The fixed point subvarieties of unipotent transformations on the flag varieties (Q1059125)

Let V be an n-dimensional vector space over a field K. For an ordered sequence \(\mu =(\mu_ 1,...,\mu_ s)\) of positive integers such that \(\mu_ 1+...+\mu_ s=n,\) let \(F_{\mu}=F_{\mu}(V)\) be the partial flag variety of type \(\mu\) defined by \(\{(W_ 1,...,W_{s-1})\in \prod_{1\leq i\leq s-1}G_{d_ i}(V);W_ i\subset W_{i+1}(1\leq i\leq s-2)\},\) where \(d_ i=\mu_ 1+...+\mu_ i\) and \(G_{d_ i}(V)\) is the Grassmann variety of all \(d_ i\)-dimensional linear subspaces in V \((i=1,2,...,s-1).\) Let u be a unipotent transformatin of V. In the paper, we consider the fixed point subvariety \(F^ u_{\mu}=\{(W_ i)\in F_{\mu};uW_ i=W_ i(1\leq i\leq s-1)\}.\) Let A be the set of all minimal semistandard \(\mu\)-tableaus of type \(\lambda\) (defined precisely in the paper), where \(\lambda\) is the Jordan type of u. For \(\alpha\in A\), let \(\lambda^ i_{\alpha} (1\leq i\leq s-1)\) be the Young diagram with \(d_ i\) squares. For \(\alpha\in A\), put \(Y_{\alpha}=\{(W_ i)\in F^ u_{\mu};\) the Jordan type of \(u| W_ i\) is \(\lambda^ i_{\alpha}(1\leq i\leq s-1)\}.\) Then we have \(F^ u_{\mu}=\amalg_{\alpha \in A}Y_{\alpha}\) (disjoint union). The main results of the paper are: (1) For \(\alpha\in A\), the variety \(Y_{\alpha}\) is an irreducible locally closed subvariety of \(F^ u_{\mu}\). - (2) For \(\alpha\in A\), the variety \(Y_{\alpha}\) has a partition \(Y_{\alpha}=\amalg_{\beta \in X_{\alpha}}S^ u_{\beta},\) where \(X_{\alpha}\) is the set of semistandard \(\mu\)- tableaus determined by \(\alpha\) and the varieties \(S^ u_{\beta}\) are the fixed point subvarieties of the Schubert cells \(S_{\beta}\). The variety \(S^ u_{\beta}\) is isomorphic to an affine space. - (3) For \(\beta\),\(\gamma\) in \(X_{\alpha}\), we have \(\beta \leq \gamma \Leftrightarrow cl S^ u_{\beta}\supseteq cl S^ u_{\gamma},\) where ''\(\leq ''\) is a partial order and cl \(S^ u_{\beta}\) (resp. cl \(S^ u_{\gamma})\) is the Zariski closure of \(S^ u_{\beta}\) (resp. \(S^ u_{\gamma})\) in \(F_{\mu}\). In particular, \(S^ u_{\alpha}\) is an open dense subvariety of \(Y_{\alpha}\). - N. Spaltenstein proved these results in the case of the full flag variety, i.e. \(\mu =(1,...,1).\) In the appendix, we study the homogeneous coordinate ring of the fixed point subvariety \(\Omega^ u\) of the Schubert variety \(\Omega\) in the Grassmann variety \(G_ d(V)\) \((=F_{(d,n-d)})\). If \(\dim \Omega^ u=\dim \Omega -1,\) we determine the defining idal of \(\Omega^ u\) and we prove that the homogeneous coordinate ring of \(\Omega^ u\) is normal and Cohen-Macaulay. Our results give an alternating proof to the fact that the minimal unipotent variety over a field K is normal and Cohen- Macaulay.