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Sep 28, 2016 at 16:28 vote accept Mikhail Borovoi
Sep 28, 2016 at 11:06 history edited Friedrich Knop CC BY-SA 3.0
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Sep 28, 2016 at 10:13 comment added Friedrich Knop Yes, that's my understanding. All automorphisms of a spherical datum come from a permutation of colors and these are all induced by an automorphism of the spherical variety. This follows from Thm. 2 and known structural properties of spherical data.
Sep 28, 2016 at 9:38 comment added Mikhail Borovoi Thank you, Friedrich! Do I understand correctly, that in the case when $\lambda=(\lambda_N,\lambda_{\mathcal D})$ is an isomorphism, there does exist an isomorphism $\varphi\colon X_1\to X_2$ inducing $\lambda$? If yes, does this follow from Losev's Theorem 1 and Theorem 2?
Sep 28, 2016 at 9:24 history edited Friedrich Knop CC BY-SA 3.0
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Sep 28, 2016 at 9:09 history answered Friedrich Knop CC BY-SA 3.0