Journal of Lie Theory 32 (2022), No. 4, 1025--1052
Copyright Heldermann Verlag 2022
Minimal Parabolic Subgroups and Automorphism Groups of Schubert Varieties
S. Senthamarai Kannan
Chennai Mathematical Institute, Siruseri, Kelambakkam, India
kannan@cmi.ac.in
Pinakinath Saha
Tata Inst. of Fundamental Research, Colaba, Mumbai, India
psaha@math.tifr.res.in
[Abstract-pdf]
Let $G$ be a simple simply-laced algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$. In this article, we show that $\omega_\alpha$ is a minuscule fundamental weight if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_{\alpha}$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w)$.
Keywords: Minuscule weights, co-minuscule roots, Schubert varieties, automorphism groups.
MSC: 14M15, 14M17.
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