Geometric Lifting

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

Last update: 07 March 2012

Geometric Lifting (Following [Mo-Ge §3.2])

Let $n_{i}^{- 1} = x_{i} (- 1) y_{i} (1) x_{i} (- 1)$ and $n_{w_{0}}^{- 1} = n_{i_{1}}^{- 1} \dots n_{i_{N}}^{- 1}$ if $w_{0} = s_{i_{1}} \dots s_{i_{N}} .$ $y_{i} (a) = x_{- α_{i}} (a) and t_{α^{\lor}} = h_{α^{\lor}} (t)$ for $a \in ℂ$ and $t \in ℂ^{\times} .$ Let $\begin{array}{rcl} x_{\underline{i}} (t_{1}, ..., t_{m}) & = & x_{i_{1}} (t_{1}) \dots x_{i_{m}} (t_{m}) \\ x_{- \underline{i}} (t_{1}, ..., t_{m}) & = & y_{i_{1}} (t_{1}) t^{- α_{i_{1}}^{\lor}} \dots y_{i_{m}} (t_{m}) t^{- α_{i_{m}}^{\lor}} \end{array}$ and define ${\tilde{R}}_{\underline{i}}^{\underline{j}} = x_{\underline{j}}^{- 1} \circ x_{\underline{i}} and {\tilde{R}}_{- \underline{i}}^{- \underline{j}} = x_{- \underline{j}}^{- 1} \circ x_{- \underline{i}} .$

For $x \in U^{-} B = U^{-} T U,$ the big cell, write $x = {[x]}_{-} {[x]}_{0} {[x]}_{+} .$ Let $^{T} : G \to G$ be the antiautomorphism given by $x_{i} {(t)}^{T} = y_{i} (t), y_{i} {(t)}^{T} = x_{i} (t) and h_{α_{i}^{\lor}} {(t)}^{T} = h_{α_{i}^{\lor}} (t) .$ Define $η^{w_{0}, e} (x) = {[{(n_{w_{0}}^{- 1} x^{T})}^{- 1}]}_{+}$

[BZ, Inv 2001] See [Mo-Ge Thm 3.2.2, and Thm 3.2.5(a)]. $\begin{array}{rcl} {[{({\tilde{R}}_{\underline{i}}^{\underline{j}})}^{\lor}]}_{Tr} & = & R_{\underline{i}}^{\underline{j}}, \\ {[{({\tilde{R}}_{\underline{i}}^{- \underline{j}})}^{\lor}]}_{Tr} & = & R_{- \underline{i}}^{- \underline{j}}, and \\ R_{- \underline{i}}^{\underline{j}} & = & {[{(x_{- \underline{i}}^{- 1} \circ η^{w_{0}, e} \circ x_{\underline{j}})}^{\lor}]}_{Tr} \end{array}$ where ${()}^{\lor}$ means that the formulas are considered in the Langlands dual of $G .$

See examples 3.2.3 and 3.2.4 in [Mo-Ge].

Geometric lifts of $R_{\underline{i}}^{\underline{j}}$

Examples 3.2.3 and 3.2.4 in [Mo-Ge]: $x_{1} (t) = (\begin{matrix} 1 & t & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) x_{2} (t) = (\begin{matrix} 1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix})$ $\begin{array}{rcccl} x_{121} (t) & = & x_{1} (t_{1}) x_{2} (t_{2}) x_{3} (t_{3}) & = & (\begin{matrix} 1 & t_{1} + t_{3} & t_{1} t_{2} \\ 0 & 1 & t_{2} \\ 0 & 0 & 1 \end{matrix}) \\ x_{212} (t) & = & x_{2} (t_{1}') x_{1} (t_{2}') x_{3} (t_{3}') & = & (\begin{matrix} 1 & t_{2}' & t_{2}' t_{3}' \\ 0 & 1 & t_{1}' + t_{3}' \\ 0 & 0 & 1 \end{matrix}) \end{array}$ So $\begin{array}{rcl} {\tilde{R}}_{121}^{212} & = & (\frac{t_{2} t_{3}}{t_{1} + t_{3}}, t_{1} + t_{3}, \frac{t_{1} t_{2}}{t_{1} + t_{3}}) \\ t^{- α_{1}^{\lor}} = (\begin{matrix} t^{- 1} & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix}) & and & t^{- α_{2}^{\lor}} = (\begin{matrix} 1 & 0 & 0 \\ 0 & t^{- 1} & 0 \\ 0 & 0 & t \end{matrix}) \end{array}$ $\begin{array}{rcl} x_{- 121} (t_{1}, t_{2}, t_{3}) & = & y_{1} (t_{1}) t_{1}^{- α_{1}^{\lor}} y_{2} (t_{2}) t_{2}^{- α_{2}^{\lor}} y_{1} (t_{3}) t_{3}^{- α_{1}^{\lor}} \\ = & (\begin{matrix} t_{1}^{- 1} t_{3}^{- 1} & 0 & 0 \\ t_{3}^{- 1} + t_{1} t_{2}^{- 1} & t_{1} t_{3} t_{2}^{- 1} & 0 \\ 1 & t_{3} & t_{2} \end{matrix}) \\ x_{- 212} (t_{1}', t_{2}', t_{3}') & = & (\begin{matrix} {t_{2}'}^{- 1} & 0 & 0 \\ {t_{1}'}^{- 1} & t_{2} {t_{1}'}^{- 1} {t_{3}'}^{- 1} & 0 \\ 1 & t_{2}' {t_{3}'}^{- 1} + {t_{1}'}^{- 1} & t_{1}' t_{3}' \end{matrix}) so that \\ {\tilde{R}}_{- 121}^{- 212} (t_{1}, t_{2}, t_{3}) & = & (\frac{t_{2} t_{3}}{t_{2} + t_{1} t_{3}}, t_{1} t_{3}, \frac{t_{1} + t_{1} t_{3}}{t_{3}}) \end{array}$ $n_{w_{0}}^{- 1} = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}), x_{- 121} (t_{1}, t_{2}, t_{3}) = (\begin{matrix} t_{1}^{- 1} t_{3}^{- 1} & 0 & 0 \\ t_{3}^{- 1} + t_{1}^{- 1} t_{2}^{- 1} & t_{1} t_{2}^{- 1} t_{3} & 0 \\ 1 & t_{3} & t_{2} \end{matrix})$ Then $\begin{array}{rcl} {(n_{w_{0}}^{- 1} x_{- 121} {(t_{1}, t_{2}, t_{3})}^{T})}^{- 1} & = & (\begin{matrix} 1 & (t_{2} + t_{1} t_{3}) t_{3}^{- 1} & t_{1} t_{3} \\ - t_{1}^{- 1} & - t_{2} t_{1}^{- 1} t_{3}^{- 1} & 0 \\ t_{2}^{- 1} & 0 & 0 \end{matrix}) \\ = & y_{1} (- t_{2} t_{1}^{- 1} (t_{2} + t_{1} t_{3})) y_{2} (- (t_{2} + t_{1} t_{3}) t_{2}^{- 1} t_{3}^{- 1}) y_{1} (- t_{3} {(t_{2} + t_{1} t_{3})}^{- 1}) \\ \cdot x_{1} (t_{1}) x_{2} (t_{3}) x_{1} (t_{2} t_{3}^{- 1}) . \end{array}$ So $R_{- 121}^{121} (t_{1}, t_{2}, t_{3}) = (t_{1}, t_{3}, t_{2} t_{3}^{- 1}) .$ Explicit formulas for rank 2 ${\tilde{R}}_{\underline{i}}^{\underline{j}}$ are found in [Mo-Ge, App.2] where they are taken from [Berenstein-Zelevinsky, Comm. Math. Helvetici, 72 (1997)].

Notes and References

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References

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