Arun Ram: Loop groups

Loop groups

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 24 June 2007. This page is the result of joint work with James Parkinson and Christoph Schwer.

Loop groups and the Iwahori subgroup

A common setup is where $𝔽$ is the field of fractions of $𝔬$ , the discrete valuation ring $𝔬$ is the ring of integers in $𝔽$ , $𝔭$ is the unique maximal ideal in $𝔬$ , and $k = 𝔬 / 𝔭$ is the residue field. The favourite examples are

\begin{array}{lll} 𝔽 = ℂ ((t)) & 𝔬 = ℂ [[t]] & k = ℂ, \\ 𝔽 = ℚ_{p} & 𝔬 = ℤ_{p} & k = 𝔽_{p}, \\ 𝔽 = 𝔽_{q} ((t)) & 𝔬 = 𝔽_{q} [[t]] & k = 𝔽_{q}, \end{array}

where

ℚ_{p}

is the field of

p

-adic numbers,

ℤ_{p}

is the ring of

p

-adic integers, and

𝔽_{q}

is the finite field with

q

elements. The diagram

\begin{matrix} 𝔽 \\ \cup ∣ \\ 𝔬 & ⟶ & k = 𝔬 / 𝔭 \end{matrix}

gives

\begin{matrix} G & = & G (𝔽) \\ \cup ∣ & \cup ∣ \\ K & = & G (𝔬) & \overset{Φ}{⟶} & G (k) \\ \cup ∣ & \cup ∣ & \cup ∣ \\ I & = & Φ^{- 1} (B (k)) & ⟶ & B (k), \end{matrix}

where

B (k) = ⟨ x_{α} (c), h_{λ^{\lor}} (d) ∣ α \in R^{+}, λ^{\lor} \in P^{\lor}, c \in k, d \in k^{\times} ⟩

. Then

I

is the Iwahori subgroup of

G

$G (k) / B (k)$ is the flag variety,
$G / I$ is the affine flag variety, and
$G / K$ is the loop Grassmanian.

Let

x_{α + k δ} (c) = x_{α} (c t^{k}) .

Then

n_{α + k δ} (c) = x_{α + k δ} (c) x_{- α - k δ} (- c^{- 1}) x_{α + k δ} (c) = n_{α} (c t^{k}) = h_{α^{\lor}} (t^{k}) n_{α} (c)

so that

n_{α + k δ} = h_{α^{\lor}} (t^{k}) n_{α}

. Then

n_{β} x_{α + k δ} (c) n_{β}^{- 1} = n_{β} x_{α} (c t^{k}) n_{β}^{- 1} = x_{s_{β} α} (c t^{k}) = x_{s_{β} α + k δ} (c),

and

h_{λ^{\lor}} (t^{k}) x_{α + ℓ δ} (c) h_{λ^{\lor}} (t^{k})^{- 1} = x_{β + ℓ δ} (t^{k ⟨ λ^{\lor}, β ⟩} c) = x_{β + (ℓ + k ⟨ λ^{\lor}, β ⟩) δ} (c) .