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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 |
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.
References:
[1] |
L. Ahlfors and A. Beurling,
Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), 101-129.
doi: 10.1007/BF02392634. |
[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. |
[3] |
F. A. Berezin,
The Method of Second Quantization, Academic Press, New York-London, 1966. |
[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. |
[5] |
P. L. Duren,
Univalent Functions, Springer-Verlag, 1983. |
[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. |
[7] |
B. L. Feigin and D. B. Fuks,
Verma modules over a Virasoro algebr, Funct. Anal. Appl., 17 (1983), 91-92.
|
[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. |
[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. |
[10] |
D. B. Fuchs,
Cohomologies of Infinite-Dimensional Lie Algebras, Moscow, 1984. |
[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. |
[12] |
G. M. Goluzin,
Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, R. I., 1969. |
[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. |
[14] |
H. Grunsky,
Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Zeitschrift, 45 (1939), 29-61.
doi: 10.1007/BF01580272. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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.
|
[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.pdf Google 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.
|
[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. |
[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. |
[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.pdf Google 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. |
[29] |
Yu. A. Neretin,
Categories of Symmetries and Infinite-Dimensional Groups, Oxford University Press, New York, 1996. |
[30] |
Yu. A. Neretin,
Lectures on Gaussian Integral Operators And Classical Groups, European Mathematical Society (EMS), 2011.
doi: 10.4171/080. |
[31] |
A. C. Schaeffer, D. C. Spencer,
Coefficient Regions for Schlicht Functions, American Mathematical Society, New York, N. Y., 1950. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
L. Ahlfors and A. Beurling,
Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), 101-129.
doi: 10.1007/BF02392634. |
[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. |
[3] |
F. A. Berezin,
The Method of Second Quantization, Academic Press, New York-London, 1966. |
[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. |
[5] |
P. L. Duren,
Univalent Functions, Springer-Verlag, 1983. |
[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. |
[7] |
B. L. Feigin and D. B. Fuks,
Verma modules over a Virasoro algebr, Funct. Anal. Appl., 17 (1983), 91-92.
|
[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. |
[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. |
[10] |
D. B. Fuchs,
Cohomologies of Infinite-Dimensional Lie Algebras, Moscow, 1984. |
[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. |
[12] |
G. M. Goluzin,
Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, R. I., 1969. |
[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. |
[14] |
H. Grunsky,
Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Zeitschrift, 45 (1939), 29-61.
doi: 10.1007/BF01580272. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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.
|
[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.pdf Google 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.
|
[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. |
[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. |
[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.pdf Google 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. |
[29] |
Yu. A. Neretin,
Categories of Symmetries and Infinite-Dimensional Groups, Oxford University Press, New York, 1996. |
[30] |
Yu. A. Neretin,
Lectures on Gaussian Integral Operators And Classical Groups, European Mathematical Society (EMS), 2011.
doi: 10.4171/080. |
[31] |
A. C. Schaeffer, D. C. Spencer,
Coefficient Regions for Schlicht Functions, American Mathematical Society, New York, N. Y., 1950. |
[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. |
[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. |
[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. |
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