The Borel-Weil-Bott theorem

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 5 May 2011

Simple $T$-modules and simple $B$-modules

In the context of Chevalley groups the data is

$\begin{array}{ccccc}G& \text{is generated by}& x_{\alpha }\left(c\right),& x_{-\alpha }\left(c\right),& h_{{\lambda }^{\vee }}\left(d\right)\\ \cup |& & & & \\ B& \text{is generated by}& x_{\alpha }\left(c\right),& & h_{{\lambda }^{\vee }}\left(d\right)\\ \cup |& & & & \\ T& \text{is generated by}& & & h_{{\lambda }^{\vee }}\left(d\right)\end{array}$

for $\alpha \in R^{+}$, ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}$, $c\in 𝔽$, $d\in {𝔽}^{\times }$, which satisfy the relations in (put this in). This data comes from

$W_0$, a $\mathbb{Z}$-reflection group acting on

${𝔥}_{\mathbb{Z}}$, a $\mathbb{Z}$-lattice,

$R^{+}$, an index set for the reflections in $W_0$.

Define

${𝔥}_{\mathbb{Z}}^{*}={\mathrm{Hom}}_{\mathbb{Z}}\left({𝔥}_{\mathbb{Z}}\right)=\{\mu :{𝔥}_{\mathbb{Z}}\to \mathbb{Z}|\mu \text{is}\mathbb{Z}\text{-linear}\}$.

The set ${𝔥}_{\mathbb{Z}}^{*}$ is an index set for the simple $T$-modules ${ℂ}_{\mu }=\text{span}\left\{v_{\mu }\right\}$ corresponding to the group homomorphisms

$\begin{array}{rccc}{X}^{\mu }:& T& ⟶& {ℂ}^{\times }\\ & h_{{\lambda }^{\vee }}\left(d\right)& ⟼& d^{⟨\mu ,{\lambda }^{\vee }⟩},\end{array}\text{in other words}\begin{array}{c}t{v}_{\mu }=X^{\mu }\left(t\right)v_{\mu }\\ h_{{\lambda }^{\vee }}\left(d\right)v_{\mu }=d^{⟨\mu ,{\lambda }^{\vee }⟩}{v}_{\mu },\end{array}$

where $⟨\mu ,{\lambda }^{\vee }⟩=\mu \left({\lambda }^{\vee }\right)$. So

$R\left(T\right)=K_T\left(\mathrm{pt}\right)=\mathrm{span}\left\{X^{\mu }\right|\mu \in {𝔥}_{\mathbb{Z}}^{*}\}\text{with}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu }$.

The set ${𝔥}_{\mathbb{Z}}^{*}$ is also an index set for the simple $B$-modules ${ℂ}_{\mu }=\text{span}\left\{v_{\mu }\right\}$ corresponding to the group homomorphisms

$\begin{array}{rccc}{X}^{\mu }:& T& ⟶& {ℂ}^{\times }\\ & h_{{\lambda }^{\vee }}\left(d\right)& ⟼& d^{⟨\mu ,{\lambda }^{\vee }⟩}\\ & x_{\alpha }\left(c\right)& ⟼& 1,\end{array}\text{so that}\begin{array}{c}b{v}_{\mu }=X^{\mu }\left(b\right)v_{\mu }\\ h_{{\lambda }^{\vee }}\left(d\right)v_{\mu }=d^{⟨\mu ,{\lambda }^{\vee }⟩}{v}_{\mu },\\ x_{\alpha }\left(c\right)v_{\mu }=v_{\mu }.\end{array}$

So

$R\left(B\right)=K_B\left(\mathrm{pt}\right)=R\left(T\right)=K_T\left(\mathrm{pt}\right)=\mathrm{span}\left\{X^{\mu }\right|\mu \in {𝔥}_{\mathbb{Z}}^{*}\}\text{with}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu }$.

The ${\mathrm{SL}}_2$-case

The group $G={\mathrm{SL}}_2$ is generated by

$x_{{\alpha }_1}\left(c\right)=\left(\begin{array}{cc}1& c\\ 0& 1\end{array}\right),x_{-{\alpha }_1}\left(c\right)=\left(\begin{array}{cc}1& 0\\ c& 1\end{array}\right),h_{\ell {\alpha }_1^{\vee }}\left(d\right)=\left(\begin{array}{cc}{d}^{\ell }& 0\\ 0& -d^{-\ell }\end{array}\right),$

and $G\supseteq B\supseteq T$ is

${\mathrm{SL}}_2\supseteq \left\{\right(\begin{array}{cc}d& c\\ 0& d^{-1}\end{array}\left)\right\}\supseteq \left\{\right(\begin{array}{cc}d& 0\\ 0& d^{-1}\end{array}\left)\right\}$.

Here

$W_0=\{1,s_{{\alpha }_1}\}\text{and}{𝔥}_{\mathbb{Z}}=\mathbb{Z}\mathrm{-span}\left\{{\alpha }_1^{\vee }\right\}\text{and}{𝔥}_{\mathbb{Z}}^{*}=\left\{{\omega }_1\right\},\text{where}⟨{\omega }_1,{\alpha }_1^{\vee }⟩.$

The irreducible $B$-modules are ${ℂ}_k=\mathrm{span}\left\{v_k\right\}$ corresponding to

$\begin{array}{rccc}{X}^{\mu }:& B& ⟶& {ℂ}^{\times }\\ & h_{\ell {\alpha }^{\vee }}\left(d\right)& ⟼& d^{k\ell }=d^{⟨k{\omega }_1,\ell {\alpha }_1^{\vee }}⟩\\ & \left(\begin{array}{cc}d& 0\\ 0& d^{-1}\end{array}\right)& ⟼& d^k\\ & \left(\begin{array}{cc}1& c\\ 0& 1\end{array}\right)& ⟼& 1,\end{array}\text{so that}\left(\begin{array}{cc}d& c\\ 0& d^{-1}\end{array}\right)v_{\mu }=d^k{v}_{\mu }.$

for $k\in \mathbb{Z}\simeq {𝔥}_{\mathbb{Z}}^{*}$. So

$R\left(T\right)=\mathrm{span}\left\{X^k\right|k\in \mathbb{Z}\},\text{with}{X}^k{X}^{\ell }=X^{k+\ell },$

giving

$R\left(T\right)=R\left(B\right)=ℂ[X,X^{-1}].$

Hermann Weyl's theorem

The irreducible integrable $G$-modules $L\left(\mu \right)$ are indexed by

$\mu \in {𝔥}_{\mathbb{Z}}^{*}\text{<em>such that</em>}⟨\mu ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\ge 0},\text{for}\alpha \in R^{+}.$

As a $B$-module, $L\left(\mu \right)$ has a unique simple submodule ${ℂ}_{\mu }$, and is characterised by this simple submodule, i.e. there is a unique (up to multiplication by a constant) vector $v_{\mu }\in L\left(\mu \right)$ such that

$x_{\alpha }\left(c\right)v_{\mu }=v_{\mu }\text{<em>and</em>}{h}_{{\lambda }^{\vee }}\left(d\right)v_{\mu }=d^{⟨\mu ,{\lambda }^{\vee }⟩}{v}_{\mu }.$

Example. In the ${\mathrm{SL}}_2$ case, the $\mu \in {𝔥}_{\mathbb{Z}}^{*}$ with $⟨\mu ,{\alpha }_1^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ are $\mu =k{\omega }_1$, for $k\in {\mathbb{Z}}_{\ge 0}$. So the irreducible ${\mathrm{SL}}_2$-modules are

$L\left(k\right)=L\left(k{\omega }_1\right),\text{for}k\in {\mathbb{Z}}_{\ge 0}$,

and $L\left(k\right)$ contains $v_k$ with

$\left(\begin{array}{cc}1& c\\ 0& 1\end{array}\right)v_k=v_k\text{and}\left(\begin{array}{cc}d& 0\\ 0& d^{-1}\end{array}\right)v_k=d^k{v}_k$.

The $G$-modules $H^0(G/B,{ℒ}_{\mu })$

Let $G$ be a group. Let $B$ be a subgroup of $G$ and let ${ℂ}_{\mu }=ℂ\text{-span}\left\{v_{\mu }\right\}$ be a one dimensional $B$-module. The line bundle ${ℒ}_{\mu }$ is

$\begin{array}{ccc}G{\times }_B{ℂ}_{\mu }& ⟶& G/B\\ (g,c{v}_{\mu })& ⟼& gB\end{array}$     where   $G{\times }_B{ℂ}_{\mu }={\displaystyle \frac{G\times ℂ}{⟨(gb,c{v}_{\mu })=(g,cb{v}_{\mu })⟩}}$

The vector space of global sections of ${ℒ}_{\mu }$ is

$H^0(G/B,{ℒ}_{\mu })=\{G{\times }_B{ℂ}_{\mu }\stackrel{s}{⟵}G/B|p\circ s={\mathrm{id}}_{G/B}\}$

Identify

$H^0(G/B,{ℒ}_{\mu })⟷\{f:G\to ℂ|f\left(gb\right)=f\left(g\right)X^{\mu }\left(b^{-1}\right)\text{for}g\in G,b\in B\}$

(coind)

by

$s\left(gB\right)=(g,f(g\left)v_{\mu }\right)$

(stof)

Note: $(g,f(g\left)v_{\mu }\right)=s\left(gB\right)=s\left(gbB\right)=(gb,f(gb\left)v_{\mu }\right)=(g,f(gb\left)b{v}_{\mu }\right)=(g,f(gb\left)X^{\mu }\right(b\left)v_{\mu }\right).$

The group $G$ acts on $H^0(G/B,{ℒ}_{\mu })$ by

$\left(gf\right)\left(h\right)=f\left(g^{-1}h\right).$

(Gact)

Note: $\left(gf\right)\left(hb\right)=f\left(g^{-1}hb\right)=f\left(g^{-1}h\right)X^{\mu }\left(b^{-1}\right)=\left(gf\right)\left(h\right)X^{\mu }\left(b^{-1}\right).$

(Borel-Weil-Bott) As $G$-modules

$$ H^0(G/B,{ℒ}_{\mu })\simeq \{\begin{array}{ll}L\left(w_0\mu \right),& \text{if}⟨w_0\mu ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}\text{for}\alpha \in R^{+},\\ 0,& \text{otherwise.}\end{array}$$

The formulation of $H^0(G/B,{ℒ}_{\mu })$ in (coind) realises $H^0(G/B,{ℒ}_{\mu })$ as the coinduction from $B$ to $G$ of a simple $B$-module. The Borel-Weil-Bott theorem makes the (remarkable) observation that coinduction from $B$ to $G$ of a simple $B$-module produces a simple $G$-module or 0.

The proof of the Borel-Weil-Bott theorem is quite simple: The left hand side is an integrable $G$-module which can be identified, by Weyl's theorem, by determining its highest weight vectors. It turns out that, if $H^0(G/B,{ℒ}_{\mu })$ is nonzero, the constant function $f$ is its unique highest weight vector. The details of this identification are below in ????.

Example ${\mathrm{SL}}_2$

$G/B=\left\{B\right\}\bigsqcup \left\{x_{{\alpha }_1}\right(c\left)n_{{\alpha }_1}^{-1}B\right|c\in ℂ\}=\{x_{-{\alpha }_1}\left(d\right)B|d\in ℂ\}\}\bigsqcup \{n_{{\alpha }_1}^{-1}B\}\simeq {\mathbb{P}}^1$,

(PUT A PICTURE HERE TO ILLUSTRATE THIS??? see The Borel-Weil-Bott theorem) since

$x_{{\alpha }_1}\left(c\right)n_{{\alpha }_1}^{-1}B=\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)B=\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}{c}^{-1}& 1\\ 0& c\end{array}\right)B=\left(\begin{array}{cc}1& 0\\ c^{-1}& 1\end{array}\right)B=x_{-{\alpha }_1}\left(c^{-1}\right)B$.

The functions $f:G\to ℂ$ such that

$f\left(gb\right)=f\left(g\right)X^{\mu }\left(b^{-1}\right)\text{for}g\in G,b\in B,$

are determined by their values on coset representatives for $G/B$. Let

$g_1\left(c\right)=f\left(\begin{array}{cc}{c}^{-1}& 1\\ 0& c\end{array}\right)\text{and}{g}_2\left(d\right)=f\left(\begin{array}{cc}1& 0\\ d& 1\end{array}\right)$.

Then (if we want polynomial functions) $g_1\in ℂ\left[c\right]$ and $g_2\in ℂ\left[d\right]$ and

$g_2\left(c^{-1}\right)=g_1\left(c\right)X^{\mu }\left(\begin{array}{cc}c& 1\\ 0& c^{-1}\end{array}\right)=g_1\left(c\right)c^k$.

Example: If $k=-7$, $g_1=c^5$ and $g_2=d^2$ then

$g_2\left(c^{-1}\right)=c^{-2}=c^5{c}^{-7}=g_1\left(c\right)c^k$.

Letting $g_2\left(d\right)=a_0+a_{1}d+\cdots +a_{\ell }{d}^{\ell }$, then $g_1\left(c\right)=c^{-\mathrm{k}}{g}_2\left(c^{-1}\right)$ is an element of $ℂ\left[c\right]$ exactly when $k\in {\mathbb{Z}}_{\le 0}$ and $\ell \le -k$. Thus

$H^0(G/B,{ℒ}_k)\simeq \{\begin{array}{ll}\text{span}\{1,c,c^2,\dots ,c^{-k}\},& \text{if}k\in {\mathbb{Z}}_{\le 0},\\ 0,& \text{if}k\in {\mathbb{Z}}_{>0}.\end{array}$

The proof of the Borel-Weil-Bott theorem for ${\mathrm{SL}}_2$: Suppose $f\in H^0(G/B,{ℒ}_k)$ is given by

$f_{\ell }\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)=c^{\ell }$.

Then

$\left(\begin{array}{cc}d& 0\\ 0& d^{-1}\end{array}\right)f_{\ell }=d^{-k-2\ell }{f}_{\ell }\text{and}\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)f_{\ell }=\sum _{j-0}^{\ell }\left(\begin{array}{c}\ell \\ j\end{array}\right){(-a)}^{\ell -j}{f}_j$.

(act)

so that $ℂf_0$ is the unique $B$-submodule in $H^0(G/B,{ℒ}_k)$. Since

$\left(\begin{array}{cc}d& 0\\ 0& d^{-1}\end{array}\right)f_0=d^{-k-2\ell }{f}_0,\text{it follows that}{H}^0(G/B,{ℒ}_k)\simeq \{\begin{array}{ll}L(-k),& \text{if}k\in {\mathbb{Z}}_{\le 0},\\ 0,& \text{if}k\in {\mathbb{Z}}_{>0}.\end{array}$,

which completes the proof.

The identities in (act) are justified by the computations

$\left(\left(\begin{array}{cc}d& 0\\ 0& d^{-1}\end{array}\right)f_{\ell }\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)=f_{\ell }\left(\left(\begin{array}{cc}{d}^{-1}& 0\\ 0& d\end{array}\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)\right)=f_{\ell }\left(\left(\begin{array}{cc}c{d}^{-2}& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}{d}^{-1}& 0\\ 0& d\end{array}\right)\right)={\left(c{d}^{-2}\right)}^{\ell }{d}^{-k}=d^{-k-2\ell }{f}_{\ell }\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)$

and

$\begin{array}{rl}\left(\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)f_{\ell }\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)& =f_{\ell }\left(\left(\begin{array}{cc}1& -a\\ 0& 1\end{array}\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)\right)=f_{\ell }\left(\begin{array}{cc}c-a& -1\\ 1& 0\end{array}\right)={(c-a)}^{\ell }\\ & =\sum _{j-0}^{\ell }\left(\begin{array}{c}\ell \\ j\end{array}\right){(-a)}^{\ell -j}{c}^j=\left(\sum _{j-0}^{\ell }\left(\begin{array}{c}\ell \\ j\end{array}\right){(-a)}^{\ell -j}{f}_j\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right).\end{array}$

Notes and References

These notes follow generally the sketch given in Segal??? [CSM LMS lecture notes]. The general type computation still needs to be put in.

References

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR??????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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