BorelBottWeilTheorem 2003-08-14 16:48:00 2004-03-20 13:17:58 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newtheorem{thm}{Theorem} \newtheorem{prop}{Proposition} \newcommand{\ab}[1]{{#1}_{\mathrm{ab}}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\Aut}{\mathrm{Aut}\,} \newcommand{\Aff}[2]{\mathrm{Aff}_{#1} #2} \newcommand{\aff}[2]{\mathfrak{aff}_{#1} #2} \newcommand{\mcB}{\mathcal{B}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bfrac}[2]{\left[\frac{#1}{#2}\\right]} \newcommand{\bkh}{\backslash} \newcommand{\Cyc}[2]{\mathcal{C}^{#1}_{#2}} \newcommand{\Cbar}[2]{\overline{\C{#1}{#2}}} \newcommand{\C}{\mathbb{C}} \newcommand{\CF}[2]{\ensuremath{\mathfrak{C}(#1,#2)}} \newcommand{\Cinf}{\EuScript{C}^{\infty}} \newcommand{\cmp}{cyclic mod $p$\xspace} \newcommand{\cp}{\mathrm{c.p.}} \newcommand{\CS}{\EuScript{CS}} \newcommand{\deck}{\EuScript{D}} \newcommand{\defl}[1]{\mathfrak{def}_{#1}} \newcommand{\Der}{\mathrm{Der}\,} \newcommand{\eH}{[X_H]-[Y_H]} \newcommand{\EL}{\mathcal{EL}} \newcommand{\End}{\mathrm{End}} \newcommand{\ES}[1]{\EuScript{#1}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\fr}[1]{\mathfrak{#1}} \newcommand{\Frat}{\mathrm{Frat}\,} \newcommand{\Gal}[1]{\Gamma(#1 |\Q)} \newcommand{\GL}[2]{\mathrm{GL}_{#1} #2} \newcommand{\g}{\mathfrak{g}} \newcommand{\gl}[2]{\mathfrak{gl}_{#1} #2} \newcommand{\GrR}[1]{a(#1 G)} \newcommand{\Gr}{\mathrm{Gr}\,} \newcommand{\mcH}{\mathcal{H}} \renewcommand{\H}{\mathbb{H}} \newcommand{\Hom}[2]{\mathrm{Hom}(#1,#2)} \newcommand{\id}{\mathrm{id}} \newcommand{\im}{\mathrm{im}} \newcommand{\ind}[2]{\mathrm{ind}^{#1}_{#2}} \newcommand{\indp}[2]{\mathfrak{ind}^{#1}_{#2}} \renewcommand{\inf}[1]{\mathfrak{inf}_{#1}} \newcommand{\inn}[1]{\langle #1\rangle} \renewcommand{\int}{\mathrm{int}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\K}{\mathcal{K}} \renewcommand{\ker}{\mathrm{ker}\,} \newcommand{\lap}[1]{\Delta_{#1}} \newcommand{\lapM}{\Delta_M} \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\lineq}{linearly equivalent\xspace} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mG}{m_G} \newcommand{\mK}{m_{\K}} \newcommand{\mindeg}[1]{\fr{md}(#1)} \newcommand{\N}{\mathbb{N}} \renewcommand{\O}{\mathcal{O}} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\Orb}{\mathrm{Orb}} \newcommand{\pad}{\hat{\Z}_p} \newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pderw}[1]{\frac{\partial}{\partial #1}} \newcommand{\pdersec}[2]{\frac{\partial^2 #1}{\partial {#2}^2}} \newcommand{\perm}[1]{\pi_{#1}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\rad}{\mathrm{rad}\,} \newcommand{\res}[2]{\mathrm{res}^{#1}_{#2}} \newcommand{\resp}[2]{\mathfrak{res}^{#1}_{#2}} \newcommand{\RG}{\EuScript{R}_G} \newcommand{\rk}{\mathrm{rk}\,} \newcommand{\V}[1]{\mathbf{#1}} \newcommand{\vp}{\varphi} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\SL}[2]{\mathrm{SL}_{#1} #2} \renewcommand{\sl}[2]{\fr{sl}_{#1} #2} \newcommand{\SO}[2]{\mathrm{SO}_{#1} #2} \newcommand{\Sp}[2]{\mathrm{Sp}_{#1} #2} \renewcommand{\sp}[2]{\fr{sp}_{#1} #2} \newcommand{\SU}[1]{\mathrm{SU}( #1)} \newcommand{\su}[1]{\fr{su}_{#1}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\Tg}{\mc{T}(\fr g)} \newcommand{\tom}{\tilde{\omega}} \newcommand{\ghtghp}{\fr g/\fr h\oplus(\fr g/\fr h^\perp)^*} \newcommand{\ghps}{(\fr g/\fr h^\perp)^*} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\Ug}{\mc{U}(\fr g)} \newcommand{\Uh}{\mc{U}(\fr h)} \renewcommand{\V}[1]{\mathbf{#1}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Zp}{\Z/p} \renewcommand{\L}{\mathcal{L}} \PMlinkescapeword{states} Let $G$ be a semisimple Lie group, and $\lambda$ be an integral weight for that group. $\lambda$ naturally defines a one-dimensional representation $C_\lambda$ of the Borel subgroup $B$ of $G$, by simply pulling back the representation on the maximal torus $T=B/U$ where $U$ is the unipotent radical of $G$. Since we can think of the projection map $\pi\colon G\to G/B$ as a \PMlinkname{principle $B$-bundle}{PrincipleBundle}, to each $C_\lambda$, we get an associated fiber bundle $\L_\lambda$ on $G/B$, which is obviously a line bundle. Identifying $\L_\lambda$ with its sheaf of holomorphic sections, we consider the sheaf cohomology groups $H^i(\L_\lambda)$. Realizing $\g$, the Lie algebra of $G$, as vector fields on $G/B$, we see that $\g$ acts on the sections of $\L_\lambda$ over any open set, and so we get an action on cohomology groups. This integrates to an action of $G$, which on $H^0(\L_\lambda)$ is simply the obvious action of the group. The Borel-Bott-Weil theorem states the following: if $(\lambda+\rho,\alpha)=0$ for any simple root $\alpha$ of $\g$, then $$H^i(\L_\lambda)=0$$ for all $i$, where $\rho$ is half the sum of all the positive roots. Otherwise, let $w\in W$, the Weyl group of $G$, be the unique element such that $w(\lambda+\rho)$ is dominant (i.e. $(w(\lambda+\rho),\alpha)>0$ for all simple roots $\alpha$). Then $$H^{\ell(w)}(\L_\lambda)\cong V_\lambda$$ where $V_\lambda$ is the unique irreducible representation of highest weight $\lambda$, and $H^i(\L_\lambda)=0$ for all other $i$. In particular, if $\lambda$ is already dominant, then $\Gamma(\L_\lambda)\cong V_\lambda$, and the higher cohomology of $\L_\lambda$ vanishes. If $\lambda$ is dominant, than $\L_\lambda$ is generated by global sections, and thus determines a map $$m_\lambda\colon G/B\to\mathbb{P}\left(\Gamma(\L_\lambda)\right).$$ This map is an obvious one, which takes the coset of $B$ to the highest weight vector $v_0$ of $V_\lambda$. This can be extended by equivariance since $B$ fixes $v_0$. This provides an alternate description of $\L_\lambda$. For example, consider $G=\mathrm{SL}_2\C$. $G/B$ is $\C P^1$, the Riemann sphere, and an integral weight is specified simply by an integer $n$, and $\rho=1$. The line bundle $\L_n$ is simply $\O(n)$, whose sections are the homogeneous polynomials of degree $n$. This gives us in one stroke the representation theory of $\mathrm{SL}_2\C$: $\Gamma(\O(1))$ is the standard representation, and $\Gamma(\O(n))$ is its $n$th symmetric power. We even have a unified decription of the action of the Lie algebra, derived from its realization as vector fields on $\C P^1$: if $H,X,Y$ are the standard generators of $\mathfrak{sl}_2\C$, then \begin{align*} H&=x\frac{d}{dx}-y\frac{d}{dy}\\ X&=x\frac{d}{dy}\\ Y&=y\frac{d}{dx}\\ \end{align*}