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Jim Humphreys
Jim Humphreys
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Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$$\phi:G\rightarrow G/B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit: This question arose from an attempt to fix a mistake in a book. The identity is indeed correct and I believe I have found the error elsewhere. Thanks Chuck and Jim!

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit: This question arose from an attempt to fix a mistake in a book. The identity is indeed correct and I believe I have found the error elsewhere. Thanks Chuck and Jim!

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit: This question arose from an attempt to fix a mistake in a book. The identity is indeed correct and I believe I have found the error elsewhere. Thanks Chuck and Jim!

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Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit2Edit: Nevermind, I think I've got it (myThis question arose from an attempt to fix a mistake after all)in a book. Thanks The identity is indeed correct and I believe I have found the error elsewhere. Thanks Chuck and Jim!

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit2: Nevermind, I think I've got it (my mistake after all). Thanks Chuck and Jim!

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit: This question arose from an attempt to fix a mistake in a book. The identity is indeed correct and I believe I have found the error elsewhere. Thanks Chuck and Jim!

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Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V^* $$$$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ (with $V^*$ the dual representation) asas $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

EditEdit2: I have decided to remove the restNevermind, as I'm afraidI think I've got it might be impolite. The question was aimed at trying to fix a(my mistake in a book, but I feel I ought to give the author more time to respondafter all). Thanks Chuck and Jim!

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V^* $$ (with $V^*$ the dual representation) as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit: I have decided to remove the rest, as I'm afraid it might be impolite. The question was aimed at trying to fix a mistake in a book, but I feel I ought to give the author more time to respond.

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\}. $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ as $G$ or $\mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit2: Nevermind, I think I've got it (my mistake after all). Thanks Chuck and Jim!

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Charles Matthews
Charles Matthews
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