Introduction to
Buildings and Combinatorial Representation Theory
American Institute of Mathematics (AIM)
March 26, 2007

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

Last update: 21 August 2014

Weyl characters

Your favourite group $G^{\vee }$ (probably ${SL}_3\left(ℂ\right)\text{)}$ corresponds to $$\begin{array}{ccc}\begin{array}{c}W=\left\{\text{chambers}\right\}\\ \text{and}\\ P^{\vee }=\left\{\text{dots}\right\}\end{array}& & \begin{array}{c} \end{array}\end{array}$$ The irreducible $G^{\vee }\text{-modules}$ $L\left({\lambda }^{\vee }\right)$ are indexed by ${\lambda }^{\vee }\in {\left(P^{\vee }\right)}^{+}$ and $$\text{char}\left(L\left({\lambda }^{\vee }\right)\right)=\sum _{{\mu }^{\vee }\in P^{\vee }}\text{Card}\left(B{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}\right)x^{{\mu }^{\vee }},$$ where $$B{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}=\left\{\text{Littelmann paths of type}\hspace{0.17em}{\lambda }^{\vee }\hspace{0.17em}\text{and end}\hspace{0.17em}{\mu }^{\vee }\right\}\text{.}$$ If $$G=G\left(ℂ\left(\left(t\right)\right)\right),K=G\left(ℂ\left[\left[t\right]\right]\right),\text{and}{U}^{-}=\left\{\left(\begin{array}{ccc}1& & 0\\ & \ddots & \\ *& & 1\end{array}\right)\right\}$$ then $G/K$ is the loop Grassmanian and $$G=\underset{{\lambda }^{\vee }\in {\left(P^{\vee }\right)}^{+}}{⨆}K{t}_{{\lambda }^{\vee }}K\text{and}G=\underset{{\mu }^{\vee }\in P^{\vee }}{⨆}{U}^{-}{t}_{{\mu }^{\vee }}K\text{.}$$ The MV cycles of type ${\lambda }^{\vee }$ and weight ${\mu }^{\vee }$ are the elements of $$MV{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}=\left\{\text{irreducible components of}\hspace{0.17em}\overline{K{t}_{{\lambda }^{\vee }}K\cap U^{-}{t}_{{\mu }^{\vee }}K}\right\},$$ and $$\text{char}\left(L\left({\lambda }^{\vee }\right)\right)=\sum _{{\mu }^{\vee }}\text{Card}\left(MV{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}\right)x^{{\mu }^{\vee }}\text{.}$$

Hecke algebras

The spherical and affine Hecke algebras are $${\stackrel{\sim }{H}}_{\text{sph}}=C(K\backslash G/K)\text{and}\stackrel{\sim }{H}=C(I\backslash G/I),$$ where $$\begin{array}{ccc}G& =& G\left(ℂ\left(\left(t\right)\right)\right)\\ \cup |& & \cup |\\ K& =& G\left(ℂ\left[\left[t\right]\right]\right)& \stackrel{\mathrm{\Phi }}{⟶}& G\left(ℂ\right)\\ \cup |& & \cup |& & \cup |\\ I& =& {\mathrm{\Phi }}^{-1}\left(B\right)& ⟶& B,\end{array}\text{where}B=\left\{\left(\begin{array}{ccc}*& & *\\ & \ddots & \\ 0& & *\end{array}\right)\right\}\text{.}$$ The Satake map is $$\begin{array}{ccc}ℂ{\left[X\right]}^W=Z\left(\stackrel{\sim }{H}\right)& \stackrel{\sim }{⟶}& Z\left(\stackrel{\sim }{H}\right)1_0=1_0\stackrel{\sim }{H}{1}_0={\stackrel{\sim }{H}}_{\text{sph}}\\ f& ⟼& f{1}_0\\ P_{{\lambda }^{\vee }}\phantom{\hspace{0.17em}=Z\left(\stackrel{\sim }{H}\right)}& \leftarrow & 1_0{X}^{{\lambda }^{\vee }}& 1_0={\chi }_{K{t}_{{\lambda }^{\vee }}K}\text{"obvious" basis}\end{array}$$ and $P_{{\lambda }^{\vee }}$ are the Hall-Littlewood polynomials. $$P_{{\lambda }^{\vee }}=\sum _{{\mu }^{\vee }\in P^{\vee }}{\text{Card}}_q\left(𝒫{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}\right)x^{{\mu }^{\vee }},$$ where $$𝒫{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}=\left\{\text{Hecke paths of type}\hspace{0.17em}{\lambda }^{\vee }\hspace{0.17em}\text{and end}\hspace{0.17em}{\mu }^{\vee }\right\}⟷\{\text{slices of}\hspace{0.17em}G/K\hspace{0.17em}\text{in}\hspace{0.17em}K{t}_{{\lambda }^{\vee }}K\cap U^{-}{t}_{{\mu }^{\vee }}K\}$$ and $${\text{Card}}_q\left(𝒫{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}\right)=\sum _{p\in 𝒫{\left({\lambda }^{\vee }\right)}_{{\mu }^{\vee }}}\text{(\# of}\hspace{0.17em}{𝔽}_q\hspace{0.17em}\text{points in slice}\hspace{0.17em}p\text{).}$$ After normalization, $$P_{{\lambda }^{\vee }}{|}_{q^{-1}=0}=\text{char}\left(L\left({\lambda }^{\vee }\right)\right)\text{.}$$

Buildings

The group $B$ is a Borel subgroup of $G=G\left(ℂ\right)$ and $$G/B=\text{flag variety}=\text{building.}$$ The cell decomposition of $G/B$ is $$G=\underset{w\in W}{⨆}BwB\text{.}$$ Idea: The points of $W$ are regions, or chambers. $$W=⟨s_1,s_2\hspace{0.17em}|\hspace{0.17em}{s}_1^2=s_2^2=1,s_1{s}_2{s}_1=s_2{s}_1{s}_2⟩\begin{array}{c} 1 s1 s2 s1s2 s2s1 s1s2s1 \end{array}$$ If $w=s_{i_1}\cdots s_{i_{\ell }}$ is a minimal length path to $w$ then $$BwB=\left\{x_{i_1}\left(c_1\right)s_{i_1}\cdots x_{i_{\ell }}\left(c_{\ell }\right)s_{i_{\ell }}B\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{\ell }\in ℂ\},\text{where}{x}_i\left(c\right)=1+c{E}_{i,i+1},$$ with $E_{i,i+1}$ the matrix with a $1$ in the $(i,i+1)$ entry and all other entries $0\text{.}$
IDEA: The points of $G/B$ are regions, or chambers. $$\begin{array}{c} x1(1)s1B x1(1)s1x2(2)s2B x1(1)s1x2(5.3)s2B x1(0)s1B x1(7)s1B x1(π/31)s1B x1(-4)s1B \end{array}$$

Just as the building of $W,$ the Coxeter complex, has relations $$s_1{s}_2{s}_1=s_2{s}_1{s}_2\begin{array}{c} \end{array}$$ the building of $G/B$ also has relations $$\begin{array}{c}{x}_1\left(c_1\right)s_1{x}_2\left(c_2\right)s_2{x}_1\left(c_3\right)s_1=x_2\left(c_3\right)s_2{x}_1(c_1{c}_3-c_2)s_1{x}_2\left(c_3\right)s_2\\ \end{array}$$ An apartment is a subbuilding of $G/B$ that looks like $W\text{.}$

The Borel subgroup of $G=G\left(ℂ\left(\left(t\right)\right)\right)$ is $I$ and $$G/I\hspace{0.17em}\text{is the}\hspace{0.17em}\text{affine flag variety}$$ with $$G=\underset{w\in \stackrel{\sim }{W}}{⨆}IwI,\text{where}\stackrel{\sim }{W}=W⋉P^{\vee }$$ $$\text{is the}\hspace{0.17em}\text{affine Weyl group}\begin{array}{c} \end{array}$$ The affine building $G/I$ has sectors $$\begin{array}{c} \end{array}\text{since}G=\underset{v\in \stackrel{\sim }{W}}{⨆}{U}^{-}vI\text{.}$$

MV polytopes

Let $$T=\left\{\left(\begin{array}{ccc}*& & 0\\ & \ddots & \\ 0& & *\end{array}\right)\right\}\text{and}\text{let}\hspace{0.17em}V\hspace{0.17em}\text{be a}\hspace{0.17em}T\text{-module}$$ with $T\text{-invariant}$ inner product $⟨,⟩$ (such that $⟨v,v⟩=0\iff v=0\text{).}$ Let $$𝔥=\text{Lie}\left(T\right)\text{and}\mathbb{P}V=\left\{\left[v\right]\hspace{0.17em}\right|\hspace{0.17em}v\in V,v\ne 0\},$$ where $\left[v\right]=\text{span}\left\{v\right\}\text{.}$ The moment map on $\mathbb{P}V$ is $$\begin{array}{cccc}\mu :& \mathbb{P}V& ⟶& {𝔥}^{*}\\ & \left[v\right]& ⟼& {\mu }_v\end{array}{\mu }_v\left(h\right)=\frac{⟨hv,v⟩}{⟨v,v⟩}\text{.}$$ Now let $V=L\left(\gamma \right)$ be a simple $G\text{-module}$ $\text{(}G=G\left(ℂ\right)\text{)}$ with highest weight vector $v^{+}\text{.}$ Then $$B\left[v^{+}\right]=\left[v^{+}\right]\text{and}G\left[v^{+}\right]\subseteq \mathbb{P}V$$ is the image of $G/B$ in $\mathbb{P}V\text{.}$ The moment map on $G/B$ (associated to $\gamma \text{)}$ is $$\begin{array}{cccccc}\mu :& G/B& ⟶& \mathbb{P}V& ⟶& {𝔥}^{*}\\ & gB& ⟼& g\left[v^{+}\right]& ⟼& {\mu }_{g{v}^{+}}\end{array}$$ Joel(Kamnitzer)'s favourite case is $G/K$ with $\gamma ={\omega }_0$ (the fundamental weight corresponding to the added node on the extended Dynkin diagram) and $$\mu \left(\text{MV cycle of type}\hspace{0.17em}{\lambda }^{\vee }\hspace{0.17em}\text{and weight}\hspace{0.17em}{\mu }^{\vee }\right)=\left(\text{MV polytope of type}\hspace{0.17em}{\lambda }^{\vee }\hspace{0.17em}\text{and weight}\hspace{0.17em}{\mu }^{\vee }\right)$$ $$\begin{array}{c} \end{array}$$

Tropicalization

Let $G=G\left(ℂ\left(\left(t\right)\right)\right)\text{.}$ $$ℂ\left(\left(t\right)\right)=\{a_{\ell }{t}^{\ell }+a_{\ell +1}{t}^{\ell +1}+\cdots \hspace{0.17em}|\hspace{0.17em}\ell \in \mathbb{Z},a_i\in ℂ\}\text{.}$$ Points of $G/I$ are $$gI,\text{where}g=\left(g_{ij}\right),g_{ij}\in ℂ\left(\left(t\right)\right)\text{.}$$ The valuation on $ℂ\left(\left(t\right)\right)$ $$v(a_{\ell }{t}^{\ell }+a_{\ell +1}{t}^{\ell +1}+\cdots )=\ell ,$$ is like log $$v\left(f_1{f}_2\right)=v\left(f_1\right)+v\left(f_2\right)\text{and}v(f_1+f_2)=\text{min}(v\left(f_1\right),v\left(f_2\right))\text{.}$$ Then $v\left(gI\right)$ is a tropical point of $v(G/I),$ the tropical flag variety. An amoeba, or tropical subvariety, is the image, under $v,$ of a subvariety of $G/I\text{.}$

Notes and References

These are a typed copy of /Volumes/Data/Users/arun/Work2007/Bites2007/aimtalk3.26.07.pdf the text of a talk at the American Institute of Mathematics in Palo Alto on March 26, 2007.

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