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The full automorphism group of T‾

Research paper by Indranil Biswas, Subramaniam Senthamarai Kannan, Donihakalu Shankar Nagaraj

Indexed on: 28 Mar '17Published on: 11 Mar '17Published in: Comptes Rendus Mathematique



Abstract

Let <img height="12" border="0" style="vertical-align:bottom" width="13" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si2.gif">G‾ be the wonderful compactification of a simple affine algebraic group G   of adjoint type defined over CC. Let <img height="14" border="0" style="vertical-align:bottom" width="43" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si12.gif">T‾⊂G‾ be the closure of a maximal torus <img height="11" border="0" style="vertical-align:bottom" width="42" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si5.gif">T⊂G. We prove that the group of all automorphisms of the variety <img height="12" border="0" style="vertical-align:bottom" width="12" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si1.gif">T‾ is the semi-direct product NG(T)⋊DNG(T)⋊D, where NG(T)NG(T) is the normalizer of T in G and D   is the group of all automorphisms of the Dynkin diagram, if G≠PSL(2,C)G≠PSL(2,C). Note that if <img height="13" border="0" style="vertical-align:bottom" width="87" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si15.gif">G=PSL(2,C), then <img height="13" border="0" style="vertical-align:bottom" width="56" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si10.gif">T‾=CP1 and so in this case <img height="16" border="0" style="vertical-align:bottom" width="119" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X17300626-si11.gif">Aut(T‾)=PSL(2,C).