# Ch.8 Continued (Monday, October 18) :::{.remark} Today: looking at more examples of Schubert varieties in detail, e.g. $\Sp_{2n}$. One can take $G' \da \GL_{2n}$ and define involutions $G' \mapsvia{\Theta} G'$. One example is $g\mapsto g^{-t}$, whose fixed points are $\Orth_{2n}$, and it's easy to check that this is an involution: \[ (\Theta')^2(g) = \Theta' ({}^t g\inv) = {}^t( {}^t g\inv)\inv = {}^t ({}^t g) = g .\] For $\Sp_{2n}$, taking \[ \theta(g) = -J^t g J \] where $J$ is the matrix \[ \begin{bmatrix} & & & & & & 1 \\ & & & & & \cdots & \\ & & & & 1 & & \\ & & & -+- & & & \\ & & -1 & & & & \\ & \cdots & & & & & \\ -1 & & & & & & \end{bmatrix} .\] We can check that this is an involution: \[ \Theta^2(g) &= \Theta(-J^t g J) \\ &= -J^t(-J^t g J)\inv J \\ &= J(Jg^{-t} J)\inv J \\ &= JJ g JJ \\ &= g .\] ::: :::{.definition title="?"} $(G')^\Theta \da \ts{g'\in G'\st \Theta(g') = g'}$ are the fixed points under the involution $\Theta$. ::: :::{.proposition title="?"} One can write \[ (G')^\Theta = \ts{g\in G' \st \omega(g' x, g' y) = \omega(x, y)} .\] for $\omega$ the associated bilinear form $\omega(x, y) = \ltranspose{x} J y$. Note that $g'x, g'y$ should be column vectors here. ::: :::{.proof title="sketch"} Write the RHS set as $\ts{g\in G \st \ltranspose{g'} J g' = J}$. Then check that if $\theta(g) = g$ for some $g\in G'$, \[ \omega(gx, gy) &= \ltranspose{(gx)} J (gy) \\ &= \ltranspose{(x\inv )} g J gy \\ &= \ltranspose{(x\inv )} g J \Theta(g) y \\ &= \ltranspose{(x\inv )} g J (-J \ltranspose{g\inv} J ) y \\ &= \ltranspose{x} Jy .\] So these two act the same on all elements $x, y$, and thus have the same matrix, yielding $\subseteq$. For the reverse containment, if $\omega(gx, gy) = \omega(x, y)$, then \[ \ltranspose{g} J g &= J \\ \implies Jg &= \ltranspose{g\inv} J \\ \implies \Theta(g) &= -J \ltranspose{g\inv } J \\ &= -JJ g \\ &= g .\] ::: :::{.remark} We can realize $\Sp_{2n}$ as $(G')^{\Theta}$. ::: :::{.fact} How do we get a Borel? It is a general fact that these can be obtained by intersecting with Borels in the ambient group, so take $B' \intersect \Sp_{2n}$ for $B' \subseteq G'$ upper triangular. Then $B'$ is $\Theta\dash$stable: ![](figures/2021-10-18_14-25-12.png) ::: :::{.remark} Let $G = (G')^\Theta$, then $G\actson G'/B'$ with finitely many orbits. So we get closure relations: ![](figures/2021-10-18_14-27-13.png) One can also fix $T' \subseteq G'$ as a maximal torus of diagonal matrices, and this is also $\Theta\dash$stable. Then $T' \intersect G$ is of the following form: \[ \begin{bmatrix} t_1 & & & & & \\ & \ddots & & & & \\ & & t_n & & & \\ & & & t_n\inv & & \\ & & & & \ddots & \\ & & & & & t_1\inv \end{bmatrix} \cong (\CC\units)\cartpower{n} .\] Writing $G'/B' = \ts{ F^\bullet \text{ complete flags}} = G' \cdot \CC^\bullet$ for the standard flag $\CC^\bullet \da (0 \subseteq \CC^1 \subseteq \CC^2 \subseteq \cdots \subseteq \CC^{2n})$. We can write this set as $\ts{F^\bullet \st (F^k)^\perp = F^{2n+1-k}}$, where $(F^k)^\perp \da \ts{ x\in \CC^{2n} \st \omega(x, y) = 0 \,\,\forall y\in F^k}$. Generally the former will be flags $\CC^{2n} =F^{2n} \to F^{2n-1} \to \cdots \to F^1\to 0$, and this says we can describe this more compactly as flags $\CC^{2n} \to F^n \to F^{n-1} \to \cdots \to F^1\to 0$ where the $F^k$ are isotropic, by inserting their orthogonal complements into the chain appropriately. ::: :::{.question} What are the Schubert varieties in $G/B$? ::: :::{.answer} For $w'\in W' = S_{2n}$, the Weyl group for $G' = \GL_{2n}$ and writing $X' = G'/B'$, the Schubert varieties are exactly $X_{w'}' \intersect G/B$. This is empty if there exists a $k$ with ???, and is $X_W$ otherwise where $W \subseteq W'$ is $\ts{ (w_1,\cdots, w_n) \st w_1 + w_{2n} = 2n+1 }$. For example, take $\sigma = (1,3,2,4) \in W$, then $X_{W'}' = (\CC^4 \to \CC^3 \to F^2 \to \CC^1)$ and $X_W = (\CC^4 \to \CC^3 \to F_2 \to \CC^1)$, where $F^2$ is a Lagrangian subspace of $\CC^4$. ::: :::{.remark} This produces a large collection of normal varieties: start with flags and add conditions. :::