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.


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