The Elliptic Weyl character formula: The ring $\widetilde{Th}$

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

Last update: 02 March 2012

The ring $\widetilde{Th}$

Let ${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}$ be a free $\mathbb{Z}-$module of rank $l$ with a positive definite symmetric bilinear form $\left(\right|):{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}\times {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}\to \mathbb{Z}.$ Let $${\stackrel{\circ }{𝔥}}_{ℝ}=ℝ{\otimes }_{\mathbb{Z}}{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}and{\stackrel{\circ }{𝔥}}_{ℂ}=ℂ{\otimes }_{\mathbb{Z}}{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}$$ and extend $\left(\right|)$ to a symmetric bilinear form on $$\begin{array}{lcccc}& & \left(\delta \right|\delta )=0,& \left(\delta \right|{\stackrel{\circ }{𝔥}}_{ℂ})=0,& \left(\delta \right|{\Lambda }_0)=1,\\ {𝔥}_{ℂ}=ℂ\delta \oplus {\stackrel{\circ }{𝔥}}_{ℂ}\oplus ℂ{\Lambda }_0& by\; setting& \left({\stackrel{\circ }{𝔥}}_{ℂ}\right|\delta )=0,& & \left({\stackrel{\circ }{𝔥}}_{ℂ}\right|{\Lambda }_0)=1,\\ & & \left({\Lambda }_0\right|\delta )=1,& \left({\Lambda }_0\right|{\stackrel{\circ }{𝔥}}_{ℂ})=0,& \left({\Lambda }_0\right|{\Lambda }_0)=0.\end{array}$$ For $\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}$ define $t_{\beta }:{𝔥}_{ℂ}\to {𝔥}_{ℂ}$ by $$\begin{array}{ll}{t}_{\beta }\left(\lambda \right)=\lambda +m\beta -(\stackrel{\_}{\lambda }+\frac{1}{2}m\beta |\beta )\delta ,if\lambda =a\delta +\stackrel{\_}{\lambda }+m{\Lambda }_0,& (RTh\; 1)\end{array}$$ with $a\in ℂ, \stackrel{\_}{\lambda }\in {\stackrel{\circ }{𝔥}}_{ℂ},$ and $m\in ℂ.$ The motivation for this formula is ??????? Let $${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*}=\{\stackrel{\_}{\lambda }\in {\stackrel{\circ }{𝔥}}_{ℝ} | \left(\stackrel{\_}{\lambda }\right|\stackrel{\_}{\beta })\in \mathbb{Z}, \; for\; \stackrel{\_}{\beta }\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}\}.$$ For $\lambda =a\delta +\stackrel{\_}{\lambda }+m{\Lambda }_0$ with $a\in ℂ, \stackrel{\_}{\lambda }\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*}, m\in {\mathbb{Z}}_{>0}$ define (see [Kac, (12.7.2) and (12.7.3)]) $$\begin{array}{rclr}{\Theta }_{\lambda }& =& {\Theta }_{a\delta +\stackrel{\_}{\lambda }+m{\Lambda }_0}=e^{-\frac{\left(\lambda \right|\lambda )\delta }{2m}}\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{t_{\beta }\left(\lambda \right)}& (RTh\; 2)\\ & =& e^{-\frac{2am+\left(\stackrel{\_}{\lambda }\right|\stackrel{\_}{\lambda })\delta }{2m}}\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{a\delta +\stackrel{\_}{\lambda }+m\beta -(\stackrel{\_}{\lambda }+\frac{1}{2}m\beta |\beta )\delta +m{\Lambda }_0}& \\ & =& e^{m{\Lambda }_0}\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{\stackrel{\_}{\lambda }+m\beta -\frac{1}{2m}((2\stackrel{\_}{\lambda }+m\beta |m\beta )+\left(\stackrel{\_}{\lambda }\right|\stackrel{\_}{\lambda }))\delta }& \\ & =& e^{m{\Lambda }_0}\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{\stackrel{\_}{\lambda }+m\beta -\frac{1}{2m}(\stackrel{\_}{\lambda }+m\beta |\stackrel{\_}{\lambda }+m\beta )\delta },& (RTh\; 3)\end{array}$$ which (modulo $\delta$) is exactly the sum over translates of $\stackrel{\_}{\lambda }$ by an $m-$dilate of the lattice ${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}.$ The expression ${\Theta }_{\lambda }$ is an element of $ℂ-\mathrm{span}\left\{e^{\lambda } \right| \lambda \in {𝔥}_{ℂ}\},$ where $e^{\lambda }$ are formal symbols indexed by $\lambda \in {𝔥}_{ℂ},$ infinite sums are allowed and $$e^{\lambda }{e}^{\mu }=e^{\lambda +\mu },for\; \lambda ,\mu \in {𝔥}_{ℂ}.$$ If $a\in ℂ$ and $\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}$ then $${\Theta }_{a\delta +\stackrel{\_}{\lambda }+m{\Lambda }_0}={\Theta }_{\stackrel{\_}{\lambda }+m{\Lambda }_0}and{\Theta }_{\stackrel{\_}{\lambda }+m\beta +m{\Lambda }_0}={\Theta }_{\stackrel{\_}{\lambda }+m{\Lambda }_0}.$$ Write $$\begin{array}{ll}\left(\lambda \right|\lambda )={\parallel \lambda \parallel }^{2}andq=e^{-\delta }so\; that{\Theta }_{\stackrel{\_}{\lambda }+m{\Lambda }_0}=e^{m{\Lambda }_0}\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{\stackrel{\_}{\lambda }+m\beta }{q}^{\frac{1}{2m}{\parallel \lambda +m\beta \parallel }^2},& (RTh\; 4)\end{array}$$

Setting $e^{\lambda }{e}^{\mu }=e^{\lambda +\mu }$ for $\lambda ,\mu \in {𝔥}_{ℂ},$ $${\Theta }_{\lambda +m{\Lambda }_0}{\Theta }_{\mu +n{\Lambda }_0}=\sum _{\gamma \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}} \mathrm{mod}(m+n){\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{d}_{\lambda +\mu +m\gamma +(m+n){\Lambda }_0}^{\lambda +m{\Lambda }_0,\mu +n{\Lambda }_0}{\Theta }_{\lambda +\mu +m\gamma +(m+n){\Lambda }_0},$$ where $$d_{\lambda +\mu +m\gamma +(m+n){\Lambda }_0}^{\lambda +m{\Lambda }_0,\mu +n{\Lambda }_0}=\sum _{\kappa \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{q}^{\frac{1}{2}\frac{mn}{(m+n)}{\parallel \frac{1}{m}\lambda -\frac{1}{n}\mu +\gamma +(m+)\kappa \parallel }^2},$$ (see [Kac, Ex. 13.1] or [KP, §13.2]). This formula gives the product structure on the graded ${\widetilde{Th}}_0-$algebra $$\widetilde{Th}=\underset{m\in {\mathbb{Z}}_{\ge 0}}{⨁}{\widetilde{Th}}_m,where\; {\widetilde{Th}}_m \; has\; {\widetilde{Th}}_0-basis\; \left\{{\Theta }_{\stackrel{\_}{\lambda }+m{\Lambda }_0} \right| \stackrel{\_}{\lambda }\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*} \mathrm{mod} m{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}\},$$ WHAT IS THE RIGHT COMBINATORIAL DESCRIPTION OF THIS BASE RING? PUT THE $G_{\mathbb{Z}}-$ACTION ON $\widetilde{Th}$ HERE?

Proof of the product formula for $\widetilde{Th}$.

		Proof.
	$$\begin{array}{rl}{\Theta }_{{\mu }_1+m_1{\Lambda }_0}& {\Theta }_{{\mu }_2+m_2{\Lambda }_0}\\ & =\left(e^{m_1{\Lambda }_0}\sum _{{\gamma }_1\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{{\mu }_1+m_1{\gamma }_1-\frac{1}{2m_1}{\parallel {\mu }_1+m_1{\gamma }_1\parallel }^2\delta }\right)\left(e^{m_2{\Lambda }_0}\sum _{{\gamma }_2\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{{\mu }_2+m_2{\gamma }_2-\frac{1}{2m_2}{\parallel {\mu }_2+m_2{\gamma }_2\parallel }^2\delta }\right)\\ & \begin{array}{rl}=e^{(m_1+m_2){\Lambda }_0}\sum _{{\gamma }_1-{\gamma }_2\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}\sum _{{\gamma }_2\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}& e^{{\mu }_1+{\mu }_2+m_1({\gamma }_1-{\gamma }_2)+(m_1+m_2){\gamma }_2}{e}^{-\frac{1}{2(m_1+m_2)}{\parallel {\mu }_1+{\mu }_2+m_1{\gamma }_1+m_2{\gamma }_2\parallel }^2\delta }\\ & \cdot e^{\frac{1}{2(m_1+m_2)}{\parallel {\mu }_1+{\mu }_2+m_1{\gamma }_1+m_2{\gamma }_2\parallel }^2\delta }{e}^{-\frac{1}{2m_1}{\parallel {\mu }_1+m_1{\gamma }_1\parallel }^2\delta }{e}^{-\frac{1}{2m_2}{\parallel {\mu }_2+m_2{\gamma }_2\parallel }^2\delta }\end{array}\\ & =\sum _{{\gamma }_1-{\gamma }_2\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{\Theta }_{{\mu }_1+{\mu }_2+m_1({\gamma }_1-{\gamma }_2)+(m_1+m_2){\Lambda }_0}{e}^{-\frac{\left(m_1{m}_2\right)}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1-\frac{1}{m_1}{\mu }_2+{\gamma }_1-{\gamma }_2\parallel }^2\delta },\end{array}$$ since $$\begin{array}{rl}\frac{1}{2(m_1+m_2)}& {\parallel {\mu }_1+{\mu }_2+m_1{\gamma }_1+m_2{\gamma }_2\parallel }^2-\frac{1}{2m_1}{\parallel {\mu }_1+m_1{\gamma }_1\parallel }^2-\frac{1}{2m_2}{\parallel {\mu }_2+m_2{\gamma }_2\parallel }^2\\ & =\frac{1}{2(m_1+m_2)}{\parallel {\mu }_1+m_1{\gamma }_1\parallel }^2+\frac{1}{2(m_1+m_2)}{\parallel {\mu }_2+m_2{\gamma }_2\parallel }^2+\frac{1}{m_1+m_2}({\mu }_1+m_1{\gamma }_1|{\mu }_2+m_2{\gamma }_2)\\ & -\frac{1}{2m_1}{\parallel {\mu }_1+m_1{\gamma }_1\parallel }^2-\frac{1}{2m_2}{\parallel {\mu }_2+m_2{\gamma }_2\parallel }^2\\ & =-\frac{m_2}{2m_1{m}_1+m_2)}{\parallel {\mu }_1+m_1{\gamma }_1\parallel }^2-\frac{m_1}{2m_2{m}_1+m_2)}{\parallel {\mu }_2+m_2{\gamma }_2\parallel }^2+\frac{1}{m_1+m_2}({\mu }_1+m_1{\gamma }_1|{\mu }_2+m_2{\gamma }_2)\\ & =-\frac{m_1{m}_2}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1+{\gamma }_1\parallel }^2-\frac{m_1{m}_2}{2(m_1+m_2)}{\parallel \frac{1}{m_2}{\mu }_2+{\gamma }_2\parallel }^2+\frac{m_1{m}_2}{m_1+m_2}(\frac{1}{m_1}{\mu }_1+{\gamma }_1|\frac{1}{m_2}{\mu }_2+{\gamma }_2)\\ & =\frac{-m_1{m}_2}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1+{\gamma }_1-\frac{1}{m_2}{\mu }_2-{\gamma }_2\parallel }^2\\ & =\frac{-m_1{m}_2}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1-\frac{1}{m_2}{\mu }_2+{\gamma }_1-{\gamma }_2\parallel }^2.\end{array}$$ Thus $$\begin{array}{rl}{\Theta }_{{\mu }_1+m_1{\Lambda }_0}& {\Theta }_{{\mu }_2+m_2{\Lambda }_0}\\ & =\sum _{{\gamma }_1-{\gamma }_2\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{\Theta }_{{\mu }_1+{\mu }_2+m_1({\gamma }_1-{\gamma }_2)+(m_1+m_2){\Lambda }_0}{e}^{-\frac{\left(m_1{m}_2\right)}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1-\frac{1}{m_2}{\mu }_2+{\gamma }_1-{\gamma }_2\parallel }^2\delta }\\ & =\sum _{\gamma \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{\Theta }_{{\mu }_1+{\mu }_2+m_1\gamma +(m_1+m_2){\Lambda }_0}{e}^{-\frac{\left(m_1{m}_2\right)}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1-\frac{1}{m_2}{\mu }_2+\gamma \parallel }^2\delta }\\ & =\sum _{\gamma \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}} \mathrm{mod}(m_1+m_2){\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{\Theta }_{{\mu }_1+{\mu }_2+m_1\gamma +(m_1+m_2){\Lambda }_0}\sum _{\kappa \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{-\frac{\left(m_1{m}_2\right)}{2(m_1+m_2)}{\parallel \frac{1}{m_1}{\mu }_1-\frac{1}{m_2}{\mu }_2+\gamma +(m_1+m_2)\kappa \parallel }^2\delta }\\ & =\sum _{\gamma \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}} \mathrm{mod}(m_1+m_2){\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{\Theta }_{{\mu }_1+{\mu }_2+m_1\gamma +(m_1+m_2){\Lambda }_0}\sum _{\kappa \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{-\frac{1}{2m_1{m}_2(m_1+m_2)}{\parallel m_2{\mu }_1-m_1{\mu }_2+m_1{m}_2\gamma +m_1{m}_2(m_1+m_2)\kappa \parallel }^2\delta }\\ & =\sum _{\gamma \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}} \mathrm{mod}(m_1+m_2){\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{\Theta }_{{\mu }_1+{\mu }_2+m_1\gamma +(m_1+m_2){\Lambda }_0}{\mathrm{ev}}_{0,0,c}\left({\Theta }_{m_2{\mu }_1-m_1{\mu }_2+m_1{m}_2\gamma +m_1{m}_2(m_1+m_2){\Lambda }_0}\right).\end{array}$$ $\square$

The subring ${\widetilde{Th}}^{W_0}$

The action of $W_0$ on $\widetilde{Th}$ induced by the action on $ℂ-\mathrm{span}\left\{e^{\lambda } \right| \lambda \in {𝔥}_{ℂ}\}$ given by $w{e}^{\lambda }=e^{w\lambda }$ is $$w{\Theta }_{\lambda }={\Theta }_{w\lambda },so\; thatw{\Theta }_{\stackrel{\_}{\lambda }+m{\Lambda }_0}={\Theta }_{w\stackrel{\_}{\lambda }+m{\Lambda }_0},$$ for $w\in W_0,$ $\stackrel{\_}{\lambda }\in {\stackrel{\circ }{𝔥}}_{ℂ}$ and $m\in {\mathbb{Z}}_{>0}.$ Let $$M_{\lambda }=\sum _{\gamma \in W_0\lambda }{\Theta }_{\lambda }and{A}_{\mu }=\sum _{w\in W_0}\det \left(w\right){\Theta }_{w\mu }.$$ Let $$\begin{array}{rcl}{\widetilde{Th}}^{W_0}& =& \{f\in \widetilde{Th} | wf=f, for w\in W_0\}and\\ {\widetilde{Th}}^{\det }& =& \{f\in \widetilde{Th} | wf=\det \left(w\right)f, for w\in W_0\}.\end{array}$$ Since $\{\lambda \in P^{+} \mathrm{mod}ℂ\delta , \lambda \left(c\right)=m\}$ is a set of representatives of the $W_0-$orbits on $\{\stackrel{\_}{\lambda }\in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*} \mathrm{mod}m{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}\}$ and $A_{\mu }=0$ if $\mu \in P^{+}-P^{++},$ it follows (as in [KP, Prop. 4.3(d-e)]) that $$\begin{array}{lcl}{\widetilde{Th}}^{W_0}& has\; basis& \left\{M_{\lambda } \right| \lambda \in P^{+} \mathrm{mod}ℂ\delta , \lambda \left(c\right)=m\},and\\ {\widetilde{Th}}^{\det }& has\; basis& \left\{A_{\lambda +\rho } \right| \lambda +\rho \in P^{++} \mathrm{mod}ℂ\delta , \lambda \left(c\right)=m\}.\end{array}$$ Because of the Weyl character formula $$\left\{\frac{A_{\lambda +\rho }}{A_{\rho }} \right| \lambda +\rho \in P^{++} \mathrm{mod}ℂ\delta , \lambda \left(c\right)=m\}is\; a\; basis\; of\; {\widetilde{Th}}^{W_0},$$ and, since $$\begin{array}{rcl}{P}^{+}& \to & P^{++}\\ \lambda & \mapsto & \lambda +\rho \end{array}is\; a\; bijection,$$ ${\widetilde{Th}}^{\det }$ is a free module of rank 1 over ${\widetilde{Th}}^{W_0},$ generated by $A_{\rho }.$ In other words, as ${\widetilde{Th}}^{W_0}-modules$ $$\begin{array}{rcl}{\widetilde{Th}}^{W_0}& \stackrel{\sim }{\to }& {\widetilde{Th}}^{\det }\\ f& \mapsto & A_{\rho }f\\ s_{\lambda }& \mapsto & A_{\lambda +\rho }\end{array}$$ BE SURE TO GET THE NORMALIZATION CONSTANT CORRECT ON THE LAST LINE!!!

Notes and References

These notes are taken from notes on the Elliptic Weyl character formula by Nora Ganter and Arun Ram.

References

References?

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