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Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For an explicit description of ${T_X}$ see e.g. here. Consider the induced bundle ${\rm GL}({T_X})\to X$ whose fiber at $x\in X$ is the automorphism group ${\rm GL}(T_x)$ of the vector space $T_x$.

Question 1. What is the group $A={\rm Aut}_X\,{T_X}$ of regular global sections of ${\rm GL}({T_X})$ over $X$?

For any $\lambda \in{\Bbb C}^\times$ we have a global section of ${\rm GL}({T_X})$ taking value $\lambda I_x$ at $x$, where $I_x$ is the identity automorphism of $T_x$. Thus we obtain a canonical embedding ${\Bbb C}^\times\hookrightarrow A$.

Question 2. Is it true that $A={\Bbb C}^\times\,$?

Question 3. In particular, is the answer to Question 2 "Yes" for $X={\rm Gr}(1,n+1)={\Bbb P}^n\,$?

I know that the answer to Question 2 is "Yes" for $X={\rm Gr}(1,2)={\Bbb P}^1$. In this case ${\rm GL}({T_X})={\Bbb C}^\times\times {\Bbb P}^1\to {\Bbb P}^1$.

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    $\begingroup$ The situation is completely uniform, and what happens for $\mathbb{P}^1$ happens for all $G/P$ (at least in type ADE), see Boralevi: On simplicity and stability of the tangent bundle of rational homogeneous varieties (mathscinet.ams.org/mathscinet-getitem?mr=3202710). So $A=\mathbb{C}^\times$ for all Grassmannians. $\endgroup$
    – pbelmans
    Commented May 9, 2020 at 14:20
  • $\begingroup$ @pbelmans: Excellent! Many thanks! Later I was going to ask a question about the tangent bundle of $G/P$ ... $\endgroup$ Commented May 9, 2020 at 14:37
  • $\begingroup$ I state here the relevant result of Ada Boralevi. Let $G$ be a simple algebraic group over $\Bbb C$ of one of the types ADE, and let $P\subset G$ be a parabolic subgroup. Set $X=G/P$. Then the tangent bundle $T_X$ of $X$ is simple, that is, ${\rm End}_X\,T_X={\Bbb C}$. This answers in the positive my Questions 2 and 3. See Ada Boralevi, On simplicity and stability of the tangent bundle of rational homogeneous varieties, Theorem 4.4. $\endgroup$ Commented May 9, 2020 at 15:15

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