[t −1 ]-submodule of rank n. In other words L is an R[t −1 ]-span of a basis of V . Any two lattices L 1 , L 2 in V are commensurable, in the sense that the quotients L 1 /(L 1 ∩ L 2 ) and we can talk about their top wedge powers. Let det(L 0 R , gL 0 R ) be the tensor product where (·) −1 = Hom R (·, R) denotes the dual vector space. Then det(L 0 R , gL 0 R ) is a one-dimensional vector space over R. Let det(L 0 R , gL 0 R ) × be the set of nonzero vectors in det(L 0 R , gL 0 R ), which form an R × -torsor. Now define the group The multiplication in the group is given by We are concerned with two subgroups of GL n (R[t, t −1 ]) together with their extensions. The first one is GL n (Z[t, t −1 ]), which plays the role of a discrete subgroup, or more precisely the so-called arithmetic subgroup. This group acts on R n = Z[t, t −1 ] n . Similarly as above, let us define a lattice in R n to be a free are free Z-modules of finite rank, and we let det(L 0 Z , gL 0 Z ) be the rank one free Z-module In other words we have the commutative diagram This construction of double cover is quite natural (see [2] §2.0).
Recall (1.2) that for any g ∈ GL n (R[t, t −1 ]) we have an inner product (, ) g on R[t, t −1 ] n . Write · g for the induced norm. The following lemma is an easy exercise.
Lemma 2.1. Fix g and for any subspace H of R[t, t −1 ] n write H for the completion with respect to · g . Then for any two lattices L 1 , Using this lemma, we may induce a metric on det(L 0 R , gL 0 R ), which we still denote by · g . The subgroup plays the role of a maximal compact subgroup. In fact K is the isometry group of (, ) 1 . Hence · gk = · g for any g ∈ GL n (R[t, t −1 ]), k ∈ K. We remark that by definition there is a family of group homomorphisms Similarly, we also have the following commutative diagram We introduce the arithmetic quotient The main result of this section is the following moduli interpretation of Q n,q . Let E be a vector bundle of rank n on X, and · be a metric on E such that after a trivialization of E it is induced by (1.2) for some g. A covolume theory of (E, · ) is a rule c which assigns a positive real number c(L) to each lattice L of H 0 (E) such that Here cov · denotes the covolume induced by the given metric · , and L R = L ⊗ Z R. Note that once we know c(L) for one lattice L, then we know c(L ′ ) for any lattice L ′ . In fact, take a lattice L ′′ ⊂ L, L ′ and define Then (2.9) does not depend on the choice of L ′′ and satisfies We remark that all covolume theories attached to (E, · ) form an R × + -torsor. Consider all the triples (E, · , c) as above. We say that two such triples Theorem 2.2. Q n,q classifies the isomorphism classes of all such triples (E, · , c).
Consider the automorphism of trivial vector bundles η γ : R n → R n , v → γv. This induces an isometry because we have Moreover we have . This proves that η γ induces an isomorphism between the triples (R n , · g , c) and (R n , · γg , c ′ ).
(2) Let (k, ω k ) ∈ K. Then associated to (gk, g . In this case the identity map induces an isometry, and . Hence the triples (R n , · g , c) and (R n , · g , c ′′ ) are isomorphic via the identity map.
We have constructed a map from Q n,q to the set of isomorphism classes of such triples. The inverse map is also clear and it is easy to prove this is a bijection. 2

Theta functions on the arithmetic quotients
Suppose we have a triple (E, · , c) as in the previous section. Via isomorphisms we may always assume that E = R n and · = · g for some g ∈ GL(R[t, t −1 ]). Let g = (g, ω g ) ∈ GL n (R[t, t −1 ]) such that g gives the triple (R n , · g , c) (see the proof of Theorem 2.2). We define where L runs over all lattices of R n ≃ H 0 (E), and the limit is taken with respect to the inclusions of lattices. To make the definition more transparent, consider the filtration of lattices Then d∈Z L d Z = R n and ϑ( g) is the value (assume the existence) which satisfies the condition that for any ε > 0, there exists d ∈ Z such that Of course, in this description we may replace {L d Z , d ∈ Z} by any exhausting filtration of lattices. Clearly ϑ( g) only depends on the isomorphism class of the triple, and from Theorem 2.2 it is a tautological fact that ϑ descends to a function on Q n,q , i.e. ϑ is left invariant under GL n (Z[t, t −1 ]) and right invariant under K.
In this section we shall prove uniform convergence properties of ϑ( g), for g varying in certain "compact" subset of GL n (R[t, t −1 ]). Equivalently, we are considering behaviors of ϑ([ g]) when [ g] lies in certain subset of the arithmetic quotient Q n,q . To this end we need the following Iwasawa decomposition: Proof of this lemma is by using standard theory of Tits systems. We also need a lemma which is similar to [12] We give some notations. Let N = {g(t) ∈ GL n (R[t −1 ])|g(∞) = 1}, and A be the group of diagonal matrices in GL n (R), then B = AN. For each d ∈ Z, let L d Note that elements of N act on L d R as blockwise unipotent linear operators. Let us define the notion of Siegel subsets of GL n (R[t, t −1 ]) which is suitable for our purpose. GL n (R[t, t −1 ]) has an ind-scheme structure. Indeed for each i ∈ N let X i be the subset of GL n (R[t, t −1 ]) consisting of elements whose entries are contained in V −i,i . Then GL n (R[t, t −1 ]) = i∈N X i , and one may choose an embedding of affine subscheme, where N (i) ∈ N depends on i. For example we may take N (i) = n 2 (2i + 1) + 2ni + 1. We call C ⊂ GL n (R[t, t −1 ]) a compact subset, if C is contained in X i for some i, and ϕ i (C) ⊂ R N (i) is compact in the real topology. It is easy to check that this notion is well-defined, i.e. does not depend on the choice of (3.5). Similarly we can define compact subsets of B. Recall from Lemma 3.1 the Iwasawa decomposition. We define a Siegel subset of GL n (R[t, t −1 ]) to be a subset of the form C Z C B K where C Z , C B are compact subsets of R × and B respectively. Now we are ready to state and prove the main result of this section. Proof. Let u ∈ N , a ∈ A, and suppose that g gives the triple (R n , · au , c). Our strategy is to first show that the sequence is bounded, and then compare ϑ L ( g) and ϑ d ( g) when L is a lattice containing L d Z . More precisely we shall prove that where o d (1) → 0 as d → ∞, and the rate of convergence depends on d but not on L. We proceed in 3 steps.
Step 1: Boundedness of subsequence. The trick is reducing to the diagonals. Let us first prove that for each d ∈ Z, Lemma 3.2 does not apply directly in this infinite dimensional case. However, consider for d 1 < d. Let π 1 , π 2 be the obvious projections Since V d1,d and L d1−1 R are orthogonal with respect to (, ) 1 , we obtain

Apply Lemma 3.2 for
On the other hand, we have the equality of covolumes with respect to the metric · au . It is easy to check that = RHS of (3.10).
Step 2: Comparison with the subsequence. Let L be a lattice containing L d Z , and let L ′ be a complement of , where δ(u) is the highest power of t −1 appearing in the entries of u. As a consequence we have Let (, ) * au be the induced inner product on V * d−2δ(u),d . Choose the Haar measure on V d−2δ(u),d such that the covolume of L d−2δ(u),d equals one (i.e. choose the ordinary Lebesgue measure on each V i = t i R ≃ R via rt i → r). Then for fixed x 0 ∈ V d−2δ(u),d , we have the Fourier transform F e −π(x+x0,x+x0)au (y) = e 2πi(x0,y)−π(y,y) * By the Poisson summation formula, Previous reasoning for (3.8) Therefore we have proved that where the rate of convergence of o d (1) does not depend on x 0 . Hence from (3.13) we obtain where again the magnitude of o d (1) does not depend on v ′ 0 . Similar arguments apply for the summation over L ′ in (3.12), and yield where the magnitude of o d (1) does not depend on L ′ , hence not on L. Combine (3.12), (3.14) and (3.15), together with the identity we obtain (3.7).
Step 3: Conclusions. After the first two steps we have proved the existence of ϑ( g). If we restrict to a Siegel subset, then by definition we have in the previous settings that c(L 0 Z ) and au vary in some compact subsets of R × + and B respectively, and moreover δ(u) is bounded. Consequently, the bound (3.11) and the quantities o d (1) appearing in Step 2 are all uniform.

Generalization to loop symplectic groups
Recall that the Siegel upper half-space H n for Sp 2n (R) is the set of all n × n complex symmetric matrices with positive definite imaginary part. Sp 2n (R) acts on H n via linear fractional transformations. L 2 (R n ) is a model for the Weil representation of Sp 2n (R), and the dense subspace S(R n ) is closed under this action.
For Ω ∈ H n the corresponding Gaussian function f Ω (x) = e πixΩx T is in S(R n ). The theta function θ : H n → C given by is automorphic for some arithmetic subgroup of Sp 2n (R). In [12] Y. Zhu has generalized this classical theory of Weil representations and theta functions to the loop group Sp 2n (R((t))). His method also works in our situation. We do not attempt to build the theory of Weil representations for our loop group in full generality. Instead we shall focus on theta functions. From now on we use row vectors instead of column vectors, and matrix group acts from the right. Define to be the isometry group of (W, , ).
Denote by V C and V * C the complexifications of V and V * ; let H be the set of Ω = X + iY ∈ Hom C (V C , V * C ) such that X, Y ∈ Hom R (V, V * ) are are self-dual and Y > 0. Note that Im(Ω) = Y induces an inner product (w, v) iY on V via (w, v) iY := (w, vY ). Let iI ∈ H such that I induces the inner product (, ) 1 defined by (1.1). In concrete terms, the operator I : V → V * maps t i e j to q −2i t −i e j , where i ∈ Z and {e 1 , . . . , e n } is the standard basis of R n . For any g acts on H by g · Ω = (αΩ + β)(γΩ + δ) −1 . Consider the diagonal embedding where g * = (g −1 ) T . Then d(g) · iI induces the inner product (, ) g . This suggests that we may generalize the theta function to the loop symplectic group.
Remark 4.1. Here our diagonal embedding differs from the usual one because in previous sections we let GL n (V ) act on V from the left. For the setting of Weil representations by convention the symplectic group acts from the right.
We first generalize the notion of covolume theory. Define a sesquilinear form (, ) Ω : V C × V C → C by (v, w) Ω = −i(v, wΩ). It is easy to check that Re(v, v) Ω > 0 for any v = 0. If L ⊂ L ′ are two lattices of R n = Z[t, t −1 ] n , the orthogonal complement L ⊥ C of L C in L ′ C with respect to the form (, ) Ω is well-defined. If {v i } is a complementary basis (i.e. {v i } span a complement of L in L ′ ), then we define the covolume of L in L ′ to be the complex number Here for the square root we take the positive branch. Similarly as in section 2, a covolume theory for Ω is a rule c assigning a nonzero complex number c(L) to each lattice L of R n such that c(L ′ ) = c(L)c(Ω, L, L ′ ) whenever L ⊂ L ′ are two lattices.
If Ω = g · iI for some g ∈ Sp 2n (R[t, t −1 ]) and c is a covolume theory for Ω, then we define Proposition 4.2. The limit defining ϑ(Ω, c) is convergent.
The proof of Proposition 4.2 relies on the Iwasawa decomposition for loop symplectic groups, which we shall recall now. Let It can be shown that K sp is the stabilizer of iI in Sp 2n (R[t, t −1 ]). Let B 0 denote the Borel subgroup of Sp 2n (R) given by and define Then we have the Iwasawa decomposition Proof of Proposition 4.2: It suffices to consider Ω = g · iI with g ∈ B sp . We use the fact that B sp is generated by elements of the form Note that in this case δ = α * ∈ GL n (R[t −1 ]), β, γ ∈ M n×n (R[t −1 ]). Moreover δ ∈ B. We may write g = g l · · · g 1 such that each g j is of the form (4.7). Hence we have a chain of transformations For each j one has either Ω j = α j Ω j−1 α T j + β j α T j or Ω j = (γ j δ T j + δ j Ω −1 j−1 δ T j ) −1 . One may keep track of these finitely many steps and apply the same method but with some variant of the arguments used in Section 3. 2

Interpretations of theta functions
We start from general theta functions with parameters. Keep the notations in the last section. Let Ω = g ·iI ∈ H , where g ∈ Sp 2n (R[t, t −1 ]). Let c be a covolume theory for Ω. For a ∈ V , b ∈ V * , z ∈ V * + V Ω = V * C = C[t, t −1 ] n , we define the theta function We have the quasi-periodicity: Let l be a positive integer and fix a covolume theory c l for lΩ. Each a ∈ Z[t, t −1 ] n gives a theta function  Following [12], let S l be the set of a ∈ Z[t, t −1 ] n such that all the coefficients in a lie in {0, 1, . . . , l − 1}. Let C |f a l (z) = 0 for all a ∈ S l }/L Ω , which is called the set of "base points" in the complex torus V * C /L Ω . Let P S l = P(C S l ) be the infinite projective space, then there is a canonical holomorphic map The well-definedness of φ l follows from quasi-periodicity (5.6), (5.7). It is known that B l = ∅ for l ≥ 2 and φ l is an embedding for l ≥ 3.
We introduce a line bundle L over V * C /L Ω . Define an action of L Ω on the trivial bundle Let L be the quotient line bundle. If f is a function on V * C satisfying (5.6) (5.7), and f is holomorphic, i.e. the restriction of f to any finite dimensional complex subspace of V * C is holomorphic, then we say f is a global section of L l . Let Γ(L l ) be the set of global sections of L l .
We claim that f a l , a ∈ S l are global sections of L l . It suffices to prove the holomorphic property. This follows from certain uniform convergence (in the variable z) results, and one may argue similarly as before. We omit the details. If we fix a Note that ( a η mod l ′ 1 l ′ 2 l 3 ) does not depend on the choice of (j 1 , j 2 ). Introduce a subset of Z[t, t −1 ] n , (5.17) S a1,a2 l1,l2 = {η ∈ Z[t, t −1 ] n |a η ∈ S l3 }. It is clear that S a1,a2 l1,l2 is a set of representatives for Z l ′ 3 [t, t −1 ] n , i.e. Similarly, for d ∈ Z such that a 1 , a 2 ∈ L d Z one has (5.19) We also need some notions from [2]. Let us denote by G (V ) the set of lattices of V and call it the semiinfinite Grassmannian of V . A dimension theory on V is a map D : G (V ) → Z such that whenever L, L ′ ∈ G (V ) and L ⊂ L ′ , we have D(L ′ ) = D(L) + dim(L ′ /L). All dimension theories on V form a Z-torsor.
Recall that we have fixed a covolume theory c j for jΩ, ∀j > 0. We claim that there exist a dimension theory D on V and a constant λ ∈ C × such that (5.20) c l1 c l2 = λ · (l ′ 3 ) −D · c l ′ 1 l ′ 2 l c l3 . Note that we view both sides of (5.20) as functions G (V ) → C × . The claim follows from the simple identity l 1 l 2 = l ′ 1 l ′ 2 l 2 3 (l ′ 3 ) −2 . The choice of (D, λ) is not unique, and we only need to fix such a pair. Now we are ready to give the asymptotic multiplication formula. such that η runs over S a1,a2 l1,l2 ∩ L d Z , and u, v run over L d Z . Then from (5.19) it follows that is a bijection. In terms of the new variables, after some heavy manipulations one can check that (5.21) equals Taking into account of (5.20), this finishes the proof. 2 Remark 5.4. When q ∈ Q, the endomorphism ring of V * C /L Ω is much larger than Z, namely, it contains Z[N t, N t −1 ] for some sufficiently large integer N . This is an interesting analogue of abelian varieties with large endomorphism rings.