## The Elliptic Weyl character formula: Examples of reductive group data

Last update: 02 March 2012

## Examples of reductive group data

Example. $G{L}_{n}:$ Let $𝔥∘ = ∑i=1n ℤεi∨ and 𝔥∘ℤ* = ∑i=1n ℤεi with ⟨εi,εj∨⟩ = δij.$ The symmetric group

Example. $S{L}_{n}:$ Let $𝔥∘ℤ = ℤ-span {ω1,...,ωn-1}, where ωi = ε1+⋯+εi-in (ε1+⋯+εn),$ with $⟨εi∨,εj⟩ = δij.$ The symmetric group

Example. $PG{L}_{n}:$ Let and let $𝔥∘ℤ = ℤ-span {ω1∨,...,ωn-1∨}, where ωi = ε1∨+⋯+εi∨-in (ε1∨+⋯+εn∨),$ with $⟨εi∨,εj⟩ = δij so that ⟨ωi∨,εj-εj+1⟩ = δij.$ The symmetric group

## Notes and References

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

References?