论文标题
真实和对称矩阵
Real and symmetric matrices
论文作者
论文摘要
我们在空间$ \ mathfrak {gl} _n'（\ mathbb c）$ n \ times n $矩阵的$ \ mathbb c）上构建了一个一系列，并用真实的特征值插入了复杂的结合和转置。我们从中得出一个分层的同态形态，在$ n \ times n $真实矩阵之间具有真实特征值和$ n \ times n $ n $对称矩阵的空间，并限制了个人$ \ naty $ \ natyrm {gl} _n（gl} _n（$ nath）$ - \ n（\ nationbb r）的真实分析等异形$ \ mathrm {o} _n（\ mathbb c）$ - 伴随轨道。我们还在经典类型和箭量品种的谎言代数的更一般环境中建立了相似的结果。为此，我们证明了关于线性空间超卡勒商人的一般结果。我们为（广义的）Kostant-Sekiguchi对应，真实和对称的跨轨道闭合的奇异性以及对真实群体和对称空间的Springer理论提供应用。
We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ of $n\times n$ matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the space of $n\times n$ real matrices with real eigenvalues and the space of $n\times n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $\mathrm{GL}_n(\mathbb R)$-adjoint orbits and $\mathrm{O}_n(\mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kähler quotients of linear spaces. We provide applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.
