I am trying to understand the tangent bundle to algebraic loop groups, particularly for $G=GL_n$, over arbitrary characteristic. Can anyone point me to existing literature related to this? In particular, I want to figure out if the tangent bundle is ample. Does anyone know of existing ideas in this regard?
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1$\begingroup$ For an affine variety, any non-zero vector bundle is ample and $GL_n$ is affine. $\endgroup$– MohanCommented Mar 17, 2017 at 13:50
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$\begingroup$ @Mohan The question is about the tangent bundle of the loop group $LGL_n$ of $GL_n$. $\endgroup$– user1504Commented Mar 17, 2017 at 15:08
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$\begingroup$ @user1504 I am ignorant and did not even know that the loop group is an algebraic variety (and I do not know what ample means otherwise). $\endgroup$– MohanCommented Mar 17, 2017 at 15:38
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$\begingroup$ According to Faltings: ems-ph.org/journals/… the algebraic loop group is an ind-affine group. Like Mohan, I do not know what an ample bundle on such a thing is. $\endgroup$– BertieCommented Mar 17, 2017 at 15:50
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$\begingroup$ It is an inductive limit of closed sub-varieties. So ample in this case means that the restriction of the bundle to every stratum is ample. $\endgroup$– user106243Commented Mar 18, 2017 at 16:02
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