# American Institute of Mathematical Sciences

June  2017, 9(2): 207-225. doi: 10.3934/jgm.2017009

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

 1 Math. Dept., University of Vienna, Oskar-Morgenstern-Platz 1,1090 Wien, Austria 2 Institute for Theoretical and Experimental Physics (Moscow), Russia 3 MechMath. Dept., Moscow State University, Institute for Information Transmission (Moscow), Russia

* Corresponding author: Yury Neretin

Received  December 2015 Revised  August 2016 Published  May 2017

Fund Project: Supported by FWF grants P25142, P28421

The group $\text{Diff}\left( {{S}^{1}} \right)$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $\text{U}(p,q)$, $\text{Sp}(2n,\mathbb{R})$, $\text{SO}^*(2n)$; 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 $\text{Diff}\left( {{S}^{1}} \right)$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.

Citation: Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009
##### References:
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Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. Reine Angew. Math., 347 (1984), 69-133. doi: 10.1515/crll.1984.347.69. Google Scholar [14] H. Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Zeitschrift, 45 (1939), 29-61. doi: 10.1007/BF01580272. Google Scholar [15] V. G. Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras, in Group Theoretical Methods in Physics (Austin, Tex., 1978), Lecture Notes in Phys., Springer, Berlin, 94 (1979), 441-445.Google Scholar [16] A. A. Kirillov, Kähler structure on the K-orbits of a group of diffeomorphisms of the circle, Funct. Anal. Appl., 21 (1987), 42-45. Google Scholar [17] A. A. Kirillov, Geometric approach to discrete series of unireps for Virasoro, J. Math. Pures Appl., 77 (1998), 735-746. doi: 10.1016/S0021-7824(98)80007-X. Google Scholar [18] A. A. Kirillov and D. V. Yur', Kähler geometry of the infinite-dimensional homogeneous space $M=\mathrm{Diff}^+(S^1)/\mathrm {Rot}(S^1)$, Funct. Anal. Appl., 21 (1987), 35-46, 96. Google Scholar [19] D. Marshall and S. Rohde, Convergence of a variant of the zipper algorithm for conformal mapping, SIAM J. Numer. Anal., 45 (2007), 2577-2609. doi: 10.1137/060659119. Google Scholar [20] K. Mimachi and Y. Yamada, Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials, Comm. Math. Phys., 174 (1995), 447-455. doi: 10.1007/BF02099610. Google Scholar [21] Yu. A. Neretin, Unitary representations with a highest weight of a group of diffeomorphisms of a circle, Funct. Anal. Appl., 17 (1983), 85-86. Google Scholar [22] Yu. A. Neretin, Unitary Highest Weight Representations of Virasoro Algebra, (Russian) Ph. D. Moscow State University, MechMath Dept., 1983. Available from http://www.mat.univie.ac.at/~neretin/phd-neretin.pdfGoogle Scholar [23] Yu. A. Neretin, On the spinor representation of $\text{O}(∞,\mathbb{C})$, Soviet Math. Dokl., 34 (1987), 71-74. Google Scholar [24] Yu. A. Neretin, On a complex semigroup containing the group of diffeomorphisms of the circle, Funct. Anal. Appl., 21 (1987), 82-83. Google Scholar [25] Yu. A. Neretin, Holomorphic continuations of representations of the group of diffeomorphisms of the circle, (Russian), Mat. Sbornik, 180 (1989), 635-657; English transl. Math. USSR-Sb., 67 (1990), 75-97. Google Scholar [26] Yu. A. Neretin, Almost invariant structures and related representations of the group of diffeomorphisms of the circle, in Representations of Lie Groups and Related Topics (eds. A. M. Vershik, D. P. Zhelobenko), Gordon & Breach, 7 (1990), 245-267. Google Scholar [27] Yu. A. Neretin, Categories Enveloping Infinite-Dimensional Groups and Representations of Category of Riemannian Surfaces, Russian doctor degree thesis, Steklov Mathematical Institute, 1991, http://www.mat.univie.ac.at/~neretin/disser/disser.pdfGoogle Scholar [28] Yu. A. Neretin, Representations of Virasoro and affine Lie algebras, In Representation Theory and Noncommutative Harmonic Analysis, I, Encyclopaedia Math. Sci., Springer, Berlin, 22 (1994), 157-234. doi: 10.1007/978-3-662-03002-8_2. Google Scholar [29] Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, Oxford University Press, New York, 1996. Google Scholar [30] Yu. A. Neretin, Lectures on Gaussian Integral Operators And Classical Groups, European Mathematical Society (EMS), 2011. doi: 10.4171/080. Google Scholar [31] A. C. Schaeffer, D. C. Spencer, Coefficient Regions for Schlicht Functions, American Mathematical Society, New York, N. Y., 1950. Google Scholar [32] G. B. Segal, The definition of conformal field theory, in Differential Geometrical Methods in Theoretical Physics, 165-171, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, 1988. Google Scholar [33] E. Sharon and D. Mumford, 2d-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2006), 55-75. doi: 10.1109/CVPR.2004.1315185. Google Scholar [34] P. Wojtaszczyk, Spaces of analytic functions with integral norm, in Handbook of the Geometry of Banach Spaces, North-Holland, Amsterdam, 2 (2003), 1671-1702. doi: 10.1016/S1874-5849(03)80046-3. Google Scholar

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##### References:
 [1] L. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), 101-129. doi: 10.1007/BF02392634. Google Scholar [2] H. Airault and Yu. A. Neretin, On the action of Virasoro algebra on the space of univalent functions, Bull. Sci. Math., 132 (2008), 27-39. doi: 10.1016/j.bulsci.2007.05.001. Google Scholar [3] F. A. Berezin, The Method of Second Quantization, Academic Press, New York-London, 1966. Google Scholar [4] M. J. Bowick and S. G. Rajeev, String theory as the Kähler geometry of loop space, Phys. Rev. Lett., 58 (1987), 535-538. doi: 10.1103/PhysRevLett.58.535. Google Scholar [5] P. L. Duren, Univalent Functions, Springer-Verlag, 1983. Google Scholar [6] B. L. Feigin and D. B. Fuks, Skew-symmetric invariant differential operators on the line and Verma modules over the Virasoro algebra, Funct. Anal. Appl., 16 (1982), 47–63, 96. Google Scholar [7] B. L. Feigin and D. B. Fuks, Verma modules over a Virasoro algebr, Funct. Anal. Appl., 17 (1983), 91-92. Google Scholar [8] B. L. Feigin and D. B. Fuchs10, Representations of the Virasoro algebra, in Representations of Lie groups and related topics (eds. A. M. Vershik, D. P. Zhelobenko), Gordon & Breach, 7 (1990), 465-554. Google Scholar [9] D. Friedan, Z. Qiu and S. Shenker, Details of the nonunitarity proof for highest weight representations of the Virasoro algebra, Comm. Math. Phys., 107 (1986), 535-542. doi: 10.1007/BF01205483. Google Scholar [10] D. B. Fuchs, Cohomologies of Infinite-Dimensional Lie Algebras, Moscow, 1984. Google Scholar [11] P. Goddard, A. Kent and D. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Comm. Math. Phys., 103 (1986), 105-119. doi: 10.1007/BF01464283. Google Scholar [12] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, R. I., 1969. Google Scholar [13] R. Goodman and N. Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. Reine Angew. Math., 347 (1984), 69-133. doi: 10.1515/crll.1984.347.69. Google Scholar [14] H. Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Zeitschrift, 45 (1939), 29-61. doi: 10.1007/BF01580272. Google Scholar [15] V. G. Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras, in Group Theoretical Methods in Physics (Austin, Tex., 1978), Lecture Notes in Phys., Springer, Berlin, 94 (1979), 441-445.Google Scholar [16] A. A. Kirillov, Kähler structure on the K-orbits of a group of diffeomorphisms of the circle, Funct. Anal. Appl., 21 (1987), 42-45. Google Scholar [17] A. A. Kirillov, Geometric approach to discrete series of unireps for Virasoro, J. Math. Pures Appl., 77 (1998), 735-746. doi: 10.1016/S0021-7824(98)80007-X. Google Scholar [18] A. A. Kirillov and D. V. Yur', Kähler geometry of the infinite-dimensional homogeneous space $M=\mathrm{Diff}^+(S^1)/\mathrm {Rot}(S^1)$, Funct. Anal. Appl., 21 (1987), 35-46, 96. Google Scholar [19] D. Marshall and S. Rohde, Convergence of a variant of the zipper algorithm for conformal mapping, SIAM J. Numer. Anal., 45 (2007), 2577-2609. doi: 10.1137/060659119. Google Scholar [20] K. Mimachi and Y. Yamada, Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials, Comm. Math. Phys., 174 (1995), 447-455. doi: 10.1007/BF02099610. Google Scholar [21] Yu. A. Neretin, Unitary representations with a highest weight of a group of diffeomorphisms of a circle, Funct. Anal. Appl., 17 (1983), 85-86. Google Scholar [22] Yu. A. Neretin, Unitary Highest Weight Representations of Virasoro Algebra, (Russian) Ph. D. Moscow State University, MechMath Dept., 1983. Available from http://www.mat.univie.ac.at/~neretin/phd-neretin.pdfGoogle Scholar [23] Yu. A. Neretin, On the spinor representation of $\text{O}(∞,\mathbb{C})$, Soviet Math. Dokl., 34 (1987), 71-74. Google Scholar [24] Yu. A. Neretin, On a complex semigroup containing the group of diffeomorphisms of the circle, Funct. Anal. Appl., 21 (1987), 82-83. Google Scholar [25] Yu. A. Neretin, Holomorphic continuations of representations of the group of diffeomorphisms of the circle, (Russian), Mat. Sbornik, 180 (1989), 635-657; English transl. Math. USSR-Sb., 67 (1990), 75-97. Google Scholar [26] Yu. A. Neretin, Almost invariant structures and related representations of the group of diffeomorphisms of the circle, in Representations of Lie Groups and Related Topics (eds. A. M. Vershik, D. P. Zhelobenko), Gordon & Breach, 7 (1990), 245-267. Google Scholar [27] Yu. A. Neretin, Categories Enveloping Infinite-Dimensional Groups and Representations of Category of Riemannian Surfaces, Russian doctor degree thesis, Steklov Mathematical Institute, 1991, http://www.mat.univie.ac.at/~neretin/disser/disser.pdfGoogle Scholar [28] Yu. A. Neretin, Representations of Virasoro and affine Lie algebras, In Representation Theory and Noncommutative Harmonic Analysis, I, Encyclopaedia Math. Sci., Springer, Berlin, 22 (1994), 157-234. doi: 10.1007/978-3-662-03002-8_2. Google Scholar [29] Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, Oxford University Press, New York, 1996. Google Scholar [30] Yu. A. Neretin, Lectures on Gaussian Integral Operators And Classical Groups, European Mathematical Society (EMS), 2011. doi: 10.4171/080. Google Scholar [31] A. C. Schaeffer, D. C. Spencer, Coefficient Regions for Schlicht Functions, American Mathematical Society, New York, N. Y., 1950. Google Scholar [32] G. B. Segal, The definition of conformal field theory, in Differential Geometrical Methods in Theoretical Physics, 165-171, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, 1988. Google Scholar [33] E. Sharon and D. Mumford, 2d-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2006), 55-75. doi: 10.1109/CVPR.2004.1315185. Google Scholar [34] P. Wojtaszczyk, Spaces of analytic functions with integral norm, in Handbook of the Geometry of Banach Spaces, North-Holland, Amsterdam, 2 (2003), 1671-1702. doi: 10.1016/S1874-5849(03)80046-3. Google Scholar
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