## The calculus of BGG operators

Last update: 16 September 2012

## The calculus of BGG operators

We work in the ring

$R=𝕃 [ [ xλ∣ λ∈𝔥ℤ* ] ] with xλ+μ=xλ+ xμ-p (xλ,xμ) xλxμ,$

with $p\left({x}_{\lambda },{x}_{\mu }\right)\in 𝕃\left[\left[{x}_{\lambda },{x}_{\mu }\right]\right]$ a power series

$p(xλ,xμ)= -a11-a12 xμ-a21xλ -a31xλ2- a22xλxμ- a13xμxλ- …,$

with ${a}_{ij}\in 𝕃$ satisfying relations such that

$x-λ+λ=x0= 0,xλ+μ, x(λ+μ)+ν= xλ+(μ+ν). (1)$

Then

$xα= -x-α 1-p(xα,x-α) x-α , 1 x-α +1xα=p (xα,x-α), (2)$

and the formula

$x-ℓα x-α =ℓ-∑j=1ℓ-1 p(x-α,x-jα) x-jα=1+ ∑j=1ℓ-1 ( 1-p (x-α,x-jα) x-jα ) , forℓ∈ℤ>0, (3)$

is proved by induction on $\ell \text{.}$ The formula

$xsiλ -xλ x-αi = ( 1-p ( xλ, x⟨λ,αi∨⟩αi ) xλ ) ( 1+ ∑ j=1 ⟨λ,α∨⟩-1 ( 1-p ( x-αi, x-jαi ) x-jαi ) ) , (4)$

for $⟨\lambda ,{\alpha }_{i}^{\vee }⟩\in {ℤ}_{\ge 0},$ generalizes one of the favourite formulas for the action of a Demazure operator REFERENCE FOR THIS!!!.

The nil affine Hecke algebra is the algebra over $𝕃$ with generators ${x}_{\lambda },$ ${y}_{\lambda },$ ${t}_{w},$ with $\lambda ,\mu \in {𝔥}_{ℤ}^{*}$ and $w\in {W}_{0},$ with relations

$xλ+μ=xλ+xμ -p(xλ,xμ) xλxμ, yλ+μ=yλ+ yμ-p (yλ,yμ) yλyμ, xλyμ=yμxλ,$

and

$tvtw=twtv, twyλ=yλtw, twxλ=xwλ tw,forv,w∈ W0,λ∈𝔥ℤ*.$

Recall from (2.13) that the pushpull operators, or BGG–Demazure operators are given by

$Ai= (1+tsi) 1x-αi, fori=1,2,…,n. (5)$

In general,

$Ai = (1+tsi) 1x-αi= 1x-αi+ 1xαi tsi= 1x-αi- 1-p ( xαi, x-αi ) x-αi x-αi tsi = 1x-αi ( 1- ( 1-p ( xαi, x-αi ) x-αi ) tsi ) = 1x-αi (1-tsi)+p ( xαi, x-αi ) tsi. (6)$

so that ${A}_{i}$ is a divided difference operator plus an extra term. As in [BE1, Prop. 3.1],

$Ai2 = (1+tsi) 1x-αi (1+tsi) 1x-αi = ( 1x-αi + 1xαitsi ) (1+tsi) 1x-αi = ( 1x-αi+ 1xαi ) (1+tsi) 1x-αi= ( 1x-αi +1xαi ) Ai,$

so that

$Ai2= ( 1x-αi+ 1xαi ) Ai=Ai ( 1x-αi +1xαi ) =Aip ( xαi, x-αi ) . (7)$

Note also that

$tsiAi = tsi (1+tsi) 1x-αi =Aiand (8) Aitsi = (1+tsi) 1x-αi tsi= (1+tsi) 1xαi= Ai x-αi xαi (9)$

If $f\in L\left[\left[{x}_{\lambda }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in {𝔥}_{ℤ}^{*}\right]\right]$ then

$fAi = f(1+tsi) 1x-αi= f1x-αi +ftsi 1x-αi and Ai(sif) = (1+tsi) sif x-αi = ( sif+f tsi ) 1x-αi,$

so that

$fAi=Ai (sif)+ ( f-sif x-αi ) ,forf∈R, (10)$

The relation (10) is the same as a key relation in the definition of the classical nil-affine Hecke algebra. (KEEP THIS COMMENT IN???) The right hand side of (10) motivates the definition of operators

$Bαf= f-sαf x-α , for rootsα. ThentwBα tw-1= Bwα, (11)$

for $w\in {W}_{0}\text{.}$ The calculus of these divided difference operators will be useful for computations in $R⋊R\left[{W}_{0}\right]\text{.}$

Next are useful, expansions of products of ${t}_{{s}_{i}}$ in terms of products of ${A}_{i}$ with $x$s on the left,

$ts1 = xα1A1- xα1 x-α1 , ts2ts1 = xs2α1 xα2A2A1 -xs2α1 xα2 x-α2 A1- xs2α1 x-s2α1 xα2A2+ xs2α1 x-s2α1 xα2 x-α2 ts1 ts2 ts1 = xs1s2α1 xs1α2 xα1A1A2 A1- xs1s2α1 xs1α2 xα1 x-α1 A2A1- xs1s2α1 x-s1s2α1 xs1α2 xα1A1A2 + xs2s1α2 x-s2s1α2 xs2α1 xα2 x-α2 A1+ xs1s2α1 x-s1s2α1 xs1α2 xα1 x-α1 A2- xs1s2α1 x-s1s2α1 xs1α2 x-s1α2 xα1 x-α1 + ( xs1α2 x-s1α2 xs1s2α1 x-s1s2α1 xα1- xs1α2 x-s1α2 xs1s2α1- xs2s1α2 x-s2s1α2 xs2α1 xα2 x-α2 ) A1 ts1 ts2 ts1 ts2 = xs2s1s2α1 xs2s1α2 xs2α1 xα2A2A1 A2A1 -xs2s1s2α1 xs2s1α2 xs2α1 xα2 x-α2 A1A2A1- xs2s1s2α1 x-s2s1s2α1 xs2s1α2 xs2α1 xα2A2A1A2 + xs2s1s2α1 x-s2s1s2α1 xs2s1α2 xs2α1 xα2 x-α2 A1A2 + ( xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1 xα2- xs2s1s2α1 xs2s1α2 x-s2s1α2 xα2- xs2s1s2α1 xs2s1α2 xs2α1 x-s2α1 ) A2A1 - ( xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1- xs2s1s2α1 xs2s1α2 x-s2s1α2 ) xα2 x-α2 A1 + ( xs2s1s2α1 x-s2s1s2α1 xs2s1α2 xs2α1 x-s2α1 - xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1 x-s2α1 xα2 ) A2 + xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1 x-s2α1 xα2 x-α2 ,$

and expansions of products of ${t}_{{s}_{i}}$ in terms of products of ${A}_{i}$ with $x$s on the right,

$ts1 = A1x-α1-1, ts1ts2 = A1A2x-α2 x-s2α1- A1x-s2α1 -A2x-α2+1, ts1ts2ts1 = A1A2A1 x-α1 x-s1α2 x-s1s2α1 -A1A2 x-s1α2 x-s1s2α1 -A2A1 x-α1 x-s1α2 +A1 x-s2α1+ A2x-s1α2 -1+A1 ( x-α1- x-s2α1- x-α1 xα1 x-s1s2α1 ) , ts1ts2 ts1ts2 = A1A2A1A2 x-α2 x-s2α1 x-s2s1α2 x-s2s1s2α1 -A1A2A1 x-s2α1 x-s2s1α2 x-s2s1s2α1 -A2A1A2 x-α2 x-s2α1 x-s2s1α2 +A1A2 ( - x-α2 xα2 x-s2s1α2 x-s2s1s2α1 -x-α2 x-s2α1 xs2α1 x-s2s1s2α1 +x-α2 x-s2α1 ) +A2A1 x-s2α1 x-s2s1α2 -A1 ( x-s2α1- x-s2α1 xs2α1 x-s2s1s2α1 ) -A2 ( x-α2- x-α2 xα2 x-s2s1α2 ) +1.$

Finally, there are expansions of products of ${A}_{i}$ in terms of products of ${t}_{{s}_{i}}:$

$A1 = (ts1+1) 1x-α1, A1A2 = (ts1+1) ( ts2 1 x-α2 x-s2α1 + 1 x-α1 x-α2 ) , A1A2A1 = (ts1+1) ( ts2ts1 1 x-α1 x-s1α2 x-s1s2α1 +ts2 1 x-α1 x-α2 x-s2α1 +1x-α1 ( 1 x-α1 x-α2 + 1 x-s1α1 x-s2α1 ) ) , A1A2A1A2 = (ts1+1) ( ts2ts1 ts2 1 x-α2 x-s2α1 x-s2s1α2 x-s2s1s2α1 + ts2ts1 1 x-α2 x-α1 x-s1α2 x-s1s2α1 +ts2 1 x-α2 x-s2α1 ( 1 x-α2 x-α1 + 1 x-s2α1 x-s2α2 + 1 x-s2s1α2 x-s2s1α1 ) + 1 x-α1 x-α2 ( 1 x-α2 x-α1 + 1 x-s2α1 x-s2α2 + 1 x-s1α2 x-s1α1 ) ) ,$

These formulas arranged so that products beginning with ${t}_{{s}_{2}}$ and ${A}_{2}$ are obtained from the above formulas by switching 1s and 2s. In particular, the "braid relations" for the operators ${A}_{i}$ are the equations given by, for example, in the case that ${s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2}$ so that ${s}_{1}{\alpha }_{2}={s}_{2}{\alpha }_{1}={\alpha }_{1}+{\alpha }_{2}$ then

$0=ts1ts2 ts1-ts2 ts1ts2$

is equivalent to

$A2A1A2- ( 1 x-α2 x-α1 - 1 x-α1 x-α3 + 1 xα2 x-α3 ) A2 = A1A2A1- ( 1 x-α1 x-α2 - 1 x-α2 x-α3 + 1 xα1 x-α3 ) A1,$

as indicated in [HLSZ, Proposition 5.7].