The Weyl character formula from the affine Hecke algebra point of view

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

Last update: 05 April 2012

Symmetric functions

The initial data is $(W_0,{𝔥}_{\mathbb{Z}}),$ a finite $\mathbb{Z}-$reflection group, i.e.

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

Example.

1. ${𝔥}_{\mathbb{Z}}=\mathrm{span}\{{\epsilon }_1,...,{\epsilon }_n\}$ with
2. $W_0=S_n$ acting by permuting ${\epsilon }_1,...,{\epsilon }_n.$

The group algebra of ${𝔥}_{\mathbb{Z}}$ is $$ℂ\left[X\right]=\mathrm{span}\left\{X^{\lambda } \right| \lambda \in {𝔥}_{\mathbb{Z}}\}with{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu }.$$ $W_0$ acts on $ℂ\left[X\right]$ by $w{X}^{\lambda }=X^{w\lambda }.$

The ring of symmetric functions is $$ℂ{\left[X\right]}^{W_0}=\{f\in ℂ\left[X\right] | wf=f\}.$$

Example: Type $G{L}_3$.
Let $z_1=X^{{\epsilon }_1},z_2=X^{{\epsilon }_2},z_3=X^{{\epsilon }_3}.$ Then $$\begin{array}{rcl}{W}_0& =& S_3,\\ ℂ\left[X\right]& =& ℂ[z_1^{\pm 1},z_2^{\pm 1},z_3^{\pm 1}]and\\ ℂ{\left[X\right]}^{W_0}& =& ℂ{[z_1^{\pm 1},z_2^{\pm 1},z_3^{\pm 1}]}^{S_3}=ℂ[e_1,e_2,e_3^{\pm 1}],\end{array}$$ where $$e_1=z_1+z_2+z_3,e_2=z_1{z}_2+z_1{z}_3+z_2{z}_3,e_3=z_1{z}_2{z}_3.$$

Weyl characters

Let $C_0$ be a fundamental region for $W_0$ acting on ${𝔥}_{ℝ}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}.$ Let $${𝔥}_{\mathbb{Z}}^{+}={𝔥}_{\mathbb{Z}}\cap \stackrel{\_}{C_0}and{𝔥}_{\mathbb{Z}}^{++}={𝔥}_{\mathbb{Z}}\cap C_0$$ where $\stackrel{\_}{C_0}$ is the closure of $C_0.$ Then $$\begin{array}{rcl}{𝔥}_{\mathbb{Z}}^{+}& \stackrel{\sim }{\to }& {𝔥}_{\mathbb{Z}}^{++}\\ \lambda & \mapsto & \rho +\lambda \end{array}as\; {𝔥}_{\mathbb{Z}}^{+}-\; modules.$$

Define $$ℂ{\left[X\right]}^{\det }=\{f\in ℂ\left[X\right] | wf=\det \left(w\right)f, \; for\; w\in W_0\}.$$

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

$$\begin{array}{crcll}& ℂ{\left[X\right]}^{W_0}& \stackrel{\sim }{\to }& ℂ{\left[X\right]}^{\det }& \; as\; ℂ{\left[X\right]}^{W_0} \; modules\\ & f& \mapsto & a_{\rho }f& \\ & s_{\lambda }& ↤& a_{\lambda +\rho }& \; "naive\; basis"\\ "naive\; basis"& m_{\lambda }& & & \end{array}$$ where $m_{\lambda }=\sum _{\gamma \in W_0\lambda }{X}^{\gamma }$ and $a_{\mu }=\sum _{w\in W_0}\det \left(w\right)X^{w\mu }.$

The Weyl character is $$s_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},for\; \lambda \in {𝔥}_{\mathbb{Z}}^{+}.$$

The affine Hecke algebra $H$

Let

1. ${𝔥}^{{\alpha }_1},...,{𝔥}^{{\alpha }_l}$ be the walls of $C_0,$
2. $s_1,...,s_l$ the corresponding reflections,

so that $s_i:{𝔥}_{\mathbb{Z}}\to {𝔥}_{\mathbb{Z}}$ is given by $$s_i\lambda =\lambda -⟨\lambda ,{\alpha }_i^{\vee }⟩{\alpha }_i,for\; i=1,...,l.$$

The affine Hecke algebra $H$ is generated by $$T_1,...,T_{l}and{X}^{\lambda }, \lambda \in {𝔥}_{\mathbb{Z}}$$ with relations $$\begin{array}{lcl}{T}_i^2=(t^{\frac{1}{2}}-t^{-\frac{1}{2}})T_i+1,& & for\; i=1,...,l,\\ {\displaystyle \underset{m_{ij} \; factors}{\underbrace{T_i{T}_j{T}_i\cdots }}}={\displaystyle \underset{m_{ij} \; factors}{\underbrace{T_j{T}_i{T}_j\cdots }}},& & for\; i\ne j \; with\; \frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i}\measuredangle {𝔥}^{{\alpha }_j},\\ X^{\lambda }{X}^{\mu }=X^{\lambda +\mu },& & for\; \lambda ,\mu \in {𝔥}_{\mathbb{Z}},\\ T_i{X}^{\lambda }=X^{s_i\lambda }{T}_i+(t^{\frac{1}{2}}-t^{-\frac{1}{2}})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}.& & \end{array}$$ Define $$T_w=T_{i_1}\cdots T_{i_l}for\; a\; reduced\; wordw=s_{i_1}\cdots s_{i_l}.$$ Then $$\left\{X^{\lambda }{T}_w \right| \lambda \in {𝔥}_{\mathbb{Z}}, w\in W_0\}is\; a\; basis\; of\; H$$ and $$ℂ\left[X\right]=\mathrm{span}\left\{X^{\lambda } \right| \lambda \in {𝔥}_{\mathbb{Z}}\}and{H}_0=\mathrm{span}\left\{T_w \right| w\in W_0\}$$ are subalgebras.

Bernstein-Satake-Lusztig isomorphisms

Let ${𝟙}_0,{\epsilon }_0\in H_0$ be such that $$\begin{array}{rcl}{𝟙}_0^2& =& {𝟙}_0\\ {\epsilon }_0^2& =& {\epsilon }_0\end{array}and\begin{array}{rcl}{T}_i{𝟙}_0& =& t^{\frac{1}{2}}{𝟙}_0\\ T_i{\epsilon }_0& =& (-t^{-\frac{1}{2}}){\epsilon }_0\end{array}for\; i=1,...,l.$$ Then $$\begin{array}{rcl}H& \to & H{𝟙}_0=ℂ\left[X\right]{𝟙}_0\\ h& \mapsto & h{𝟙}_0\end{array}$$ makes $ℂ\left[X\right]$ into an $H-$module (the polynomial representation). Then $$\begin{array}{cccccccc}& Bernstein& & Satake& & Lusztig& & \\ ℂ{\left[X\right]}^{W_0}& =& Z\left(H\right)& \stackrel{\sim }{\to }& {𝟙}_{0}H{𝟙}_0& \stackrel{\sim }{\to }& {\epsilon }_{0}H{𝟙}_0& \\ f& \mapsto & f& \mapsto & f{𝟙}_0& \mapsto & A_{\rho }f{𝟙}_0& \\ s_{\lambda }& \multicolumn{3}{c}{↤}& C_{\lambda }& ↤& A_{\lambda +\rho }& \} \; "naive\; bases"\\ P_{\lambda }(0,t)& \multicolumn{3}{c}{↤}& M_{\lambda }& & \end{array}$$ where $$M_{\lambda }={𝟙}_0{X}^{\lambda }{𝟙}_{0}and{A}_{\mu }={\epsilon }_0{X}^{\mu }{𝟙}_0.$$ $C_{\lambda }$ is the Kazhdan-Lusztig basis of the spherical Hecke algebra ${𝟙}_{0}H{𝟙}_0=K_0\left(\mathrm{Perv}(G/K)\right)$ the Grothendieck group of the category $\mathrm{Perv}(G/K)$ of perverse sheaves on the loop Grassmannian $G/K.$
$P_{\lambda }(0,t)$ is Macdonald's spherical function for $G\left({\mathbb{Q}}_p\right)/G\left({\mathbb{Z}}_p\right).$

Weyl's Theorems

Let $G\left(ℂ\right)$ be the reductive algebraic group corresponding to $(W_0,{𝔥}_{\mathbb{Z}}).$

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

$q-$Weyl denominator

$$A_{\rho }=\prod _{\alpha \in R^{+}}(t^{\frac{1}{2}}{X}^{\frac{\alpha }{2}}-t^{-\frac{1}{2}}{X}^{-\frac{\alpha }{2}})$$

Notes and References

These notes are from lecture notes of Arun Ram (CBMS Lecture 2). It was also a lecture for the 2011 semester 2 Representation Theory course at the University of Melbourne (05/08/2011).

References

References?

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