The group of diffeomorphisms of the circle: reproducing kernels and analogs of spherical functions

The group $Diff$ 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 $\Xi$ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of $Diff$ in the space of holomorphic functionals on $\Xi$, reproducing kernels on $\Xi$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.


Introduction
1.1. The purpose of the paper. The group Diff(S 1 ) 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 [29], 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(S 1 ).
The purpose of this paper is to present details of a strange calculation sketched at the end of my paper [24], 1989. It also was included to my thesis [26], 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(S 1 ). Generally, it is a projective representation. Operators ρ(g) are determined up to constant factors. We normalize them by the condition where v is a highest weight vector 2 . Then we have ρ(g 1 )ρ(g 2 ) = c(g 1 , g 2 ) ρ(g 1 g 2 ).
where the expression c(g 1 , g 2 ) (the canonical cocycle) is canonically determined by the representation ρ. In this paper, we derive a formula for c(g 1 , g 2 ). 1 Supported by the grants FWF, projects P25142, P28421. 2 Since the representation ρ is projective, the operators ρ(g) are defined up to scalar factors. If ρ(g)v, v = 0, then we can write corrected operators ρ(g) := ρ(g)v, v −1 ρ(g). The property ρ(g)v, v = 0 holds for all g ∈ Diff(S 1 ). This follows from explicit calculations given below. Also, it is possible a priory proof from formula (3.3). Of course, the new operators ρ(g) are not unitary.
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(S 1 ) on the space univalent functions and for invariant reproducing kernels on the space of holomorphic functionals on the space of univalent functions.
1.2. Virasoro algebra and highest weight modules. For details, see, e.g., [27]. Denote by Vect R and Vect C respectively the Lie algebras of real (respectively complex) vector fields on the circle. Choosing a basis L n := e inϕ ∂/i∂ϕ in Vect C , we get the commutation relations Recall that the Virasoro algebra Vir is a Lie algebra with a basis L n , ζ, where n ranges in Z, and commutation relations Let h, c ∈ C. A module with the highest weight (h, c) over Vir is a module containing a vector v such that 1) L 0 v = hv, ζv = cv; 2) L −n v = 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).
By [15], [6], a Verma module M (h, c) is reducible if and only if (h, c) ∈ C 2 satisfies at least one equation of the form 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 L n = −L * −n . A module L(h, c) is unitarizable iff (h, c) ∈ R 2 satisfies one of the following conditions: where α, β, p ∈ Z, p 2, 1 α p, 1 β p − 1.
3) There is an explicit construction for representations (1.2) with c = 1/2 (possible values of h are 0, 1/16, 1/2), see [20], [28], 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 h < (c − 1)/24 are unknown. ⊠ 1.4. Standard boson realizations. Consider the space F of polynomials of variables z 1 , z 2 , . . . . Define the creation and annihilation operators a n , where n ranges in Z \ 0, by a n f (z) = √ nz n f (z), for n > 0; (−n) · ∂ ∂zn f (z), for n < 0.
Next, we write formulas for representations of the Virasoro algebra in F . Fix the parameters α, β ∈ C. For n = 0 we set L n := 1 2 k,l: k+l=n a k a l + (α + inβ)a n ; L 0 := n>0 a n a −n + 1 2 (α 2 + β 2 ); Then we have L −n 1 = 0 for n < 0 and In this way, we get a representation of Vir, whose Jordan-Hölder series coincides with the Jordan-Hölder series of M (h, c) with For points (α, β) in a general position we get a Verma module.
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 uknown. ⊠ 1.5. The group of diffeomorphisms of the circle. Denote by Diff(S 1 ) the group of smooth orientation preserving diffeomorphisms of the circle. Its Lie algebra is Vect R .
Any unitarizable highest weight representation of Vir can be integrated to a projective unitary representation of Diff(S 1 ), see [13]. A module L(h, c) with an arbitrary highest weight (h, c) ∈ C 2 can be integrated to a projective representation of Diff(S 1 ) by bounded operators in a Frechet space, see [22]. This follows from the universal fermionic construction, see [28], Sect. VII.3.
1.6. The welding. Denote by C = C ∪ ∞ the Riemann sphere. Denote by D + the disk |z| 1 on C, by D − the disk |z| 1 on C. Denote by D • ± their interiors. Denote by S 1 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 up to the boundary.
Let γ ∈ Diff(S 1 ). Let us glue the disks D + and D − identifying the points z ∈ S 1 ⊂ D + with γ(z) ∈ S 1 ⊂ D − . We get a two-dimensional real manifold with complex structure on the images of D + and D − . According [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 Then we have a diffeomorphism p −1 − • p + : S 1 → S 1 . 1.7. The semigroup Γ. An element of the semigroup 3 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, are univalent up to the boundary, and Two triples (R, r + , r − ) and (R ′ , r ′ + , r ′ − ) are equivalent if there is a biholomorphic 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) and glue them together identifying points We get a Riemann sphere and two univalent functions, i.e., an element of Γ.
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 Frechet 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(S 1 ), see [24], [28], Sect. 7.4. c) Standard boson represantations of Virasoro algebra (see Subsect. 1.4) can be integrated to holomorphic representations of Γ by bounded operators, see [24]. Our calculation is based on explicit formulas for such representations.
1.8. The complex domain Diff(S 1 )/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 = (S, s). We glue P \ p − (D 0 − ) and S \ s(D • + ) identifying points p − (z) ∈ P \ p − (D 0 − ) and s(z) ∈ S \ s(D • + ), where z ranges S 1 . We get a Riemann sphere, an univalent map from D + to the Riemann sphere, and a distinguished point.
Below we assume Remark. The space Ξ of univalent functions s(z) = z + a 2 z 2 + . . . 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 Ξ were explicitly evaluated. Infortunately, 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. Unforunately, this happened before Ξ and Ξ became a topic of the representation theory [16], [23], [4], [24] in 1986-1989. ⊠ 1.9. The cocycles λ and µ. Let 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 + ) determines an element Diff(S 1 ), denote it 4 by γ r . Take a function p − : D − → C such that (C, p + , p − ) determines an element of Diff(S 1 ), denote it by γ p . Consider the product γ r γ p and take the corresponding triple We define two functions 5 . 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 where a constant κ h,c (R, P) ∈ C is a canonically defined.
2) The group of continuous cohomologies H 2 (Diff(S 1 ), R) is generated by two cocyles (see [10], Theorem 3.4.4). The first one is the Bott cocycle To define the second cocycle, we write the multivalued expression and choose its continuous branch on Diff(S 1 ) × Diff(S 1 ) such that c 1 (e, e) = 0 (this cocycle can be reduced to a Z-cocycle determining the universal covering group of Diff(S 1 )). The cocycles µ, λ are equivalent to c 1 , c 2 in the group 3) I never met a theorem that all measurable R/2πZ-central extensions of Diff(S 1 ) are reduced to cocycles c 1 , c 2 . In our context, cocycles are continuous by construction exposed below, see (2.16). However, the question has some interest from the point of view of representation theory. ⊠ 1.11. Realization of highest weight representations in the space of holomorphic functionals on the domain Ξ.
determines a representation of Γ in the space of holomorphic functionals on Ξ. Remark. Formulas for the underlying action of the Lie algebra Vect on Ξ were firstly obtained in by Scheffer [30], 1948, and rediscovered in [18]. The action of Vir corresponding (1.7) was present in [23] without a proof, for a proof, see [17]. Considering an action in the space of polynomials in Taylor coefficients of p(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 independent on this. To avoid a scholastic discussion, we note that a construction [24], 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., [29], Sect. 7.1.  by (1.7). The corresponding representation of Virasoro algebra is L(h, c). Moreover, for q ∈ Diff(S 1 ) these operators are unitary.
Next, consider the union of the chain of Hilbert spaces Its completion is the Fock space F ∞ with infinite number degrees of freedom. For a = (a 1 , a 2 , . . . ) ∈ ℓ 2 we define an element ϕ a ∈ F by ϕ a (z) := exp n j=1 z j a j := lim n→∞ ϕ (a1,...,an) (z 1 , . . . , z n ), 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 ℓ 2 by We identify F ∞ with this space of holomorphic functions. Since any infinite-dimensional Hilbert space H is isomorphic ℓ 2 , we can define a Fock space F (H) as a space of functions on H determined by the reproducing kernel On Hilbert spaces determined by reproducing kernels, see, e.g., [29], Sect. 7.1.
Then for any function f ∈ F ∞ , we have where k z (u) := K(z, u).
Let A, B be bounded operators, let L, K be their kernels, let k z (u) := K(z, u), l z (u) := L(z, u) then the kernel M of BA is M (z, u) = k u , l z F∞ .

Gaussian operators.
See [28], Sect. V.4, Sect. VI. 2-4, [29], Sect.5-6. Consider a block symmetric matrix 6 S = K L L t M of size ∞ + ∞ and two vectors λ, µ ∈ ℓ 2 . Define a Gaussian operator in F ∞ as an operator with the kernel Here the vectors z, u, λ, µ ∈ ℓ 2 are regarded as vectors-rows, z t , u t , λ t , µ t are vector-columns. The expression in the curly brackets is a quadratic expression in z, u.
If also S < 1, then the operator B[S|σ] is bounded. More on conditions of boundedness, see [28], Sect.V.2-4. Product of two Gaussian operators is given by the formula (2.2) 6 The symbol L t denotes the transposed matrix.
where the constant σ(. . . ) is given by which are symplectic in the following sense: 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 ℓ 2 ⊕ ℓ 2 generated by the group Sp(2∞, R) and shifts by vectors of the form (h, h). According [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 The shifts (h, h) act by the operators Let g ∈ ASp(2n, R), let B[S|σ] be the corresponding Gaussian operator. By the formula (2.2), 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 f (z) = c k z k on the circle S 1 defined up to an additive constant. Define the inner product in V smooth by Denote by V the completion of V smooth with respect to this inner product. Notice that all elements of V are L 2 -functions on S 1 . Denote by V + (resp. V − ) the subspace consisting of series k>0 c k z k (resp. k<0 c k z k ). These subspaces consist of functions admitting holomorphic continuations to the disks D • + and D • − respectively. We define a skew-symmetric bilinear form on V by the formula 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 For a diffeomorphism γ ∈ Diff(S 1 ) we define the linear operator in V by Evidentely, our form is Diff(S 1 )-invariant: 2.7. Construction of highest weight representations of Diff(S 1 ). Consider the Fock space F (V − ) corresponding to the Hilbert space V − . We wish to construct projective representations of Diff(S 1 ) 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. Fix α, β ∈ R. For γ ∈ Diff(S 1 ) we consider the affine transformation of V given by the formula (2.13) 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 alement of V and we can multiply it by a constant α.
⊠ The formula (2.13) determines an embedding of the group Diff(S 1 ) to the group ASp(2∞, R). Restricting the Weil representation to Diff(S 1 ) we get a unitary projective representation of Diff(S 1 ). On the level of the Lie algebra we obtain the representation determined in Subsection 1.4. See [28], Sect. VII. 2 Next, we wish to write the matrix (2.7). Applying (2.10) we represent the block Φ in (2.7) as Formally, the right-hand side is well-defined for real-analytic f and real analytic γ ∈ Diff(S 1 ). But it makes sense for any distribution f and any smooth γ. For instance, we can decompose f (γ −1 (z)) into the Fourier series k∈Z c k z k and take Φf (z) := k>0 c k z k .
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 action of Γ. For details and proofs, see [24], [28], Sect. VII.4-5. Let α, β ∈ C. The semigroup Γ acts in the space F (V − ) by Gaussian operators where operators Consider a function f ∈ V − . Then the function f • (r + ) −1 is defined on the contour r + (S 1 ). Let us decompose 7 it as F 1 + F 2 , where F 1 is holomorphic in the domain r + (D 0 + ), and F 2 is holomorphic in C \ r + (D + ). This decomposition is determined up to an additive constant, In a similar way, we take a function f ∈ V + , decompose Recall that the subspaces V + and V − are dual one to another with respect to the pairing (2.9). It can be shown that the operators L : V − → V − and L t : V + → V + are dual one to another. Also, the operators M : Finally, we set 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].
⊠ It remains to explain the meaning of the expression (2.1). For 8 f t ± ∈ V ± , we set where the form {·, ·} is defined by (2.9), also we omit the similar expression with m 1 , m 2 . Also we note that under this normalization we have π(N α,β (R)1) = 1, (2.16) where π is the projection operator in F (V − ) to the line C · 1.
Comparing this with (3.4), we come to (3.8) we have Also we have a similar expression for µ(r − , p + ).

3.4.
Step 4. Next, we evaluate the following product by formula (2.2) Keeping in the mind identity we observe that the expressions in curly brackets in (3.11) and (3.10) coincide. We express the term in the curly bracket from (3.11) and substitute to (3.10). Also, the expressions in the square brackets in (3.12) and (3.9) coincide. We express the term in the square brackets from (3.12) and substitute it to (3.9). In this way, we get 3.5.
Step 5. Now we wish to evaluate L −1 (r + , r − ). For this aim consider the Gaussian operator, corresponding to the diffeomorphism γ r , On the other hand applying formula (2.8) to the same γ r we get an expression of the form Therefore L(r + , r − ) = Φ t−1 , i.e. L −1 = Φ t . Thus, where P + is the projection operators to V + . In the same way, we obtain L(p + , p − ) t−1 ℓ t 1 = P − T (γ p )ℓ t 1 .
It remains to expain Corollaries 1.2-1.3 from the theorem. .
In this way, we send the Fock space to the space of holomrphic functionals on Ξ, this gives us the desired realization (see [24], Subs. 4.12).