Holomorphic Functions and the Heat Kernel Measure on an Infinite Dimensional Complex Orthogonal Group

HOLOMORPHIC FUNCTIONS AND THE HEAT KERNEL MEASURE ON AN INFINITE DIMENSIONAL COMPLEX ORTHOGONAL GROUP

MARIA GORDINA Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect 2 to the heat kernel measure. The closure of these polynomials, HL (SOHS , µt), is one of two spaces of holomorphic functions we consider. The second space, HL2(SO( )), consists of functions which are holomorphic on an analog of the ∞ Cameron-Martin subspace for the group. It is proved that there is an isometry from the first space to the second one. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from HL2(SO( )) into the Hilbert space ∞ associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. All the results of this paper are formulated for one concrete group, the Hilbert-Schmidt complex orthogonal group, though our methods can be applied in more general situations.

1. Introduction In this paper we will study Hilbert spaces of holomorphic functions over a particular infinite dimensional complex group G. We will choose an infinite dimensional Lie algebra naturally associated with the group, and fix a Hermitian inner product on it. As usual, one may view the universal enveloping algebra as a space of left- invariant differential operators on the group. A holomorphic function on G then determines an element, α, of the dual of the universal enveloping algebra by means of the identity α, β = (βf)(e) for all left-invariant differential operators β, where e is the identityh elementi of the group. Thus, α is just the set of Taylor coefficients of f at the identity. When G is finite dimensional, the Taylor map, f α, is known (cf. [7]) to be an isometry from the Hilbert space of holomorphic functions,7→ square integrable with respect to a heat kernel measure on G, to a subspace of the dual of the universal enveloping algebra. An outline of these finite dimensional results of B. Driver and L. Gross [7] will be given later in the introduction. The objective of the present work is to establish a corresponding isometry for an infinite dimensional group.

1991 Mathematics Subject Classification. Primary 46E50, 60H15; Secondary 22E30, 22E65, 58G32. Key words and phrases. Heat kernel measure, infinite dimensional group, stochastic differential equation. 1 2 MARIA GORDINA

Two fundamental differences between the finite dimensional and infinite dimensional cases must be addressed. First, we need to have an analog of the heat kernel measure on an infinite dimensional, non locally compact, non commutative group. For a finite dimensional group the heat kernel measure has a density with respect to Haar measure, making it possible to apply techniques from partial differential equations. However, on the group to be considered in this paper there is no Haar measure to help analyze a heat kernel density. Partial differential equation techniques for studying such measures are not available. Instead, we will use stochastic differential equations in infinite dimensions to construct such a measure. The theory of such equations has been developed for Hilbert and Banach spaces, not infinite dimensional manifolds or groups. Moreover, since there are no previous examples to guide our intuition for a general theory, we will focus in this paper on one particular case. We will take the group to be the group of complex Hilbert-Schmidt orthogonal operators over an infinite dimensional Hilbert space. This Hilbert-Schmidt complex orthogonal group, SOHS, will be described in Section 2. The fact that the group SOHS can be embedded in a Hilbert space, namely, the space of Hilbert-Schmidt operators, plays an important role in the construction of the heat kernel measure. To construct the heat kernel measure we will use a diffusion on the group. We will construct the diffusion on G by first constructing a diffusion on this ambient Hilbert space. The stochastic differential equation determining the diffusion is nonlinear with Lipshitz nonlinearities, and its coefficients are chosen in such a way that the stochastic process actually lives in the group. Then we will use finite dimensional approximations to this diffusion to obtain some of our results. We can identify elements of the group with infinite matrices, which allows us to make significant use of the polynomials in the matrix entries. Using properties of the stochastic process we will prove that these naturally defined holomorphic polynomials are square integrable with respect to the measure determined by the process. A second difference between the finite and infinite dimensional cases arises from the need to find a viable notion of holomorphic function on the group SOHS. We will consider two spaces of holomorphic functions. One of the spaces, 2 HL (SOHS, µt), is defined as the closure of the holomorphic polynomials in 2 L (µt)-norm. However, such a function might not have a version which is differentiable on the whole group. Therefore we also consider a Hilbert space HL2(SO( )) of functions differentiable on a subset SO( ) of the group. This subset plays∞ the role of the Cameron-Martin subspace. The∞ main theorem is that the Taylor map, f α, is an isometry from HL2(SO( )) into a Hilbert space contained in the dual of7→ the universal enveloping algebra. This∞ will imply, in particular, that any function in HL2(SO( )) is uniquely determined by its derivatives at the identity. In addition, there is∞ a natural isometry from the closure of the holomorphic polynomials 2 2 in L (µt)-norm to HL (SO( )). This isometry is an extension of the inclusion of these polynomials into this second∞ space. When G is finite dimensional, the heat kernel measure is defined as follows. There t∆/4 is a unique probability measure satisfying e f = µt f, where the Laplacian ∆ is a naturally defined second order differential operator∗ associated with a given Hermitian inner product on the Lie algebra of G. In the finite dimensional case the measure µt = µt(z)dz has the heat kernel µt(z) as its density with respect to Haar n measure on G. Note that if G = C then the heat kernel measure µt is a Gaussian 2 measure. Denote by HL (G, µt(z)dz) the space of holomorphic functions over G which are square integrable with respect to the heat kernel measure µt. B. Driver HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 3 and L. Gross [7] have shown that the Taylor map, f α, is an isomorphism 2 0 7→ between HL (G, µt(z)dz) and a subspace, Jt , of the algebraic dual of the universal enveloping algebra of the Lie algebra of G, when G is a connected simply connected 0 complex (finite dimensional) Lie group. Jt is a Hilbert space with respect to a norm which is naturally associated with the given inner product on the Lie algebra of G. In finite dimensions, when G happens to be the complexification of a compact 2 Lie group K, there is, in addition to the two Hilbert spaces, HL (G, µt(z)dz) 0 and Jt , discussed above, a third naturally arising Hilbert space. The third space 2 is L (K, ρt(x)dx), where ρt(x)dx is the heat kernel measure associated to a bi- invariant Laplacian on K. B. Hall [11], [12] has described a natural unitary iso- 2 2 morphism from L (K, ρt(x)dx) to HL (G, µt(z)dz). It is worth noting that one should not expect an analog of Hall’s transform for our group because his proof requires Ad-invariance of the inner product on Lie(K), and this condition seems 2 0 to be really essential. By contrast, the isomorphism from HL (G, µt(z)dz) to Jt does not require Ad-invariance of the Hermitian inner product on the Lie algebra of G. This is important in infinite dimensions since the norm we need to use is not Ad-invariant. Though there is an invariant inner product on the Lie algebra 0 of SOHS, namely, the Hilbert-Schmidt inner product, Jt turns out to be trivial in this case. The same happens for some other infinite dimensional groups whose Lie algebras have Ad-invariant inner product. We will show this in Section 8 for the Hilbert-Schmidt complex orthogonal group, SOHS. 2 0 L. Gross proved in [9] the isomorphism between L (K, ρt(x)dx) and Jt for a compact connected Lie group K using probabilistic methods. Later O. Hijab in [14], [15] found an analytical proof of this isomorphism. This isometry also depends on the AdK-invariance of the inner product on Lie(K) and seems unlikely to have a natural analog for our infinite dimensional orthogonal group. An extensive exposition of the subject can be found in [10]. In conclusion, we should note that in the case when G is replaced by a separable 0 Hilbert space, Jt is a bosonic Fock space. In this situation the isomorphisms between these three spaces give three different representations of the Fock space. The isomorphisms were studied by V.Bargmann, I. Segal, P. Kréeand others (see, for example, [1], [2], [17], [24], [25]). Recently such spaces have been studied in case of a complex abstract Wiener space by several authors including I. Shigekawa [26] and H. Sugita [28], [29]. Our results are precise analogs of some of the linear results of I. Shigekawa and H. Sugita, though the methods are entirely different. Not much is known for infinite dimensional groups. However, B. Hall and A. Sen- 2 gupta [13] have extended the isomorphism between L (K, ρt(x)dx) and 2 HL (G, µt(x)dx) to the group of paths in a compact (finite dimensional) Lie group. There have been a number of works about properties of infinite dimensional orthogonal and unitary groups. The representation theory of these groups has been studied, for example, in [16], [20], [27]. P. da la Harpe, R.J. Plymen and R.F. Streater have described topological properties of infinite dimensional orthogonal groups in [4], [5], [19] in connection with spinors in Hilbert space. One of the groups they have considered is the Hilbert-Schmidt orthogonal group. Acknowledgement. I wish to thank Professor L. Gross for his invaluable guid- ance throughout the process of preparation of this work. It is a pleasure to acknowl- edge stimulating discussions with Professor E. Dynkin and Professor B. Driver. 4 MARIA GORDINA

2. Notation and main results First of all, we describe the group we will consider in this paper. This group is represented as a group of operators in a Hilbert space. Let Hr be a real separable Hilbert space, H = H iH its complexification and I the identity operator on H. r ⊕ r Denote by HSr the Hilbert-Schmidt operators on Hr. Then HSr is a real Hilbert space equipped with the Hilbert-Schmidt inner product A, B HSr = T r(B∗A) for any A, B HS . h i ∈ r Consider the complexification of HSr denoted by HS = HSr iHSr. HS can be identified with the space of complex linear Hilbert-Schmidt⊕ operators on H. Any element A HS can be written as a sum of ReA and i ImA, where ReA and ImA HS . The∈ product of elements of HS is defined naturally by ∈ r AB = (ReA ReB ImA ImB) + i (ReA ImB + ImA ReB). − The space HS is a real Hilbert space equipped with the following inner product: def A, B = ReT r(A∗B) = ReA, ReB + ImA, ImB . h iHS h iHSr h iHSr Definition 2.1. The Hilbert-Schmidt complex orthogonal group SOHS is the connected component containing the identity I of the group OHS = B : B I HS, BT B = BBT = I , where BT means the transpose of the operator{ B−, i. e.∈ T } B = (ReB)∗ + i(ImB)∗.

Proposition 4.4 shows why OHS is not connected. The fact that SOHS I HS gives two essential advantages: first of all, it helps to define an inner product− on⊂ a Lie algebra of SOHS, and second, it will allow us to construct the heat kernel measure on SOHS. The Lie algebra of skew-symmetric Hilbert-Schmidt operators soHS = A : A HS, AT = A plays the role of a Lie algebra of SO . The group SO { ∈ − } HS HS is the complexification of the Hilbert-Schmidt orthogonal group, SOHSr , and soHS is the complexification of its Lie algebra. Note that , is an Ad-HS -invariant h· ·iHS r inner product on soHS. Later we will consider another inner product on a dense subspace of soHS (which is not Ad-invariant) since the isometry we are trying to establish is trivial in the invariant case. 2 2 0 We will consider three Hilbert spaces HL (SOHS, µt), HL (SO( )) and Jt . 0 ∞ The first two spaces are spaces of functions, while Jt has an algebraic nature. 2 2 The first space, HL (SOHS, µt), is the closure in L (SOHS, µt) of the space of holomorphic polynomials (in the matrix entries) on SOHS. Here µt is the heat kernel measure on SOHS which will be defined in Section 3 as the transition probability of a diffusion on SOHS. Note that this definition uses the fact that all holomorphic polynomials are square-integrable with respect to the heat kernel measure. The latter will be proved in Section 5. To define HL2(SO( )) we consider SO( ), a subset of SO , which is the ∞ ∞ HS closure of the direct limit of a sequence of finite dimensional subgroups of SOHS in a Riemannian metric. More precisely, fix an orthonormal basis f ∞ of H . { m}m=1 r Let HSn n = A : Afm, fk = 0 if max(m, k) > n . Take a basis ek ∞k=1 of HS × {2n2 h i } { } such that ek k=1 is a basis of HSn n. Note that SOHS(n) = B SOHS,B I { } × { ∈ − ∈ HSn n is a group isomorphic to the complexification of the special orthogonal × } n T group of R . Then gn = Lie(SOHS(n)) = soHS(n) = A HSn n,A = A is { ∈ × − } its Lie algebra. Note that g = ngn is a Lie subalgebra of soHS. We will choose a new inner product , on g∪so that the corresponding norm is stronger than the Hilbert-Schmidt norm.hh· ·ii This new inner product defines a Riemannian metric HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 5 on nSOHS(n). SO( ) is the closure of nSOHS(n) in this Riemannian metric. Then∪ HL2(SO( )) denotes∞ a certain space∪ of holomorphic functions on SO( ) with a certain direct∞ limit-type norm. More detailed definitions of these objects∞ are given in Section 7. 0 Finally, we need the following notation to describe Jt . Notation 2.2. Let g be a complex Lie algebra with a Hermitian inner product on it. Then T (g) will denote the algebraic tensor algebra over g as a complex vector space and T 0(g) will denote the algebraic dual of T (g). Define a norm on T (g) by n n 2 k! 2 k β = β , β = β , β g⊗ , k = 0, 1, 2, ..., t > 0(2.1) | |t tk | k| k k ∈ kX=0 kX=0 k k Here βk is the cross norm on g⊗ arising from the inner product on g⊗ determined | | k! by the given inner product on g. The coefficients tk are related to the heat kernel. Tt(g) will denote the completion of T in this norm. The topological dual of Tt(g) may be identified with the subspace Tt∗(g) of T 0(g) consisting of such α T 0(g) that the norm ∈ k 2 ∞ t 2 ∞ k α = α , α = α , α (g⊗ )∗, k = 0, 1, 2, ..., t > 0(2.2) | |t k! | k| k k ∈ kX=0 kX=0 k k is finite. Here α is the norm on (g⊗ )∗ dual to the cross norm on g⊗ . | k| There is a natural pairing for any α T 0(g) and β T (g) denoted by ∈ ∈ n ∞ ∞ k k α, β = α , β , α = α , β = β , α (g⊗ )∗, β g⊗ , k = 0, 1, 2, ... h i h k ki k k k ∈ k ∈ kX=0 kX=0 kX=0 Denote by J(g) the two-sided ideal in T (g) generated by ξ η η ξ [ξ, η], ξ, η g . 0 0 { ⊗ − ⊗ −0 ∈ } Let J (g) = α T 0(g): α(J) = 0 . Finally, let J (g) = T ∗(g) J (g). { ∈ } t t ∩ The first natural choice for g for our group is soHS with the Hilbert-Schmidt inner product, which is Ad-HSr-invariant. The problem with this inner product 0 is that in this case Jt is trivial (see Section 8). Therefore we will use another Lie algebra, g = gn = soHS(n), as described above together with an appropriate inner product∪ on g. ∪ The inner product on g will be non Ad-invariant. Namely, the inner product 1/2 1/2 is defined by A, B = Q− A, Q− B , where Q is a symmetric positive hh ii h iHS trace class operator on soHS such that all soHS(n) are invariant subspaces of Q. Actually, the closure of g in the corresponding norm is the subspace of soHS defined 1/2 1/2 as U = Q (so ). This norm is denoted by X = Q− X . 0 HS k kU0 k kHS In some sense U0 determines the directions in which derivatives of functions on SOHS will be taken. If we choose too large a set of directions we might not have nonconstant holomorphic functions. Moreover, if we define informally the Laplacian as

1 ∞ ∆ = (∂ )2 Q 2 ξn n=1 X where ξn n∞=1 is an orthonormal (in , ) basis of U0, then the stronger inner product{ gives} a weaker Laplacian. Inhh· Section·ii 3.3 we show that the heat kernel measure µt solves the heat equation with the Laplacian ∆Q. The weaker Laplacian 6 MARIA GORDINA allows µt to live on SOHS. Note that SO( ) is an analog of the Cameron-Martin subspace associated with any inﬁnite dimensional∞ Wiener space. 0 0 For the Lie algebra g we denote Jt = Jt (g). The latter can be called a space of Taylor coeﬃcients for functions in HL2(SO( )). Indeed, our main result is ∞ 2 0 that the Taylor series determines an isometry from HL (SO( )) into Jt . The 0 ∞ fact that Jt is not trivial (for the Lie algebra g) follows from the existence of this 2 0 2 isometry from HL (SOHS, µt) to Jt and the fact that HL (SOHS, µt) contains all holomorphic polynomials on SOHS. For each function f in HL2(SO( )) we use a natural notation ∞ 1 ∞ n (1 D)− f = D f − e e n=1 X for the series of its derivatives at the identity considered as an element of T 0. Then our results (Theorem 7.6, Theorem 7.5 and Theorem 7.4) can be summarized as follows. 1. HL2(SO( )) is a Hilbert space. 1∞ 2 0 2. (1 D)e− is an isometry from HL (SO( )) into Jt . 3. The− embedding of the space of holomorphic∞ polynomials HP into the Hilbert 2 2 space HL (SO( )) can be extended to an isometry from HL (SOHS, µt) into HL2(SO( ∞)). ∞ The following diagram describes these isometries:

HP inclusion HL2(SO , µ ) Theorem 7.4 HL2(SO( )) Theorem 7.5 J 0 −−−−−→ HS t −−−−−−−−→ ∞ −−−−−−−−→ t

3. Construction of the heat kernel measure

The goal of this section is to define the heat kernel measure on SOHS. To achieve it we will use a diffusion on the group. If we consider a stochastic process in SOHS Xt :Ω R SOHS, then Yt = Xt I is an element of HS. This space is a Hilbert× space,−→ which enables us to use the− machinery of the stochastic differential equations in Hilbert spaces developed in recent years. 3.1. Existence and uniqueness of a stochastic process. We begin with the definition of the process Yt. Denote by U the space soHS with the inner product , HS. Let Wt be a U-valued Wiener process with a covariance operator Q : U h·U.·i We assume that Q is a symmetric positive trace-class operator. Sometimes−→ we will identify Q with its extension by zero to the orthogonal complement of U in HS. 1 1/2 1/2 Let U0 = Q 2 (U) as before with the inner product u, v 0 = Q− u, Q− v HS. 0 h i h i Denote by L2 = L2(U0,HS) the space of the Hilbert-Schmidt operators from U0 2 to HS with the (Hilbert-Schmidt) norm Ψ L0 = T r[ΨQΨ∗]. k k 2 We need to define the coefficients of the stochastic differential equation before we formulate Theorem 3.3.

0 Notation 3.1. B : HS L2, B(Y )U = U(Y + I) for U U0, F : HS HS, 1 1/2 −→T 1/2 ∈ −→ F (Y ) = 2 n(Q en) (Q en)(Y + I), where en ∞n=1 is an orthonormal basis of HS as− a real space. { } P 1/2 T 1/2 Proposition 3.2. m(Q em) (Q em) does not depend on the choice of the 1/2 T 1/2 T basis e ∞. In addition, (Q e ) (Q e ) = e Qe . { m}1 P m m m m m m P P HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 7

Proof. Deﬁne a bilinear real form on HS HS by L(f, g) = Λ((Q1/2f)T (Q1/2g)), where Λ is a real bounded linear functional× on HS. Then f L(f, g) is a 7→ bounded linear functional on HS and so L(f, g) = f, g˜ HS for someg ˜ HS. There exists a linear operator on HS such that L(f,h g)i = f, Bg . Now∈ we h iHS can see that since trace is independent of a basis, T rB = m em, Bem HS = 1/2 T 1/2 h i m Λ(Q em) (Q em) does not depend on the choice of em m∞=1. To prove the T { P} second half of the statement, we note that e Qe is independent of e ∞ P m m m { m}m=1 by similar reasons. At the same time, if we choose em m∞=1 in which Q is diagonal, 1/2 T P1/2 { } T T say, Qem = λmem, then m(Q em) (Q em) = m λmemem = m emQem. Since both sums are independent of the choice of the basis, they are equal for any P P P basis e ∞ . { m}m=1 Theorem 3.3. 1. The stochastic diﬀerential equation

dYt = B(Yt)dWt + F (Yt)dt, (3.1) Y0 = 0 has a unique solution, up to equivalence, among the processes satisfying T P Y 2 ds < = 1. k skHS ∞ Z0 ! 2. For any p > 2 there exists a constant Cp,T > 0 such that p E sup Yt HS 6 Cp,T . t [0,T ] k k ∈ Proof of Theorem 3.3. To prove this theorem we will use Theorem 7.4, p.186 from the book by G.DaPrato and J.Zabczyk [3]. It is enough to check that 1. F is a measurable mapping from HS to HS. 2. F (Y1) F (Y2) HS 6 C Y1 Y2 HS for any Y1,Y2 HS. k −2 k k2 − k ∈ 3. F (Y ) HS 6 K(1 + Y HS) for any Y HS. k k k k ∈ 0 4. B(Y ) is a measurable mapping from HS to L2. 5. B(Y ) B(Y ) 0 C Y Y for any Y ,Y HS. 1 2 L2 6 1 2 HS 1 2 k −2 k 2k − k ∈ 6. B(Y ) L0 6 K(1 + Y HS) for any Y HS. k k 2 k k ∈ Proof of 1. Let us check that F (Y ) is in HS for any Y HS. ∈ 1 F (Y ) (Q1/2e )T (Q1/2e ) ( Y + 1) < , k kHS 6 2k n n kHS k kHS ∞ n X since (Q1/2e )T (Q1/2e ) (Q1/2e )T (Q1/2e ) k n n kHS 6 k n n kHS 6 n n X X Q1/2e 2 = T rQ < 6 k nkHS ∞ n X by the assumption. The measurability is trivial. Proof of 2. 1 F (Y ) F (Y ) (Q1/2e )T Q1/2e Y Y k 1 − 2 kHS 6 2k n nkHSk 1 − 2kHS 6 n X T rQ Y Y 6 k 1 − 2kHS 8 MARIA GORDINA as in the proof of 1. Proof of 3.

1 F (Y ) (Q1/2e )T (Q1/2e )Y + k kHS 6 2k n n kHS n X 1 1 + (Q1/2e )T (Q1/2e ) T rQ( Y + 1). 2k n n kHS 6 2 k kHS n X Finally, F (Y ) 2 1 (T rQ)2(1 + Y 2 ). k kHS 6 2 k kHS 0 Proof of 4. We want to check that B(Y ) is in L2 for any Y from HS. First of all, B(Y )U HS, for any U U0. Indeed, B(Y )U = U(Y + I) = UY + U HS, since U and V∈are in HS. ∈ ∈ Now let us verify that B(Y ) L2. Consider the Hilbert-Schmidt norm of B as ∈ 0 an operator from U0 to HS. Take an orthonormal basis um m∞=1 of U0. Then 1/2 { } Q− um ∞m=1 is an orthonormal basis of U and the Hilbert-Schmidt norm of B can{ be found} as follows:

2 ∞ ∞ B(Y ) L2 = B(Y )um,B(Y )um HS = um(Y + I), um(Y + I) HS 6 k k 0 h i h i m=1 m=1 X X ∞ Y + I 2 u , u = Y + I 2T rQ < , k k h m miHS k k ∞ m=1 X since the operator norm Y + I is ﬁnite. k k Proof of 5. Similarly to the previous proof we have 1/2 B(Y1) B(Y2) L0 6 Y1 Y2 HS(T rQ) . k − k 2 k − k Proof of 6. Use the estimate we have got in the proof of 4: 1/2 1/2 B(Y ) 0 (T rQ) Y + I (T rQ) ( Y + 1), so L2 6 6 HS k k2 k k2 k k B(Y ) L0 6 2(T rQ)(1 + Y HS). k k 2 k k 3.2. Process Yt + I lives in SOHS.

Theorem 3.4. Yt + I lies in SOHS for any t > 0 with probability 1. T T Proof of Theorem 3.4. We need to check that (Yt +I) (Yt +I) = (Yt +I)(Yt +I) = I with probability 1 for any t > 0. To achieve this goal, we will apply Itô’sformula T T to G(Yt), where G is defined as follows: G(Y ) = Λ(Y Y + Y + Y ), Λ is a nonzero linear real bounded functional from HS to R. Then (Y + I)T (Y + I) = I if and only if Λ(Y T Y + Y T + Y ) = 0 for any Λ. In order to use Itô’sformula we must verify several properties of the process Yt and the mapping G: 0 1. B(Ys) is an L2-valued process stochastically integrable on [0,T ] 2. F (Ys) is an HS-valued predictable process Bochner integrable on [0,T ] P -a.s. 3. G and the derivatives Gt,GY ,GYY are uniformly continuous on bounded subsets of [0,T ] HS. × Proof of 1. See 4 in the proof of Theorem 3.3. Proof of 2. See 1 in the proof of Theorem 3.3.

Proof of 3. Let us calculate Gt,GY ,GYY . First of all, Gt = 0. HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 9

G(Y + εS) G(Y ) T T T GY (Y )(S) = lim − = Λ(S Y + Y S + S + S) = ε 0 → ε = Λ(ST (Y + I) + (Y + I)T S) for any S HS. ∈

GY (Y + εT )(S) GY (Y )(S) T T GYY (Y )(S T ) = lim − = Λ(S T + T S) ε 0 ⊗ → ε for any S, T HS. Thus condition 3 is satisﬁed. We will use∈ the following notation:

G (Y )(S) = G¯ (Y ),S , Y h Y iHS G (Y )(S T ) = G¯ (Y )S, T , YY ⊗ h YY iHS where G¯Y is an element of HS and G¯YY is an operator on HS corresponding to the functionals G HS∗ and G (HS HS)∗. Y ∈ YY ∈ ⊗ Now we can apply Itˆo’sformula to G(Yt):

t t G(Yt) = G¯Y (Ys),B(Ys)dWs HS + G¯Y (Ys),F (Ys) HSds + 0 h i 0 h i (3.2) Z t Z 1 1/2 1/2 + T r[G (Y )(B(Y )Q )(B(Y )Q )∗]ds 2 YY s s s Z0 Let us calculate the three integrands in (3.2) separately. The ﬁrst integrand is

G¯ (Y ),B(Y )dW = G¯ (Y ), dW (Y + I) = h Y s s siHS h Y s s s iHS T T Λ((dWs(Ys + I)) (Ys + I) + (Ys + I) dWs(Ys + I)) = Λ( (Y + I)T dW (Y + I) + (Y + I)T dW (Y + I)) = 0, − s s s s s s since Wt is an soHS-valued process. The second integrand is

G¯ (Y ),F (Y ) = Λ((F (Y ))T (Y + I) + (Y + I)T F (Y )) = h Y s s iHS s s s s 1 = Λ (Y + I)T (Q1/2e )T (Q1/2e )(Y + I) + −2 s n n s n X T 1/2 T 1/2 + (Ys + I) (Q en) (Q en)(Ys + I) = = Λ (Y + I)T (Q1/2e )T (Q1/2e )(Y + I) . − s n n s n X 10 MARIA GORDINA

The third integrand is

1 1/2 1/2 T r[G (Y )(B(Y )Q )(B(Y )Q )∗] = 2 YY s s s 1 = G¯ (Y )B(Y )Q1/2e ,B(Y )Q1/2e ) = 2 h YY s s n s n iHS n X 1 = Λ((B(Y )Q1/2e )T (B(Y )Q1/2e ) + (B(Y )Q1/2e )T (B(Y )Q1/2e )) = 2 s n s n s n s n n X 1/2 T 1/2 = Λ(Q en(Ys + I)) (Q en(Ys + I)) = n X T 1/2 T 1/2 = Λ(Ys + I) (Q en) (Q en)(Ys + I). n X 1 1/2 1/2 Therefore G¯ (Y ),F (Y ) + T r[G¯ (Y )(B(Y )Q )(B(Y )Q )∗] = 0. This h Y s s iHS 2 YY s s s shows that the stochastic diﬀerential of G is zero, so G(Yt) = 0 for any t > 0. T By the Fredholm alternative Yt +I has an inverse, therefore it has to be (Yt +I) . T T Thus (Yt + I) (Yt + I) = (Yt + I)(Yt + I) = I for any t > 0.

3.3. Deﬁnition of the heat kernel measure. Let us deﬁne µt as follows:

f(X)µt(dX) = Ef(Xt(I)) = Pt,0f(I) ZSOHS for any bounded Borel function f on SOHS.

Deﬁnition 3.5. µt is called the heat kernel measure on SOHS.

Now we will present a motivation for such a name for µt. What follows will not be used to prove the main results of this paper. Note that F and B depend only on Y HS, therefore Ps,tf(Y ) = Ef(Y (t, s; Y )) = Pt sf(Y ). According to Theorem 9.16∈ from the book by G.DaPrato and J.Zabczyk [3],− p. 258, the following is true: 2 For any ϕ Cb (HS) and Y HS function v(t, Y ) = Ptϕ(Y ) is a unique ∈ 1,2 ∈ strict solution from Cb (HS) for the parabolic type equation called Kolmogorov’s backward equation:

∂ 1 1/2 1/2 v(t, Y ) = T r[v (t, Y )(B(Y )Q )(B(Y )Q )∗] + F (Y ), v (t, Y ) (3.3) ∂t 2 YY h Y iHS v(0,Y ) = ϕ(Y ), t > 0,Y HS. ∈

n Here Cb (HS) denotes the space of all functions from HS to R that are n-times continuously Frechet differentiable with all derivatives up to order n bounded and k,n Cb (HS) denotes the space of all functions from [0,T ] HS to R that are k- times continuously Frechet differentiable with respect to t and× n-times continuously Frechet differentiable with respect to Y with all partial derivatives continuous in [0,T ] HS and bounded. × HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 11

Let us rewrite Equation (3.3) as the heat equation. First of all

1/2 1/2 T r[vYY (t, Y )(B(Y )Q )(B(Y )Q )∗] = v (t, Y )(B(Y )Q1/2)e , (B(Y )Q1/2)e = h YY n niHS n X = v (t, Y )(Q1/2e )(Y + I), (Q1/2e )(Y + I) h YY n n iHS n X and 1 F (Y ), v (t, Y ) = (Q1/2e )T (Q1/2e )(Y + I), v (t, Y ) . h Y i −2 h n n Y iHS n X Now change Y to X I. Then we get that for any smooth bounded function − ϕ(X): HS + I R, function v(t, X) = Ptϕ(X) satisﬁes this equation which can be considered as→ the heat equation:

∂ v(t, X) = L v(t, X) (3.4) ∂t 1 v(0,X) = ϕ(X), t > 0,X HS + I, ∈ 1,2 where the differential operator L1 on the space Cb (HS + I) is defined by 1 L v def= [ v (X)(Q1/2e )X, (Q1/2e )X 1 2 h XX n n iHS − n X (Q1/2e )T (Q1/2e )X, v (X) ]. − h n n X iHS Our goal is to show that L1 is a Laplacian on SOHS in a sense. More precisely, L1 is a half of sum of second derivatives in the directions of an orthonormal basis of an analog of Lie algebra of SOHS. We begin with description of an analog of a Lie 1/2 algebra for SOHS and an inner product on it. Recall that U0 = Q (soHS) with 1/2 1/2 the inner product A, B = Q− A, Q− B . Note that U is a subspace hh ii h iHS 0 of soHS, but it is not a Lie subalgebra. As we will see later though, under some assumptions on Q space U has a dense (in the norm determined by , ) subspace 0 hh· ·ii which is a Lie algebra. Let ξn n∞=1 be an orthonormal basis of U0 and define the Laplacian by { }

1 ∞ (3.5) (∆v)(X) = (ξ˜ ξ˜ v)(X), 2 n n n=1 X ˜ d ˜ where (ξnv)(X) = t=0 v(exp(tξn)X) for a function v : SOHS R and so ξn is dt | → the right-invariant vector ﬁeld on SOHS corresponding to ξn Let us calculate derivatives of v : I + HS R in the direction of ξn: → d (ξ˜ v)(X) = v (X) (exp(tξ )X) = v (X)(ξ X) n X dt |t=0 n X n and therefore ˜ ˜ 2 (ξnξnv)(X) = vXX (X)(ξnX, ξnX) + vX (ξnX). 12 MARIA GORDINA

Using ξT = ξ we can rewrite the Laplacian n − n 1 (∆v)(X) = [v (X)(ξ X, ξ X) + v (ξ2 X)] = 2 XX n n X n n X 1 = [v (X)(ξ X, ξ X) v (ξT ξ X)]. 2 XX n n − X n n n X Define L2v(X) = (∆v)(X). Since ξñ is a right-invariant vector field, L2 is a right-invariant differential operator. It is known that L2 does not depend on choice of the basis ξn n∞=1. Let ξn = 1/2 { } Q en, where en ∞n=1 is an orthonormal basis of U with , HS as a inner product. { } 2 h· ·i Then we see that L1v = L2v on Cb (HS + I).

4. Approximation of the process 1/2 Let F be a subspace of U0 = Q (soHS) and PF a projection onto F. Then F KerQ = 0 , and therefore PFQPF is positive and invertible on F. Indeed, ∩ { } 1/2 1/2 for any f F we have PFQPFf, f HS = Qf, f HS = Q f, Q f HS > 0 if Qf = 0. ∈ h i h i h i Consider6 an equation

dYF = BF(YF)dWt + FF(YF)dt, YF(0) = 0, where 1 F (Y ) = ((P QP )1/2e )T (P QP )1/2e (Y + I), F −2 F F m F F m m X BF(Y )U = (PFU)(Y + I). This equation has a unique solution by the same arguments as in Section 3.1. Denote QF = PFQPF.

Lemma 4.1. YF is a solution of the equation

dYF = BF(YF)dWF,t + FF(YF)dt, (4.1) YF(0) = 0, where W = P W . In addition, I + Y SO a.s. F,t F t F,t ∈ HS Proof. The ﬁrst part is easy to check. Now check that PFQPF is the covariance operator of WF. By the deﬁnition of a covariance operator we know that f, Qg = E f, W g, W , therefore h iHS h 1iHSh 1iHS

f, Q g = P f, QP g = E P f, W P g, W = h F iHS h F F iHS h F 1iHSh F 1iHS

= E f, P W g, P W . h F 1iHSh F 1iHS Thus Equation 4.1 is actually the same as Equation 3.1 with the Wiener process that has the covariance QF instead of Q. Recall that we chose F and B so that the solution of Equation 3.1 plus the identity is in SOHS. Therefore I + YF,t SOHS a.s. ∈ HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 13

1/2 Choose a sequence of subspaces gn of U0 = Q soHS so that gn gn+1 and the closure of g in the metric of U is equal to U . Denote Y =⊂Y ,B = ∪n n 0 0 n,t gn,t n Bgn ,Fn = Fgn ,Qn = Qgn ,Pn = Pgn . Lemma 4.2. 1. T rQF 6 T rQ. 2. PnQ Q, −−−→n →∞ 3. QPn Q, −−−→n →∞ 4. PnQPn Q, −−−→n →∞ where convergence is in the trace class norm.

Proof of 1. First of all, QF is nonnegative. Thus T rQ = Q e , e Qe , e = T rQ. F h F n niHS 6 h n niHS n n X X Proof of 2. According to [21],v.1, Theorems VI.17,VI.21 there are orthonormal systems φ ∞ , ψ ∞ in HS and positive numbers λ such that { m}m=1 { m}m=1 m λ < , m ∞ m X Q( ) = λ , ψ φ . · mh· miHS m n X Consider an operator Ay = x, ψ φ. Then A = ψ φ . Assume h iHS k kT rCl k kHSk kHS ﬁrst that ψ HS = 1, φ HS = 1. Note that A y = y, ψ HSψ, since A∗y = k k k 2k | | h i φ, y HSψ and A∗A = A . Take such an orthonormal basis en ∞m=1 of HS that h i | | 2 { } 2 e1 = ψ. Then A T rCl = n A en, en HS = n en, ψ HS = ψ, ψ HS = 1. Using this we see thatk k h| | i h i h i P P

, ψ φ = 1, kh· miHS mkT rCl , ψ P φ 1, kh· miHS n mkT rCl 6 , ψm HS(Pnφm φm) T rCl 0. kh· i − k −−−→n →∞ In addition, PnQ Q T rCl 6 m λm < . Then PnQ Q T rCl 0 by the k − k ∞ k − k −−−→n →∞ Dominated Convergence Theorem.P The proof of 3 and 4 is similar.

Theorem 4.3. Denote by H2 the space of equivalence classes of HS-valued predictable processes with the norm: 2 1/2 Y 2 = ( sup E Y (t) HS) . ||| ||| t [0,T ] k k ∈ Then

Yn Y 2 0. ||| − ||| −−−→n →∞ Proof of Theorem 4.3. Let us apply the local inversion theorem (see, for example, Lemma 9.2, from the book by G.DaPrato and J.Zabczyk [3], p. 244) to t t K(y, Y ) = y + 0 B(Y )dWt + 0 F (Y )dt, where y is the initial value of Y and Y = Y (y, t) is an HS-valued predictable process. Analogously we deﬁne K (y, Y ) = R R n 14 MARIA GORDINA

t t y + 0 Bn(Y )dWt + 0 Fn(Y )dt. To apply this lemma we need to check that K and K satisfy the following conditions: n R R 1. For any Y1(t) and Y2(t) from H2

2 2 sup E K(y, Y1) K(y, Y2) HS 6 α sup E Y1(t) Y2(t) HS, t [0,T ] k − k t [0,T ] k − k ∈ ∈ where 0 6 α < 1 2. For any Y1(t) and Y2(t) from H2

2 2 sup E Kn(y, Y1) Kn(y, Y2) HS 6 α sup E Y1(t) Y2(t) HS, t [0,T ] k − k t [0,T ] k − k ∈ ∈ where 0 6 α < 1 3. limn Kn(y, Y ) = K(y, Y ) in H2. →∞ Proof of 1. In what follows we use (5.2) to estimate the part with the stochastic diﬀerential.

E K(y, Y ) K(y, Y ) 2 = k 1 − 2 kHS t t 2 = E (F (Y1) F (Y2))ds + (B(Y1) B(Y2))dWs HS 6 k 0 − 0 − k t Z Z t 2 2 6 2E( F (Y1) F (Y2) HSds) + 2E B(Y1) B(Y2)dWs HS 6 0 k − k k 0 − k Z t Z t 2 2 2 6 2(T rQ) E( Y1 Y2 HSds) + 8E B(Y1) B(Y2) L0 ds 6 0 k − k 0 k − k 2 Z t Z t 2 2 2 6 2(T rQ) E( Y1 Y2 HSds) + 8T rQE Y1 Y2 HSds 6 0 k − k 0 k − k Z t Z t 2 2 2 6 2(T rQ) tE Y1 Y2 HSds + 8T rQE Y1 Y2 HSds 6 0 k − k 0 k − k Z 2 Z 2 6 (2(T rQ) t + 8T rQ)t sup E Y1 Y2 HS t [0,T ] k − k ∈ Note that for small t we can make (2(T rQ)2t+8T rQ)t as small as we wish, therefore 1 holds.

Proof of 2.

1 T F (Y ) F (Y ) = (Q1/2e ) (Q1/2e )(Y Y ) k n 1 − n 2 kHS 2k n m n m 1 − 2 kHS 6 m X

1 T 1 (Q1/2e ) Q1/2e Y Y Q1/2e 2 Y Y = 6 2 k n m n mkHSk 1 − 2kHS 6 2 k n mkHSk 1 − 2kHS m m X X = T rQ Y Y T rQ Y Y nk 1 − 2kHS 6 k 1 − 2kHS by Lemma 4.2. Similarly to the proof of condition 5 in the proof of Theorem 3.3 we have that HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 15

2 Bn(Y1) Bn(Y2) L = Pn( )(Y1 Y2) U0 HS 6 k − k 0 k · − k → 1/2 1/2 1/2 1/2 T r((P Q )∗P Q ) Y Y = T r(Q P Q ) Y Y = n n k 1 − 2kHS n k 1 − 2kHS = T r(QP ) Y Y T rQ Y Y . n k 1 − 2kHS 6 k 1 − 2kHS Now use the same estimates as in 1 to see that 2 holds. Proof of 3. Here again we will use (5.2) to estimate the part with the stochastic diﬀerential. t t K (y, Y ) K(y, Y ) 2 = (B (Y ) B(Y ))dW + (F (Y ) F (Y ))dt 2 = ||| n − |||2 ||| n − t n − |||2 Z0 Z0 t t 2 sup E (Bn(Y ) B(Y ))dWs + (Fn(Y ) F (Y ))ds HS 6 t [0,T ] k 0 − 0 − k ∈ Z Z

t t 2 2 sup 2E( Fn(Y ) F (Y ) HSds) + 2E Bn(Y ) B(Y )dWs HS 6 t [0,T ] 0 k − k k 0 − k ∈ Z Z

t t 2 2 6 sup 2E( Fn(Y ) F (Y ) HSds) + 8E Bn(Y ) B(Y ) L0 ds 0. t [0,T ] 0 k − k 0 k − k 2 −−−→n ∈ Z Z →∞ Indeed,

1 T T F (Y ) F (Y ) = [(Q1/2e ) (Q1/2e ) (Q1/2e ) (Q1/2e )](Y + I) = n − 2 n m n m − m m m X 1 1 = [e T Q e e T Qe ](Y + I) = (e T (Q Q)e )(Y + I) 2 m n m − m m 2 m n − m m m X X by Proposition 3.2. Note that Qn Q is a self-adjoint trace class operator, thus (n) − (n) (n) (n) there exists a basis em m∞=1 in which Qn Q is diagonal: (Qn Q)em = λm em . Thus by Lemma 4.2{ } − −

1 (n) 1 Fn(Y ) F (Y ) HS 6 Y + I HS λm = T r(Q Qn) Y + I HS 0. k − k k k 2 | | 2 − k k −−−→n m →∞ X 2 Now let us estimate B(Y ) Bn(Y ) L0 . Choose an orthonormal basis k − k 2 u ∞ in U . Then { m}m=1 0

2 ∞ 2 B(Y ) Bn(Y ) L0 = (B(Y ) Bn(Y ))um HS = k − k 2 k − k m=1 X ∞ ∞ = (I P )u (Y + I) 2 Y + I 2 (I P )u 2 k − n m kHS 6 k k k − n mkHS 6 m=1 m=1 X X Y + I 2T r[Q1/2(I P )Q1/2] = Y + I 2T r[(I P )Q1/2Q1/2] = 6 k k − n k k − n 2 = Y + I T r[(I Pn)Q] 0. k k − −−−→n →∞ by Lemma 4.2. 16 MARIA GORDINA

We know that there are unique elements Y,Yn in the space H2 such that Y = K(y, Y ), Yn = Kn(y, Yn) and therefore limn Yn = Y for any y by the local inversion lemma. →∞

Now we will consider a concrete sequence of gn depending on an orthonormal basis fm m∞=1 of Hr. Recall that HSn n = A : Afm, fk = 0 if max(m, k) > { } × { 2n2 h i n . Take a basis ek k∞=1 of HS such that ek k=1 is a basis of HSn n. } { } { } × We also defined groups SOHS(n) = B SOHS,B I HSn n isomorphic { ∈ − ∈ × } to the complexification of the special orthogonal group of Rn and Lie algebras T gn = Lie(SOHS(n)) = soHS(n) = A HSn n,A = A with an inner product 1/2 { ∈ 1/2 × − } A, B = (P QP )− A, (P QP )− B . Here P = P and P QP hh iin h n n n n iHS n soHS (n) n n is considered as an operator on gn, where it is invertible and positive. Proposition 4.4. Define an operator K on HS by Kf = f , Kf = f , for 1 − 1 m m m > 1. Then for any A, B SOHS(n) we have A KB > B , where is the operator norm. ∈ k − k k k k · k

1 1 1 1 1 Proof. A KB > AB− K B− − = AB− K B − , since B k − k k − 1 kk k k − kk k ∈ SOHS(n). We know that AB− SOHS(n), therefore we can use the Cartan 1 ∈ decomposition to write AB− in the form UV , where U, V SOHS, U is real 1 ∈ orthogonal and V is self-adjoint and positive. We have AB− K = UV K = k − k k − k V U ∗K . Note that U ∗K is real orthogonal and detU ∗K = 1. This means k − k − that at least one eigenvalue of U ∗K is equal to 1. All the eigenvalues of V are − real and positive. Therefore V U ∗K 1 which completes the proof. k − k >

Corollary 4.5. The group OHS consists of two connected components, namely O = SO K(SO ). HS HS ∪ HS From now we assume that soHS(n) is an invariant subspace of operator Q for all n. There is a plenty of such operators Q. For example, all nonnegative (positive on soHS) trace class operators which are diagonal in the basis ek k∞=1. Under this 1/2 1/2 { } condition, in particular, A, B n = Q− A, Q− B HS. n hh ii h ni n Denote Pt f(Y ) = Ef(Yn(t, Y )), Y HSn n and v (t, Y ) = Pt f(Y ). Then the following theorem holds. ∈ ×

2 n Theorem 4.6. For any f Cb (HSn n) the function v (t, Y ) is a unique strict 1,2 ∈ × solution from C (HSn n) for the parabolic type equation: b ×

∂ n 1 n 1/2 1/2 n v (t, Y ) = T r[v (B (Y )Q )(B (Y )Q )∗] + F (Y ), v ∂t 2 YY n n h n Y iHS n v (0,Y ) = f(Y ), t > 0,Y HSn n. ∈ × n We want to show that Pt corresponds to the heat kernel measure deﬁned on SOHS(n) as on a Lie group (i. e. as in the ﬁnite dimensional case). Note that for any Y HSn n ∈ ×

n 1/2 1/2 T r[vYY (t, Y )(Bn(Y )Q )(Bn(Y )Q )∗] = vn (t, Y )P (Q1/2e )(Y + I),P (Q1/2e )(Y + I) . h YY n m n m iHS m X HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 17

F (Y ), vn (t, Y ) = h n Y iHS 1 ((P QP )1/2e )T ((P QP )1/2e )(Y + I), vn (t, Y ) . − 2 h n n m n n m Y iHS m X 1/2 2n2 Note that (PnQPn) em m=1 is an orthonormal basis of gn since all gn are invariant subspaces{ of Q, so }

1 n 1/2 1/2 n L v = T r[v (t, Y )(B (Y )Q )(B (Y )Q )∗] + F (Y ), v (t, Y ) 1 2 YY n n h n Y iHS n is equal to the Laplacian ∆ on SOHS(n) deﬁned similarly to (3.5). Thus, the n n transition probability Pt = µt (dX), where the latter is the heat kernel measure on SOHS(n) deﬁned in the usual way.

Notation 4.7. f L2(SO (n),µn) = f t,n, f L2(SO ,µ ) = f t. k k HS t k k k k HS t k k

5. Properties of the stochastic process and holomorphic polynomials The following Proposition is a reﬁnement of the second part of Theorem 3.3. It might be useful for the estimates of the L2-norms of holomorphic polynomials on SOHS. Proposition 5.1. For any p > 2, t > 0

p 1 t E Yt HS < (e Cp,t 1), k k Cp,t −

p 1 1 p p 1 p 1 p p 1 where Cp,t = 2 − max (T rQ) t − ,C p 2 − (T rQ) 2 t 2 − . { 2 2 } t p Proof. First of all, let us estimate E( 0 F (Ys) HSds) . From the proof of con- k k1 dition 3 of Theorem 3.3 we have F (Y ) HS 6 2 T rQ( Y HS + 1). Use H¨older’s 1 k 1 R k k k t t p inequality f(s)ds ( f ds) p (t p0 ) for f(s) = F (Y ) to get: 0 6 0 k s kHS R R t t p p 1 p (5.1) E( F (Ys) HSds) 6 t − E F (Ys) HSds 6 0 k k 0 k k Z Z t p 1 1 p p t − ( T rQ) E ( Y + 1) ds 2 k kHS Z0 t p Now estimate E 0 B(Ys)dWs HS. From part 6 of the proof of Theorem 3.3 k 2 k 2 we know that B(Y ) L0 6 2T rQ( Y HS + 1). In addition we will use Lemma k Rk 2 k k 7.2, p.182 from the book by G.DaPrato and J.Zabczyk [3]: for any r > 1 and for 0 arbitrary L2-valued predictable process Φ(t),

s t 2r 2 r (5.2) E( sup Φ(u)dW (u) HS) 6 CrE( Φ(s) L0 ds) , t [0,T ], s [0,t] k 0 k 0 k k 2 ∈ ∈ Z Z r 2r 2r2 where Cr = (r(2r 1)) ( 2r 1 ) . Then − − 18 MARIA GORDINA

t t p 2 p (5.3) E B(Y )dW C p E( B(Y ) 0 ds) 2 s s HS 6 2 s L 6 k 0 k 0 k k 2 Z t Z t p p p p p p 2 2 2 2 2 2 1 2 2 C p (2T rQ) E( (( Y HS + 1)ds) 6 C p 2 (T rQ) t − E ( Y HS + 1) ds 2 k k 2 k k Z0 Z0 q q 1 q Now we can use inequality (x+1) 6 2 − (x +1) for any x > 0 for the estimates (5.1) and (5.3):

t t p p 1 1 p p 1 p E( F (Ys) HSds) 6 t − ( T rQ) 2 − E (1 + Y HS)ds = 0 k k 2 0 k k Z Z t 1 p 1 p p = t − (T rQ) (t + E Y ds) 2 k kHS Z0 and

t t p p p p 1 p 1 p E B(Y )dW C p 2 2 (T rQ) 2 t 2 − 2 2 − E (1 + Y )ds = s s HS 6 2 s HS k 0 k 0 k k Z Z t p p 1 p 1 p = C p (T rQ) 2 2 − t 2 − (t + E Y ds) 2 k kHS Z0 Finally,

t t p p 1 p p E Yt HS 6 2 − [E( F (Ys) HSds) + E B(Ys)dWs HS] k k 0 k k k 0 k Z Z t C (t + E Y p ds), 6 p,t k skHS Z0 p 1 1 p p 1 p 1 p p 1 where C = 2 max (T rQ) t − ,C p 2 − (T rQ) 2 t 2 − . p,t − 2 2 p 1 { t } Thus, E Yt < (e Cp,t 1) by Gronwall’s lemma. k kHS Cp,t −

Notation 5.2. Suppose f is a function from SOHS to C. Let DX f = (Df)(X) denote a unique element of U0∗ such that d (D f)(ξ) = (ξf˜ )(X) = f(exp(tξ)X), ξ U = Q1/2so ,X SO , X dt ∈ 0 HS ∈ HS t=0 k k if the derivative exists. Similarly DX f = (D f)(X) denotes a unique element of k (U0⊗ )∗ such that k k (D f)(β) = (βf˜ )(X), β U ⊗ ,X SO X ∈ 0 ∈ HS k k k and Dn,X f = (Dnf)(X) denotes a unique element of (gn∗ )⊗ such that k k (D f)(β) = (βf˜ )(X), β g ⊗ ,X SO (n). n,X ∈ n ∈ HS Later we will use the following notation

1 ∞ k 1 ∞ k (1 D)− f = D f and (1 D)− = D f. − X X − n,X n,X kX=0 kX=0 HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 19

Recall that we have fixed f ∞ , an orthonormal basis of H . { k}k=1 r Definition 5.3. Let p : SOHS C. p is called a holomorphic polynomial, −→ m,l if p is a complex linear combination of finite products of monomials pk (X) = k ( ReXfm, fl + i ImXfm, fl ) . We will denote the space of all such polynomials byh HP. i h i Theorem 5.4. 1. HP Lp, for any p > 1. ⊂ k 2. The kth derivative (DX p)(β) exists and is complex linear for any p P, k 1/2 k ∈ X SO , β (U )⊗ = (Q so )⊗ . ∈ HS ∈ 0 HS Proof.

pm,l(X) pµ (dX) = E pm,l(Y + I) p = E (Y ) + δ ) pk | k | t | k t | | t ml ml | 6 ZSOHS

pk 1 pk pk 1 pk E2 − ( (Y ) + 1) 2 − E( Y + 1) < . | t ml| 6 k tkHS ∞ m,l p p Which means that pk L for any p > 1. Thus HP L (SOHS, µt). ∈ m,l⊂ To prove the second part is enough to check that pk is holomorphic. Let ξ be m,l any element of U0, then the derivative of pk in the direction of ξ can be calculated by the formula

d (ξp˜ m,l)(X) = pm,l(exp(tξ)X) = k dt k t=0 m,l m,l m,l = k( Re(ξX )fm, fl + i Im(ξX)fm, fl )pk 1(X) = kp1 (ξX)pk 1(X). h i h i − − m,l m,l ˜ m,l To prove that pk is holomorphic all we need is to check that Dpk (ξ) = (ξpk ) is complex linear. Indeed, for any α C we have ∈ ˜ m,l m,l m,l m,l m,l ˜ m,l ((αξ)pk )(X) = kp1 (αξX)pk 1(X) = kαp1 (ξX)pk 1(X) = α(ξpk )(X) − −

Remark 5.5. Any polynomial p HP can be written in the following form p(X) = m nm ∈ k=1 l=1 T r(AklX) for some Akl HS. The converse is not true in general, 2 ∈ m nm but the closure in L (SOHS, µt) of all functions of the form k=1 l=1 T r(AklX) coincidesP Q with the closure of holomorphic polynomials. Therefore the next definition P Q is basis-independent, though the definition of HP depends on the choice of f ∞ . { k}k=1 2 Definition 5.6. The closure of all holomorphic polynomials in L (SOHS, µt) is 2 called HL (SOHS, µt). Lemma 5.7. Let f : SO [0, ] be a continuous function in L2(SO , µ ). HS −→ ∞ HS t If f t,n 6 C < , then Ef(Xn) Ef(X). k k ∞ −−−→n →∞

Proof. Note that there exists a subsequence Xnk such that Xnk X a.s. { } −−−→k →∞ We will prove ﬁrst that if f t,nk 6 C < , then Ef(Xnk ) Ef(X). Denote k k ∞ −−−→k g (ω) = f(X (ω)), g(ω) = f(X(ω)), ω Ω. →∞ k nk ∈ Our goal is to prove that Ω gk(ω)dP Ω g(ω)dP as k . Deﬁne fl(X) = min f(X), l for l > 0 and g (ω) = f−→(X (ω)), g (ω) = f→(X ∞(ω)). Then g { } R k,l l Rnk l l k,l 6 20 MARIA GORDINA l, gl 6 l for any ω Ω, so Ω gk,l(ω)dP Ω gl(ω)dP by the Dominated ∈ −−−→k →∞ Convergence Theorem since f Ris continuous. R

(g (ω) g (ω))dP = (f(X ) l)dP k − k,l nk − 6 ΩZ ω:f(XZ ) l { nk > } 1 2 6 f(Xnk )1 ω:f(Xn ) l dP 6 f t,nk (P ω : f(Xnk ) > l ) 6 { k > } k k { } ΩZ E f(X ) f | nk | 6 k kt,nk l by Chebyshev’s inequality. Thus C2 0 (g (ω) g (ω))dP . 6 k − k,l 6 l ZΩ Similarly 1 0 (g(ω) g (ω))dP f E f(X) . 6 − l 6 l k kt | | ZΩ

Therefore Ef(Xnk ) Ef(X) as k . To complete the proof−→ suppose that→ the ∞ conclusion is not true. Then there is a subsequence X such that Ef(X Ef(X) > ε for any k. However, we always nk | nk − | can choose a subsequence Xnk such that Xnk X a.s. and therefore m m −−−−→m →∞ Ef(Xnk ) Ef(X). Contradiction. m −−−−→m →∞ Corollary 5.8. f t,n f t for any f HP. k k −−−→n k k ∈ →∞ Proof. From the estimates on E Y p we can ﬁnd C(p, t) such that p 2 k tkHS k| | kt,n(k) 6 C(p, t) for any k. Then apply Lemma 5.7 to f = p 2. | | 6. Estimates of derivatives of holomorphic functions

Let SO( ) denote the closure of nSOHS(n) in the following Riemannian ∞ 1 1 dh ∪ metric: d(A, B) = inf 0 h(s)− ds U0 ds , where h : [0, 1] SOHS, h(0) = A, h(1) = B . Note that{ gk = g(nk) is a} Lie algebra and U→is the closure of } R ∪n 0 g in the inner product on U0. The following estimate was proved by B. Driver and L. Gross in [7] for f 2 n ∈ HL (SOHS(n), µt ): 2 ˜ 2 2 k! β g 2/s k (βf)(g) g ) k 6 f t,n | | e| | , for g SOHS(n), r > 0, s + r 6 t, β g⊗n , | |( n∗ ⊗ k k rk ∈ ∈ ˜ k k where (βf)(g) ( ) k is (TgG(n)∗)⊗ -norm (which can be identified with (gn∗ )⊗ ). | | gn∗ ⊗ We will need a slight modification of this estimate. Taking supremum over all k β g⊗ , β = 1, we get ∈ n | | k 2 2 k! g 2/s (6.1) (D f)(g) f e| | , | n | 6 k kt,n rk k where Dn is defined for SOHS(n) and gn by Notation 5.2. Note that if f t,n are uniformly bounded, (6.1) gives us a uniform bound, i. e. independent ofk kn. The k following estimates can be proved for Dn: HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 21

Lemma 6.1. Let r > 0, q + r t, X, Y SO (n), f HL2(SO (n), µn). Then 6 ∈ HS ∈ HS t k k k (Dnf)(X) (Dnf)(Y ) (g )⊗ 6 f t,nKk+1d(X,Y ), | − | n∗ k k 1/2 k! X 2+d(X,Y )2/q where Kk = Kk(X,Y ) = rk e| | . Proof. Take h : [0, 1] SO (n) such that h(0) = X, h(1) = Y . Then by (6.1) −→ HS

1 k k d k k k Dnf(Xi) Dnf(Y ) (g )⊗ = (Dnf)(h(s))ds (g∗ )⊗ 6 | − | n∗ | ds | n Z0

1 1 d k k d k k (Dnf)(h(s)) (g∗ )⊗ ds 6 D(Dnf)(h(s))( h(s)) (g∗ )⊗ ds 6 |ds | n | ds | n Z0 Z0

1 k d k+1 D((Dnf)(h(s)) (g )⊗ h(s) Th(s)SOHS (n)ds 6 | | n∗ |ds | Z0

1/2 1 (k + 1)! h(u) 2/q d f sup e| | h(s) ds t,n k+1 Th(s)SOHS (n) 6 k k r u [0,1] 0 |ds | ∈ Z

1/2 1 (k + 1)! h(0) 2+d(h(0),h(u))2/q d f sup e| | h(s) ds t,n k+1 Th(s)SOHS (n) k k r u [0,1] 0 |ds | ∈ Z Taking inﬁmum over all such h we see that

1/2 k k (k + 1)! X 2+d(X,Y )2/q k Dnf(X) Dnf(Y ) (g )⊗ 6 f t,n e| | d(X,Y ). | − | ∗n k k rk+1

Lemma 6.2. Let X SO (n), f, g HL2(SO (n), µn), t > 0. Then ∈ HS ∈ HS t

k k k Dnf(X) Dng(X) (g∗ )⊗ 6 Mk f g t,n, | − | n k − k 1/2 k! X 2/t where Mk = Mk(X, t) = (t/2)k e| | . Proof. From (6.1) we have that for r > 0, q + r 6 t

k k 2 2 k! X 2/q Dnf(X) Dng(X) (g ) k 6 f g t,n e| | . | − | n∗ ⊗ k − k rk Now take q = r = t/2 to get what we claimed.

2 n Lemma 6.3. Let X SOHS(n), ξ gn, f HL (SOHS(n), µt ). Then there is a constant C = C(X, ξ,∈ t) > 0 such that∈ ∈ f(euξX) f(X) − (D f)(ξ) f Cu | u − X | 6 k kt,n for small enough u > 0. 22 MARIA GORDINA

sξ Proof. Let h(s) = e X, 0 6 s 6 u.

f(euξX) f(X) 1 u d − (D f)(ξ) = f(h(s))ds (D f)(ξ) = u − X u ds − X Z0 1 u d 1 u d = ( f(h(s)) (D f)(ξ))ds = (D f)( h(s)) (D f)(ξ))ds = u ds − X u h(s) ds − X Z0 Z0 1 u = (D f)(ξ) (D f)(ξ))ds. u h(s) − X Z0 Thus by Lemma 6.1 for any r > 0, q + r 6 t

f(euξX) f(X) 1 u − (D f)(ξ) (D f)(ξ) (D f)(ξ)) ds | u − X | 6 u | h(s) − X | 6 Z0 1 u f K (h(s),X)d(h(s),X) ξ ds u k kt,n 2 k kU0 6 Z0 u 1 √2 ( X 2+ ξ 2 s2)/q 2 f e | | k kU0 s ξ ds = 6 u k kt,n r k kU0 Z0 u √2 X 2/q 2 1 ( ξ 2 /q)s2 k kU0 = f t,n e| | ξ U0 e sds = k k r k k u 0 Z 2 2 ( ξ U /q)u q X 2/q e k k 0 1 = f t,n e| | − 6 f t,nCu k k √2r u k k for small u. Theorem 6.4. Let f be a function on SO (n). Suppose that f is ∪n HS |SOHS (n) holomorphic for any n and f t,n 6 Ct < . Then there exists a unique continuous function g on SO( ) suchk thatk g ∞= f for any n. ∞ |SOHS (n) Proof. Take X SO( ). We would like to deﬁne g by ∈ ∞ d g(X) = lim f(Xn),Xn SOHS(n),Xn X. n ∈ −−−→n →∞ →∞ Let us check that the limit exist. Assume that l 6 n. By Lemma 6.1 for X ,X SO (n) n l ∈ HS

k k k D f(Xn) D f(Xl) (g )⊗ 6 f t,nKk+1(Xn,Xl)d(Xn,Xl) | − | n∗ k k 6 CtKk+1(Xn,Xl)d(Xn,Xl).

Thus f(Xn) is a Cauchy sequence and therefore the limit exists. The uniqueness of the extension follows from this simple argument: g (X) g (X) g (X) g (X ) + g (X ) g (X ) + g (X ) g (X) = | 1 − 2 | 6 | 1 − 1 n | | 1 n − 2 n | | 2 n − 2 |

= g1(X) g1(Xn) + g2(Xn) g2(X) 0 | − | | − | −−−→n →∞ d for Xn SOHS(n), X SO( ),Xn X. ∈ ∈ ∞ −−−→n →∞ HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 23

7. The isometries In Section 6 we noted that g = g = so(n) is a Lie algebra which is dense ∪n n ∪n in U0. For this Lie algebra ( with the inner product , ) and its Lie subalgebras 0 0 hh· ·ii gn we can consider T,T 0,Tt∗, J, J ,Jt deﬁned in Section 2. Notation 7.1.

T (g) = T,T 0(g) = T 0,Tt∗(g) = Tt∗,T (g(n)) = Tn,T 0(g(n)) = Tn0 ,Tt∗(g(n)) = Tt,n∗ , 0 0 0 0 0 0 0 0 J(g) = J, J (g) = J ,Jt (g) = Jt ,J(g(n)) = Jn,J (g(n)) = Jn,Jt (g(n)) = Jt,n. The norm deﬁned by Equation (2.2) will be denoted by for g and by for | · |t | · |t,n gn.

2 n Lemma 7.2. Suppose f SOHS (n) HL (SOHS(n), µt ) for all n. Then f t,n 6 f for any n. | ∈ k k k kt,n+1 2 1 2 Proof. First of all, f t,n = (1 D)n,e− f t,n by the Driver-Gross isomorphism, 1 k2 k | −k 2| where (1 D )− f = ∞ (D f)(e) . Note that | − n e |t,n | k=0 n |t,n (Dk fP)(e) 2 = ξ˜ ...ξ˜ f(e) 2 | n | | i1 ik | 16imX6dimgn dimgn k 2 k 2 for any orthonormal basis ξl l=1 of gn. Therefore (Dnf)(e) 6 (Dn+1f)(e) , so the claim holds. { } | | | |

Theorem 6.4 allows us to introduce the following deﬁnition. Deﬁnition 7.3. HL2(SO( )) is a space of continuous functions on SO( ) such ∞ ∞ that their restrictions to SOHS(n) are holomorphic for every n and f t, =supn f t,n = limn f t,n < . k k ∞ {k k } →∞ k k ∞ Theorem 7.4. The embedding of HP into HL2(SO( )) can be extended to an isometry from HL2(SO , µ ) into HL2(SO( )). ∞ HS t ∞ Proof. By Theorem 5.4 HP L2(SO , µ ). In addition, by Corollary 5.8 ⊂ HS t p t,n p t, p HP. k k −−−→n k k ∈ →∞ Therefore p t, = p t and so the embedding is an isometry. 2 k k ∞ k k HL (SOHS, µt) is the closure of HP, therefore the isometry extends to it from HP.

1 2 0 Theorem 7.5. (1 D)− is an isometry from HL (SO( )) into J . − e ∞ t

Proof. Tn is a subalgebra of T . Note that Tn0 can be easily identiﬁed with a subspace of T 0. Namely, for any α T 0 we can deﬁne α as follows: n ∈ n α on T α = n n (0 on Tn⊥

0 Therefore Tn0 = (Tn⊥) . Deﬁne Πn to be an orthogonal projection from T to Tn. Let Π0n denote the following map from T 0 to Tn0 : (Πn0 α)(x) = α(Πnx), α T 0, x T. 1 2 1 ∈ ∈ Then Π0 (1 D)− : HL (SO , µ ) T 0 is equal to (1 D)− . Indeed, note n ◦ − e HS t −→ n − n,e 24 MARIA GORDINA that if we choose an orthonormal basis of g so that ξ m=dimgn is an orthonormal { m}m=1 basis of gn, then Πn can be described explicitly

0 if ks > dim gn for some 1 6 s 6 l Πn(ξk ξk ... ξk ) = 1 ⊗ 2 ⊗ ⊗ l ξ ξ ... ξ otherwise ( k1 ⊗ k2 ⊗ ⊗ kl 1 1 And Π0 (1 D)− f, ξ ξ ... ξ = (1 D)− f, Π (ξ ξ ... ξ ) = h n ◦ − e k1 ⊗ k2 ⊗ ⊗ kl i h − e n k1 ⊗ k2 ⊗ ⊗ kl i 0 if ks > dim gn for some 1 6 s 6 l 1 (1 D)− f, ξ ξ ... ξ otherwise (h − e k1 ⊗ k2 ⊗ ⊗ kl i 1 1 So Π0n (1 D)−e = (1 D)n,e− . B. Driver and L. Gross proved in [7] that Πn0 (1 1 ◦ − − 2 n 0 ◦ − D)e− is an isometry from HL (SOHS(n), µt ) into Jt,n. Let us deﬁne a restriction 2 2 n map Rn: HL (SO( )) HL (SOHS(n), µt ) by f f SOHS (n). Thus we have a commutative diagram:∞ −→ 7→ |

1 (1 D)− HL2(SO( )) − e J 0 ∞ −−−−−−→ t

Rn Πn0

1  (1 D)−  2  n − n,e 0 HL (SOHSy (n), µt ) Jyt,n −−−−−−−−→Driver-Gross 1 Now we can prove that (1 D)− is an isometry. − e

f t, = lim f t,n = lim Rnf t,n ∞ n n k k →∞ k k →∞ k k by Proposition 7.2. It is clear that Πn0 α t = Πn0 α t,n α t for any α | | | | −−−→n | | ∈ 1 1 →∞ T 0. In particular, Π0n (1 D)e− f t,n (1 D)e− f t. At the same time | ◦ − | −−−→n | − | 1 1 →∞ Πn0 (1 D)e− f t,n = (1 D)e,n− Rnf t,n = Rnf t,n f t, by the | ◦ − | | − ◦ | k k −−−→n k k ∞ Driver-Gross isomorphism. →∞ Theorem 7.6. HL2(SO( )) is a Hilbert space. ∞ Proof. It is clear that t, is a seminorm. Suppose that f t, = 0. Then f = 0 for any n, t >k ·0. k We∞ know that HL2(SO (n), µn)k isk a∞ Hilbert space, k kt,n HS t therefore f SO (n) = 0 for all n. By Lemma 6.4 we have that f SO ( ) = 0. | HS | HS ∞ Thus t, is a norm. k · k ∞ 2 Let us now show that HL (SO( )) is a complete space. Suppose fm m∞=1 is a 2 ∞ { } Cauchy sequence in HL (SO( )). Then f ∞ is a Cauchy sequence ∞ { m |SOHS (n)}m=1 in HL2(SO (n), µn) for all n. Therefore there exists g HL2(SO (n)) such HS t n ∈ HS that fm SOHS (n) gn. Note that gn+m SOHS (n) = gn. In addition, | −−−−→m | →∞

(7.1) g f + f g k nkt,n 6 k m|SOHS (n)kt,n k m|SOHS (n) − nkt,n 6 fm t, + fm SO (n) gn t,n. k k ∞ k | HS − k Note that fm t, m∞=1 is again a Cauchy sequence, so it has a (ﬁnite) limit as {k k ∞} m . Taking a limit in (7.1) as m we get that gn t,n n∞=1 are uniformly bounded.→ ∞ → ∞ {k k } By Lemma 6.4 there exists a continuous function g on SOHS( ) such that 2 ∞ g SOHS (n)= gn. Thus g HL (SO( )). Now we need to prove that fm g | ∈ ∞ −−−−→m →∞ HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 25

2 1 1 in HL (SO( )). Let α = (1 D)−e g, αm = (1 D)e− fm. By Theorem 7.5 1 ∞ − − 0 (1 D)e− is an isometry, so αm is a Cauchy sequence in Jt . Thus there exists − 0 0 α0 Jt such that αm α0 (in Jt ). The question is whether α = α0. ∈ −−−−→m Using the same notation→∞ as in the proof of Theorem 7.5 we see that 1 1 Π0 α = (1 D)− g and Π0 α = (1 D)− f . n − n,e n m − n,e m 2 n We know that HL (SOHS(n), µt ) is a Hilbert space, therefore Πn0 α = Πn0 α0 for any n. Thus α = α0, which completes the proof.

Remark 7.7. Note that SOHS(n) are not simply connected and therefore the map 1 (1 D)e− may not be surjective. It is known that if a connected (finite dimensional) − 1 Lie group G is simply connected, then the map (1 D)e− is surjective. At the same time if G is not simply connected, then this map is− not onto. Our proof of Theorem 7.5 did not rely on any specific properties of groups SOHS(n), therefore one can prove an analog of this theorem in a more abstract situation. In particular, if we consider a sequence of finite dimensional connected simply connected Lie groups 1 G ∞ , the surjectivity of the map (1 D)− can be proved. { n}n=1 − e

0 8. Triviality of Jt for an invariant inner product 0 If we have an isometry described above then the size of Jt gives us information about how many square-integrable holomorphic functions might exist. Indeed, if 0 Jt is isomorphic to C, then the only such functions can be constants. This is exactly the situation that happens in the case of an Ad-invariant inner product on Lie(SOHS). To see this we will use the following theorem. Theorem 8.1. Suppose g is a Lie algebra with an inner product , . Assume that h· ·i there is an orthonormal basis ξ ∞ of g such that for any k there are nonzero { k}k=1 αk C and an inﬁnite set of distinct pairs (im, jm) satisfying ξk = αk[ξim , ξjm ]. ∈ 0 Then Jt is isomorphic to C.

Proof of Theorem 8.1. First we prove that ξk lies in the completion of J in the norm deﬁned by (2.1) for any k, t > 0. Indeed, denote η = ξ ξ ξ ξ . m im ⊗ jm − jm ⊗ im Then ηm ηl for any m = l and ηm t = ηl t for any m, l. From the assump- ⊥ 6 1 k k k k 1 n tions on g we know that ηm ξk J for all m. If we deﬁne µn = ηm, − αk ∈ n m=1 then µ + ξ = 1 n (η + ξ ) lies in J. Note that n k n m=1 m k P P n 1 2 1 µn t = ηm t = η1 t 0. k k nv k k √nk k −−−→n um=1 →∞ u X t This shows that µn + ξk ξk and therefore ξk lies in J¯ . −−−→n 1< →∞ Denote by Tt a subspace of Tt containing tensors of order greater than 1, then 1< 1< 0 0 Tt = T C. By the above J is dense in T for any t > 0 and J = T ∗ J is t ⊕ t t t ∩ isomorphic to C for such g. Corollary 8.2. J 0 is trivial for g = so with , = , . t HS h· ·i h· ·iHS 26 MARIA GORDINA

Proof. Take an orthonormal basis fm m∞=1 in Hr. Then bij, i < j defined by b f = f , b f = f , b f = 0 for{m }= i, j is an orthogonal basis of so . One ij i − j ij j i ij m 6 HS can think of bij as the following infinite matrices: i j ...... i ... 0 ...... 1 ... b = ij ......  j ... 0 1 ......   −  ......      Operators bij satisfy the following identity: [bik, bjk] = bij for k = i, k = j. 0 − 6 6 Therefore Jt is trivial by Theorem 8.1. References [1] V.Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Communications of Pure and Applied Mathematics, 24, 1961, 187-214. [2] V.Bargmann, Remarks on a Hilbert space of analytic functions, Proc. of the National Acad- emy of Sciences, 48, 1962, 199-204. [3] G.DaPrato and J.Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univer- sity Press,Cambridge,1992. [4] P. de la Harpe, Some properties of infinite-dimensional orthogonal groups, Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972) , Vol. II , Internat. Atomic Energy Agency, Vienna, 1974, 71-77. [5] P. de la Harpe, The Clifford algebra and the spinor group of a Hilbert space, Compositio Math. 25 , 1972, 245-261. [6] B.Driver, On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall isomorphisms, J. of Funct. Anal., 133,1995,69-128. [7] B.Driver and L.Gross, Hilbert spaces of holomorphic functions on complex Lie groups, Proceedings of the 1994 Taniguchi International Workshop (K. D. Elworthy, S. Kusuoka, I. Shigekawa, Eds.), World Scientific Publishing Co. Pte. Ltd, Singapore/New Jer- sey/London/Hong Kong, 1997. [8] L. Gross, Existence and uniqueness of physical ground states, J. Funct. Anal., 10, 1972, 52-109. [9] L. Gross, Uniqueness of ground states for Schrödingeroperators over loop groups, J. Funct. Anal., 112, 1993, 373-441. [10] L. Gross and P. Malliavin, Hall’s transform and the Segal-Bargmann map, Itô’sStochastic Calculus and Probability Theory, (M.Fukushima, N. Ikeda, H. Kunita and S. Watanabe, Eds.), Springer-Verlag, New York/Berlin, 1996. [11] B.Hall, Quantum Mechanics in Phase Space, to appear in Proc. of the Summer Research Conf. on Quantization, (L. Coburn and M. Rieffel, Eds.), 1997. [12] B.Hall, The Segal-Bargmann ’coherent state’ transform for compact Lie groups, J. of Funct. Anal., 122, 1994, 103-151. [13] B.Hall and A. Sengupta, The Segal-Bargmann transform for paths in a compact group, in preparation, 1997. [14] O. Hijab, Hermite functions on compact Lie groups I, J. of Funct. Anal., 125, 1994, 480-492. [15] O. Hijab, Hermite functions on compact Lie groups II, J. of Funct. Anal., 133, 1995, 41-49. [16] A. A. Kirillov, Representations of the infinite-dimensional unitary group, Dokl. Akad. Nauk. SSSR, 212, 1973, 288-290. [17] P. Krée, Calcul d’intégrales et de dérivéesen dimension infinite, J. of Funct. Anal., 31, 1979, 150-186. [18] E. Nelson, Analytic vectors, Annals of Mathematics, 70, 1959, 572-615. [19] R. J. Plymen, R. F. Streater, A model of the universal covering group of SO(H)2, Bull. London Math. Soc. 7, 1975, no. 3, 283-288. [20] A. Pressley, G. Segal, Loop groups, Clarendon Press, Oxford University Press, New York, 1986. [21] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, 1972. HOLOMORPHIC FUNCTIONS ON AN INFINITE DIMENSIONAL GROUP 27

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Department of Mathematics, Cornell University, Ithaca, New York 14853 E-mail address: [email protected]