Introduction to moment maps on flag varieties

Arun Ram
Department of Mathematics
University of Wisconsin
Madison, WI 53706 USA
ram@math.wisc.edu

Moment maps-distilled

Let $V$ be a vector space. Projective space is

$\mathbb{P}V=\left\{\text{ lines in }V\right\}=\left\{\right[v]\mid v\in V,v\ne 0\}$

where $\left[v\right]=\text{span}\left\{v\right\}$.

Let $⟨,⟩$ be an inner product on $V$. The map

$\begin{array}{cccc}\mu :& V& ⟶& V^{*}\\ & v& \mapsto & ⟨v,\cdot ⟩\end{array}\text{gives}\begin{array}{cccc}\mu :& \mathbb{P}V& ⟶& V^{*}\\ & \left[v\right]& \mapsto & \frac{⟨v,\cdot ⟩}{⟨v,v{⟩}^{1/2}}\end{array}$

Let $T$ be a torus,

$T=\{\left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)\mid x_i\ne 0\}$

acting on $V$ and assume $⟨,⟩$ is $T$-invariant

$⟨t{v}_1,t{v}_2⟩=⟨v_1,v_2⟩,\text{for}t\in T,v_1,v_2\in V.$

Let $𝔥=\mathrm{Lie}\left(T\right)$. Then the moment map on projective space is

$\begin{array}{cccc}\mu :& \mathbb{P}V& ⟶& {𝔥}^{*}\\ & \left[v\right]& \mapsto & {\mu }_v\end{array}\text{where}{\mu }_v={\displaystyle \frac{⟨hv,v⟩}{⟨v,v⟩}}\text{,}\text{for }h\in 𝔥\text{.}$

Weights and convexity

Let $\lambda \in {𝔥}^{*}$. A weight vector of weight $\lambda$ is

$v\in V$ such that $hv=\lambda \left(h\right)v,$ for all $h\in 𝔥$.

Then ${\mu }_v\left(h\right)=\frac{⟨\mathrm{hv},v⟩}{⟨v,v⟩}=\lambda \left(h\right)\text{,}$ so that

$v\stackrel{\mu }{\mapsto }\left(\text{weight of }v\right)\text{,}$ for weight vectors.

Let $v_{\lambda },v_{\gamma }$ be weight vectors of weights $\lambda ,\gamma$, respectively. If $v\in \text{span}\{v_{\lambda },v_{\gamma }\}$ then

$v=a{v}_{\lambda }+b{v}_{\gamma },\mathrm{and}\left|v_{\lambda }\right|=1,\left|v_{\gamma }\right|=1,\left|v\right|=1,$

is a reasonable assumption. Then

${\mu }_v\left(h\right)=\frac{⟨h(a{v}_{\lambda }+b{v}_{\gamma }),a{v}_{\lambda }+b{v}_{\gamma }⟩}{⟨v,v⟩}=a^2\lambda \left(h\right)+b^2\gamma \left(h\right)=(a^2\lambda +(1–a^2\left)\gamma \right)\left(h\right)$

since distinct weights are orthogonal. So

$\text{span}\{v_{\lambda },v_{\gamma }\}\stackrel{\mu }{\mapsto }$

More generally, given weight vectors $v_1,\dots ,v_k$ with weights ${\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(k\right)}$ then

$\mathrm{span}\{v_1,\dots ,v_k\}\stackrel{\mu }{\mapsto }\mathrm{Conv}({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(k\right)})=$

the convex hull of the points ${\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(k\right)}$.

Flag varieties

Let $B$ be a Borel subgroup of a Kac-Moody group $G$. The coset space

$G/B$ is the flag variety.

Let $\lambda \in P^{+}$, $V=L\left(\lambda \right)$ the simple $G$-module of highest weight $\lambda$. Let $T$ be the maximal torus of $G$. Then

${\mathrm{Res}}_T^G\left(L\right(\lambda \left)\right)=\underset{\mu \in P}{⨁}L(\lambda {)}_{\mu },\text{where }L(\lambda {)}_{\mu }=\{m\in L(\lambda )\mid hm=\mu (h)m,\text{for all }h\in 𝔥\}.$

The group $T$ acts on $G/B$ and the $T$-fixed points of $G/B$ are $\mathrm{wB}$, $w\in W$. Then $B\left[v^{+}\right]=\left[v^{+}\right]$ and the image of $G/B$ in $\mathbb{P}V$ is

$G\left[v^{+}\right]\subseteq \mathbb{P}V\text{and}G/B=G\left[v^{+}\right]\text{ if }\lambda \text{ is regular}.$

The moment map on $G/B$ associated to $\lambda$ is the restriction of $\mu :\mathbb{P}V\to {𝔥}^{*}$ to $G/B$,

$\begin{array}{ccccc}G/B& ⟶& G\left[v^{+}\right]& ⟶& {𝔥}^{*}\\ \mathrm{gB}& \mapsto & g\left[v^{+}\right]& \mapsto & {\mu }_{g\left[v^{+}\right]}\end{array}\text{and}\mathrm{wB}\mapsto w\left[v^{+}\right]\mapsto w\lambda$

since $w{v}^{+}$ has weight $w\lambda$.

The Schubert variety

$X_w=\overline{BwB}=\bigsqcup _{z\le w}BzB$

where $\le$ is the Bruhat-Chevalley order on $W$. So

$X_w\stackrel{\mu }{\mapsto }\mathrm{Conv}(z\in W\mid z\le w).$

The Demazure module is

$L(\lambda {)}_w=\text{span}\{X_w\left[v^{+}\right]\},$

a $B$-submodule, but not a $G$-submodule of $L\left(\lambda \right)$.

Loop groups

$\begin{array}{ccccc}G& =& G\left(ℂ\right(\left(t\right)\left)\right)& & \\ \cup |& & \cup |& & \\ K& =& G\left(ℂ\right[\left[t\right]\left]\right)& \stackrel{\Phi }{⟶}& G\left(ℂ\right)\\ \cup |& & \cup \mid & & \cup |\\ I& =& {\Phi }^{–1}\left(B\right)& ⟶& B\end{array}\text{and}{U}^{–}=\{\left(\begin{array}{ccc}1& & 0\\ & \ddots & \\ *& & 1\end{array}\right)\mid *\in ℂ\left(\right(t\left)\right)\}$

Then $G/I$ is controlled by

$G=\bigsqcup _{w\in \stackrel{\sim }{W}}IwI\text{and}\text{G =}\bigsqcup _{v\in \stackrel{\sim }{W}}{U}^{–}vI$,

where $\stackrel{\sim }{W}$ is the affine Weyl group. Points of $G/I$ in $IwI$ are indexed by

labeled paths $1\stackrel{c_1}{⟶}\cdots \stackrel{c_{\ell }}{⟶}w,c_i\in ℂ$.

There is a "folding map" on paths so that

$IwI\cap U^{–}vI=\left\{\text{paths }1\stackrel{c_1}{\to }\cdots \stackrel{c_{\ell }}{\to }w\text{ whose folding ends in }v\right\}.$

Let $p$ be a folded path without labels. Let

$S_p=\left\{\text{paths }1\stackrel{c_1}{\to }\cdots \stackrel{c_{\ell }}{\to }w\text{ whose folding ends in }p\right\}.$

Question: What is the moment map image of $\overline{S_p}$.

The closed slices $\overline{S_p}$ are generalizations of the MV-polytopes studied by Anderson, Kogan and Kamnitzer.