The Elliptic Weyl character formula: The ring Th˜

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 02 March 2012

The ring Th˜

Let 𝔥 be a free -module of rank l with a positive definite symmetric bilinear form (|): 𝔥×𝔥 . Let 𝔥 = 𝔥 and 𝔥 = 𝔥 and extend (|) to a symmetric bilinear form on (δ|δ) =0, (δ|𝔥)=0, (δ|Λ0)=1, 𝔥 = δ 𝔥 Λ0 by setting (𝔥|δ)=0, (𝔥|Λ0)=1, (Λ0|δ)=1, (Λ0|𝔥)=0, (Λ0|Λ0)=0. For β𝔥 define tβ: 𝔥𝔥 by tβ(λ) = λ+mβ- (λ_+ 12mβ|β)δ, if λ=aδ+λ_+mΛ0, (RTh 1) with a,  λ_𝔥, and m. The motivation for this formula is ??????? Let 𝔥* = {λ_𝔥  |  (λ_|β_),  for   β_𝔥}. For λ=aδ+λ_+mΛ0 with a,  λ_𝔥*,  m>0 define (see [Kac, (12.7.2) and (12.7.3)]) Θλ = Θaδ+λ_+mΛ0 = e- (λ|λ)δ 2m β𝔥 etβ(λ) (RTh 2) = e- 2am+(λ_|λ_)δ 2m β𝔥 e aδ+λ_+mβ- (λ_+ 1 2 mβ|β)δ+mΛ0 = emΛ0 β𝔥 e λ_+mβ- 1 2m ( (2λ_+mβ|mβ) +(λ_|λ_) )δ = emΛ0 β𝔥 e λ_+mβ- 1 2m ( λ_+mβ|λ_+mβ )δ , (RTh 3) which (modulo δ) is exactly the sum over translates of λ_ by an m-dilate of the lattice 𝔥. The expression Θλ is an element of -span {eλ  |  λ𝔥}, where eλ are formal symbols indexed by λ𝔥, infinite sums are allowed and eλeμ = eλ+μ, for   λ,μ𝔥. If a and β𝔥 then Θ aδ+λ_+mΛ0 = Θ λ_+mΛ0 and Θ λ_+mβ+mΛ0 = Θ λ_+mΛ0 . Write (λ|λ) = λ2 and q=e-δ so that Θ λ_+mΛ0 = emΛ0 β𝔥 eλ_+mβ q 1 2m λ+mβ2 , (RTh 4)

Setting eλeμ = eλ+μ for λ,μ𝔥, Θλ+mΛ0 Θμ+nΛ0 = γ𝔥 mod(m+n)𝔥 d λ+μ+mγ+(m+n)Λ0 λ+mΛ0, μ+nΛ0 Θ λ+μ+mγ+(m+n)Λ0 , where d λ+μ+mγ+(m+n)Λ0 λ+mΛ0, μ+nΛ0 = κ𝔥 q 12 mn (m+n) 1mλ- 1nμ+ γ+(m+)κ 2 , (see [Kac, Ex. 13.1] or [KP, §13.2]). This formula gives the product structure on the graded Th˜0-algebra Th˜ = m0 Th˜m, where   Th˜m   has   Th˜0-basis   {Θλ_+mΛ0  |  λ_𝔥*  mod m𝔥}, WHAT IS THE RIGHT COMBINATORIAL DESCRIPTION OF THIS BASE RING? PUT THE G-ACTION ON Th˜ HERE?

Proof of the product formula for Th˜.

Θμ1+m1Λ0 Θμ2+m2Λ0 =( em1Λ0 γ1𝔥 e μ1+m1γ1- 1 2m1 μ1+m1γ12δ ) ( em2Λ0 γ2𝔥 e μ2+m2γ2- 1 2m2 μ2+m2γ22δ ) =e(m1+m2)Λ0 γ1-γ2𝔥 γ2𝔥 e μ1+μ2+m1(γ1-γ2)+(m1+m2)γ2 e - 1 2(m1+m2) μ1+μ2+m1γ1+m2γ22δ e 1 2(m1+m2) μ1+μ2+m1γ1+m2γ22δ e - 1 2m1 μ1+m1γ12δ e - 1 2m2 μ2+m2γ22δ = γ1-γ2𝔥 Θ μ1+μ2+m1(γ1-γ2)+(m1+m2)Λ0 e - (m1m2) 2(m1+m2) 1 m1 μ1- 1 m1 μ2+γ1-γ22δ , since 1 2(m1+m2) μ1+μ2+m1γ1+m2γ22 - 1 2m1 μ1+m1γ12 - 1 2m2 μ2+m2γ22 = 1 2(m1+m2) μ1+m1γ12 + 1 2(m1+m2) μ2+m2γ22 + 1 m1+m2 (μ1+m1γ1|μ2+m2γ2) - 1 2m1 μ1+m1γ12 - 1 2m2 μ2+m2γ22 =- m2 2m1m1+m2) μ1+m1γ12 - m1 2m2m1+m2) μ2+m2γ22 + 1 m1+m2 (μ1+m1γ1|μ2+m2γ2) =- m1m2 2(m1+m2) 1 m1 μ1+γ12 - m1m2 2(m1+m2) 1 m2 μ2+γ22 + m1m2 m1+m2 ( 1 m1 μ1+γ1| 1 m2 μ2+γ2 ) = -m1m2 2(m1+m2) 1 m1 μ1+γ1- 1 m2 μ2-γ22 = -m1m2 2(m1+m2) 1 m1 μ1 - 1 m2 μ2 +γ1 -γ22 . Thus Θμ1+m1Λ0 Θμ2+m2Λ0 = γ1-γ2𝔥 Θ μ1+μ2+m1(γ1-γ2)+(m1+m2)Λ0 e - (m1m2) 2(m1+m2) 1 m1 μ1- 1 m2 μ2+γ1-γ22δ = γ𝔥 Θ μ1+μ2+m1γ+(m1+m2)Λ0 e - (m1m2) 2(m1+m2) 1 m1 μ1- 1 m2 μ2+γ2δ = γ𝔥 mod(m1+m2)𝔥 Θ μ1+μ2+m1γ+(m1+m2)Λ0 κ𝔥 e - (m1m2) 2(m1+m2) 1 m1 μ1- 1 m2 μ2+γ+(m1+m2)κ2δ = γ𝔥 mod(m1+m2)𝔥 Θ μ1+μ2+m1γ+(m1+m2)Λ0 κ𝔥 e - 1 2m1m2(m1+m2) m2μ1-m1μ2+m1m2γ+m1m2(m1+m2)κ2δ = γ𝔥 mod(m1+m2)𝔥 Θ μ1+μ2+m1γ+(m1+m2)Λ0 ev0,0,c ( Θ m2μ1-m1μ2+m1m2γ+m1m2(m1+m2)Λ0 ).

The subring Th˜W0

The action of W0 on Th˜ induced by the action on -span {eλ  |  λ𝔥} given by weλ = ewλ is wΘλ =Θwλ, so that wΘλ_+mΛ0 = Θwλ_+mΛ0, for wW0, λ_𝔥 and m>0. Let Mλ = γW0λ Θλ and Aμ = wW0 det(w)Θwμ. Let Th˜W0 = {fTh˜  |  wf=f, for  wW0} and Th˜det = {fTh˜  |  wf=det(w)f, for  wW0}. Since {λP+ modδ, λ(c)=m} is a set of representatives of the W0-orbits on {λ_𝔥* modm𝔥} and Aμ=0 if μP+-P++, it follows (as in [KP, Prop. 4.3(d-e)]) that Th˜W0 has basis {Mλ  |  λP+ modδ, λ(c)=m}, and Th˜det has basis {Aλ+ρ  |  λ+ρP++ modδ, λ(c)=m}. Because of the Weyl character formula { Aλ+ρ Aρ  |  λ+ρP++ modδ, λ(c)=m } is a basis of   Th˜W0, and, since P+ P++ λ λ+ρ is a bijection, Th˜det is a free module of rank 1 over Th˜W0, generated by Aρ. In other words, as Th˜W0-modules Th˜W0 Th˜det f Aρf sλ Aλ+ρ 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.



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