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This may be a dumb question.
I want to prove a following lemma.

Let $V,V'$ be vector spaces of dimension $r$ over $\mathbb{C}$.
Assume there is an isomorphism $\Phi$ between the flag varieties, $\Phi:\operatorname{Fl}(V)\rightarrow \operatorname{Fl}(V')$.
Then, $\Phi$ induces an isomorphism as Lie algera, $\Psi: \operatorname{End}_{0}(V)\rightarrow \operatorname{End}_{0}(V')$.
( $\operatorname{End}_{0}(V)$ is the Lie algebra consisting of trace-zero endomorphisms. )

I don't know how to define $\Psi$ explicitly and why it is an isomorphism as Lie algebra.
Thanks in advance.

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1 Answer 1

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Let $T_{Fl(V)}$ be the tangent bundle; then $$ H^0(Fl(V), T_{Fl(V)}) \cong End_0(V). $$ Therefore, $\Psi$ can be defined as the differential of $\Phi$. To be more precise, the differential of $\Phi$ defines an isomorphism $$ T_{Fl(V)} \stackrel{d\Phi}\to \Phi^*T_{Fl(V')}, $$ hence the induced morphism on global sections gives $$ End_0(V) \cong H^0(Fl(V), T_{Fl(V)}) \stackrel{d\Phi}\to H^0(Fl(V),\Phi^*T_{Fl(V')}) \cong H^0(Fl(V'),T_{Fl(V')}) \cong End_0(V'). $$

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  • $\begingroup$ Thanks. This is very helpful for me. But why $H^0(\operatorname{Fl}(V),\operatorname{T}_{Fl(V)}) \cong \operatorname{End}_{0}(V)$? If it is not easy to explain, could you tell me some reference? $\endgroup$
    – Aoki
    Oct 2, 2020 at 17:18
  • $\begingroup$ It is a standard fact that $H^0(G/P,T) = \mathfrak{g}$ for any semisimple algebraic group $G$ and its parabolic subgroup $P$; this should be in any textbook on algebraic groups. In the case $G = \mathrm{SL}(V)$ and $P = B$ a Borel subgroup, this result reduces to what you want. $\endgroup$
    – Sasha
    Oct 2, 2020 at 17:46

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