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. $G{L}_n:$ Let $$\stackrel{\circ }{𝔥}=\sum _{i=1}^n\mathbb{Z}{\epsilon }_i^{\vee }and{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*}=\sum _{i=1}^n\mathbb{Z}{\epsilon }_{i}with⟨{\epsilon }_i,{\epsilon }_j^{\vee }⟩={\delta }_{ij}.$$ The symmetric group $$W_0=S_{n}acts\; on\; {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*} \; by\; permuting\; {\epsilon }_1,...,{\epsilon }_n .$$

Example. $S{L}_n:$ Let $${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}=\mathbb{Z}-span\{{\omega }_1,...,{\omega }_{n-1}\},where{\omega }_i={\epsilon }_1+\cdots +{\epsilon }_i-\frac{i}{n}({\epsilon }_1+\cdots +{\epsilon }_n),$$ with $$⟨{\epsilon }_i^{\vee },{\epsilon }_j⟩={\delta }_{ij}.$$ The symmetric group $$W_0=S_{n}acts\; on\; {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*} \; by\; permuting\; {\epsilon }_1,...,{\epsilon }_n .$$

Example. $PG{L}_n:$ Let $${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*}=\{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n | {\lambda }_i\in \mathbb{Z} \; and\; {\lambda }_1+\cdots +{\lambda }_n=0\}so\; that{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}^{*}=\sum _{i=1}^{n-1}\mathbb{Z}({\epsilon }_1-{\epsilon }_{i+1})$$ and let $${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}=\mathbb{Z}-span\{{\omega }_1^{\vee },...,{\omega }_{n-1}^{\vee }\},where{\omega }_i={\epsilon }_1^{\vee }+\cdots +{\epsilon }_i^{\vee }-\frac{i}{n}({\epsilon }_1^{\vee }+\cdots +{\epsilon }_n^{\vee }),$$ with $$⟨{\epsilon }_i^{\vee },{\epsilon }_j⟩={\delta }_{ij}so\; that⟨{\omega }_i^{\vee },{\epsilon }_j-{\epsilon }_{j+1}⟩={\delta }_{ij}.$$ The symmetric group $$W_0=S_{n}acts\; on\; {\stackrel{\circ }{𝔥}}_{\mathbb{Z}} \; by\; permuting\; {\epsilon }_1^{\vee },...,{\epsilon }_n^{\vee } .$$

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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