Affine root systems of classical type

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

Last update: 26 January 2013

Affine root systems

The notation for affine root systems is $W_{S} = {s_{a} | a \in S} and S = (S_{1}, S_{2}),$ where $S_{1} = {a \in S | \frac{1}{2} a \notin S} and S_{2} = {a \in S | 2 a \notin S} .$

Define $\begin{array}{rclrcl} O_{1} & = & {\pm ε_{i} + r | 1 \leq i \leq n, r \in ℤ} & O_{2} & = & {\pm 2 ε_{i} + 2 r | 1 \leq i \leq n, r \in ℤ} \\ O_{3} & = & {\pm ε_{i} + \frac{1}{2} (2 r + 1) | 1 \leq i \leq n, r \in ℤ} & O_{4} & = & {\pm 2 ε_{i} + 2 r + 1 | 1 \leq i \leq n, r \in ℤ} \\ O_{5}^{-} & = & {\pm (ε_{i} - ε_{j}) + r | 1 \leq i < j \leq n, r \in ℤ} & O_{5}^{+} & = & {\pm (ε_{i} + ε_{j}) + r | 1 \leq i < j \leq n, r \in ℤ} \end{array}$ and $O_{5} = O_{5}^{-} ⊔ O_{5}^{+}$ so that the affine root systems of classical type are missing Note that $ε_{n} \in O_{1}, 2 ε_{n} \in O_{2}, ε_{1} + \frac{1}{2} δ \in O_{3}, 2 ε_{1} + δ \in O_{4},$ and $ε_{i} - ε_{i + 1} \in O_{5}, for i = 1, ..., n - 1 .$

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

W_{S} = {\begin{cases} W_{C_{n}}, & for types  C_{n}, C_{n}^{\lor}, B C_{n}, (B C_{n}, C_{n}), (C_{n}^{\lor}, B C_{n}), (C_{n}^{\lor}, C_{n}) , \\ W_{B_{n}}, & for types  B_{n}, B_{n}^{\lor}, (B_{n}, B_{n}^{\lor}) , \\ W_{D_{n}}, & for type D_{n} \end{cases}

where

W_{C_{n}} = W_{0} ⋉ 𝔥_{ℤ} \supseteq W_{B_{n}} = W_{0} ⋉ {\overline{𝔥}}_{ℤ} \supseteq W_{D_{n}} = \overline{W_{0}} ⋉ {\overline{𝔥}}_{ℤ}

with

W_{0} = G_{2, 1, n}, \overline{W_{0}} = G_{2, 2, n}

and

𝔥_{ℤ} = \sum_{i = 1}^{n} ℤ ε_{i}, and {\overline{𝔥}}_{ℤ} = {λ_{1} ε_{1} + \dots + λ_{n} ε_{n} | λ_{1} + \dots + λ_{n} = 0 \mod 2},

SAY SOMETHING ABOUT

W_{S}

orbits on

S .

HOW DO THESE COMPARE TO KAC AND SAHI-ION???

When $n = 2$ define $\begin{array}{rclrcl} O_{1} & = & {\pm ε_{i} + r | 1 \leq i \leq 2, r \in ℤ} & O_{2} & = & {\pm 2 ε_{i} + 2 r | 1 \leq i \leq 2, r \in ℤ} \\ O_{3} & = & {\pm ε_{i} + \frac{1}{2} (2 r + 1) | 1 \leq i \leq 2, r \in ℤ} & O_{4} & = & {\pm 2 ε_{i} + 2 r + 1 | 1 \leq i \leq 2, r \in ℤ} \\ O_{5}^{-} & = & {\pm (ε_{1} - ε_{2}) + r | r \in ℤ} & O_{5}^{+} & = & {\pm (ε_{1} + ε_{2}) + r | r \in ℤ} \end{array}$ Then the "classical" affine root systems of rank 2 are

When

n = 1

define

\begin{array}{rclrcl} O_{1} & = & {\pm ε_{1} + r | r \in ℤ} & O_{2} & = & {\pm 2 ε_{1} + 2 r | r \in ℤ} \\ O_{3} & = & {\pm ε_{1} + \frac{1}{2} (2 r + 1) | r \in ℤ} & O_{4} & = & {\pm 2 ε_{1} + 2 r + 1 | r \in ℤ} \end{array}

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

References?

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