Vector Fields on the Space of Functions Univalent Inside the Unit Disk via Faber Polynomials

We obtain the Kirillov vector fields on the set of functions $f$ univalent inside the unit disk, in terms of the Faber polynomials of $1/f(1/z)$. Our construction relies on the generating function for Faber polynomials.


Introduction
The Virasoro algebra has a representation in the tangent bundle over the space of functions univalent in the unit disk which are smooth on its boundary. This realization was obtained by A.A. Kirillov and D.V. Yur'ev [4,5] as first-order differential operators. Following [4], consider f (z) a holomorphic function univalent in the unit disc D = {z ∈ C; |z| ≤ 1}, smooth up to the boundary of the disc and normalized by the conditions f (0) = 0 and f ′ (0) = 1, thus f (z) = z 1 + n≥1 c n z n (1.1) and the series (1.1) converges to f (z) on D. By De Branges theorem proving the Bieberbach conjecture, the coefficients (c n ) n≥1 lie in the infinite-dimensional domain |c n | < n + 1, for n ≥ 1.
In the following, we shall call this infinite domain the manifold of coefficients. On the other hand, we denote a function univalent outside the unit disc. In order to study the representations of the Virasoro algebra [6], A.A. Kirillov considered the action of vector fields on the set of the diffeomorphisms of the circle by perturbing the equation f • γ = g, where γ is a diffeomorphism of the circle. He obtained a sequence of vector fields L p , (p being positive or negative integer) acting on the set of functions univalent inside the unit disk. These vector fields are expressed as of the contour integral, it is not difficult to see that φ −n (z) vanishes in (1.2). The term φ p (z) comes from the residue at the pole t = 0 and z 1−p f ′ (z) comes from the residue at t = z. We have φ 0 (z) = −f (z) and L 0 f (z) = zf ′ (z) − f (z). When p > 0 the evaluation of the integral (1.2) is more delicate since we have a non-vanishing residue at zero. For p ≥ 0, we find where the coefficients (c j ) j≥1 are the coefficients of f (z) given by (1.1). With the residue calculus, it has been made explicit in [1] The coefficients of Λ p (u) depend on the (c j ) j≥1 and have been calculated in [3,Proposition 3.2]. We note that φ p (z) is obtained by eliminating the powers of z n , n ≤ 1 in z 1−p f ′ (z); the elimination is done by expanding z 1−p f ′ (z) in powers of f (z), then substracting the part φ p (z) of the series with powers f (z) n , n ≤ 1. This method is analogous to the elimination of terms in power series developed by Schiffer for Faber polynomials [8]. Let h(z) be a univalent function holomorphic outside the unit disc except for a pole with residue equal to 1 at infinity, thus for |z| > 1, Let t ∈ C, at a neighborhood of z = ∞, we have the expansion The function F n (t) is a polynomial of degree n in the variable t and is called the n th Faber polynomial of the function h. Schiffer showed in [8] that the polynomial F n (t) is the unique polynomial in t of degree m such that The objective of this paper is to show that with a method analogous to that of M. Schiffer for obtaining Faber polynomials, we can recover Kirillov vector fields L −p when p ≥ 0. For this, let f (z) be a univalent function as in (1.1), for p ≥ 0, we start from z 1−p f ′ (z), it expands in D as nc n z n−p + terms in z n (n ≥ 2).
In Section 2, the function Λ p (u) for p ≥ 0, is constructed in such a way that expands in powers z n with n ≥ 2. In fact, the function Λ p (t) with respect to z 1−p f ′ (z) plays the same role as the Faber polynomials F m (t) with respect to h(z) m . Then we prove that z 1−p f ′ (z)+Λ p (f (z)) is equal to the expression (1.2) of the vector field L −p f (z) found by Kirillov and Yur'ev. Note that the method of elimination of terms in power series as developed in [8] for Faber polynomials is a formal calculation on series and does not require smoothness assumptions for f (z) at the boundary of the unit disk. Thus, it is conceivable to extend the calculations of the vector fields for functions f (z) which present a singularity at the boundary of D. This is our main motivation for adapting Schiffer's method to the formal series z 1−p f ′ (z). The regularity assumptions on f (z) are stronger in the case of the variational approaches developed in [4] or [7] since in the variational case, it is assumed that f (z) is smooth up to the boundary of the unit disc.
Let Λ p (u) as in (1.3), we give an expression of Λ p (u), p > 0, in terms of the Faber polynomials F n (w) of the function h(z) = 1/f (1/z). We have where F n (w) are the Faber polynomials associated to the function h. In terms of f , We find that the functions u → Λ p (u) are determined by the expansion at a neighborhood of ξ = 0, and u is a complex number. For p ≥ 2, we show that we can calculate the coefficient Λ p (u) of ξ p in the expansion (1.5) as follows, Then, we recover (1.2) as Corollary, see (2.21). In Section 3, we put We prove that for any u and v in the unit disc, there holds k≥1 p≥0 We say that the right hand side of (1.8) is a generating function for the homogeneous polynomials A p k . Note that the right hand side in (1.8) has a meaning when u → v. All series obtained as expansions of a function are convergent inside their disc of convergence which is determined by the singularities of the function. In Section 4, we identify as in [4] the vector fields (L k ) k≥1 with first order differential operators on the manifold of coefficients of functions univalent on D; as quoted before, this manifold comes from De Branges theorem, the coefficients (c n ) n≥1 lie in the infinite-dimensional domain |c n | < n + 1, for n ≥ 1. Some properties of this infinite-dimensional manifold have been investigated in [4,5], for example Kähler structure or Ricci curvature. Here, we shall not develop the properties of this manifold. We only examine the action of the (L k ) k≥1 on the functions Λ p (u). The functions Λ p (u) have their coefficients in this manifold. We find that L k (Λ p+k (u)) = (2k + p)Λ p (u) for p ≥ 1. In Section 5, we consider the reverse series 2 Elimination of terms in power series and the vector fields L −p f (z) in terms of the Faber polynomials of 1/f (1/z) Let f (z) as in (1.1), there exists a unique sequence of rational functions (Λ p ) p≥0 of the form such that To prove the existence of Λ p (u), we expand z 1−p f ′ (z) in powers of f (z). Then −Λ p (f (z)) is the sum of terms with powers of f (z) n such that n ≤ 1. The unicity of the function Λ p (u) satisfying (2.1), (2.2) results from matching equal powers of z in the expansions (2.1), (2.2). We can calculate directly For p = 0, we have and Λ 0 (u) = −u.

Theorem 1. Λ p (w) is expressed in terms of the Faber polynomials of
The functions (Λ p (w)) p≥0 are given by Proof . Consider the function As in [8], we have where F n (w) are the Faber polynomials associated to the function h. Since h(z) = 1/f (1/z), we have (1.4). If we take the derivative of (1.4) with respect to w and then integrate with respect to z, we obtain where the β n,k are the Grunsky coefficients of h, see [8].
We write (2.12) as (2.14) At this step we have eliminated all the negative powers and the constant term in such a way that the series on the right side of (2.13) has only terms in z n with n ≥ 1. To eliminate the term in z in order to have only terms in z n , n ≥ 2, we put for p ≥ 1, From (2.13), we see that the coefficients of z, a p p are determined with a 1 1 = 2c 1 and if p > 1, With the convention c 0 = 1, B p 0 = a p p , we have A p 0 = 0. Now, we prove that (a p p ) is given by (1.6) or (2.6). For this, we consider (2.10) with n = 1, it gives Because of the symmetry (2.9) in the Grunsky coefficients, taking the derivative with respect to z in (2.18), we obtain, Then we multiply the power series f ′ (z) = 1 + 2c 1 z + 3c 2 z 2 + · · · by the power series in (2.19), it gives (2.16) on one side and (1.6) or (2.6) on the other side. To prove (1.5) or (2.7), we put then with (2.5) and (2.6), and from (2.15), we see that We put Λ p (w) = M p ( 1 w ). We obtain (2.7) or (1.5).
Consider the homogeneous polynomials B p k given by (2.14) for p > 1 and B 1 k = (k + 2)c k+1 . We rewrite (3.4) as k≥0 p≥2 Moreover, since B 1 k = (k + 2)c k+1 , we can write the sum in (3.5), starting from p = 1. On the other hand, Finally, we obtain k≥0 p≥1 Since A p k = B p k − a p p c k as in (2.17), A p 0 = 0 and since (2.6) is true, we have k≥1 p≥1 We divide by v. Since k≥1 p≥0 we obtain (1.8).
4 The differential operators (L k ) k≥1 on the functions Λ p (u) We identify the set of functions univalent on the unit disk with the set of their coefficients via the map For k ≥ 1, we put ∂ k = ∂ ∂c k . Following [4], we consider the partial differential operators We have z 1+k f ′ (z) = L k [f (z)] and ∂ n L j = L j ∂ n + (n + 1)∂ n+j . We put L 0 = n≥1 nc n ∂ n and for k ≥ 1 The operators ∂ k are related to the (L k+p ) p≥0 as follows, see [2], where the (B n ) n≥1 are independent of k. We calculate the B n , n ≥ 0, with Proof . We verify (4.2) on f (z). We have we have to prove We divide by z k+1 and we obtain (4.2).
By considering expansions as in [1], we obtain with (1.5), the action of (L k ) k≥1 on the functions (Λ p (u)) p≥0 .
5 The (L k ) k≥1 and the reverse series of f (z) As in Section 4, we consider the differential operators (L k ) k≥1 given by (4.1). We denote f −1 (z) the inverse function of f (z), (f −1 • f = Identity), we say also reverse series of f (z). For any integer q, consider the series where δ q n are homogeneous polynomials in the variables (c 1 , c 2 , . . . , c n , . . . ), the coefficients of f (z). Then L p δ q n z n .