The Weyl Character formula

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

Last update: 8 October 2012

Symmetric functions

Initial data:

1. $(W_0,{𝔥}_{\mathbb{Z}}^{*})$ a finite $\mathbb{Z}$-reflection group
2. ${𝔥}_{\mathbb{Z}}^{*}$ is a free $\mathbb{Z}$-modules
3. $W_0$ a finite subgroup of $GL\left({𝔥}_{\mathbb{Z}}^{*}\right)$ generated by reflections

Example: Type $G{L}_n$

$$\begin{array}{c}{𝔥}_{\mathbb{Z}}^{*}=\mathbb{Z}\text{-span}\{{\epsilon }_1,\dots ,{\epsilon }_n\}\\ W_0=S_n\text{acting by permuting}{\epsilon }_1,\dots ,{\epsilon }_n\text{.}\end{array}$$

The group algebra of ${𝔥}_{\mathbb{Z}}^{*}$ is

$$ℂ\left[X\right]=\text{span}\{X^{\lambda }\mid \lambda \in {𝔥}_{\mathbb{Z}}^{*}\}\text{with}{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu }\text{.}$$

$W_0$ acts on $ℂ\left[X\right]$ by $w{X}^{\lambda }=X^{w\lambda }\text{.}$

The ring of symmetric functions is

$$ℂ{\left[X\right]}^{W_0}=\{f\in ℂ\left[X\right]\mid wf=f\text{for}w\in W_0\}\text{.}$$

Example: Type $G{L}_3$

Let $z_1=X^{{\epsilon }_1},$ $z_2=X^{{\epsilon }_2},$ $z_3=X^{{\epsilon }_3},$ $W_0=S_3,$ $ℂ\left[X\right]=ℂ[z_1^{\pm 1},z_2^{\pm 1},z_3^{\pm 1}]$ and

$$ℂ{\left[X\right]}^{W_0}=ℂ{[z_1^{\pm 1},z_2^{\pm 1},z_3^{\pm 1}]}^{S_3}=ℂ[e_1,e_2,e_3^{\pm 1}]$$

where

$$e_1=z_1+z_2+z_3,e_2=z_1{z}_2+z_1{z}_3+z_2{z}_3,e_3=z_1{z}_2{z}_3\text{.}$$

Weyl characters

Let $C_0$ be a fundamental region for $W_0$ acting on ${𝔥}_{ℝ}\text{.}$

$${𝔥}_{\mathbb{Z}}^{+}={𝔥}_{\mathbb{Z}}\cap {\bar{C}}_0\text{and}{𝔥}_{\mathbb{Z}}^{++}={𝔥}_{\mathbb{Z}}\cap C_0$$

where ${\bar{C}}_0$ is the closure of $C_0\text{.}$ Then

$$\begin{array}{ccc}{𝔥}_{\mathbb{Z}}^{+}& \stackrel{\sim }{⟶}& {𝔥}_{\mathbb{Z}}^{++}\\ \lambda & ⟼& \rho +\lambda \end{array}\text{as}{𝔥}_{\mathbb{Z}}^{+}\text{-modules}$$

Define

$$ℂ{\left[X\right]}^{\text{det}}=\{f\in ℂ\left[X\right]\mid wf=\text{det}\left(w\right)f\text{for}w\in W_0\}\text{.}$$

$ℂ{\left[X\right]}^{\text{det}}$ is a free $ℂ{\left[X\right]}^{W_0}$-module of rank 1

$$\begin{array}{ccccc}& ℂ{\left[X\right]}^{W_0}& \stackrel{\sim }{⟶}& ℂ{\left[X\right]}^{\text{det}}& \text{as}ℂ{\left[X\right]}^{W_0}\text{-modules}\\ & f& ⟼& a_{\rho }f\\ \text{"naive basis"}& m_{\lambda }\\ & s_{\lambda }& ↤& a_{\lambda +\rho }& \text{"naive basis"}\end{array}$$

where ${\displaystyle m_{\lambda }=\sum _{\gamma \in W_0\lambda }{X}^{\gamma }}$ and ${\displaystyle a_{\mu }=\sum _{w\in W_0}\text{det}\left(w\right)X^{W\mu }\text{.}}$

The Weyl character is

$$s_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},\text{for}\lambda \in {𝔥}_{\mathbb{Z}}^{+}\text{.}$$

Weyl's Theorems

Let $G^{\vee }\left(ℂ\right)$ be the reductive algebraic group corresponding to $(W_0,{𝔥}_{\mathbb{Z}}),$ $T^{\vee }a$ a maximal torus of $G^{\vee }\text{.}$

1. The simple $T^{\vee }$-modules $X^{\lambda }$ are indexed by $\lambda \in {𝔥}_{\mathbb{Z}}$
2. The simple $G^{\vee }$-modules $L\left(\lambda \right)$ are indexed by $\lambda \in {𝔥}_{\mathbb{Z}}^{+}$
3. The character of $L\left(\lambda \right)$ is $${\text{Res}}_T^G\left(L\left(\lambda \right)\right)=s_{\lambda }$$
4. ${\displaystyle a_{\rho }=X^{\rho }\prod _{\alpha \in R^{+}}(1-X^{-\alpha })}$
   where $R^{+}$ is an index set for the reflections in $W_0$ such that $$s_{\alpha }\mu =\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha \text{for}\alpha \in R^{+},\mu \in {𝔥}_{\mathbb{Z}}\text{.}$$

$$C0 s1C0 s2C0 ρ s1s2C0 s2s1C0 s1s2s1C0=s2s1s2C0$$

The affine Hecke algebra $H$

Let ${𝔥}^{{\alpha }_1},\dots ,{𝔥}^{{\alpha }_{\ell }}$ be the walls of $C_0,$ $s_1,\dots ,s_{\ell }$ the corresponding reflections, so that

$$s_i:{𝔥}_{\mathbb{Z}}⟶{𝔥}_{\mathbb{Z}}\text{with}{s}_i\lambda =\lambda -⟨\lambda ,{\alpha }_i^{\vee }⟩{\alpha }_i\text{.}$$

The affine Hecke algebra $H$ is generated by

$$T_1,\dots ,T_{\ell }\text{and}{X}^{\lambda },\lambda \in {𝔥}_{\mathbb{Z}}$$

with relations

$$\begin{array}{c}\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_i{T}_j{T}_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_j{T}_i{T}_j\dots },\text{for}i\ne j\text{with}\frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i}\nless {𝔥}^{{\alpha }_j}\\ T_i^2=(t^{\frac{1}{2}}-t^{-\frac{1}{2}})T_i+1,\text{for}i=1,\dots ,\ell \\ X^{\lambda }{X}^{\mu }=X^{\lambda +\mu },\text{for}\lambda ,\mu \in {𝔥}_{\mathbb{Z}}\\ T_i{X}^{\lambda }=X^{s_i\lambda }{T}_i+(t^{\frac{1}{2}}-t^{-\frac{1}{2}})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}\end{array}$$

Define

$$T_w=T_{i_1}\dots T_{u_{\ell }}\text{for a reduced word}w=s_{i_1}\dots s_{i_{\ell }}\text{.}$$

Then

$$\{X^{\lambda }{T}_w\mid \lambda \in {𝔥}_{\mathbb{Z}},w\in W_0\}\text{is a basis of}H$$

and

$$ℂ\left[X\right]=\text{span}\{X^{\lambda }\mid \lambda \in {𝔥}_{\mathbb{Z}}\}\text{and}{H}_0=\text{span}\{T_w\mid w\in W_0\}$$

are subalgebras.

Geometric Langlands

Let $1_0,{\epsilon }_0\in H_0$ be such that

$$\begin{array}{c}{1}_0^2=1_0\\ {\epsilon }_0^2={\epsilon }_0\end{array}\text{and}\begin{array}{c}{T}_i{1}_0=t^{\frac{1}{2}}{1}_0\\ T_i{\epsilon }_0=(-t^{-\frac{1}{2}}){\epsilon }_0\end{array}\text{for}i=1,\dots ,\ell$$

Then

$$\begin{array}{ccc}H& ⟶& H{1}_0=ℂ\left[X\right]1_0\\ h& ⟼& h{1}_0\end{array}\text{makes}ℂ\left[X\right]\text{an}H\text{-module,}$$

the polynomial representation of $H\text{.}$ Then

$$\begin{array}{ccccccc}ℂ{\left[X\right]}^{W_0}& =& Z\left(H\right)& \stackrel{\sim }{⟶}& 1_{0}H{1}_0& \stackrel{\sim }{⟶}& {\epsilon }_{0}H{1}_0\\ f& ⟼& f& ⟼& f{1}_0& ⟼& A_{\rho }f{1}_0\\ s_{\lambda }& \multicolumn{3}{c}{\stackrel{\phantom{aa}}{↤}}& C_{\lambda }& \stackrel{\phantom{aa}}{↤}& A_{\lambda +\rho }\\ P_{\lambda }(0,t)& \multicolumn{3}{c}{\stackrel{\phantom{aa}}{↤}}& M_{\lambda }\end{array}$$

where

$$M_{\lambda }=1_0{X}^{\lambda }{1}_0\text{and}{A}_{\mu }={\epsilon }_0{X}^{\mu }{1}_0,$$

$c_{\lambda }$ is the Kazhdan-Lusztig basis of the spherical Hecke algebra $1_{0}H{1}_0=\text{?????}\text{Perv}\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\right)$ the Grothendieck group of the category of perverse sheaves on the loop Grassmannian $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$k$}\right.\text{.}$

$P_{\lambda }(0,t)$ is Macdonald's spherical function for $\raisebox{1ex}{$G\left({\mathbb{Q}}_p\right)$}\!\left/ \!\raisebox{-1ex}{$G\left({\mathbb{Z}}_p\right)$}\right.\text{.}$

Notes and References

These are a typed version of lecture notes for the third in a series of lectures given at the Brazil Algebra Conference, Salvador, 18 July 2012.

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