## A Generalized Cartan Decomposition for the Double Coset Space {\large$SU(2n+1)\backslash SL(2n+1,\mathbb{C})/Sp(n,\mathbb{C})$}

J. Math. Sci. Univ. Tokyo
Vol. 17 (2010), No. 2, Page 201--215.

Sasaki, Atsumu
A Generalized Cartan Decomposition for the Double Coset Space {\large$SU(2n+1)\backslash SL(2n+1,\mathbb{C})/Sp(n,\mathbb{C})$}
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Abstract:
This paper gives a generalization of the Cartan decomposition for the non-symmetric space $SL(2n+1,\mathbb{C})/Sp(n,\mathbb{C})$. Our method uses the herringbone stitch introduced by T. Kobayashi \cite{cartan}, and as a corollary, we prove that $SU(2n+1)$ acts on the spherical variety $SL(2n+1,\mathbb{C})/Sp(n,\mathbb{C})$ in a strongly visible fashion with slice of real dimension $2n$.

Keywords: Cartan decomposition, visible action, symmetric space, herringbone stitch, spherical variety.

Mathematics Subject Classification (2010): Primary 32M10; Secondary 53C35,
Received: 2010-03-19