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: 16 September 2012

The calculus of BGG operators

We work in the ring

$$R=𝕃\left[[x_{\lambda }\mid \lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right]\text{with}{x}_{\lambda +\mu }=x_{\lambda }+x_{\mu }-p(x_{\lambda },x_{\mu })x_{\lambda }{x}_{\mu },$$

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

$$p(x_{\lambda },x_{\mu })=-a_{11}-a_{12}{x}_{\mu }-a_{21}{x}_{\lambda }-a_{31}{x}_{\lambda }^2-a_{22}{x}_{\lambda }{x}_{\mu }-a_{13}{x}_{\mu }{x}_{\lambda }-\dots ,$$

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

$$\begin{array}{cc}{x}_{-\lambda +\lambda }=x_0=0,x_{\lambda +\mu },x_{(\lambda +\mu )+\nu }=x_{\lambda +(\mu +\nu )}\text{.}& \text{(1)}\end{array}$$

Then

$$\begin{array}{cc}{x}_{\alpha }=\frac{-x_{-\alpha }}{1-p(x_{\alpha },x_{-\alpha })x_{-\alpha }},\frac{1}{x_{-\alpha }}+\frac{1}{x_{\alpha }}=p(x_{\alpha },x_{-\alpha }),& \text{(2)}\end{array}$$

and the formula

$$\begin{array}{cc}\frac{x_{-\ell \alpha }}{x_{-\alpha }}=\ell -\sum _{j=1}^{\ell -1}p(x_{-\alpha },x_{-j\alpha })x_{-j\alpha }=1+\sum _{j=1}^{\ell -1}(1-p(x_{-\alpha },x_{-j\alpha })x_{-j\alpha }),\text{for}\ell \in {\mathbb{Z}}_{>0},& \text{(3)}\end{array}$$

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

$$\begin{array}{cc}\frac{x_{s_i\lambda }-x_{\lambda }}{x_{-{\alpha }_i}}=(1-p(x_{\lambda },x_{⟨\lambda ,{\alpha }_i^{\vee }⟩{\alpha }_i})x_{\lambda })(1+\sum _{j=1}^{⟨\lambda ,{\alpha }^{\vee }⟩-1}(1-p(x_{-{\alpha }_i},x_{-j{\alpha }_i})x_{-j{\alpha }_i})),& \text{(4)}\end{array}$$

for $⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{\ge 0},$ 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 $\lambda ,\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and $w\in W_0,$ with relations

$$x_{\lambda +\mu }=x_{\lambda }+x_{\mu }-p(x_{\lambda },x_{\mu })x_{\lambda }{x}_{\mu },y_{\lambda +\mu }=y_{\lambda }+y_{\mu }-p(y_{\lambda },y_{\mu })y_{\lambda }{y}_{\mu },x_{\lambda }{y}_{\mu }=y_{\mu }{x}_{\lambda },$$

and

$$t_v{t}_w=t_w{t}_v,t_w{y}_{\lambda }=y_{\lambda }{t}_w,t_w{x}_{\lambda }=x_{w\lambda }{t}_w,\text{for}v,w\in W_0,\lambda \in {𝔥}_{\mathbb{Z}}^{*}\text{.}$$

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

$$\begin{array}{cc}{A}_i=(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}},\text{for}i=1,2,\dots ,n\text{.}& \text{(5)}\end{array}$$

In general,

$$\begin{array}{ccc}{A}_i& =& (1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}=\frac{1}{x_{-{\alpha }_i}}+\frac{1}{x_{{\alpha }_i}}{t}_{s_i}=\frac{1}{x_{-{\alpha }_i}}-\frac{1-p(x_{{\alpha }_i},x_{-{\alpha }_i})x_{-{\alpha }_i}}{x_{-{\alpha }_i}}{t}_{s_i}\\ & =& \frac{1}{x_{-{\alpha }_i}}(1-(1-p(x_{{\alpha }_i},x_{-\alpha i})x_{-\alpha i})t_{s_i})=\frac{1}{x_{-{\alpha }_i}}(1-t_{s_i})+p(x_{{\alpha }_i},x_{-{\alpha }_i})t_{s_i}\text{.}& \text{(6)}\end{array}$$

so that $A_i$ is a divided difference operator plus an extra term. As in [BE1, Prop. 3.1],

$$\begin{array}{ccc}{A}_i^2& =& (1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}=(\frac{1}{x_{-{\alpha }_i}}+\frac{1}{x_{{\alpha }_i}}{t}_{s_i})(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}\\ & =& (\frac{1}{x_{-{\alpha }_i}}+\frac{1}{x_{{\alpha }_i}})(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}=(\frac{1}{x_{-{\alpha }_i}}+\frac{1}{x_{{\alpha }_i}})A_i,\end{array}$$

so that

$$\begin{array}{cc}{A}_i^2=(\frac{1}{x_{-{\alpha }_i}}+\frac{1}{x_{{\alpha }_i}})A_i=A_i(\frac{1}{x_{-{\alpha }_i}}+\frac{1}{x_{{\alpha }_i}})=A_{i}p(x_{{\alpha }_i},x_{-{\alpha }_i})\text{.}& \text{(7)}\end{array}$$

Note also that

$$\begin{array}{cccc}{t}_{s_i}{A}_i& =& t_{s_i}(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}=A_i\text{and}& \text{(8)}\\ A_i{t}_{s_i}& =& (1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}{t}_{s_i}=(1+t_{s_i})\frac{1}{x_{{\alpha }_i}}=A_i\frac{x_{-{\alpha }_i}}{x_{{\alpha }_i}}& \text{(9)}\end{array}$$

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

$$\begin{array}{ccc}f{A}_i& =& f(1+t_{s_i})\frac{1}{x_{-{\alpha }_i}}=f\frac{1}{x_{-{\alpha }_i}}+f{t}_{s_i}\frac{1}{x_{-{\alpha }_i}}\text{and}\\ A_i\left(s_{i}f\right)& =& (1+t_{s_i})\frac{s_{i}f}{x_{-{\alpha }_i}}=(s_{i}f+f{t}_{s_i})\frac{1}{x_{-{\alpha }_i}},\end{array}$$

so that

$$\begin{array}{cc}f{A}_i=A_i\left(s_{i}f\right)+\left(\frac{f-s_{i}f}{x_{-{\alpha }_i}}\right),\text{for}f\in R,& \text{(10)}\end{array}$$

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

$$\begin{array}{cc}{B}_{\alpha }f=\frac{f-s_{\alpha }f}{x_{-\alpha }},\text{for roots}\alpha \text{.}\text{Then}{t}_w{B}_{\alpha }{t}_{w^{-1}}=B_{w\alpha },& \text{(11)}\end{array}$$

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,

$$\begin{array}{ccc}{t}_{s_1}& =& x_{{\alpha }_1}{A}_1-\frac{x_{{\alpha }_1}}{x_{-{\alpha }_1}},\\ t_{s_2}{t}_{s_1}& =& x_{s_2{\alpha }_1}{x}_{{\alpha }_2}{A}_2{A}_1-x_{s_2{\alpha }_1}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}}{A}_1-\frac{x_{s_2{\alpha }_1}}{x_{-s_2{\alpha }_1}}{x}_{{\alpha }_2}{A}_2+\frac{x_{s_2{\alpha }_1}}{x_{-s_2{\alpha }_1}}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}}\\ t_{s_1}{t}_{s_2}{t}_{s_1}& =& x_{s_1{s}_2{\alpha }_1}{x}_{s_1{\alpha }_2}{x}_{{\alpha }_1}{A}_1{A}_2{A}_1-x_{s_1{s}_2{\alpha }_1}{x}_{s_1{\alpha }_2}\frac{x_{{\alpha }_1}}{x_{-{\alpha }_1}}{A}_2{A}_1-\frac{x_{s_1{s}_2{\alpha }_1}}{x_{-s_1{s}_2{\alpha }_1}}{x}_{s_1{\alpha }_2}{x}_{{\alpha }_1}{A}_1{A}_2\\ & & +\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}{x}_{s_2{\alpha }_1}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}}{A}_1+\frac{x_{s_1{s}_2{\alpha }_1}}{x_{-s_1{s}_2{\alpha }_1}}{x}_{s_1{\alpha }_2}\frac{x_{{\alpha }_1}}{x_{-{\alpha }_1}}{A}_2-\frac{x_{s_1{s}_2{\alpha }_1}}{x_{-s_1{s}_2{\alpha }_1}}\frac{x_{s_1{\alpha }_2}}{x_{-s_1{\alpha }_2}}\frac{x_{{\alpha }_1}}{x_{-{\alpha }_1}}\\ & & +(\frac{x_{s_1{\alpha }_2}}{x_{-s_1{\alpha }_2}}\frac{x_{s_1{s}_2{\alpha }_1}}{x_{-s_1{s}_2{\alpha }_1}}{x}_{{\alpha }_1}-\frac{x_{s_1{\alpha }_2}}{x_{-s_1{\alpha }_2}}{x}_{s_1{s}_2{\alpha }_1}-\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}{x}_{s_2{\alpha }_1}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}})A_1\\ t_{s_1}{t}_{s_2}{t}_{s_1}{t}_{s_2}& =& x_{s_2{s}_1{s}_2{\alpha }_1}{x}_{s_2{s}_1{\alpha }_2}{x}_{s_2{\alpha }_1}{x}_{{\alpha }_2}{A}_2{A}_1{A}_2{A}_1\\ & & -x_{s_2{s}_1{s}_2{\alpha }_1}{x}_{s_2{s}_1{\alpha }_2}{x}_{s_2{\alpha }_1}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}}{A}_1{A}_2{A}_1-\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}{x}_{s_2{s}_1{\alpha }_2}{x}_{s_2{\alpha }_1}{x}_{{\alpha }_2}{A}_2{A}_1{A}_2\\ & & +\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}{x}_{s_2{s}_1{\alpha }_2}{x}_{s_2{\alpha }_1}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}}{A}_1{A}_2\\ & & +(\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}{x}_{s_2{\alpha }_1}{x}_{{\alpha }_2}-x_{s_2{s}_1{s}_2{\alpha }_1}\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}{x}_{{\alpha }_2}-x_{s_2{s}_1{s}_2{\alpha }_1}{x}_{s_2{s}_1{\alpha }_2}\frac{x_{s_2{\alpha }_1}}{x_{-s_2{\alpha }_1}})A_2{A}_1\\ & & -(\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}{x}_{s_2{\alpha }_1}-x_{s_2{s}_1{s}_2{\alpha }_1}\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}})\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}}{A}_1\\ & & +(\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}{x}_{s_2{s}_1{\alpha }_2}\frac{x_{s_2{\alpha }_1}}{x_{-s_2{\alpha }_1}}-\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}\frac{x_{s_2{\alpha }_1}}{x_{-s_2{\alpha }_1}}{x}_{{\alpha }_2})A_2\\ & & +\frac{x_{s_2{s}_1{s}_2{\alpha }_1}}{x_{-s_2{s}_1{s}_2{\alpha }_1}}\frac{x_{s_2{s}_1{\alpha }_2}}{x_{-s_2{s}_1{\alpha }_2}}\frac{x_{s_2{\alpha }_1}}{x_{-s_2{\alpha }_1}}\frac{x_{{\alpha }_2}}{x_{-{\alpha }_2}},\end{array}$$

and expansions of products of $t_{s_i}$ in terms of products of $A_i$ with $x$s on the right,

$$\begin{array}{ccc}{t}_{s_1}& =& A_1{x}_{-{\alpha }_1}-1,\\ t_{s_1}{t}_{s_2}& =& A_1{A}_2{x}_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}-A_1{x}_{-s_2{\alpha }_1}-A_2{x}_{-{\alpha }_2}+1,\\ t_{s_1}{t}_{s_2}{t}_{s_1}& =& A_1{A}_2{A}_1{x}_{-{\alpha }_1}{x}_{-s_1{\alpha }_2}{x}_{-s_1{s}_2{\alpha }_1}-A_1{A}_2{x}_{-s_1{\alpha }_2}{x}_{-s_1{s}_2{\alpha }_1}-A_2{A}_1{x}_{-{\alpha }_1}{x}_{-s_1{\alpha }_2}\\ & & +A_1{x}_{-s_2{\alpha }_1}+A_2{x}_{-s_1{\alpha }_2}-1+A_1(x_{-{\alpha }_1}-x_{-s_2{\alpha }_1}-\frac{x_{-{\alpha }_1}}{x_{{\alpha }_1}}{x}_{-s_1{s}_2{\alpha }_1}),\\ t_{s_1}{t}_{s_2}{t}_{s_1}{t}_{s_2}& =& A_1{A}_2{A}_1{A}_2{x}_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}{x}_{-s_2{s}_1{\alpha }_2}{x}_{-s_2{s}_1{s}_2{\alpha }_1}\\ & & -A_1{A}_2{A}_1{x}_{-s_2{\alpha }_1}{x}_{-s_2{s}_1{\alpha }_2}{x}_{-s_2{s}_1{s}_2{\alpha }_1}-A_2{A}_1{A}_2{x}_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}{x}_{-s_2{s}_1{\alpha }_2}\\ & & +A_1{A}_2(-\frac{x_{-{\alpha }_2}}{x_{{\alpha }_2}}{x}_{-s_2{s}_1{\alpha }_2}{x}_{-s_2{s}_1{s}_2{\alpha }_1}-x_{-{\alpha }_2}\frac{x_{-s_2{\alpha }_1}}{x_{s_2{\alpha }_1}}{x}_{-s_2{s}_1{s}_2{\alpha }_1}+x_{-{\alpha }_2}{x}_{-s_2{\alpha }_1})\\ & & +A_2{A}_1{x}_{-s_2{\alpha }_1}{x}_{-s_2{s}_1{\alpha }_2}\\ & & -A_1(x_{-s_2{\alpha }_1}-\frac{x_{-s_2{\alpha }_1}}{x_{s_2{\alpha }_1}}{x}_{-s_2{s}_1{s}_2{\alpha }_1})-A_2(x_{-{\alpha }_2}-\frac{x_{-{\alpha }_2}}{x_{{\alpha }_2}}{x}_{-s_2{s}_1{\alpha }_2})+1\text{.}\end{array}$$

Finally, there are expansions of products of $A_i$ in terms of products of $t_{s_i}:$

$$\begin{array}{ccc}{A}_1& =& (t_{s_1}+1)\frac{1}{x_{-{\alpha }_1}},\\ A_1{A}_2& =& (t_{s_1}+1)(t_{s_2}\frac{1}{x_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}}+\frac{1}{x_{-{\alpha }_1}{x}_{-{\alpha }_2}}),\\ A_1{A}_2{A}_1& =& (t_{s_1}+1)\left(\begin{array}{c}{t}_{s_2}{t}_{s_1}\frac{1}{x_{-{\alpha }_1}{x}_{-s_1{\alpha }_2}{x}_{-s_1{s}_2{\alpha }_1}}+t_{s_2}\frac{1}{x_{-{\alpha }_1}{x}_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}}\\ +\frac{1}{x_{-{\alpha }_1}}(\frac{1}{x_{-{\alpha }_1}{x}_{-{\alpha }_2}}+\frac{1}{x_{-s_1{\alpha }_1}{x}_{-s_2{\alpha }_1}})\end{array}\right),\\ A_1{A}_2{A}_1{A}_2& =& (t_{s_1}+1)\left(\begin{array}{c}{t}_{s_2}{t}_{s_1}{t}_{s_2}\frac{1}{x_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}{x}_{-s_2{s}_1{\alpha }_2}{x}_{-s_2{s}_1{s}_2{\alpha }_1}}\\ +t_{s_2}{t}_{s_1}\frac{1}{x_{-{\alpha }_2}{x}_{-{\alpha }_1}{x}_{-s_1{\alpha }_2}{x}_{-s_1{s}_2{\alpha }_1}}\\ +t_{s_2}\frac{1}{x_{-{\alpha }_2}{x}_{-s_2{\alpha }_1}}(\frac{1}{x_{-{\alpha }_2}{x}_{-{\alpha }_1}}+\frac{1}{x_{-s_2{\alpha }_1}{x}_{-s_2{\alpha }_2}}+\frac{1}{x_{-s_2{s}_1{\alpha }_2}{x}_{-s_2{s}_1{\alpha }_1}})\\ +\frac{1}{x_{-{\alpha }_1}{x}_{-{\alpha }_2}}(\frac{1}{x_{-{\alpha }_2}{x}_{-{\alpha }_1}}+\frac{1}{x_{-s_2{\alpha }_1}{x}_{-s_2{\alpha }_2}}+\frac{1}{x_{-s_1{\alpha }_2}{x}_{-s_1{\alpha }_1}})\end{array}\right),\end{array}$$

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

$$0=t_{s_1}{t}_{s_2}{t}_{s_1}-t_{s_2}{t}_{s_1}{t}_{s_2}$$

is equivalent to

$$\begin{array}{ccc}\multicolumn{3}{c}{A_2{A}_1{A}_2-(\frac{1}{x_{-{\alpha }_2}{x}_{-{\alpha }_1}}-\frac{1}{x_{-{\alpha }_1}{x}_{-{\alpha }_3}}+\frac{1}{x_{{\alpha }_2}{x}_{-{\alpha }_3}})A_2}\\ & =& A_1{A}_2{A}_1-(\frac{1}{x_{-{\alpha }_1}{x}_{-{\alpha }_2}}-\frac{1}{x_{-{\alpha }_2}{x}_{-{\alpha }_3}}+\frac{1}{x_{{\alpha }_1}{x}_{-{\alpha }_3}})A_1,\end{array}$$

as indicated in [HLSZ, Proposition 5.7].

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