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e-Functions and the Campbell-Hausdorff Formula

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Symmetric Spaces and the Kashiwara-Vergne Method

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2115))

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Abstract

The goal of this chapter is to adapt the Kashiwara-Vergne method to an arbitrary symmetric space. We show that it admits an e-function, which can be constructed by means of the Campbell-Hausdorff formula. We prove several properties of this function which reflect, by Chap. 3, on invariant analysis on the space. The results extend to line bundles.

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Notes

  1. 1.

    Applying \(\mathop{\mathrm{ch}}\nolimits y\), resp. \(\mathop{\mathrm{sh}}\nolimits y\), instead and adding eliminates \(\overline{A}\) and gives back the expression (4.6) of \(\overline{Z}\).

  2. 2.

    See e.g. Bourbaki, Intégration, chap. VII, Sect 2, no. 6.

  3. 3.

    An explicit expression of U 3 and U 5 is given by Lemma 4.30.

  4. 4.

    The result also follows from (4.37) directly, since we have here \(\mathop{\mathrm{tr}}\nolimits _{\mathfrak{h}}\mathop{ \mathrm{ad}}\nolimits \mathfrak{h}_{{\ast}} = 0\) and \(\mathop{\mathrm{tr}}\nolimits _{\mathfrak{h}}[y,\partial _{Y }C] = 0\), C being even.

  5. 5.

    See also the correction in [15].

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  1. Laboratoire J. A. Dieudonné, UMR 7351 CNRS Université de Nice - Sophia Antipolis, Parc Valrose, France

    François Rouvière

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Rouvière, F. (2014). e-Functions and the Campbell-Hausdorff Formula. In: Symmetric Spaces and the Kashiwara-Vergne Method. Lecture Notes in Mathematics, vol 2115. Springer, Cham. https://doi.org/10.1007/978-3-319-09773-2_4

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