## The calculus of BGG operators

Last update: 17 February 2013

## The calculus of BGG operators

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

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

and

$tvtw=tvw, twyλ=yλtw, twxλ=xwλ tw,for v,w∈W0 ,λ∈𝔥ℤ*.$

Recall from (4.2) that the pushpull operators, or BGG-Demazure operators are given by

$Ai=(1+tsi) 1x-αi, for i=1,2,…,n. (8.1)$

In general,

$Ai = (1+tsi) 1x-αi= 1x-αi+ 1xαi tsi= 1x-αi- 1-p ( xαi, x-αi ) x-αi x-αi tsi = 1x-αi ( 1- ( 1-p ( xαi, x-αi ) x-αi ) tsi ) =1x-αi (1-tsi)+p ( xαi, x-αi ) tsi. (8.2)$

so that ${A}_{i}$ is a divided difference operator plus an extra term. As in [BEv0968883, Prop. 3.1],

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

so that

$Ai2= ( 1x-αi- 1xαi ) Ai=ai ( 1x-αi- 1xαi ) =Aip ( xαi, x-αi ) . (8.3)$

Note also that

$tsiAi = tsi (1+tsi) 1x-αi =Aiand (8.4) Aitsi = (1+tsi) 1x-αi tsi= (1+tsi) 1xαi =Ai x-αi xαi . (8.5)$

If $f\in 𝕃\left[\left[{x}_{\lambda } \mid \lambda \in {𝔥}_{ℤ}^{*}\right]\right]$ then

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

so that

$fAi=Ai (sif)+ ( f-sif x-αi ) . (8.6)$

The relation (8.6) is the analogue, for this setting, of a key relation in the definition of the classical nil-affine Hecke algebra (see [CGi1433132, Lemma 7.1.10] or [GRa0405333, (1.3)]).

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

$ts1 = xα1A1- xα1 x-αi , ts2 ts1 = xs2α1 xα2 A2A1- xs2α1 xα2 x-α2 A1- xs2α1 x-s2α1 xα2 A2+ xs2α1 x-s2α1 xα2 x-α2 ts1 ts2 ts1 = xs1s2α1 xs1α2 xα1 A1A2A1- xs1s2α1 xs1α2 xα1 x-α1 A2A1- xs1s2α1 x-s1s2α1 xs1α2 xα1A1A2 + xs2s1α2 x-s2s1α2 xs2α1 xα2 x-α2 A1+ xs1s2α1 x-s1s2α1 xs1α2 xα1 x-α1 A2- xs1s2α1 x-s1s2α1 xs1α2 x-s1α2 xα1 x-α1 + ( xs1α2 x-s1α2 xs1s2α1 x-s1s2α1 xα1- xs1α2 x-s1α2 xs1s2α1- xs2s1α2 x-s2s1α2 xs2α1 xα2 x-α2 ) A1 ts1ts2 ts1ts2 = xs2s1s2α1 xs2s1α2 xs2α1 xα2A2A1 A2A1 - xs2s1s2α1 xs2s1α2 xs2α1 xα2 x-α2 A1A2A1- xs2s1s2α1 x-s2s1s2α1 xs2s1α2 xs2α1 xα2 A2A1A2 + xs2s1s2α1 x-s2s1s2α1 xs2s1α2 xs2α1 xα2 x-α2 A1A2 + ( xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1 xα2- xs2s1s2α1 xs2s1α2 x-s2s1α2 xα2- xs2s1s2α1 xs2s1α2 xs2α1 x-s2α1 ) A2A1 - ( xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1- xs2s1s2α1 xs2s1α2 x-s2s1α2 ) xα2 x-α2 A1 + ( xs2s1s2α1 x-s2s1s2α1 xs2s1α2 xs2α1 x-s2α1 - xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1 x-s2α1 xα2 ) A2 + xs2s1s2α1 x-s2s1s2α1 xs2s1α2 x-s2s1α2 xs2α1 x-s2α1 xα2 x-α2 ,$

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

$ts1 = A1x-α1-1, ts1ts2 = A1A2 x-α2 x-s2α1- A1 x-s2α1- A2x-α2+1, ts1 ts2 ts1 = A1A2A1 x-α1 x-s1α2 x-s1s2α1 -A1A2 x-s1α2 x-s1s2α1 -A2A1 x-α1 x-s1α2 + A1x-s2α1 +A2x-s1α2 -1+A1 ( x-α1- x-s2α1- x-α1 xα1 x-s1s2α1 ) , ts1ts2 ts1ts2 = A1A2 A1A2 x-α2 x-s2α1 x-s2s1α2 x-s2s1s2α1 - A1A2A1 x-s2α1 x-s2s1α2 x-s2s1s2α1- A2A1A2 x-α2 x-s2α1 x-s2s1α2 + A1A2 ( - x-α2 xα2 x-s2s1α2 x-s2s1s2α1- x-α2 x-s2α1 xs2α1 x-s2s1s2α1+ x-α2 x-s2α1 ) + A2A1 x-s2α1 x-s2s1α2 - A1 ( x-s2α1- x-s2α1 xs2α1 x-s2s1s2α1 ) -A2 ( x-α2- x-α2 xα2 x-s2s1α2 ) +1.$

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

$A1 = (ts1+1) 1x-α1, A1A2 = (ts1+1) ( ts2 1 x-α2 x-s2α1 + 1 x-α1 x-α2 ) , A1A2A1 = (ts1+1) ( ts2ts1 1 x-α1 x-s1α2 x-s1s2α1 +ts2 1 x-α1 x-α2 x-s2α1 + 1x-α1 ( 1 x-α1 x-α2 + 1 x-s1α1 x-s2α1 ) ) , A1A2 A1A2 = (ts1+1) ( ts2ts1ts2 1 x-α2 x-s2α1 x-s2s1α2 x-s2s1s2α1 +ts2ts1 1 x-α2 x-α1 x-s1α2 x-s1s2α1 +ts2 1 x-α2 x-s2α1 ( 1 x-α2 x-α1 + 1 x-s2α1 x-s2α2 + 1 x-s2s1α2 x-s2s1α1 ) + 1 x-α1 x-α2 ( 1 x-α2 x-α1 + 1 x-s2α1 x-s2α2 + 1 x-s1α2 x-s1α1 ) ) .$

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 [HLS1208.4114, Proposition 5.7].

## Notes and References

This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.