# Coordinate ring of an equivariant embedding of a homogeneous projective variety

Lie algebra: Let $$G$$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $$B$$. Let $$P$$ be a parabolic subgroup containing $$B$$. Let $$\lambda$$ be a dominant integral weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $$A_\lambda(G/P)$$ is a sum of highest weight representations. Namely, \begin{align*} A_\lambda(G/P) = \oplus_{n \in \mathbb{N}} V(n\lambda). \end{align*}

Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $$G$$ be a simply connected analytic supergroup with $$\mathfrak{g} = Lie(G)$$ (a basic classical Lie superalgebra) and a Borel subsupergroup $$B$$. Let $$P$$ be a subsupergroup containing $$B$$. I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $$\lambda$$ is a typical weight.