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Rafael Mrden
Rafael Mrden
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Here is a less direct, but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is a Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating the Kazhdan–Lusztig polynomialpolynomials and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category $\mathcal O_S$, Communications in Algebra, 37:1, 323-356.

Here is a less direct, but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is a Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating the Kazhdan–Lusztig polynomial and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category $\mathcal O_S$, Communications in Algebra, 37:1, 323-356.

Here is a less direct, but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is a Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating the Kazhdan–Lusztig polynomials and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category $\mathcal O_S$, Communications in Algebra, 37:1, 323-356.

Inlined link; PDF -> abs; link to more direct, but longer, proof; mild proofreading
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LSpice
LSpice
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Here is a less directless direct, but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is a Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating Kazhdan-Luszigthe Kazhdan–Lusztig polynomial and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category 𝒪 S Kostant Modules in Blocks of Category $\mathcal O_S$, Communications in Algebra, 37:1, 323-356, ( https://arxiv.org/pdf/math/0604336.pdf ).

Here is a less direct, but shorter proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating Kazhdan-Luszig polynomial and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category 𝒪 S , Communications in Algebra, 37:1, 323-356, ( https://arxiv.org/pdf/math/0604336.pdf )

Here is a less direct, but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is a Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating the Kazhdan–Lusztig polynomial and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category $\mathcal O_S$, Communications in Algebra, 37:1, 323-356.

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Rafael Mrden
Rafael Mrden
  • 1.4k
  • 12
  • 17

Here is a less direct, but shorter proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L_w$.

By the construction in [1, 4.2] we have a chain complex (called generalized BGG complex): $$0 \to M_{sts} \to M_{st} \oplus M_{ts} \to M_{t} \to L_t \to 0, $$ where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L_t$ is Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating Kazhdan-Luszig polynomial and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.

[1]. Brian D. Boe & Markus Hunziker (2009) Kostant Modules in Blocks of Category 𝒪 S , Communications in Algebra, 37:1, 323-356, ( https://arxiv.org/pdf/math/0604336.pdf )