Holomorphic functions and hulls of Matsuki-dual orbits of SU(p,q) in complex Grassmannians

Abstract

Let G₀ be a real form of a complex semisimple group acting on a flag manifold Z, with maximal compact subgroup K₀ and its complexification K. Matsuki duality pairs each G₀-orbit γ with a K-orbit κ, and M=κ∩γ is a compact K₀-orbit. For lower-dimensional γ, M is a compact CR manifold, and we ask which functions are holomorphic near M and what its holomorphic hull is. In general, we show that K₀-finite holomorphic functions near M are regular functions on K-orbits, and that M carries non-constant holomorphic germs if some K-stable prime divisor of Z misses κ, and, under a connectedness hypothesis, only constants otherwise. For G₀=SU(p,q) acting on any Grassmannian Gr_k(ℂ^p+q) we give a complete answer. Holomorphic functions near M are constant unless M is of holomorphic type: k∈{p,q} and M lies in the Harish-Chandra chart of the bounded symmetric domain 𝒟. Then M is the manifold of rank-c partial isometries, and a holomorphic hull of M is 𝒟̅∩{rank≤ c}. This agrees with the polynomial hull of Kaup and Zaitsev except in one totally real case, where it is strictly smaller because det⁻¹ is holomorphic near U(p). The same dichotomy decides when global CR functions on M are constant and when holomorphic sections of line bundles near M are global; its exceptions differ from the Gindikin-Matsuki exceptions for cycle spaces, except at the base point of 𝒟. This settles the descent question left open in earlier work with K. Shabbir on rank-one orbits. We also determine which K-orbits carry non-constant regular functions, prove constancy via chains of rational curves for small c, and compute Levi forms in examples.

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BibTeX RIS

BibTeX

@misc{ullah2026holomorphic,
  title = {{Holomorphic functions and hulls of Matsuki-dual orbits of $SU(p,q)$ in complex Grassmannians}},
  author = {Irfan Ullah},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41828},
  eprint = {2610.41828},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41828},
}

RIS

TY  - UNPB
TI  - Holomorphic functions and hulls of Matsuki-dual orbits of SU(p,q) in complex Grassmannians
AU  - Irfan Ullah
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41828
UR  - https://arxiv.org/abs/2610.41828
AB  - Let G_0 be a real form of a complex semisimple group acting on a flag manifold Z, with maximal compact subgroup K_0 and its complexification K. Matsuki duality pairs each G_0-orbit γ with a K-orbit κ, and M=κ∩γ is a compact K_0-orbit. For lower-dimensional γ, M is a compact CR manifold, and we ask which functions are holomorphic near M and what its holomorphic hull is. In general, we show that K_0-finite holomorphic functions near M are regular functions on K-orbits, and that M carries non-constant holomorphic germs if some K-stable prime divisor of Z misses κ, and, under a connectedness hypothesis, only constants otherwise. For G_0=SU(p,q) acting on any Grassmannian Gr_k(ℂ^p+q) we give a complete answer. Holomorphic functions near M are constant unless M is of holomorphic type: k∈{p,q} and M lies in the Harish-Chandra chart of the bounded symmetric domain 𝒟. Then M is the manifold of rank-c partial isometries, and a holomorphic hull of M is 𝒟̅∩{rank≤ c}. This agrees with the polynomial hull of Kaup and Zaitsev except in one totally real case, where it is strictly smaller because det^−1 is holomorphic near U(p). The same dichotomy decides when global CR functions on M are constant and when holomorphic sections of line bundles near M are global; its exceptions differ from the Gindikin-Matsuki exceptions for cycle spaces, except at the base point of 𝒟. This settles the descent question left open in earlier work with K. Shabbir on rank-one orbits. We also determine which K-orbits carry non-constant regular functions, prove constancy via chains of rational curves for small c, and compute Levi forms in examples.
ER  -

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