On F. A. Berezin and his work on representations of current groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 5-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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We tell about F. A. Berezin and his outstanding role in the development of mathematical physics. We discuss the connections between Berezin's work on representations of current groups with values in the universal covering of the group of motions of the complex ball and the works by Vershik, Gelfand, and Graev on the multiplicative integral of representations.
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A. M. Vershik. On F. A. Berezin and his work on representations of current groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 5-14. http://geodesic.mathdoc.fr/item/ZNSL_2006_331_a1/

[1] F. A. Berezin, “Predstavleniya nepreryvnogo pryamogo proizvedeniya universalnykh nakryvayuschikh gruppy dvizhenii kompleksnogo shara”, Trudy Mosk. mat. obschestva, 36, 1978, 275–293 | MR | Zbl

[2] F. A. Berezin, “Representations of the infinite direct product of universal coverings of isometry groups of the complex ball”, Rep. Math. Phys., 9:1 (1976), 15–30 | DOI | MR | Zbl

[3] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Predstavleniya gruppy $\operatorname{SL}(2,R)$, gde $R$ – koltso funktsii”, Uspekhi mat. nauk, 28:5 (1973), 83–128 | MR

[4] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Neprivodimye predstavleniya gruppy $G^X$ i kogomologii”, Funkts. anal. i prilozh., 8:3 (1974), 67–68 | MR

[5] A. M. Vershik, M. I. Graev, “Kommutativnaya model predstavleniya gruppy $O(n,1)^X$ i obobschennaya mera Lebega v prostranstve raspredelenii”, Funkts. anal. i prilozh., 39:2 (2005), 1–12 | MR | Zbl

[6] A. M. Vershik, M. I. Graev, “Struktura dopolnitelnykh serii i osobykh predstavlenii grupp $O(n,1)$ i $U(n,1)$”, Uspekhi mat. nauk, 61:5 (2006), 3–88 | MR | Zbl

[7] A. M. Vershik, M. Ior, N. V. Tsilevich, “O tozhdestve Markova–Kreina i kvaziinvariantnosti gamma-protsessa”, Zapiski nauchn. semin. POMI, 283, 2001, 21–36 | MR | Zbl

[8] A. M. Vershik, S. I. Karpushev, “Kogomologii grupp v unitarnykh predstavleniyakh, okrestnost edinitsy i uslovno polozhitelno opredelennye funktsii”, Mat. sb., 119:4 (1982), 521–533 | MR | Zbl

[9] C. Delaroche, A. Kirillov, “Sur les relations entre l'espace dual d'un groupe et la structure de ses sous-groupes fermés”, Seminaire Bourbaki, Expose 243, 1968

[10] I. M. Gelfand, M. I. Graev, A. M. Vershik, “Models of representations of current groups”, Representations of Lie Groups and Lie Algebras, ed. A. A. Kirillov, Akadémiai Kiadó, Budapest, 1985, 121–179 | MR

[11] M. I. Graev, A. M. Vershik, “The basic representation of the current group $O(n,1)^X$ in the $L^2$ space over the generalized Lebesgue measure”, Indag. Math., 16:3–4 (2005), 499–529 | DOI | MR | Zbl

[12] V. P. Maslov, M. A. Shubin, A. M. Vershik, N. D. Vvedenskaya, “Alik Berezin in the recollections of friends”, Amer. Math. Soc. Transl. Ser. 2, 175, Am. Math. Soc., Providence, RI, 1996, 225–236 | MR | Zbl

[13] Y. Shalom, “Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group”, Ann. Math., 152 (2000), 113–182 | DOI | MR | Zbl

[14] N. Tsilevich, A. Vershik, M. Yor, “An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process”, J. Funct. Anal., 185 (2001), 274–296 | DOI | MR | Zbl