*It is a copy from MSE the place the query didn’t entice a lot consideration.*

I am working over $mathbb{C}$ right here. Let $G=mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic comparable to the $1$st node within the Kind $B_n$ Dynkin diagram, following Bourbaki notation- I imply the endpoint which is adjoining to a doubled edge. (It is a *minuscule* node.) Then $G/P$ needs to be what known as the *(maximal) orthogonal Grassmannian* $mathrm{OG}(n,2n+1)$: these are the isotropic subspaces (with respect to a nondegenerate symmetric bilinear kind) of maximal dimension in $mathbb{C}^{2n+1}$.

The Borel-Weil theorem says that the $m$th homogeneous element of the coordinate ring of $G/P=mathrm{OG}(n,2n+1)$ needs to be isomorphic to the irreducible illustration $V^{momega_1}$, the place $omega_1$ is the corresponding basic weight. This could maintain at the least at say the extent of representations of the Lie algebra $mathfrak{g}=mathfrak{so}(2n+1)$. Truly, it may be that we get the *contragredient* illustration $(V^{momega_1})^*$ this fashion (as a result of we’re performing on features). However in Kind B negation belongs to the Weyl group so I feel we must always have $(V^{lambda})^*simeq V^{lambda}$ for any irreducible illustration.

So particularly, the linear a part of the coordinate ring of $mathrm{OG}(n,2n+1)$ is the $mathfrak{g}$ illustration $V^{omega_1}$. Now, the linear a part of this coordinate ring additionally looks as if a wonderfully good $G$ illustration to me. And I’d guess that it’s the irreducible illustration $V^{omega_1}$. However that may’t be proper: $V^{omega_1}$ *shouldn’t be realizable* as an $mathrm{SO}(2n+1)$ illustration, due to the truth that $mathrm{SO}(2n+1)$ isn’t merely linked; to get this illustration we’re presupposed to must take the merely linked double cowl $widetilde{mathrm{SO}}(2n+1)$, which can also be referred to as the *spin group* $mathrm{Spin}(2n+1)$. (This illustration $V^{omega_1}$ is usually referred to as the *spin illustration*.)

**Query**: the place am I getting confused right here? What’s (the linear a part of) the coordinate ring of the orthogonal Grassmannian as a illustration of the particular orthogonal group?