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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 infi- nite 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 realiza- tions 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 repre- sentations 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 the- ory includes realizations of highest weight representations in spaces of holomor- phic (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 '(g) = hρ(g)v; vi be the spherical function. Then '(g1g2) c(g1; g2) = : '(g1) '(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 in' complex) vector fields on the circle. Choosing a basis Ln := e @=i@' 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 2 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) 2 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 2 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) 2 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 2 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 n 0, by (p 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 2 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) 2 C2 can be integrated to a projective representation of Diff(S1) by bounded operators in a Fr´echet space, see [23]. This follows from the universal fermionic construction, see [29], Sect. VII.3. 1.6. The welding. Denote by C = C [ 1 the Riemann sphere. Denote by D+ ◦ the disk jzj 6 1 on C, by D− the disk jzj > 1 on C. Denote by D± their interiors. Denote by S1 the circle jzj = 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 2 Diff(S ). Let us glue the disks D+ and D− identifying the points 1 1 z 2 S ⊂ D+ with γ(z) 2 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]).
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