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With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible action for holomorphic actions of Lie groups on complex manifolds. In this article we show that the restriction of the unitary completion of the Zuckerman derived functor module \(A_{\mathfrak {q}}(\lambda )\) of a connected real linear semisimple Lie group \(G_{\mathbb {R}}\) to its symmetric subgroup \(H_{\mathbb {R}}\) is multiplicity-free if the flag variety \(G /Q\) is an \(H \) -spherical variety. From this result we find that a conjecture of Kobayashi on the multiplicity-freeness property of \(A_{\mathfrak {q}}(\lambda )\) holds true.

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Multiplicity-Freeness Property of the Restriction of  \(A_{\mathfrak {q}}(\lambda )\) and Visible Actions on Spherical Flag Varieties for Symmetric Pairs

  • 会议论文

  • 发表日期：

    2026年

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  • Yuichiro Tanaka

摘要

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible action for holomorphic actions of Lie groups on complex manifolds. In this article we show that the restriction of the unitary completion of the Zuckerman derived functor module \(A_{\mathfrak {q}}(\lambda )\) of a connected real linear semisimple Lie group \(G_{\mathbb {R}}\) to its symmetric subgroup \(H_{\mathbb {R}}\) is multiplicity-free if the flag variety \(G /Q\) is an \(H \) -spherical variety. From this result we find that a conjecture of Kobayashi on the multiplicity-freeness property of \(A_{\mathfrak {q}}(\lambda )\) holds true.