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This is not really an answer to your question -- more of a long comment, I suppose -- but there is a somewhat intuitive way to understand what these modules look like, representation-theoretically speaking. It's easier to describe what the duals to these modules look like, so let me do that. First, for any weight $\mu$, let $\mu^+$ denote the unique dominant element in the Weyl group orbit of $-\mu$ and let $V(\mu^+)$ denote the Weyl module for $G$ of highest weight $\mu^+$ (i.e., $V(\mu^+) = H^0(-\mu^+)^*$)H^0(-w_0\mu^+)^*$). Then the Joseph module $P(\mu)^*$ is just the $B$-submodule of $V(\mu^+)$ generated by any nonzero weight vector of weight $-\mu$. (Dually, this now describes the surjection $H^0(-\mu^+) H^0(-w_0\mu^+) \twoheadrightarrow P(\mu)$).

Now set $$\lbrace \mu_1, \ldots, \mu_r \rbrace := \lbrace s \mu : s \in W \textrm{ is a simple reflection and } s \mu < \mu \rbrace .$$ Let $I(\mu)$ be the $B$-submodule of $V(\mu^+)$ generated by $P(\mu_i)^*$, $1 \leq i \leq r$. Equivalently, $I(\mu)$ is the $B$-submodule of $V(\mu^+)$ generated by nonzero weight vectors of weights $-\mu_1, \ldots, -\mu_r$. Then $Q(\mu)^*$ fits into an exact sequence $$ 0 \to I(\mu) \to P(\mu)^* \to Q(\mu)^* \to 0 . $$ Remark that we can also describe $I(\mu)$ as the submodule of $V(\mu^+)$ generated by all Joseph modules properly contained in $P(\mu)^*$.

As for the evaluation map $\varepsilon$, it can be described as follows. Let $V \subseteq V(\mu^+)$ denote the highest weight subspace of weight $\mu^+$. Then $\varepsilon : P(\mu) \twoheadrightarrow k_{-\mu^+}$ is dual to the inclusion $V \hookrightarrow P(\mu)^*$ of the highest weight subspace. (Remark also that $V = P(-\mu^+)^*$).
show/hide this revision's text 1

This is not really an answer to your question -- more of a long comment, I suppose -- but there is a somewhat intuitive way to understand what these modules look like, representation-theoretically speaking. It's easier to describe what the duals to these modules look like, so let me do that. First, for any weight $\mu$, let $\mu^+$ denote the unique dominant element in the Weyl group orbit of $-\mu$ and let $V(\mu^+)$ denote the Weyl module for $G$ of highest weight $\mu^+$ (i.e., $V(\mu^+) = H^0(-\mu^+)^*$). Then the Joseph module $P(\mu)^*$ is just the $B$-submodule of $V(\mu^+)$ generated by any nonzero weight vector of weight $-\mu$. (Dually, this now describes the surjection $H^0(-\mu^+) \twoheadrightarrow P(\mu)$).

Now set $$\lbrace \mu_1, \ldots, \mu_r \rbrace := \lbrace s \mu : s \in W \textrm{ is a simple reflection and } s \mu < \mu \rbrace .$$ Let $I(\mu)$ be the $B$-submodule of $V(\mu^+)$ generated by $P(\mu_i)^*$, $1 \leq i \leq r$. Equivalently, $I(\mu)$ is the $B$-submodule of $V(\mu^+)$ generated by nonzero weight vectors of weights $-\mu_1, \ldots, -\mu_r$. Then $Q(\mu)^*$ fits into an exact sequence $$ 0 \to I(\mu) \to P(\mu)^* \to Q(\mu)^* \to 0 . $$ Remark that we can also describe $I(\mu)$ as the submodule of $V(\mu^+)$ generated by all Joseph modules properly contained in $P(\mu)^*$.

As for the evaluation map $\varepsilon$, it can be described as follows. Let $V \subseteq V(\mu^+)$ denote the highest weight subspace of weight $\mu^+$. Then $\varepsilon : P(\mu) \twoheadrightarrow k_{-\mu^+}$ is dual to the inclusion $V \hookrightarrow P(\mu)^*$ of the highest weight subspace. (Remark also that $V = P(-\mu^+)^*$).