Fix an integer $k$. Let $X=G/P$ be a complex rational homogeneous variety. I assume here $G$ is a simply connected semi simple complex Lie group and $P=P_k$ is a maximal parabolic subgroup defined by dropping the $k$-th simple root (Assume we have chosen and fixed a Borel subgroup to avoid ambiguity, and we use Bourbaki convention about the order of simple roots).

I want to compute the cohomology of the homogeneous bundle $T_X\otimes L^{-\lambda_k}$ over $X$. Here, $T_X$ is the tangent bundle while $L^{-\lambda_k}$ is the line bundle corresponding to the 1-dimensional $P$-representation with character induced by $\lambda_k$. For example, if $X$ is a Grassmannian variety embedded into a projective space using Plücker embedding, then $L^{\lambda_k}$ is just $\mathcal O(-1)$.

My strategy is to use Borel Weil Bott theorem. But $T_X$ is not irreducible in general (i.e. The $P$-representation $\mathfrak{g/p}$ is not irreducible), so is not $T_X\otimes L^{-\lambda_k}$. Hence, I have to find a $P$-representation filtration of $mathfrak{g/p}$, say $0\subset s_1\subset s_2\subset\ldots\subset s_r=\mathfrak{g/p}$ with quotients $T_i$ irreducible $P$-representations. We shall use the same notation for the homogeneous vector bundles corresponding to $s_i, T_i$. My plan is: STEP I. compute the cohomology of $T_i\otimes L^{\lambda_k}$ using Borel Weil Bott theorem for all $i$ since they are irreducible. STEP II. Using the filtration and step I to get the cohomology of $T_X\otimes L^{\lambda_k}$.

STEP I is easily done by prudent computation, and STEP II can be done in most cases. But I meet some difficulties in step II for some special cases: I need to write the connection morphism in these cases. I will use the following example to demonstrate my dilemma here.

From now on, let $G$ be the simply connected Lie group of type $B_l$ and $P=P_2$ be a maximal parabolic subgroup defined by dropping the second simple root. The the filtration of the tangent bundle of $X=G/P$ is $0\subset s_1\subset s_2=T_X$, and hence we have a short exact sequence $$ 0\to s_1\to T_X\to s_2/s_1\to 0$$ Tensoring $L^{-\lambda_2}$, we get another short exact sequence which we simply write as $$0\to s_1(-1)\to T_X(-1)\to s_2/s_1(-1)\to 0$$from which we have a long exact sequence. According to my computation using Borel Weil Bott theorem $H^q(s_1(-1))=\mathbb C, q=1; 0, q\ne 1$ $H^q(s_2/s_1(-1))=\mathbb C, q=0; 0, q\ne 0$.

Hence, my long exact sequence looks like $$ 0\to H^0(T_X(-1))\to \mathbb C \stackrel{\delta}{\to} \mathbb C \to H^1(T_X)\to 0$$ Therefore, to determine what I want: $H^q(T_X(-1))$ I need to write down $\delta$ explicitly. Maybe there are other method that can determine the cohomology of $T_X(-1)$ directly without $\delta$, so any idea is welcome and appreciated!

Fix an integer $k$. Let $X=G/P$ be a complex rational homogeneous variety. I assume here $G$ is a simply connected semi simple complex Lie group and $P=P_k$ is a maximal parabolic subgroup defined by dropping the $k$-th simple root (Assume we have chosen and fixed a Borel subgroup to avoid ambiguity, and we use Bourbaki convention about the order of simple roots).

I want to compute the cohomology of the homogeneous bundle $T_X\otimes L^{-\lambda_k}$ over $X$. Here, $T_X$ is the tangent bundle while $L^{-\lambda_k}$ is the line bundle corresponding to the 1-dimensional $P$-representation with character induced by $\lambda_k$. For example, if $X$ is a Grassmannian variety embedded into a projective space using Plücker embedding, then $L^{-\lambda_k}$ is just $\mathcal O(-1)$.

My strategy is to use Borel Weil Bott theorem. But $T_X$ is not irreducible in general (i.e. The $P$-representation $\mathfrak{g/p}$ is not irreducible), so is not $T_X\otimes L^{-\lambda_k}$. Hence, I have to find a $P$-representation filtration of $\mathfrak{g/p}$, say $0\subset s_1\subset s_2\subset\ldots\subset s_r=\mathfrak{g/p}$ with quotients $T_i$ irreducible $P$-representations. We shall use the same notation for the homogeneous vector bundles corresponding to $s_i, T_i$. My plan is: STEP I. compute the cohomology of $T_i\otimes L^{-\lambda_k}$ using Borel Weil Bott theorem for all $i$ since they are irreducible. STEP II. Using the filtration and step I to get the cohomology of $T_X\otimes L^{-\lambda_k}$.

STEP I is easily done by prudent computation, and STEP II can be done in most cases. But I meet some difficulties in step II for some special cases: I need to write the connection morphism down explicitly in these cases. I will use the following example to demonstrate my dilemma here.

From now on, let $G$ be the simply connected Lie group of type $B_l$ and $P=P_2$ be a maximal parabolic subgroup defined by dropping the second simple root. The the filtration of the tangent bundle of $X=G/P$ is $0\subset s_1\subset s_2=T_X$, and hence we have a short exact sequence $$ 0\to s_1\to T_X\to s_2/s_1\to 0$$ Tensoring $L^{-\lambda_2}$, we get another short exact sequence which we simply write as $$0\to s_1(-1)\to T_X(-1)\to s_2/s_1(-1)\to 0$$from which we have a long exact sequence. According to my computation using Borel Weil Bott theorem $$H^q(s_1(-1))=\mathbb C, q=1; 0, q\ne 1$$ $$H^q(s_2/s_1(-1))=\mathbb C, q=0; 0, q\ne 0$$

Hence, my long exact sequence looks like $$ 0\to H^0(T_X(-1))\to \mathbb C \stackrel{\delta}{\to} \mathbb C \to H^1(T_X)\to 0$$ Therefore, to determine what I want: $H^q(T_X(-1))$, I need to write down $\delta$ explicitly. Maybe there are other methods that can determine the cohomology of $T_X(-1)$ directly without $\delta$, so any idea is welcome and appreciated!