Decomposition of complex hyperbolic isometries by involutions

详细信息    查看全文

• 作者：Krishnendu Gongopadhyay ; ² ; ^{krishnendug@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ^{krishnendu@iisermohali.ac.in" class="auth_mail" title="E-mail the corresponding author} ; Cigole Thomas¹ ; ³ ; ^{cigolethomas@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 51M10 ; secondary ; 51F25
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 July 2016
• 年：2016
• 卷：500
• 期：Complete
• 页码：63-76
• 全文大小：364 K

文摘

A k-reflection of the n  -dimensional complex hyperbolic space ${\mathrm{H}}_{\mathbb{C}}^n$ is an element in U(n,1)$\mathrm{U}(n,1)$ with negative type eigenvalue λ  , |λ|=1$|\lambda |=1$, of multiplicity k+1$k+1$ and positive type eigenvalue 1 of multiplicity n−k$n-k$. We prove that a holomorphic isometry of ${\mathrm{H}}_{\mathbb{C}}^n$ is a product of at most four involutions and a complex k  -reflection, k≤2$k\le 2$. Along the way, we prove that every element in SU(n)$\mathrm{SU}(n)$ is a product of four or five involutions according as $n\not\equiv 2\mathrm{mod}4$ or $n\equiv 2\mathrm{mod}4$. We also give a short proof of the well-known result that every holomorphic isometry of ${\mathrm{H}}_{\mathbb{C}}^n$ is a product of two anti-holomorphic involutions.