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We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists ofstrata $\Omega_{\mathbf{\Lambda}}$ labeled by unordered sets$\mathbf{\Lambda}=(\lambda^{(1)},\dots,\lambda^{(n)})$ of nonzero partitionswith at most $N$ parts, satisfying a condition depending on $d$, and such that$(\otimes_{i=1}^n V_{\lambda^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here$V_{\lambda^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highestweight $\lambda^{(i)}$. We show that the closure of a stratum$\Omega_{\mathbf{\Lambda}}$ is the union of the strata $\Omega_{\mathbf\Xi}$,$\mathbf{\Xi}=(\xi^{(1)},\dots,\xi^{(m)})$, 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$i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agreeswith the Wronski map.  We introduce and study the new object: the self-dual Grassmannian$\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar$\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed byrepresentation theory of the Lie algebra $\mathfrak{g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrakg_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$.

Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$