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I recently ran into a 30+ years old literature by AndersonAndersen and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson[Andersen-Jantzen] saysays that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0.$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\ 0 & \text{otherwise} \end{cases} $$$$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i, \\ 0 & \text{otherwise.} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

I recently ran into a 30+ years old literature by Anderson and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson-Jantzen] say that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\ 0 & \text{otherwise} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

I recently ran into a 30+ years old literature by Andersen and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Andersen-Jantzen] says that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0.$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i, \\ 0 & \text{otherwise.} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

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I recently ran into a 30+ years old literature by Anderson and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic GroupsCohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson-Jantzen] say that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\ 0 & \text{otherwise} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

I recently ran into a 30+ years old literature by Anderson and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson-Jantzen] say that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\ 0 & \text{otherwise} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

I recently ran into a 30+ years old literature by Anderson and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson-Jantzen] say that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\ 0 & \text{otherwise} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.

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cohomology of flag variety

I recently ran into a 30+ years old literature by Anderson and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the setting:

$G$ complex simple Lie group, $B = HN$ a Borel subgroup corresponding to the positive roots of $\mathfrak{g}$, and $\mathfrak{n} = Lie(N)$. Theorem 3.6(a) of [Anderson-Jantzen] say that for all $i > 0$ and $n \geq 0$,

$$H^i(G/B, S^n(\mathfrak{n}^*)) = 0$$

I tried an example for $G = SL(3)$ and $n = 2$, where $\alpha_1 = (1,-1,0)$ and $\alpha_2 = (0,1,-1)$, $\alpha_1 + \alpha_2 = (1,0,-1)$ are the positive roots in $\mathfrak{h}^*$. The weights of $S^2(\mathfrak{n}^*)$ are given by $$S^2(\mathfrak{n}^*) = \mathbb{C}_{(-2,2,0)} \oplus \mathbb{C}_{(0,-2,2)} \oplus \mathbb{C}_{(-2,0,2)} \oplus \mathbb{C}_{(-1,0,1)} \oplus \mathbb{C}_{(-2,1,1)} \oplus \mathbb{C}_{(-1,-1,2)}.$$

Then I tried to apply Bott-Borel-Weil: $$H^i(G/B, \mathbb{C}_{\lambda}^*) = \begin{cases} V_{\mu}^* &\text{if }w(\lambda + \rho) = \mu + \rho\ \text{for some}\ w \in W\ \text{with}\ l(w) = i \\ 0 & \text{otherwise} \end{cases} $$ Here $\mu$ must be a dominant weight of $G$.

For instance, if $\mathbb{C}_{(-2,2,0)} = \mathbb{C}_{(2,-2,0)}^*$, then $$(2,-2,0)+\rho = (2,-2,0) + (1,0,-1) = (3,-2,-1).$$

Let $w \in W$ be the transposition of the last two entries with $l(w) = 1$, then $w((2,-2,0)+\rho) = (3,-1,-2) = (2,-1,-1) + \rho$.

Does it imply that $H^1(G/B,S^2(\mathfrak{n}^*))$ contains a copy of $V_{(2,-1,-1)}^* = V_{(1,1,-2)}$, which is non-trivial? Any insight would be appreciated.