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Harish-Chandra’s c-function; 50 years later
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 385-402.

Nous discutons différents aspects de la fonction c de Harish-Chandra, en soulignant ses interactions avec la transformée horosphérique.

We discuss different aspects of the c-function of Harish-Chandra with focus on its connection with the horospherical transform.

Publié le : 2016-07-11
DOI : https://doi.org/10.5802/afst.1498
@article{AFST_2016_6_25_2-3_385_0,
     author = {Simon Gindikin},
     title = {Harish-Chandra's $c$-function; 50 years later},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {2-3},
     year = {2016},
     pages = {385-402},
     doi = {10.5802/afst.1498},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2016_6_25_2-3_385_0/}
}
Simon Gindikin. Harish-Chandra’s $c$-function; 50 years later. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 385-402. doi : 10.5802/afst.1498. https://afst.centre-mersenne.org/item/AFST_2016_6_25_2-3_385_0/

[1] Akhiezer (D.N.), Gindikin (S.G.).— On Stein extensions of real symmetric spaces, Math. Ann., 286 p. 1-12 (1990).

[2] Beerends (R.J.).— The Fourier transform of Harish-Chandra’s c-function and inversion of Abel transform, Math. Ann., 277 1 p. 1-23 (1987).

[3] Braverman (A.), Finkelberg (M.), Kazhdan (D.).— Affine Gindikin-Karpelevich formula via Uhlenbeck spaces, arXiv:0912.5132 [math.RT] (2009).

[4] Braverman (A.), Garland (H.), Kazhdan (D.), Patnaik (M.).— An affine Gindikin-Karpelevich formula, arXiv:1212.6473 [math.RT] (2012).

[5] DeConcini (C.),Procesi (C.).— Complete symmetric varieties, Lecture Notes in Math., 996 p. 1-44 (1983).

[6] Gelfand (I.M.), Graev (M.I.).— Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry, Amer. Math. Transl. (2), 37 p. 351-429 (1964).

[7] Gindikin (S.).— Integral geometry on symmetric manifolds, Amer. Math. Transl. (2), 148 p. 29-37 (1991).

[8] Gindikin (S.).— Holomorphic language for ¯-cohomology and representations of real semisimple Lie groups, The Penrose Transform and Analytic Cohomology in Representation Theory (M. Eastwood, J. Wolf, R. Zierau, eds), 154, Cont. Math., Amer. Math. Soc., p. 103-115 (1993).

[9] Gindikin (S.).— Integral geometry on SL(2,), Math. Research Letters, 7 p. 1-15 (2000).

[10] Gindikin (S.).— Product-formula for c-function and inverse horospherical transform, Amer. Math. Soc. Transl. (2), 210 p. 125-134 (2003).

[11] Gindikin (S.).— Harmonic analysis on symmetric manifolds from the point of view of complex analysis, Japanese J. Math., 1, 1 p. 87-105 (2006).

[12] Gindikin (S.).— The integral Cauchy formula on symmetric Stein manifolds, Colloquium de Giorgi, p. 19-28, Edizione della Normale (2006).

[13] Gindikin (S.).— Horospherical transform on Riemannian symmetric manifolds of noncompact type, Funct. Anal. Appl., 42, 4 p. 1-11 (2008).

[14] Gindikin (S.).— Helgason’s conjecture in complex analytical interior, Representation Theory, Complex Analysis and Integral Geometry (B. Krotz, O. Offen, E. Sayag, eds), Birkhauser p. 87-96 (2010).

[15] Gindikin (S.).— Harmonic analysis on symmetric spaces as complex analysis, Automorphic Forms and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro (Cogdell, Shahidi, and Soudry, eds), AMS, Contemporary Math., 614 p. 69-80 (2014).

[16] Gindikin (S.).— Local inversion formulas for horospherical transforms, Moscow Math. J., 13, 2 p. 267-280 (2013).

[17] Gindikin (S.).— Intermediate horospherical transforms, wonderful compactification and c-functions, to appear.

[18] Gindikin (S.), Goodman (R.).— Restricted roots and restricted form of the Weyl dimension formula for spherical varieties, J. Lie Theory, 13, 1 p. 257-311 (2013).

[19] Gindikin (S.G.), Karpelevich (F.I.).— Plancherel measure for symmetric spaces of non-positive curvature, Sovjet Math. Dokl., 3 p. 962-965 (1962).

[20] Gindikin (S.G.), Karpelevich (F.I.).— A problem of integral geometry, In memoriam: N. G. Chebotarev, Izdat. Kazan. Univ. p. 30-43 (1964), English transl. in, Selecta Math. Sov., 1p. 169-184 (1981).

[21] Gindikin (S.G.), Karpelevich (F.I.).— On a integral connected with symmetric Riemann spaces of of nonpositive curvature, Izv. Akad Nauk SSSR Ser. Mat., 30 p. 1147-1156 (1966), English transl. in, Amer. Math. Soc. Transl. (2), 85 p. 249-258 (1969).

[22] Harish-Chandra.— Spherical functions on a semisimple Lie group. I,II, Americamn J. Math., 80 p. 241-310, 553-613 (1958).

[23] Helgason (S.).— Groups and Geometric Analysis, Academic Press (1984).

[24] Hilgert (J.), Pasquale (A.), Vinberg (E.).— The dual horospherical Radon transform for polynomials, Moscow Math. J., 2 p. 113-126 (2002).

[25] Knapp (A.W.), Stein (E.M.).— Intertwining operators for semisimple groups, Annals Math., 93 p. 489-578 (1971).

[26] Macdonald (I.G.).— Spherical functions on a group of p-adic type, Ramanujan Institute lecture notes 2. Madras (1971).

[27] Olafsson (G.), Pasquale (A.).— Ramanujan’s Master Theorem for the hypergeometric Fourier transform on root systems, J. Fourier Anal. Appl., 19 6 p. 1150-1183 (2013).

[28] Oshima (T.).— A realization of of Riemannian symmetric spaces, J. Math Soc. Japan, 30 (1978) p. 117-132.

[29] Vinberg (E.).— On reductive algebraic semigroups, Amer.Math.Soc. Transl., 169 p. 145-182 (1995).