The calculus of BGG operators

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

Last update: 17 February 2013

The calculus of BGG operators

The nil affine Hecke algebra is the algebra over 𝕃 with generators xλ, yλ, tw, with λ,μ𝔥* and wW0, 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,forv,wW0 ,λ𝔥*.

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

Ai=(1+tsi) 1x-αi, fori=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 Ai 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 f𝕃 [ [ xλ λ𝔥* ] ] 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 tsi in terms of products of Ai with xs 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 tsi in terms of products of Ai with xs 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 Ai in terms of products of tsi:

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 ts2 and A2 are obtained from the above formulas by switching 1s and 2s. In particular, the “braid relations” for the operators Ai are the equations given by, for example, in the case that s1s2s1= s2s1s2 so that s1α2=s2α1 =α1+α2 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.

page history