Self-dual Grassmannian, Wronski map, and representations of ${\mathfrak{g}\mathfrak{l}}_N$, ${\mathfrak{s}\mathfrak{p}}_{2r}$, ${\mathfrak{s}\mathfrak{o}}_{2r+1}$

We define a ${\mathfrak{g}\mathfrak{l}}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{G}\mathrm{r}(N,d)$. The ${\mathfrak{g}\mathfrak{l}}_N$-stratification consists of strata ${\mathrm{\Omega }}_{\mathbf{\Lambda }}$ labeled by unordered sets $\mathbf{\Lambda }=({\lambda }^{(1)},\dots ,{\lambda }^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $({\otimes }_{i=1}^n{V}_{{\lambda }^{(i)}}{)}^{{\mathfrak{s}\mathfrak{l}}_N}\mathrm{\ne }0$. Here $V_{{\lambda }^{(i)}}$ is the irreducible ${\mathfrak{g}\mathfrak{l}}_N$-module with highest weight ${\lambda }^{(i)}$. We show that the closure of a stratum ${\mathrm{\Omega }}_{\mathbf{\Lambda }}$ is the union of the strata ${\mathrm{\Omega }}_{\mathbf{\Xi }}$, $\mathbf{\Xi }=({\xi }^{(1)},\dots ,{\xi }^{(m)})$, such that there is a partition $\{I_1,\dots ,I_m\}$ of $\{1,2,\dots ,n\}$ with $ {\rm {Hom}}{\mathfrak{gl}N} (V{\xi^{(i)}}, \otimes{j\in I_i}V_{\lambda^{(j)}}\big)\neq 0$for$i=1,\dots,m$\mathrm{.}The$\mathfrak{gl}N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $\mathrm{s}\mathrm{G}\mathrm{r}(N,d)\subset \mathrm{G}\mathrm{r}(N,d)$. Our main result is a similar ${\mathfrak{g}}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $\mathfrak {g}{2r+1}:=\mathfrak{sp}{2r}$if$N=2r+1$andoftheLiealgebra$\mathfrak g{2r}:=\mathfrak{so}_{2r+1}$if$N=2r$.