Geodesic
Integral Models of Representations of Current Groups
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 1, pp. 22-32.

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We suggest a new construction of nonlocal representations of the current group. Instead of the Fock space, which is usually used in this situation, we consider the direct integral of countable tensor products of representations over the trajectories of some stochastic process. The construction substantially uses the invariance of the so-called infinite-dimensional Lebesgue measure.
Keywords: current group, summable representation, integral of tensor products. 
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A. M. Vershik; M. I. Graev. Integral Models of Representations of Current Groups. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 1, pp. 22-32. http://geodesic.mathdoc.fr/item/FAA_2008_42_1_a1/

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

[2] 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

[3] I. M. Gelfand, M. I. Graev, A. M. Vershik, “Models of representations of current groups”, Representations of Lie groups and Lie algebras, Akad. Kiadó, Budapest, 1985, 121–179 | MR

[4] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Neprivodimye predstavleniya gruppy $G^X$ i kogomologii”, Funkts. analiz i ego pril., 8:2 (1974), 67–69 | MR | Zbl

[5] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Predstavleniya gruppy $SL(2,R)$, gde $R$ — koltso funktsii”, UMN, 28:5 (1973), 83–128 | MR | Zbl

[6] A. M. Vershik, I. M. Gelfand, M. I. Graev, “Kommutativnaya model predstavleniya gruppy tokov $SL(2,\mathbb{R})^X$, svyazannaya s unipotentnoi podgruppoi”, Funkts. analiz i ego pril., 17:2 (1983), 70–72 | MR | Zbl

[7] A. M. Vershik, “Suschestvuet li mera Lebega v beskonechnomernom prostranstve?”, Analiz i osobennosti. Chast 2, K 70-letiyu so dnya rozhdeniya akademika Vladimira Igorevicha Arnolda, Tr. MIAN, 259, Nauka, M., 2007, 256–281

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

[9] A. M. Vershik, N. V. Tsilevich, “Fokovskie faktorizatsii i razlozheniya prostranstva $L^2$ nad obschimi protsessami Levi”, UMN, 58:3 (351) (2003), 3–50 | DOI | MR | Zbl

[10] N. Tsilevich, A. Vershik, M. Yor, “An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process”, J. Funct. Anal., 185:1 (2001), 274–296 | DOI | MR | Zbl

[11] H. Araki, “Factorisable representations of the current algebra”, Publ. RIMS Kyoto Univ. Ser. A, 5:3 (1970), 361–422 | DOI | MR | Zbl

[12] A. M. Perelomov, Obobschennye kogerentnye sostoyaniya i ikh primeneniya, M., Nauka, 1987 | MR

[13] G. Beitmen, L. Erdeii, Vysshie transtsendentnye funktsii, t. 2, Fizmatgiz, M., 1974