## Affine root systems of classical type

Last update: 26 January 2013

## Affine root systems

The notation for affine root systems is where

Define and ${O}_{5}={O}_{5}^{-}\bigsqcup {O}_{5}^{+}$ so that the affine root systems of classical type are missing Note that and

Where the left notation is the notation in [M03, §1.3] and the right notation is that of [BT]. The Weyl groups of these are where $WCn = W0⋉𝔥ℤ ⊇ WBn = W0⋉𝔥_ℤ ⊇ WDn = W0_⋉𝔥_ℤ$ with $W0=G2,1,n, W0_=G2,2,n$ and SAY SOMETHING ABOUT ${W}_{S}$ orbits on $S.$ HOW DO THESE COMPARE TO KAC AND SAHI-ION???

When $n=2$ define Then the "classical" affine root systems of rank 2 are

When $n=1$ define Then the "classical" affine root systems of rank 1 are

## Notes and References

These notes are from a study of Macdonald polynomials of type C in 2008-2009.

References?