## 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 $\alpha \in {R}^{+}$, ${\lambda }^{\vee }\in {𝔥}_{ℤ}$, $c\in 𝔽$, $d\in {𝔽}^{×}$, which satisfy the relations in (put this in). This data comes from
${W}_{0}$, a $ℤ$-reflection group acting on
${𝔥}_{ℤ}$, a $ℤ$-lattice,
${R}^{+}$, an index set for the reflections in ${W}_{0}$.
Define
 ${𝔥}_{ℤ}^{*}={\mathrm{Hom}}_{ℤ}\left({𝔥}_{ℤ}\right)=\left\{\mu :{𝔥}_{ℤ}\to ℤ\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mu \phantom{\rule{.5em}{0ex}}\text{is}\phantom{\rule{.5em}{0ex}}ℤ\text{-linear}\right\}$.
The set ${𝔥}_{ℤ}^{*}$ is an index set for the simple $T$-modules ${ℂ}_{\mu }=\text{span}\left\{{v}_{\mu }\right\}$ corresponding to the group homomorphisms
 $\begin{array}{rccc}{X}^{\mu }:& T& ⟶& {ℂ}^{×}\\ & {h}_{{\lambda }^{\vee }}\left(d\right)& ⟼& {d}^{⟨\mu ,{\lambda }^{\vee }⟩},\end{array}\phantom{\rule{2em}{0ex}}\text{in other words}\phantom{\rule{2em}{0ex}}\begin{array}{c}t{v}_{\mu }={X}^{\mu }\left(t\right){v}_{\mu }\\ {h}_{{\lambda }^{\vee }}\left(d\right){v}_{\mu }={d}^{⟨\mu ,{\lambda }^{\vee }⟩}{v}_{\mu },\end{array}$
where $⟨\mu ,{\lambda }^{\vee }⟩=\mu \left({\lambda }^{\vee }\right)$. So
 $R\left(T\right)={K}_{T}\left(\mathrm{pt}\right)=\mathrm{span}\left\{{X}^{\mu }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mu \in {𝔥}_{ℤ}^{*}\right\}\phantom{\rule{2em}{0ex}}\text{with}\phantom{\rule{2em}{0ex}}{X}^{\mu }{X}^{\nu }={X}^{\mu +\nu }$.
The set ${𝔥}_{ℤ}^{*}$ is also an index set for the simple $B$-modules ${ℂ}_{\mu }=\text{span}\left\{{v}_{\mu }\right\}$ corresponding to the group homomorphisms
 $\begin{array}{rccc}{X}^{\mu }:& T& ⟶& {ℂ}^{×}\\ & {h}_{{\lambda }^{\vee }}\left(d\right)& ⟼& {d}^{⟨\mu ,{\lambda }^{\vee }⟩}\\ & {x}_{\alpha }\left(c\right)& ⟼& 1,\end{array}\phantom{\rule{2em}{0ex}}\text{so that}\phantom{\rule{2em}{0ex}}\begin{array}{c}b{v}_{\mu }={X}^{\mu }\left(b\right){v}_{\mu }\\ {h}_{{\lambda }^{\vee }}\left(d\right){v}_{\mu }={d}^{⟨\mu ,{\lambda }^{\vee }⟩}{v}_{\mu },\\ {x}_{\alpha }\left(c\right){v}_{\mu }={v}_{\mu }.\end{array}$
So
 $R\left(B\right)={K}_{B}\left(\mathrm{pt}\right)=R\left(T\right)={K}_{T}\left(\mathrm{pt}\right)=\mathrm{span}\left\{{X}^{\mu }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mu \in {𝔥}_{ℤ}^{*}\right\}\phantom{\rule{2em}{0ex}}\text{with}\phantom{\rule{2em}{0ex}}{X}^{\mu }{X}^{\nu }={X}^{\mu +\nu }$.

## The ${\mathrm{SL}}_{2}$-case

The group $G={\mathrm{SL}}_{2}$ is generated by
 ${x}_{{\alpha }_{1}}\left(c\right)=\left(\begin{array}{cc}1& c\\ 0& 1\end{array}\right),\phantom{\rule{2em}{0ex}}{x}_{-{\alpha }_{1}}\left(c\right)=\left(\begin{array}{cc}1& 0\\ c& 1\end{array}\right),\phantom{\rule{2em}{0ex}}{h}_{\ell {\alpha }_{1}^{\vee }}\left(d\right)=\left(\begin{array}{cc}{d}^{\ell }& 0\\ 0& -{d}^{-\ell }\end{array}\right),$
and $G\supseteq B\supseteq T$ is
 ${\mathrm{SL}}_{2}\supseteq \left\{\left(\begin{array}{cc}d& c\\ 0& {d}^{-1}\end{array}\right)\right\}\supseteq \left\{\left(\begin{array}{cc}d& 0\\ 0& {d}^{-1}\end{array}\right)\right\}$.
Here
 ${W}_{0}=\left\{1,{s}_{{\alpha }_{1}}\right\}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{𝔥}_{ℤ}=ℤ\mathrm{-span}\left\{{\alpha }_{1}^{\vee }\right\}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{𝔥}_{ℤ}^{*}=\left\{{\omega }_{1}\right\},\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}⟨{\omega }_{1},{\alpha }_{1}^{\vee }⟩.$
The irreducible $B$-modules are ${ℂ}_{k}=\mathrm{span}\left\{{v}_{k}\right\}$ corresponding to
 $\begin{array}{rccc}{X}^{\mu }:& B& ⟶& {ℂ}^{×}\\ & {h}_{\ell {\alpha }^{\vee }}\left(d\right)& ⟼& {d}^{k\ell }={d}^{⟨k{\omega }_{1},\ell {\alpha }_{1}^{\vee }}⟩\\ & \left(\begin{array}{cc}d& 0\\ 0& {d}^{-1}\end{array}\right)& ⟼& {d}^{k}\\ & \left(\begin{array}{cc}1& c\\ 0& 1\end{array}\right)& ⟼& 1,\end{array}\phantom{\rule{2em}{0ex}}\text{so that}\phantom{\rule{2em}{0ex}}\left(\begin{array}{cc}d& c\\ 0& {d}^{-1}\end{array}\right){v}_{\mu }={d}^{k}{v}_{\mu }.$
for $k\in ℤ\simeq {𝔥}_{ℤ}^{*}$. So
 $R\left(T\right)=\mathrm{span}\left\{{X}^{k}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}k\in ℤ\right\},\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{X}^{k}{X}^{\ell }={X}^{k+\ell },$
giving
 $R\left(T\right)=R\left(B\right)=ℂ\left[X,{X}^{-1}\right].$

## Hermann Weyl's theorem

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

 $\mu \in {𝔥}_{ℤ}^{*}\phantom{\rule{1em}{0ex}}\text{such that}\phantom{\rule{1em}{0ex}}⟨\mu ,{\alpha }^{\vee }⟩\in {ℤ}_{\ge 0},\phantom{\rule{.5em}{0ex}}\text{for}\phantom{\rule{.5em}{0ex}}\alpha \in {R}^{+}.$
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
 ${x}_{\alpha }\left(c\right){v}_{\mu }={v}_{\mu }\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{h}_{{\lambda }^{\vee }}\left(d\right){v}_{\mu }={d}^{⟨\mu ,{\lambda }^{\vee }⟩}{v}_{\mu }.$

Example. In the ${\mathrm{SL}}_{2}$ case, the $\mu \in {𝔥}_{ℤ}^{*}$ with $⟨\mu ,{\alpha }_{1}^{\vee }⟩\in {ℤ}_{\ge 0}$ are $\mu =k{\omega }_{1}$, for $k\in {ℤ}_{\ge 0}$. So the irreducible ${\mathrm{SL}}_{2}$-modules are

 $L\left(k\right)=L\left(k{\omega }_{1}\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{.5em}{0ex}}k\in {ℤ}_{\ge 0}$,
and $L\left(k\right)$ contains ${v}_{k}$ with
 $\left(\begin{array}{cc}1& c\\ 0& 1\end{array}\right){v}_{k}={v}_{k}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{cc}d& 0\\ 0& {d}^{-1}\end{array}\right){v}_{k}={d}^{k}{v}_{k}$.

## The $G$-modules ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)$

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

 $\begin{array}{ccc}G{×}_{B}{ℂ}_{\mu }& ⟶& G/B\\ \left(g,c{v}_{\mu }\right)& ⟼& gB\end{array}$     where   $G{×}_{B}{ℂ}_{\mu }=\frac{G×ℂ}{⟨\left(gb,c{v}_{\mu }\right)=\left(g,cb{v}_{\mu }\right)⟩}$
The vector space of global sections of ${ℒ}_{\mu }$ is
 ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)=\left\{G{×}_{B}{ℂ}_{\mu }\stackrel{s}{⟵}G/B\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}p\circ s={\mathrm{id}}_{G/B}\right\}$
Identify
 ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)\phantom{\rule{2em}{0ex}}⟷\phantom{\rule{2em}{0ex}}\left\{f:G\to ℂ\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}f\left(gb\right)=f\left(g\right){X}^{\mu }\left({b}^{-1}\right)\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}g\in G,\phantom{\rule{0.2em}{0ex}}b\in B\right\}$ (coind)
by
 $s\left(gB\right)=\left(g,f\left(g\right){v}_{\mu }\right)$ (stof)

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

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

 ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)\simeq \left\{\begin{array}{ll}L\left({w}_{0}\mu \right),& \text{if}\phantom{\rule{0.5em}{0ex}}⟨{w}_{0}\mu ,{\alpha }^{\vee }⟩\in {ℤ}_{\ge 0}\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\alpha \in {R}^{+},\\ 0,& \text{otherwise.}\end{array}$

The formulation of ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)$ in (coind) realises ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)$ 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 ${H}^{0}\left(G/B,{ℒ}_{\mu }\right)$ is nonzero, the constant function $f$ is its unique highest weight vector. The details of this identification are below in ????.

## Example ${\mathrm{SL}}_{2}$

 $G/B=\left\{B\right\}\bigsqcup \left\{{x}_{{\alpha }_{1}}\left(c\right){n}_{{\alpha }_{1}}^{-1}B\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}c\in ℂ\right\}=\left\{{x}_{-{\alpha }_{1}}\left(d\right)B\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}d\in ℂ\right\}\right\}\bigsqcup \left\{{n}_{{\alpha }_{1}}^{-1}B\right\}\simeq {ℙ}^{1}$,
(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
 $f\left(gb\right)=f\left(g\right){X}^{\mu }\left({b}^{-1}\right)\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}g\in G,\phantom{\rule{0.2em}{0ex}}b\in B,$
are determined by their values on coset representatives for $G/B$. Let
 ${g}_{1}\left(c\right)=f\left(\begin{array}{cc}{c}^{-1}& 1\\ 0& c\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{g}_{2}\left(d\right)=f\left(\begin{array}{cc}1& 0\\ d& 1\end{array}\right)$.
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 ${g}_{1}\left(c\right)={c}^{-k}{g}_{2}\left({c}^{-1}\right)$ is an element of $ℂ\left[c\right]$ exactly when $k\in {ℤ}_{\le 0}$ and $\ell \le -k$. Thus
 ${H}^{0}\left(G/B,{ℒ}_{k}\right)\simeq \left\{\begin{array}{ll}\text{span}\left\{1,c,{c}^{2},\dots ,{c}^{-k}\right\},& \text{if}\phantom{\rule{0.5em}{0ex}}k\in {ℤ}_{\le 0},\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}k\in {ℤ}_{>0}.\end{array}$

The proof of the Borel-Weil-Bott theorem for ${\mathrm{SL}}_{2}$: Suppose $f\in {H}^{0}\left(G/B,{ℒ}_{k}\right)$ is given by

 ${f}_{\ell }\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)={c}^{\ell }$.
Then
 $\left(\begin{array}{cc}d& 0\\ 0& {d}^{-1}\end{array}\right){f}_{\ell }={d}^{-k-2\ell }{f}_{\ell }\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right){f}_{\ell }=\sum _{j-0}^{\ell }\left(\begin{array}{c}\ell \\ j\end{array}\right){\left(-a\right)}^{\ell -j}{f}_{j}$. (act)
so that $ℂ{f}_{0}$ is the unique $B$-submodule in ${H}^{0}\left(G/B,{ℒ}_{k}\right)$. Since
 $\left(\begin{array}{cc}d& 0\\ 0& {d}^{-1}\end{array}\right){f}_{0}={d}^{-k-2\ell }{f}_{0},\phantom{\rule{2em}{0ex}}\text{it follows that}\phantom{\rule{2em}{0ex}}{H}^{0}\left(G/B,{ℒ}_{k}\right)\simeq \left\{\begin{array}{ll}L\left(-k\right),& \text{if}\phantom{\rule{0.5em}{0ex}}k\in {ℤ}_{\le 0},\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}k\in {ℤ}_{>0}.\end{array}$,
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
 $\begin{array}{rl}\left(\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right){f}_{\ell }\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)& ={f}_{\ell }\left(\left(\begin{array}{cc}1& -a\\ 0& 1\end{array}\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right)\right)={f}_{\ell }\left(\begin{array}{cc}c-a& -1\\ 1& 0\end{array}\right)={\left(c-a\right)}^{\ell }\\ & =\sum _{j-0}^{\ell }\left(\begin{array}{c}\ell \\ j\end{array}\right){\left(-a\right)}^{\ell -j}{c}^{j}=\left(\sum _{j-0}^{\ell }\left(\begin{array}{c}\ell \\ j\end{array}\right){\left(-a\right)}^{\ell -j}{f}_{j}\right)\left(\begin{array}{cc}c& -1\\ 1& 0\end{array}\right).\end{array}$

## 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.