A Borsuk-Ulam Theorem for compact Lie group actions by Biasi C., de Mattos D. PDF

By Biasi C., de Mattos D.

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B) If L is of limit circle type at b, then there exists a function χb ∈ Db such that Eλ ∩ Db (χb ) = Cϕb,λ . 4. For (b), assume that L is of limit Weyl, eigenfunction expansions, symmetric spaces 33 circle type at b. Then Eλ ⊂ Db . Take χb = ϕb,λ . Then the space on the left-hand side of the equality equals the space of f ∈ Eλ with [f, χb ]c = 0. The latter space is one dimensional since [ · , · ]b is non-degenerate on Eλ . 3, and the result follows. To make the treatment as uniform as possible, we agree to always use the dummy boundary datum χb = 0 in case L is of limit point type at b.

Clerc, Fonctions sphériques des espaces symétriques compacts, Trans. Amer. Math. Soc. 306 (1988), 421-431. [Fr-Ma] L. A. Frota-Mattos, The complex-analytic extension of the Fourier series on Lie groups, in Proceedings of Symposia in Pure Mathematics, Volume 30, Part 2 (1977), 279-282. [Gin1] S. Gindikin, Holomorphic horospherical duality “sphere-cone”, Indag. , 16 (2005), 487-497. [Gin2] S. Gindikin, Horospherical Cauchy-Radon transform on compact symmetric spaces, Mosc. Math. J. 6 (2006), no. 2, 299-305, 406.

1) b(ν) = b(−ν). If ν is real and non-zero, then Φν and Φ−ν form a basis of Eν 2 and from the asymptotic behavior of the (constant) Wronskian [Φν , Φ−ν ]t one reads off that b(ν) a(−ν) − a(ν) b(−ν) = 2iν, (ν ∈ R \ {0}). 2) it follows that Im a(ν)b(ν) = −ν, (ν ∈ R \ {0}). 3) In particular, a and b do not vanish anywhere on R \ {0}. We can now determine the spectral matrix for this problem. 4). From this we conclude that the spectral matrix P (λ) has zero entries except for the one in the upper left corner, which we denote by ρ(λ).

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