The Elliptic Weyl character formula: Examples of reductive group data

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

Last update: 02 March 2012

Examples of reductive group data

Example. GLn: Let 𝔥 = i=1n εi and 𝔥* = i=1n εi with εi,εj = δij. The symmetric group W0=Sn acts on   𝔥*   by permuting   ε1...εn.

Example. SLn: Let 𝔥 = -span {ω1,...,ωn-1}, where ωi = ε1++εi-in (ε1++εn), with εi,εj = δij. The symmetric group W0=Sn acts on   𝔥*   by permuting   ε1...εn.

Example. PGLn: Let 𝔥* = {λ1ε1++λnεn  |  λi   and   λ1++λn=0} so that 𝔥* = i=1n-1 (ε1-εi+1) and let 𝔥 = -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 W0=Sn acts on   𝔥   by permuting   ε1...εn.

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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