## Geometric Lifting

Last update: 07 March 2012

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

Let ${n}_{i}^{-1}={x}_{i}\left(-1\right){y}_{i}\left(1\right){x}_{i}\left(-1\right)$ and ${n}_{{w}_{0}}^{-1}={n}_{{i}_{1}}^{-1}\cdots {n}_{{i}_{N}}^{-1}$ if ${w}_{0}={s}_{{i}_{1}}\cdots {s}_{{i}_{N}}.$ $yi(a) =x-αi(a) and tα∨ = hα∨(t)$ for $a\in ℂ$ and $t\in {ℂ}^{×}.$ Let $x i _ (t1,...,tm) = xi1(t1) ⋯ xim(tm) x- i _ (t1,...,tm) = yi1(t1)t-αi1∨ ⋯ yim(tm)t-αim∨$ and define $R˜ i _ j _ = x j _ -1 ∘ x i _ and R˜- i _ - j _ = x- j _ -1 ∘ x- i _ .$

For $x∈U-B = U-TU,$ the big cell, write $x=[x]- [x]0 [x]+.$ Let ${\phantom{0}}^{T}:G\to G$ be the antiautomorphism given by Define $ηw0,e(x) = [ ( nw0-1xT )-1 ]+$

[BZ, Inv 2001] See [Mo-Ge Thm 3.2.2, and Thm 3.2.5(a)]. where 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 ${R}_{\underset{_}{i}}^{\underset{_}{j}}$

Examples 3.2.3 and 3.2.4 in [Mo-Ge]: $x1(t) = 1 t 0 0 1 0 0 0 1 x2(t) = 1 0 0 0 t 0 0 0 1$ $x121(t) = x1(t1) x2(t2) x3(t3) = 1 t1+t3 t1t2 0 1 t2 0 0 1 x212(t) = x2(t1') x1(t2') x3(t3') = 1 t2' t2't3' 0 1 t1'+t3' 0 0 1$ So $x-121(t1,t2,t3) = y1(t1) t1-α1∨ y2(t2) t2-α2∨ y1(t3) t3-α1∨ = t1-1t3-1 0 0 t3-1+t1t2-1 t1t3t2-1 0 1 t3 t2 x-212(t1',t2',t3') = t2'-1 0 0 t1'-1 t2t1'-1t3'-1 0 1 t2't3'-1+t1'-1 t1't3' so that R˜-121-212 (t1,t2,t3) = ( t2t3 t2+t1t3 , t1t3, t1+t1t3 t3 )$ $nw0-1 = 0 0 1 0 -1 0 1 0 0 , x-121 (t1,t2,t3) = t1-1t3-1 0 0 t3-1+t1-1t2-1 t1t2-1t3 0 1 t3 t2$ Then $( nw0-1x-121 ( t1,t2,t3 )T )-1 = 1 (t2+t1t3)t3-1 t1t3 -t1-1 -t2t1-1t3-1 0 t2-1 0 0 = y1 ( -t2t1-1(t2+t1t3) ) y2 ( -(t2+t1t3)t2-1t3-1 ) y1 ( -t3( t2+t1t3 )-1 ) ⋅ x1(t1) x2(t3) x1(t2t3-1).$ So $R-121121 (t1,t2,t3) = (t1,t3,t2t3-1).$ Explicit formulas for rank 2 ${\stackrel{˜}{R}}_{\underset{_}{i}}^{\underset{_}{j}}$ are found in [Mo-Ge, App.2] where they are taken from [Berenstein-Zelevinsky, Comm. Math. Helvetici, 72 (1997)].

## Notes and References

Where are these from?

References?
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