Reproducing Kernels and Analogs of Spherical Functions

JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2017009 c American Institute of Mathematical Sciences Volume 9, Number 2, June 2017 pp. 207–225

THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE: REPRODUCING KERNELS AND ANALOGS OF SPHERICAL FUNCTIONS

Yury Neretin∗

Math. Dept., University of Vienna Oskar-Morgenstern-Platz 1, 1090 Wien, Austria Institute for Theoretical and Experimental Physics (Moscow), Russia MechMath. Dept., Moscow State University Institute for Information Transmission (Moscow), Russia

Abstract. The group Diff(S1) of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups U(p, q), Sp(2n, R), SO∗(2n); the space Ξ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of Diff(S1) in the space of holomorphic functionals on Ξ, reproducing kernels on Ξ determining inner products, and expressions (‘canonical cocycles’) replacing spherical functions.

1. Introduction.

1.1. The purpose of the paper. The group Diff(S1) of orientation preserving diffeomorphisms of the circle has unitary projective highest weight representations. There is a well-developed representation theory of unitary highest weight representations of real semi-simple Lie groups (see, e.g., an elementary introduction in [30], Chapter 7, and further references in this book). In particular, this theory includes realizations of highest weight representations in spaces of holomorphic (vector-valued) functions on Hermitian symmetric spaces, reproducing kernels, Berezin–Wallach sets, Olshanski semigroups, Berezin–Guichardet–Wigner formulas for central extensions. All these phenomena exist for the group Diff(S1). The purpose of this paper is to present details of a strange calculation sketched at the end of my paper [25], 1989. It also was included to my thesis [27], 1992. Later the calculation and its result never were exposed or repeated. Let us consider a unitary highest weight representation ρ of the group Diff(S1). Generally, it is a projective representation. Operators ρ(g) are determined up to constant factors. We normalize them by the condition hρ(g)v, vi = 1,

2010 Mathematics Subject Classiﬁcation. Primary: 81R10, 46E22; Secondary: 30C99, 43A90. Key words and phrases. Univalent functions, welding, reproducing kernel, Fock space, Virasoro algebra, highest weight representation, spherical function. Supported by FWF grants P25142, P28421. ∗ Corresponding author: Yury Neretin.

207 208 YURY NERETIN where v is a highest weight vector1. Then we have

ρ(g1)ρ(g2) = c(g1, g2) ρ(g1g2). where the expression c(g1, g2) (the canonical cocycle) is canonically determined by the representation ρ. In this paper, we derive a formula for c(g1, g2). Remark. Let G be a semisimple group, K a maximal compact subgroup and v a K-ﬁxed vector, let ϕ(g) = hρ(g)v, vi be the spherical function. Then

ϕ(g1g2) c(g1, g2) = . ϕ(g1) ϕ(g2) In our case, spherical functions are not well-deﬁned but the ‘canonical cocycles’, which are hybrids of central extensions and spherical functions, exist. A formula for canonical cocycles implies some automatic corollaries: we can write explicit formulas for realizations of highest weight representations of Diﬀ(S1) on the space of univalent functions and for invariant reproducing kernels on the space of univalent functions. 1.2. Virasoro algebra and highest weight modules. For details, see, e.g.,

[28]. Denote by VectR and VectC respectively the Lie algebras of real (respectively inϕ complex) vector ﬁelds on the circle. Choosing a basis Ln := e ∂/i∂ϕ in VectC, we get the commutation relations

[Ln,Lm] = (m − n)Lm+n.

Recall that the Virasoro algebra Vir is a Lie algebra with a basis Ln, ζ, where n ranges in Z, and commutation relations n3 − n [L ,L ] = (m − n)L + δ ζ, [L , ζ] = 0. n m m+n 12 n+m,0 n Let h, c ∈ C.A module with the highest weight (h, c) over Vir is a module containing a vector v such that 1) L0v = hv, ζv = cv; 2) L−nv = 0 for n < 0; 3) the vector v is cyclic. There exists a unique irreducible highest weight module L(h, c) with a given highest weight (h, c), also there is a universal highest weight module M(h, c) (a Verma module), such that any module with highest weight (h, c) is a quotient of the Verma module M(h, c). In particular, if M(h, c) is irreducible, we have M(h, c) = L(h, c). By [15], [6], a Verma module M(h, c) is reducible if and only if (h, c) ∈ C2 satisﬁes at least one equation of the form 1 1 1 1 h − (α2 − 1)(c − 13) + (α2 − 1) h − (β2 − 1)(c − 13) + (β2 − 1) + 24 2 24 2 1 + (α2 − β2)2 = 0, where α, β range in . 16 N

1 Since the representation ρ is projective, the operators ρ(g) are deﬁned up to scalar factors. If hρ(g)v, vi 6= 0, then we can write corrected operators −1 ρe(g) := hρ(g)v, vi ρ(g). The property hρ(g)v, vi 6= 0 holds for all g ∈ Diﬀ(S1). This follows from explicit calculations given below. Also, it is possible a priory proof, see footnote 9. Of course, the new operators ρe(g) are not unitary. THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 209

If M(h, c) is reducible, its composition series is ﬁnite or countable, it is described in [7], [8].

1.3. Unitarizability. A module L(h, c) is called unitarizable if it admits a positive ∗ definite inner product such that Ln = −L−n. A module L(h, c) is unitarizable iff (h, c) ∈ R2 satisfies one of the following conditions: 1. (continuous series) h > 0, c > 1; (1) 2 6 αp − β(p + 1) − 1 2. (discrete series) c = 1 − , h = , (2) p(p + 1) 4p(p + 1) where α, β, p ∈ Z, p > 2, 1 6 α 6 p, 1 6 β 6 p − 1. For sufficiency of these conditions, see [11], for necessity, see [9], [22], [26]. Remarks. 1) For h > 0, c > 1 we have L(h, c) = M(h, c). For (h, c) of the form (2) modules M(h, c) have countable composition series. 2) There are simple explicit constructions for representations L(h, c) for c > 1, h > (c − 1)/24. The present paper is based on a such construction, for another (fermionic) construction, see [22], [29], Sect. 7.3). 3) There is an explicit construction for representations (2) with c = 1/2 (possible values of h are 0, 1/16, 1/2), see [21], [29], Sect. 7.3. The module L(0, 0) is the trivial one-dimensional module. 4) As far as I know, transparent constructions (that visualize the action of the algebra/group, the space of representation and the inner product) for the other points of discrete series and for the domain 0 6 h < (c − 1)/24 are unknown. 1.4. Standard boson realizations. Consider the space F of polynomials of vari- ables z1, z2, . . . . Define the creation and annihilation operators an, where n ranges in Z \ 0, by (√ nznf(z), for n > 0; anf(z) = p(−n) · ∂ f(z), for n < 0. ∂zn Next, we write formulas for representations of the Virasoro algebra in F. Fix the parameters α, β ∈ C. For n 6= 0 we set 1 X L := a a + (α + inβ)a ; n 2 k l n k,l: k+l=n X 1 L := a a + (α2 + β2); 0 n −n 2 n>0 ζ := 1 + 12β2.

Then we have L−n1 = 0 for n < 0 and 1 L · 1 = (α2 + β2), ζ · 1 := 1 + 12β2. 0 2 In this way, we get a representation of Vir, whose Jordan–H¨olderseries coincides with the Jordan–H¨olderseries of M(h, c) with 1 h = (α2 + β2), c = 1 + 12β2. 2 For points (α, β) in a general position we get a Verma module. 210 YURY NERETIN

Remarks. 1) For formulas for singular vectors, see [20]. 2) For α and β in a general position representations with parameters (α, β) and (α, −β) are equivalent. As far as I know explicit formula for the intertwining operator remains to be unknown. 1.5. The group of diffeomorphisms of the circle. Denote by Diff(S1) the group of smooth orientation preserving diffeomorphisms of the circle. Its Lie algebra is VectR. Any unitarizable highest weight representation of Vir can be integrated to a projective unitary representation of Diff(S1), see [13]. A module L(h, c) with an arbitrary highest weight (h, c) ∈ C2 can be integrated to a projective representation of Diff(S1) by bounded operators in a Fréchet space, see [23]. This follows from the universal fermionic construction, see [29], Sect. VII.3.

1.6. The welding. Denote by C = C ∪ ∞ the Riemann sphere. Denote by D+ ◦ the disk |z| 6 1 on C, by D− the disk |z| > 1 on C. Denote by D± their interiors. Denote by S1 the circle |z| = 1. We say that a function f : D± → C is univalent up to the boundary if f is an embedding, which is holomorphic in the interior of the disk and is smooth with a nonvanishig Jacobian up to the boundary. 1 Let γ ∈ Diﬀ(S ). Let us glue the disks D+ and D− identifying the points 1 1 z ∈ S ⊂ D+ with γ(z) ∈ S ⊂ D−. We get a two-dimensional real manifold with complex structure on the images of D+ and D−. According to [1] this structure admits a unique extension to the separating contour (in fact, the smoothness of γ is redundant, for a minimal condition, see [1]). Thus we get a one-dimensional complex manifold, i.e., a Riemann sphere C, and a pair of univalent maps D± → C. Conversely, consider a Riemann sphere C and a pair of maps p− : D− → C, p+ : D+ → C univalent up to a boundary such that ◦ ◦ p−(D−) ∩ p+(D+) = ∅ p−(D−) ∪ p+(D+) = C. −1 1 1 Then we have a diﬀeomorphism p− ◦ p+ : S → S . This correspondence is highly nontrivial, on computational aspects and applica- tions to pattern recognition, see, e.g., [33], [19].

2 1.7. The semigroup Γ. An element of the semigroup Γ is a triple R := (R, r+, r−), where R is the Riemann surface (a one-dimensional complex manifold) equivalent to the Riemann sphere C,

r− : D− → R, r+ : D+ → R are univalent up to the boundary, and

r−(D−) ∩ r+(D+) = ∅. 0 0 0 Two triples (R, r+, r−) and (R , r+, r−) are equivalent if there is a biholomorphic 0 0 map ϕ : R → R such that r± = ϕ ◦ r±. Deﬁne a product R = PQ of two elements P := (P, p+, p−), Q := (Q, q+, q−). We take surfaces (closed disks) ◦ ◦ P\ p−(D−), Q\ q+(D+)

2This semigroup is an analog of Olshanski semigroups, see, e.g., [30], Add. to Chapter 3. THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 211

◦ and glue them together identifying points p−(z) ∈ P \ p−(D−) and q+(z) ∈ Q \ ◦ 1 q+(D+), where z ranges in S . We get a Riemann sphere and two univalent functions, i.e., an element of Γ. It is clear that diffeomorphisms γ ∈ Diff(S1) can be regarded as limit points of Γ. We also define the semigroup Γ = Γ ∪ Diff(S1). The product in Γ is defined by the same procedure of gluing, on Diff(S1) it coincides with the composition of diffeomorphisms. This semigroup was discovered in [24], [32]; for details, see [25], [29], Sect VII.4. There are the following statements about representations of Γ: a) Any module L(h, c) can be integrated to a projective holomorphic representation of the semigroup Γ by bounded operators in a Fréchet space. b) Any unitarizable module L(h, c) can be integrated to a representation of Γ, which is holomorphic on Γ and unitary up to scalars on Diff(S1), see [25], [29], Sect. 7.4. c) Standard boson representations of Virasoro algebra (see Subsect. 1.4) can be integrated to holomorphic representations of Γ by bounded operators for any α, β ∈ C, see [25]. Our calculation is based on explicit formulas for such representations. 1.8. The complex domain Diff(S1)/T. Next, consider the space Ξ, whose points are pairs (S, s, σ), where S is a Riemann surface equivalent to Riemannian sphere and s : D+ → S is a map univalent up to the boundary, and σ is a point in S\s(D+). The semigroup Γ acts on Ξ in the following way. Let P := (P, p+, p−) ∈ Γ, S = 0 ◦ 0 (S, s). We glue P\ p−(D−) and S\ s(D+) identifying points p−(z) ∈ P \ p−(D−) ◦ 1 and s(z) ∈ S \ s(D+), where z ranges S . We get a Riemann sphere, an univalent map from D+ to the Riemann sphere, and a distinguished point. Below we assume S = C, s(0) = 0, σ = ∞. 2 ◦ Remark. The space Ξe of univalent functions s(z) = z + a2z + ... from D+ to C was a subject of a wide and interesting literature (see books [12], [5]). In 1907 Koebe showed that this space is compact, in next 80 years extrema of numerous functionals on Ξe were explicitly evaluated. Unfortunately, this science declared the Biebarbach conjecture as the main purpose. The proof of the conjecture (De Branges, 1985) implied a decay of interest to the subject. Unfortunately, this happened before Ξ and Ξe became a topic of the representation theory [16], [24], [4], [25] in 1986-1989. 1.9. The cocycles λ and µ. Let

r− : D− → C \ 0, p+ : D+ → C \ ∞ be functions univalent up to the boundary. We assume that r−(∞) = ∞, p+(0) = 0 (there are no extra conditions). + + + Let us take a function r : D+ → C such that r (0) = 0 and (C, r−, r ) 1 3 − determines an element Diﬀ(S ), denote it by γr. Take a function p : D− → C − 1 such that (C, p+, p ) determines an element of Diﬀ(S ), denote it by γp. Consider the product γrγp and take the corresponding triple

γrγp = (Q, q+, q−).

3There is a one-parametric family of such diﬀeomorphisms, we must choose a map of a conformal disk to a conformal disk sending a distinguished point to a distinguished point. 212 YURY NERETIN

We can assume Q = C, q+(0) = 0, q−(∞) = ∞. We deﬁne two functions4 Z r−(z) q+(γr(z)) p+(γp(z)) q−(z) λ(r−, p+) = ln d ln + − ln d ln − . (3) |z|=1 z r (γr(z)) γp(z) p (z)

µ(r−, p+) Z 0 0 1 0 q+(γr(z)) 0 q−(z) (4) = − ln r−(z) d ln + 0 + ln p+(γp(z)) d ln − 0 . 24 |z|=1 (r ) (γr(z)) (p ) (z)

1.10. Canonical cocycles. Consider a highest weight representation L(h, c) of Vir. Denote by N h,c the corresponding representation of the semigroup Γ. Since our representation is projective, operators N h,c(R) are deﬁned up to scalar factors. Denote by v a highest weight vector in the completed L(h, c), it is determined up to a scalar factor. The projection operator π to the line Cv is determined canonically (its kernel is a sum of all weight subspaces with weight diﬀerent from h). The condition π N h,c(R)v = v determines an operator N h,c(R) canonically. Now we have

h,c h,c h,c h,c N (R) N (P) = κ (R, P) N (R ◦ P), (5) where the constant κh,c(R, P) ∈ C is canonically deﬁned. Theorem 1.1.

h,c κ (R, P) = exp hλ(r−, p+) · exp{cµ(r−, p+)}. (6)

Remarks. 1) By construction, the functions λ, µ : Diff(S1) × Diff(S1) → C deter- mine C-central extensions of Diff(S1). In other words they represent classes of the second group cohomologies H2(Diff(S1), C). 2) The group of continuous cohomologies H2(Diff(S1), R) is generated by two cocycles (see [10], Theorem 3.4.4). The first one is the Bott cocycle Z 1 0 0 c1(γ1, γ2) = ln γ1(γ2(z)) d ln γ2(z). 2 |z|=1 To define the second cocycle, we write the multivalued expression

c2(γ1, γ2) = ln(γ1(γ2(z))) − ln(γ2(z)) − ln γ1(z)

1 1 and choose its continuous branch on Diff(S ) × Diff(S ) such that c1(e, e) = 0 (this cocycle can be reduced to a Z-cocycle determining the universal covering group of 1 2 1 Diff(S )). The cocycles µ, λ are equivalent to c1, c2 in the group H (Diff(S ), C). 3) I never met a theorem that all measurable R/2πZ-central extensions of Diff(S1) are reduced to cocycles c1, c2. In our context, cocycles are continuous by construction exposed below, see (25). However, the question has some interest from the point of view of representation theory.

4The formulas in [25] contain a mistake in rational coeﬃcients in the front of the integrals. This happened in manipulations with ‘logarithmic forms’ f(z) + ln dz. We have ln dz = iϕ + ln dϕ, the summand iϕ [25] was lost. THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 213

1.11. Realization of highest weight representations in the space of holomorphic functionals on the domain Ξ.

Corollary 1. Let f(S) = (C, s, ∞) ∈ Ξ, Q := (C, q+, q−) ∈ Γ, q+(0) = 0, q−(0) = 0. The formula ρh,c(Q) f(S) = f(SQ) exp h λ(q−, s) · exp{c µ(q−, s)} (7) determines a representation of Γ in the space of holomorphic functionals on Ξ. For (h, c) in a general position (if the Verma module M(h, c) is irreducible ), the corresponding module over the Virasoro algebra is M(h, c). Proof is contained in Subsect. 3.8. Remark. Formulas for the underlying action of the Lie algebra Vect on Ξ were firstly obtained in the book by Schaeffer and Spencer, [31], 1950, and rediscovered in [18]. The action of Vir corresponding (7) was present in [24] without a proof, for a proof, see [17]. Considering an action in the space of polynomials in Taylor coefficients of s(z), we get a module dual to the Verma module. Some modifications of formulas are contained in [2]. Remark. Of course, we can invent many different definitions of holomorphic functions on Ξ and of spaces of holomorphic functions. Our statement is almost inde- pendent on this. To avoid a scholastic discussion, we note that a construction [25], Subs. 4.12 gives an operator sending Fock space to the space of functions on Ξ. 1.12. Reproducing kernel. On Hilbert spaces determined by reproducing kernels see, e.g., [30], Sect. 7.1. Corollary 2. Let L(h, c) be unitarizable. Then the kernel −1 Kh,c(r, p) := exphλ(r∗, p) · exp{cµ(r∗, p)}, where r∗(z) := r(z−1) , (8) determines a Hilbert space of holomorphic functions invariant with respect to the operators ρh,c(Q) defined by (7). The corresponding representation of Virasoro algebra is L(h, c). Moreover, for q ∈ Diff(S1) these operators are unitary. Proof is contained in Subsect. 3.9.

2. Preliminaries. Explicit formulas for representations of Γ.

2.1. The Fock space. See [29], Sect. V.3, or [30], Sect. 4.1, for details of deﬁnition and proofs. The boson Fock space with n degrees of freedom, denoted by Fn, is a Hilbert space of holomorphic functions on Cn with inner product Z n 1 −|z|2 Y hf, gi = n f(z)g(z)e d Re zj d Im zj. π n C j=1

n (n) For a = (a1, . . . , an) ∈ C we deﬁne an element ϕa ∈ Fn by n nX o ϕa(z) := exp zjaj . j=1 Then we have a reproducing property

f(a) = hf, ϕaiFn . 214 YURY NERETIN

Consider an isometric embedding Jn : Fn → Fn+1 determined by

Jnf(z1, . . . , zn, zn+1) = f(z1, . . . , zn). Next, consider the union of the chain of Hilbert spaces

· · · −→ Fn −→ Fn+1 −→ ....

Its completion is the Fock space F∞ with an infinite number degrees of freedom. For a = (a1, a2,... ) ∈ `2 we define an element ϕa ∈ F∞ by ∞ nX o ϕa(z) := exp zjaj := lim ϕ(a ,...,a )(z1, . . . , zn), n→∞ 1 n j=1 the limit is a limit in the sense of the Hilbert space F∞. Then for any element h ∈ F∞ we define a holomorphic function on `2 by

h(u) = hh, ϕuiF∞ .

We identify F∞ with this space of holomorphic functions. Since any infinite-dimensional Hilbert space H is isomorphic `2, we can define a Fock space F (H) as a space of functions on H determined by the reproducing kernel R(h1, h2) = exp hh1, h2iH . On Hilbert spaces determined by reproducing kernels, see, e.g., [30], Sect. 7.1. 2.2. Symbols of operators. For details, see [29], Sect. V.3, VI.1, [30], Sect.4.1, Sect.7.1. Let A be a bounded linear operator in F∞. Following F.A.Berezin, define its kernel by

K(z, u) = hAϕu, ϕziF∞ , where z, u ∈ `2. Then for any function f ∈ F∞, we have

Af(z) = hf, kziF∞ , where kz(u) := K(z, u). Let A, B be bounded operators, let L, K be their kernels, let kz(u) := K(z, u), lz(u) := L(z, u) then the kernel M of BA is

M(z, u) = hku, lziF∞ . 2.3. Gaussian operators. See [29], Sect. V.4, Sect. VI. 2-4, [30], Sect.5-6. Con- KL sider a block symmetric matrix5 S = of size ∞ + ∞ and two vectors λ, Lt M µ ∈ `2. Deﬁne a Gaussian operator t KL λ B Lt M µt in F∞ as an operator with the kernel 1 KL zt exp z u + zλt + uµt . (9) 2 Lt M ut t t t t Here the vectors z, u, λ, µ ∈ `2 are regarded as vectors-rows, z , u , λ , µ are vector-columns. The expression in the curly brackets is a quadratic expression in z, u. If an operator B[... ] is bounded, then

5The symbol Lt denotes the transposed matrix. THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 215

1. kSk 6 1, kKk < 1, kMk < 1; 2. K, M are Hilbert-Schmidt operators. λt If also kSk < 1, then for any σ = the operator B[S|σ] is bounded. µt For more details on the boundedness conditions, see [29], Sect.V.2-4. The product of two Gaussian operators is given by the formula t t KL λ PQ π B t t B t t L M µ Q R κ = σ(M,P ; µ, π) (10) −1 t −1 t −1 t t K + LP (1 − MP ) L L(1 − PM) Q λ + L(1 − PM) (π + P µ ) × B t −1 t −1 t t −1 t t , Q (1 − MP ) LR + Q (1 − MP ) MQ κ + Q (1 − MP ) (Mπ + µ ) where the constant σ(... ) is given by

( −1 t) −1/2 1 −P 1 π σ(M,P ; µ, π) = det(1 − MP ) exp π µ . (11) 2 1 −M µt

Remarks. 1) The conditions kMk < 1, kP k < 1 imply kMP k < 1. Hence we can evaluate (1 − MP )−1/2 as 1 3 (1 − MP )−1/2 = 1 + MP − (MP )2 + ... 2 8 Next, M, P are Hilbert–Schmidt operators. Therefore MP is a trace class operator. It is easy to show that the series for −1+(1−MP )−1/2 converges in the trace norm, therefore the determinant in (11) is well-deﬁned. 2) For Hilbert–Schmidt operators M, P with norm < 1 the following identity holds: det(1 − MP )s = det(1 − PM)s. 3) The big matrix in the right-hand side of (10) admits a transparent geometric interpretation, see [29], Theorem 5.4.3, Sect. VI.5. 2.4. Gaussian vectors. Deﬁne Gaussian vectors b[P |πt], where P is a Hilbert- Schmidt symmetric matrix with kP k < 1 and π ∈ `2 by n1 o b[P |π](z) := exp zP zt + zπt . 2 t Then b[P |π ] ∈ F∞. Also, t KL λ t B b[P |π ] Lt M µt = σ(M,P ; µ, π) (12) × b[K + LP (1 − MP )−1Lt|λt + L(1 − PM)−1(πt + P µt)], where σ(... ) is given by (11). Notice that K + LP (1 − MP )−1Lt and λt + L(1 − PM)−1(πt + P µt) also are elements of the right-hand side of (10). Inner products of vectors b[... ] are given by t t t hb[K|µ ], b[P, π ]iF∞ = σ(M, P ; µ, π ). (13) Remark. We have b[O|0] = 1. Obviously, for any Gaussian operator B[S|σ], we have hB[S|σ] · 1, ·1i = 1. (14) 216 YURY NERETIN

2.5. The Weil representation. Now consider the space H = `2 ⊕`2 and consider the group of bounded operators ΦΨ , (15) Ψ Φ which are symplectic in the following sense: t ΦΨ 0 1 ΦΨ 0 1 = . Ψ Φ −1 0 Ψ Φ −1 0 We denote by Sp(2∞, R) the group of such matrices satisfying an additional condition: — Ψ is a Hilbert–Schmidt operator. Next, we consider the group ASp(2∞, R) of affine transformations of `2 ⊕ `2 generated by the group Sp(2∞, R) and shifts by vectors of the form (h, h). According to [3], §4, Theorem 3, the group ASp(2∞, R) has a standard projective unitary representation in F∞ by unitary Gaussian operators. Elements of Sp(2∞, R) act by operators −1 t −1 ΨΦ (Φ ) 0 τ · B . (16) Φ−1 −Φ−1Ψ 0 The shifts (h, h) act by the operators 0 1 h B . 1 0 −h Let g ∈ ASp(2n, R), let B[S|σ] be the corresponding Gaussian operator. By the formula (10), the matrix S depends only on the linear part of g. 2.6. The Hilbert space V . First, we consider a space V smooth of smooth functions P k 1 f(z) = k∈Z ckz on the circle S defined up to an additive constant. Define the inner product in V smooth by k k k l hz , z iV smooth = |k|, hz , z iV smooth = 0 if l 6= k. Denote by V the completion of V smooth with respect to this inner product (this is a Sobolev space H1/2(S1)). Notice that all elements of V are L2-functions on 1 P k S . Denote by V+ (resp. V−) the subspace consisting of series k>0 ckz (resp. P k k<0 ckz ). These subspaces consist of functions admitting holomorphic continu- ◦ ◦ ations to the disks D+ and D− respectively. We define a skew-symmetric bilinear form on V by the formula Z {f, g} = f(z) dg(z). (17) |z|=1

The subspaces V+, V− are isotropic and dual one two another with respect to this form. The projection operators to V± are given by the Cauchy integral 1 Z f(u) P±f(z) = ± lim . (18) ε→0∓ 2πi |u|=1+ε u − z For a diffeomorphism γ ∈ Diff(S1) we define the linear operator in V by T (γ)f(z) = fγ−1(z). (19) Evidently, our form is Diff(S1)-invariant: −1 T (γ)f1,T (γ)f2 = f1, f2 or T (γ)f1, f2 = f1,T (γ) f2 . (20) THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 217

2.7. Construction of highest weight representations of Diff(S1). Consider the Fock space F (V−) corresponding to the Hilbert space V−. We wish to construct 1 projective representations of Diff(S ) in F (V−) corresponding to the standard boson realization of representations of Vir. Let us regard the circle as the quotient R/2πZ with coordinate ϕ. Lemma 2.1. Let γ ∈ Diff(S1). Then the operator T (γ−1)f(ϕ) = f(γ(ϕ)) is contained in the group Sp(2∞, R). Proof. Consider the Hilbert transform 1 Z 2π ϕ − ψ Hf(ϕ) := · p.v. cot f(ψ) dψ π 0 2 It send einϕ to i sgn(n)einϕ. In other words, its block matrix with respect to the i 0 decomposition V = V ⊕ V is . Therefore, it is sufficient to prove that a + − 0 −i commutator T (γ−1),H is a Hilbert–Schmidt operator. We have

1 Z 2π ϕ − ψ HT γ−1f(ϕ) = cot f(γ(ψ)) dψ; π 0 2 1 Z 2π γ(ϕ) − ψ T γ−1Hf(ϕ) = cot f(ψ) dψ. π 0 2 Hence

T (γ−1),Hf(ψ) = 1 Z 2π γ(ϕ) − γ(ψ) ϕ − ψ = cot γ0(ψ) − cot f(γ(ψ)) dψ. π 0 2 2 Let us show that the expression in square brackets is a smooth function on the torus R/2πZ × R/2πZ. We have h i γ(ϕ) − γ(ψ) ... = cot γ0(ψ) 2 (21) 1 γ(ϕ) − γ(ψ) ϕ − ψ × 1 − tan cot γ0(ψ) 2 2

γ(ϕ)−γ(ψ) ϕ−ψ The product Q(ϕ, ψ) = tan 2 cot 2 has a removable singularity on the diagonal ϕ = ψ. Assuming Q(ψ, ψ) = γ0(ψ) we get a smooth function. Therefore the expression in the curly brackets in (21) is smooth and has zero on the diagonal ϕ = ψ. Hence the whole expression (21) has a removable singularity on the diagonal and is smooth. Therefore its Fourier coeﬃcients rapidly decrease and the operator T (γ−1),H is contained in all Schatten classes.

Fix α, β ∈ R. For γ ∈ Diﬀ(S1) we consider the aﬃne transformation of V given by the formula

−1 0 Tα,β(γ )f(ϕ) = f γ(ϕ) + α γ(ϕ) − ϕ + β ln γ (ϕ). (22) 218 YURY NERETIN

Remark. Recall that ϕ, γ(ϕ) are elements of R/2πZ. We take an arbitrary continuous R-valued branch of γ(ϕ). Then the function γ(ϕ) − ϕ is well defined up to an additive constant and it is an element of V and we can multiply it by a constant α. The formula (22) determines an embedding of the group Diff(S1) to the group ASp(2∞, R). Restricting the Weil representation to Diff(S1) we get a unitary projective representation of Diff(S1). On the level of the Lie algebra we obtain the representation determined in Subsection 1.4. See [29], Sect. VII.2 Next, we wish to write the matrix (15). Applying (18) we represent the block Φ in (15) as 1 Z f(γ−1(z)) dz Φf(z) = lim . ε→0− 2πi |z|=1+ε z − u Formally, the right-hand side is well-defined for real-analytic f and real analytic γ ∈ Diff(S1). But it makes sense for any distribution6 f and any smooth γ. For −1 P k instance, we can decompose f(γ (z)) into the Fourier series ckz and take k∈Z X k Φf(z) := ckz . k>0 In the same way, we write an expression for Ψ. In the next subsection, we will express the operators Φ−1, ΨΦ−1 in the terms of welding. 2.8. Formulas for the action of Γ. For details and proofs, see [25], [29], Sect. VII.4-5. Let α, β ∈ C. The semigroup Γ acts in the space F (V−) by Gaussian operators t t K(r+) L(r+, r−) | −(β + iα) `1(r+) + β m1(r+) Nα,β(R) = B t t t , L(r+, r−) M(r−) | −(β + iα) `2(r−) + β m2(r−) where operators t K : V− → V+,L : V− → V−,L : V+ → V+,M : V+ → V− and vectors t t t t `1, m1 ∈ V+, `2, m2 ∈ V− will be defined now. −1 Consider a function f ∈ V−. Then the function f ◦ (r+) is defined on the 1 7 contour r+(S ). Let us decompose it as F1 + F2, where F1 is holomorphic in 0 the domain r+(D+), and F2 is holomorphic in C \ r+(D+). This decomposition is determined up to an additive constant, F1 7→ F1 + c, F2 7→ F2 − c. We set

Kf := F1 ◦ r+, Lf = F2 ◦ r−.

In a similar way, we take a function f ∈ V+, decompose

−1 f ◦ (r−) = F1 + F2, 1 r−(S ) ◦ where F1 is holomorphic in r−(D−) and F2 is holomorphic in C \ r−(D−). We set t Mf := F1 ◦ r−,L f = F2 ◦ r+

6 2 2 Our space V is contained in L and V+ is contained in the Hardy space H , on Riesz projector for Hardy spaces, see, e.g., [34]. 7This decomposition is given by the Sokhotski–Plemelj integral formula. THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 219

Recall that the subspaces V+ and V− are dual one to another with respect to the t pairing (17). It can be shown that the operators L : V− → V− and L : V+ → V+ are dual one to another. Also, the operators M : V+ → V−, K : V− → V+ are dual to themselves. Finally, we set r (z) r (z) `t (r ) = ln + , `t (r ) = ln − , (23) 1 + z 2 − z t 0 t 0 m1(r+) = ln r+(z), m2(r−) = ln r−(z). (24)

Remark. The operator K(r+) is the Grunsky matrix of the univalent function r+, it is a fundamental object of theory of univalent functions, see [14], [12], [5].

8 t It remains to explain the meaning of the expression (9). For f± ∈ V±, we set t KL (f−) t t t t t t f− f+ t t := {f−, Kf−} + 2{f+, Lf−} + {f+, Mf+}, L M (f+) where the form {·, ·} is deﬁned by (17), also t `1 t t t f− f+ t := {f−, `1} + {f+, `2}, `2 we omit the similar expression with m1, m2. Also we note that under this normalization we have

π(Nα,β(R)1) = 1, (25) where π is the projection operator in F (V−) to the line C · 1. The operators Nα,β(R) are bounded in the Fock space for arbitrary α, β ∈ C for arbitrary R ∈ Γ. For α, β ∈ R, R ∈ Diﬀ(S1) these operators are unitary up to scalar factors.

1 Remarks. 1) If α, β∈ / R, R ∈ Diﬀ(S ) \ PSL(2, R), then Nα,β(R) is unbounded. 2) For R ∈ Γ and α, β ∈ C operators Nα,β(R) are contained in the trace class. Their traces (characters of representations) were evaluated in [25].

3. The calculation. Here we prove Theorem 1.1, i.e. evaluate canonical cocycles.

3.1. Step 1. Consider

R = (C, r+, r−), P = (C, p+, p−) ∈ Γ such that

r+(0) = 0, r−(∞) = ∞, p+(0) = 0, p−(∞) = ∞. + − + − 1 Next, we choose r , p such that (C, r , r−), (C, p+, p ) ∈ Diﬀ(S ) as it was discussed in Subsection 1.9. Take the corresponding γr, γp, and represent γrγp as (C, q+, q−). Formulas (10), (11) allow to multiply Gaussian operators Nα,β(R) Nα,β(P). We get Nα,β(R) Nα,β(P) = κα,β(R, P) Nα,β(R ◦ P),

8We write f t, `t, . . . instead of f, `, . . . to symbolize that in our formulas functions correspond to vector-columns (we apply an operator to a function from the left). 220 YURY NERETIN where the canonical cocycle κα,β is given by κα,β(R, P) = det[(1 − MK)]−1/2 −1 1 −K 1 × exp −(iα + β)` + βm −(iα + β)` + βm (26) 2 1 1 2 2 1 −M t t −(iα + β) `1 + β m1 × t t . −(iα + β) `2 + β m2 where

K := K(p+),M := M(r−),

`1 := `1(p+), `2 := `2(r−),

m1 := m1(p+), m2 := m2(r−). In particular, we see that the cocycle

κα,β(R, P) = κα,β(r−, p+) does not depend on p−, q+ and is holomorphic in α, β. Keeping in mind further manipulations we represent (26) in the form

κα,β(r−, p+) = det[(1 − MK)]−1/2 −1 1 −K 1 `t × exp − α2 ` ` 1 2 1 2 1 −M `t 2 (27) −1 t t −K 1 −`1 + m1 − iαβ `1 `2 t t 1 −M −`2 + m2 −1 t t 1 2 −K 1 −`1 + m1 + β −`1 + m1 −`2 + m2 t t . 2 1 −M −`2 + m2 3.2. Step 2. We also use the notation of Subsection 1.10. In particular, we have another notation for the canonical cocycles κh,c. By deﬁnition, 1 (α2+β2),1+12β2 κα,β(r−, p+) = κ 2 (r−, p+). Setting α = β = 0, we get 0,1 −1/2 κ0,0(r−, p+) = κ (r−, p+) = det (1 − K(p+) M(r−)) . (28) Obviously9, h ,c h ,c h +h ,c +c κ 1 1 (R, P) κ 2 2 (R, P) = κ 1 2 1 2 (R, P). (29) Keeping in mind the holomorphy, we get that κh,c has the form h,c κ (r−, p+) = exp hλ(r−, p+) + cµ(r−, p+) ,

9 Let (h1, c1), (h2, c2) be in the domain of unitarity (1). Let v1, v2 be the highest vectors in L(h1, c1), L(h2, c2). The cyclic span of v1 ⊗ v2 ∈ L(h1, c1) ⊗ L(h2, c2) is the module L(h1 + h1, c1 + c2). This implies the statement for (h1, c1), (h2, c2) satisfying (1). It remains to apply the holomorphy. The same reasoning shows that matrix elements of a highest weight vector hρh,c(g)v, vi are 1 non-vanishing for all unitary representations of Diﬀ(S ). Indeed, if hρh0,c0 (g0)v, vi = 0, then 0 0 0 0 hρh0+h ,c0+c (g)v, vi = 0, where h > 0, c > 1. Since formulas of representations are a priory holomorphic in (h, c), we get hρh,c(g0)v, vi = 0 for all (h, c). But for trivial one-dimensional representation ρ0,0 this is false. THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 221 where λ(r−, p+), µ(r−, p+) are some functions. In another notation, n1 o (r , p ) = exp (α2 + β2)λ(r , p ) + (1 + 12β2)µ(r , p ) κα,β − + 2 − + − + −1/2 = det 1 − K(p+)M(r−) (30) n1 o × exp (α2 + β2)λ(r , p ) + 12β2µ(r , p ) . 2 − + − + (we applied 28).

3.3. Step 3. The exponentials in (26) and (30) must coincide. This implies that the term with αβ in (27) is absent. This means that

−1 t −1 t −K 1 `1 −K 1 m1 `1 `2 t = `1 `2 t . (31) 1 −M `2 1 −M m2 Next, we transform the coeﬃcient in front of β2 in (27) killing the mixed term with both l and m using (31),

−1 t t 1 −K 1 −`1 + m1 −`1 + m1 `2 + m2 t t 2 1 −M −`2 + m2 −1 t −1 t 1 −K 1 m1 1 −K 1 `1 = m1 m2 t − `1 `2 t 2 1 −M m2 2 1 −M `2 Therefore, we can reduce (27) to the form

−1/2 κα,β(r−, p+) = det[(1 − MK)] −1 t 1 2 2 −K 1 `1 × exp − (α + β ) `1 `2 t 2 1 −M `2 (32) −1 t 1 2 −K 1 m1 + β m1 m2 t . 2 1 −M m2 Comparing this with (30), we come to

−1 t −K 1 `1 λ(r−, p+) = − `1 `2 t ; (33) 1 −M `2 −1 t 1 −K 1 m1 µ(r−, p+) = m1 m2 t . (34) 24 1 −M m2 Since −1 −K 1 M(1 − KM)−1 (1 − MK)−1 = , 1 −M (1 − KM)−1 K(1 − MK)−1 we have

h −1 t −1 t i λ(r−, p+) = `1 M(1 − KM) `1 + (1 − MK) `2 (35)

n −1 t −1 t o + `2 (1 − KM) `1 + K(1 − MK) `2 (36)

Also we have a similar expression for µ(r−, p+). 222 YURY NERETIN

3.4. Step 4. Next, we evaluate the following product by formula (10)

Nα,β(γr) Nα,β(γp) = const ·Nα,β(γrγp). The formula involves 3 operators of the type B[... |.]. This implies an identity involving 3 matrices [... |.]. Let us substitute β = 0 and write the expression for the last column in the matrix [... |.] corresponding to γrγp. We get

t `1(q+) (37) t + + n −1 t t o = `1(r ) + L(r , r−) 1 − K(p+)M(r−) `1(p+) + K(p+)`2(r−) ; and t `2(q−) (38) t − t − h −1 t t i = `2(p ) + L (p+, p ) 1 − M(r−)K(p+) `2(r−) + M(r−)`1(p+) .

Keeping in mind the identities M(1 − KM)−1 = (1 − MK)−1M, K(1 − MK)−1 = (1 − KM)−1K, we observe that the expressions in curly brackets in (37) and (36) coincide. We express the term in the curly bracket from (37)

n o + −1 t t + ... = L(r , r−) `1(q+) − `1(r ) and substitute to (36). Also, the expressions in the square brackets in (38) and (35) coincide. We express the term in the square brackets from (38) and substitute it to (35). In this way, we get

λ(r−, p+) + −1 t t + =`2(r−)L(r , r−) `1(q+) − `1(r ) (39) t − −1 t t − + `1(p+)L (p+, p ) `2(q−) − `2(p ) .

−1 + 3.5. Step 5. Now we wish to evaluate L (r , r−). For this aim consider the Gaussian operator, corresponding to the diﬀeomorphism γr, + + K(r ) L(r , r−) ∗ B t + L (r , r−) M(r−) ∗

On the other hand applying formula (16) to the same γr we get an expression of the form −1 t −1 ΨΦ (Φ ) ∗ B . Φ−1 −Φ−1Ψ ∗ + t −1 t Therefore L(r , r−) = (Φ )−1, i.e. L = Φ . Thus,

+ t −1 t t (L(r , r−) ) `2 = P+T (γr)`2, (40) where P+ is the projection operators to V+. In the same way, we obtain

− t −1 t t (L(p+, p ) ) `1 = P−T (γp)`1. (41) THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE 223

3.6. Step 6. Evidently, for g ∈ V−, we have Z Z gd(P+f) = g df = f, g . (42) |z|=1 |z|=1

In the same way, for f ∈ V+, Z Z gd(P−f) = g df = f, g . |z|=1 |z|=1

Now we can transform two summands in formula (39) for λ(r−, p+). The ﬁrst summand equals to

+ −1 t t + `2(r−)L(r , r−) `1(q+) − `1(r ) + + t −1 t = `1(q+) − `1(r ) (L(r , r−) ) `2(r−) + t = `1(q+) − `1(r ) P+T (γr)`2(r−) n t t + t o = `1(q+) − `1(r ) ,P+T (γr)`2(r−)

n t t + t o = `1(q+) − `1(r ) ,T (γr)`2(r−)

n −1 t t + t o = T (γr) `1(q+) − `1(r ) , `2(r−) .

We applied (40), (42), and the invariance (20). A similar calculation gives us the second summand of(39),

t −1 t t `1(p+)L (p+, p−) `2(q−) − `2(p−) n t t t o = `1(p+),P−T (γp) `2(q−) − `2(p−)

n t t t o = `1(p+),T (γp) `2(q−) − `2(p−)

n −1 t t t o = T (γp) `1(p+), `2(q−) − `2(p−) .

Applying explicit expressions (23) for `1, `2 we get formula (3) for λ(r−, p+).

3.7. Final remarks. The derivation of formula (4) µ(r−, p+) is similar, we simply change `1, `2 to m1, m2. Our calculation of λ(r−, p+), µ(r−, p+) is valid for c > 1, h > (c − 1)/24. Due to the holomorphy we extend the result to arbitrary (h, c) ∈ C2. It remains to explain Corollaries 1.2–1.3 from the theorem.

3.8. The action of Diﬀ(S1) in the space of holomorphic functions. For an univalent function s, we denote s◦ := s(z). For any element f of the Fock space F (V−) we assign a holomorphic functional on Ξ given by D E F (s) := f, bK(s◦)|(−iα + β)`(s◦) + βm(s◦) . F (V−) In this way, we send the Fock space to the space of holomorphic functionals on Ξ, this gives us the desired realization (see [25], Subs. 4.12). 224 YURY NERETIN

3.9. Reproducing kernels. Let c > 1, h > (c − 1)/24. The problem can be easily reduced to evaluation of inner products

hNα,β(γ1) · 1 , Nα,β(γ2) · 1iF (V−), (43) 1 where γ1, γ2 ∈ Diﬀ(S ). Since operators Nα,β(γ) are unitary up to a constant, we have ∗ −1 Nα,β(γ) = s ·Nα,β(γ ). for some s ∈ C. On the other hand, −1 hNα,β(γ) · 1, 1iF (V−) = h1, Nα,β(γ ) · 1iF (V−) = 1 and this implies s = 1. Therefore, (43) equals to −1 hNα,β(γ2 ) Nα,β(γ1) · 1 , 1iF (V−) and we reduce our problem to the evaluation of the canonical cocycle. Next, we have a priory identity (see footnote 9) for unitary representations L(h1, c1), L(h2, c2): Kh1,c1 (r, p) Kh2,c2 (r, p) = Kh1+h2,c1+c2 (r, p).

h1,c1 Taking arbitrary L(h1, c1), c2 > 1, and suﬃciently large h2 we can obtain K (r, p) as Kh1+h2,c1+c2 (r, p) Kh1,c1 (r, p) = . Kh2,c2 (r, p) This also shows that the formula for canonical cocycles remains to be valid for all unitary representations of Diﬀ(S1).

Acknowledgments. The work was upported by the grants FWF, projects P25142, P28421.

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