Duke Mathematical Journal

Hua-type integrals over unitary groups and over projective limits of unitary groups

Yurii A. Neretin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We discuss some natural maps from a unitary group ${\rm U}(n)$ to a smaller group ${\rm U}(n-m)$. (These maps are versions of the Livšic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces. (Values of the integrals are products of $\Gamma$-functions.) These integrals generalize Hua integrals. We construct inverse limits of unitary groups equipped with analogues of Haar measure and evaluate some integrals over these inverse limits.

Article information

Source
Duke Math. J., Volume 114, Number 2 (2002), 239-266.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575410

Digital Object Identifier
doi:10.1215/S0012-7094-02-11423-9

Mathematical Reviews number (MathSciNet)
MR1920189

Zentralblatt MATH identifier
1019.43008

Subjects
Primary: 43A85: Analysis on homogeneous spaces

Citation

Neretin, Yurii A. Hua-type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114 (2002), no. 2, 239--266. doi:10.1215/S0012-7094-02-11423-9. https://projecteuclid.org/euclid.dmj/1087575410


Export citation

References

  • G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge Univ. Press, Cambridge, 1999.
  • J. Arazy and G. Zhang, ``Invariant mean value and harmonicity in Cartan and Siegel domains'' in Interaction between Functional Analysis, Harmonic Analysis, and Probability (Columbia, Mo., 1994), Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, 19--40.
  • A. Borodin and G. Olshanski, Point processes and the infinite symmetric group, Math. Res. Lett. 5 (1998), 799--816.
  • --. --. --. --., Distributions on partitions, point processes, and the hypergeometric kernel, Comm. Math. Phys. 211 (2000), 335--358.
  • --. --. --. --., Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), 87--123. \CMP1 860 761
  • --------, Correlation kernels arising from the infinite-dimensional unitary groups and its representations, to appear.
  • J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., Oxford Univ. Press, New York, 1994.
  • M. V. Fedoryuk, Asymptotics: Integrals and Series (in Russian), Spravochn. Mat. Bibl., ``Nauka,'' Moscow, 1987.
  • F. R. Gantmakher, The Theory of Matrices (in Russian), 4th ed., ``Nauka,'' Moscow, 1988; ; English translation: The Theory of Matrices, Vols. 1, 2, Chelsea, New York, 1959.
  • S. G. Gindikin, Analysis on homogeneous spaces (in Russian), Uspekhi Mat. Nauk, 19, no. 4 (1964), 3--92.
  • S. Helgason, Differential Geometry and Symmetric Spaces, Pure Appl. Math. 12, Academic Press, New York, 1962.
  • Hua Loo Keng [L.-K. Hua], Harmonic Analysis of Functions of Several Complex Variables in Classical Domains (in Chinese), Science Press, Peking, 1958, ; Russian translation: Izdat. Inostr. Lit., Moscow, 1959, ; English translation: Amer. Math. Soc., Providence, 1963.
  • S. V. Kerov, Subordinators and permutation actions with quasi-invariant measure (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223 (1995), 181--218.; English translation in J. Math. Sci. (New York) 87 (1997), 4094--4117.
  • S. Kerov, G. Olshanski, and A. Vershik, Harmonic analysis on the infinite symmetric group: A deformation of regular representation, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 773--778.
  • M. S. Livšic, On spectral decomposition of linear non self-adjoint operators (in Russian), Mat. Sbornik N.S. 34 (1954), 145--199., ; English translation in Amer. Math. Soc. Transl. Ser. 2 5 (1957), 67--114.
  • Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, London Math. Soc. Monogr. (N.S.) 16, Oxford Univ. Press, New York, 1996. ; Russian translation: Editorial URSS, Moscow, 1998.
  • --. --. --. --., Conformal geometry of symmetric spaces and Kreĭn-Shmulian generalized linear fractional mappings (in Russian), Mat. Sb. 190, no. 2 (1999), 93--122.; English translation in Sb. Math. 190 (1999), 255--283.
  • --. --. --. --., Matrix analogues of $B$-function and Plancherel formula for Berezin kernel representations (in Russian), Mat. Sb. 191, no. 5 (2000), 67--100.; English translation in Sb. Math. 191 (2000), 683--715.
  • --. --. --. --., On separation of spectra in harmonic analysis of Berezin kernels (in Russian), Funktsional. Anal. i Prilozhen. 34, no. 3 (2000), 49--62.; English translation in Funct. Anal. Appl. 34 (2000), 197--207.
  • --. --. --. --., Plancherel formula for Berezin deformation of $L^2$ on Riemannian symmetric space, J. Funct. Anal. 189 (2002), 336--408. \CMP1 891 853
  • Yu. A. Neretin and G. I. Olshanskiĭ, Boundary values of holomorphic functions, singular unitary representations of groups $\Oh(p,q)$ and their limits as $q\to\infty$ (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223 (1995), 9--91.; English translation in J. Math. Sci. (New York) 87 (1997), 3983--4035.
  • N. K. Nikolskiĭ, Treatise on Shift Operator (in Russian), ``Nauka,'' Moscow, 1980, ; English translation: Grundlehren Math. Wiss. 273, Springer, Berlin, 1986.
  • G. I. Olshanskiĭ, ``Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe'' in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math. 7, Gordon and Breach, New York, 1990, 269--463.
  • --------, An introduction to harmonic analysis on infinite-dimensional unitary group, preprint, 2001.
  • --------, Inverse limits of symmetric spaces, unpublished notes.
  • D. Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70 (1987), 323--356.
  • H. Shimomura, On the construction of invariant measure over the orthogonal group on the Hilbert space by the method of Cayley transformation, Publ. Res. Inst. Math. Sci. 10 (1974/75), 413--424.
  • A. N. Shiryaev, Probability (in Russian), ``Nauka,'' Moscow, 1980; ; English translation: Probability, 2d ed., Grad. Texts in Math. 95, Springer, New York, 1996,
  • A. Unterberger and H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (1994), 563--597.