Borel Weil and Borel Weil Bott — Examples

Given: The Axiom of Choice, $G=S{L}_2(\mathbb{C})$, its upper triangular Borel, the flag variety $X=G\mathrm{/}B\cong {\mathbb{P}}^1$, the fundamental weight ${\omega }_1$ and Weyl vector $\rho ={\omega }_1$, the simple root $\alpha$ with reflection $s$, and the bundle ${\mathcal{L}}_{-{\omega }_1}$.

In rank one $\alpha =2{\omega }_1$ and $\rho =\alpha \mathrm{/}2={\omega }_1$. Thus for $\lambda =-{\omega }_1$ one has $\lambda +\rho =0$, $s\cdot \lambda =s(0)-\rho =-{\omega }_1$, and $⟨\lambda ,{\alpha }^{\vee }⟩=-1$. The shifted weight is therefore not regular (Fundamental weights for a chosen simple root system, The Weyl vector, Rank-one cohomology shifts across a simple wall).

For $G=S{L}_2$ the unique simple root $\alpha$ has $P_{\alpha }=G$, so the fibre $F=P_{\alpha }\mathrm{/}B$ is all of $X=G\mathrm{/}B$, and under the fixed identification of the fibre with the two-affine projective line ${\mathbb{P}}^1$ the restriction ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^1}(⟨\lambda ,{\alpha }^{\vee }⟩)$; in particular ${\mathcal{L}}_{-{\omega }_1}\cong \mathcal{O}(-1)$ (A minimal-parabolic flag projection is a projective-line bundle, Flag line-bundle degree on a minimal-parabolic fiber, Two-affine projective line and its twists).

On ${\mathbb{P}}^1$ over $\mathbb{C}$ one has $H^q(\mathcal{O}(d))=0$ unless $q=0$ or $q=1$; $H^0(\mathcal{O}(d))=0$ for $d<0$, and $H^1(\mathcal{O}(d))$ is nonzero exactly for $d\le -2$. In particular at $d=-1$ both $H^0$ and $H^1$ vanish (Cohomology of O(d) on projective space).

The wall-crossing isomorphism: since the pairing is $-1$, $H^i(X,{\mathcal{L}}_{-{\omega }_1})\cong H^{i+1}(X,{\mathcal{L}}_{s\cdot (-{\omega }_1)})=H^{i+1}(X,{\mathcal{L}}_{-{\omega }_1})$ for all $i\ge 0$; and all cohomology vanishes by the singular-weight lemma (Rank-one cohomology shifts across a simple wall, Singular dot weights have zero line-bundle cohomology, The Borel-Weil-Bott theorem).

$X$ is smooth projective of dimension $1$, so $H^q(X,{\mathcal{L}}_{-{\omega }_1})=0$ for $q>1$ (Serre duality for locally free sheaves on a smooth projective variety).

Given: The Axiom of Choice, $G=S{L}_2(\mathbb{C})$ with upper triangular Borel, $X=G\mathrm{/}B\cong {\mathbb{P}}^1$, the fundamental weight ${\omega }_1$, $\rho ={\omega }_1$, the simple reflection $s$, and the bundles ${\mathcal{L}}_{m{\omega }_1}$ for $m\in \mathbb{Z}$.

Borel-Weil-Bott: if $\nu +\rho$ is singular all cohomology of ${\mathcal{L}}_{\nu }$ vanishes, and if $\nu +\rho$ is regular with Weyl element $v$, then $H^{\mathrm{\ell }(v)}(X,{\mathcal{L}}_{\nu })\cong L(v\cdot \nu {)}^{\ast }$ and all other cohomology vanishes (The Borel-Weil-Bott theorem).

In rank one $\rho ={\omega }_1$, so $m{\omega }_1+\rho =(m+1){\omega }_1$ is regular exactly for $m\mathrm{\ne }-1$; for $m\ge 0$ the Weyl element is $v=1$ with $1\cdot (m{\omega }_1)=m{\omega }_1$, while for $m\le -2$ the Weyl element is $s$ and $s(\lambda +\rho )-\rho$ with $\lambda =m{\omega }_1$ equals $-(m+2){\omega }_1$: indeed $s((m+1){\omega }_1)=-(m+1){\omega }_1$, so $s\cdot (m{\omega }_1)=-(m+1){\omega }_1-{\omega }_1=-(m+2){\omega }_1$ (Fundamental weights for a chosen simple root system, The Weyl vector, Length and longest Weyl-group element).

For $G=S{L}_2$ the unique simple root $\alpha$ has $P_{\alpha }=G$, so the minimal parabolic fibre is all of $X=G\mathrm{/}B$, and under the fixed identification of $X$ with ${\mathbb{P}}^1$ the bundle ${\mathcal{L}}_{m{\omega }_1}$ restricts to $\mathcal{O}(m)$ by the degree computation $⟨m{\omega }_1,{\alpha }^{\vee }⟩=m$; the singular boundary case $m=-1$ has all cohomology vanishing (A minimal-parabolic flag projection is a projective-line bundle, Minimal parabolic from one negative simple root, Flag line-bundle degree on a minimal-parabolic fiber, Two-affine projective line and its twists, The sl2 singular weight has no cohomology).

On ${\mathbb{P}}^1$ over $\mathbb{C}$: $H^0(\mathcal{O}(m))$ has dimension $m+1$ for $m\ge 0$ and vanishes for $m<0$; $H^1(\mathcal{O}(m))$ vanishes for $m\ge -1$ and has dimension $-m-1$ for $m\le -2$; all other cohomology vanishes (Cohomology of O(d) on projective space). The Weyl dimension formula for ${\mathfrak{s}\mathfrak{l}}_2$, whose single positive root pairs with $\rho$ by $1$, gives $\dim L(k{\omega }_1)=k+1$ for $k\ge 0$ (The Weyl dimension formula).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$ with standard Cartan and Borel, simple roots ${\alpha }_1,{\alpha }_2$, fundamental weights ${\omega }_1,{\omega }_2$, $\rho ={\omega }_1+{\omega }_2$, the simple reflection $s_1$, the longest element $w_0$ of the Weyl group, and $\lambda =s_1\cdot \rho =\rho -2{\alpha }_1=-3{\omega }_1+3{\omega }_2$.

In $A_2$ the simple roots are ${\alpha }_1,{\alpha }_2$, the positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$, and $\rho ={\omega }_1+{\omega }_2$ satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$; the simple reflection acts by $s_1(\nu )=\nu -⟨\nu ,{\alpha }_1^{\vee }⟩{\alpha }_1$, and the dot action is $w\cdot \nu =w(\nu +\rho )-\rho$ with $\mathrm{\ell }(s_1)=1$ (Classical complex matrix Lie algebras, Root systems of the classical complex Lie algebras, Fundamental weights for a chosen simple root system, The Weyl vector in fundamental coordinates, The Weyl vector, Root reflections and the Weyl group action).

Borel-Weil-Bott: if $\nu +\rho$ is regular with Weyl element $v$, then $H^{\mathrm{\ell }(v)}(X,{\mathcal{L}}_{\nu })\cong L(v\cdot \nu {)}^{\ast }$ and all other cohomology vanishes (The Borel-Weil-Bott theorem).

The Weyl dimension formula gives $\dim L(\rho )={\prod }_{\beta \in {\mathrm{\Phi }}^{+}}(2\rho ,\beta )\mathrm{/}(\rho ,\beta )=2^{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}=2^3=8$, since $(2\rho ,\beta )=2(\rho ,\beta )$ and $(\rho ,\beta )>0$ for the three positive roots; in particular $H^1\cong L(\rho {)}^{\ast }$ is eight-dimensional (The Weyl dimension formula, Root systems of the classical complex Lie algebras, The Weyl vector in fundamental coordinates).

Euler-characteristic form of Borel-Weil-Bott: ${\sum }_i(-1{)}^i\mathrm{c}\mathrm{h}{H}^i(X,{\mathcal{L}}_{\nu })=(-1{)}^{\mathrm{\ell }(v)}\mathrm{c}\mathrm{h}L(v\cdot \nu {)}^{\ast }$ when $\nu +\rho$ is regular with Weyl element $v$ (The Borel-Weil-Bott Euler character is a signed dual Weyl character).

For a dominant integral weight $\nu$ the dual $L(\nu {)}^{\ast }$ is irreducible of highest weight $-w_0\nu$; since $w_0$ maps ${\mathrm{\Phi }}^{+}$ onto ${\mathrm{\Phi }}^{-}$, it sends $2\rho ={\sum }_{\alpha >0}\alpha$ to $-2\rho$, so $w_0\rho =-\rho$ and $-w_0\rho =\rho$, hence $L(\rho {)}^{\ast }$ is irreducible of highest weight $\rho$ and therefore $L(\rho {)}^{\ast }\cong L(\rho )$ by the classification of finite-dimensional irreducibles, with $\mathrm{c}\mathrm{h}L(\rho {)}^{\ast }=\mathrm{c}\mathrm{h}L(\rho )$ (Highest weight of the dual representation, Length and longest Weyl-group element, Highest-weight classification).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$ with its standard Cartan, simple roots ${\alpha }_1,{\alpha }_2$, fundamental weights ${\omega }_1,{\omega }_2$, $\rho ={\omega }_1+{\omega }_2$, the flag variety $X=G\mathrm{/}B$ of dimension $N=3$, the longest element $w_0$ and the weight $\lambda =-3\rho$.

In $A_2$ the simple roots are ${\alpha }_1,{\alpha }_2$, the positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$, and the Weyl vector is $\rho ={\omega }_1+{\omega }_2$ with $⟨\rho ,{\alpha }_i^{\vee }⟩=1$; the longest element satisfies $w_0{\mathrm{\Phi }}^{+}={\mathrm{\Phi }}^{-}$, hence $w_0\rho =-\rho$, and $\mathrm{\ell }(w_0)=N=3=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, while the dot action is $w\cdot \nu =w(\nu +\rho )-\rho$ (Classical complex matrix Lie algebras, Root systems of the classical complex Lie algebras, Fundamental weights for a chosen simple root system, The Weyl vector in fundamental coordinates, The Weyl vector, Length and longest Weyl-group element).

Borel-Weil-Bott: for regular $\nu +\rho$ with Weyl element $v$, $H^{\mathrm{\ell }(v)}(X,{\mathcal{L}}_{\nu })\cong L(v\cdot \nu {)}^{\ast }$ and all other cohomology vanishes (The Borel-Weil-Bott theorem).

Compatibility with Serre duality: for regular $\lambda$ with Weyl element $w$ and $\mu =-\lambda -2\rho$, the Weyl element of $\mu$ is $w_{0}w$ and the Serre pairing identifies $H^{\mathrm{\ell }(w)}(X,{\mathcal{L}}_{\lambda }{)}^{\vee }$ with $H^{N-\mathrm{\ell }(w)}(X,{\mathcal{L}}_{\mu })$, matching $L(w\cdot \lambda {)}^{\ast }$ with $L(w\cdot \lambda )$ (Borel-Weil-Bott is compatible with Serre duality).

${\omega }_X=K_X\cong {\mathcal{L}}_{-2\rho }$ and Serre duality gives a perfect pairing $H^3(X,{\mathcal{L}}_{-3\rho })\times H^0(X,{\mathcal{L}}_{\rho })\to H^3(X,K_X)\cong \mathbb{C}$; moreover $\dim L(\rho )=8$: the Weyl dimension formula is $\dim L(\rho )={\prod }_{\beta \in {\mathrm{\Phi }}^{+}}(2\rho ,\beta )\mathrm{/}(\rho ,\beta )=2^{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}=2^3$, since $(2\rho ,\beta )=2(\rho ,\beta )$ and $(\rho ,\beta )>0$ for the three positive roots (Canonical weight of a flag variety, Serre duality for locally free sheaves on a smooth projective variety, The Weyl dimension formula, Root systems of the classical complex Lie algebras, The Weyl vector in fundamental coordinates).