Polynomials and symmetric functions

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

Last update: 25 February 2012

Satake isomorphism

Recall, from (3.6) of [Ram], that the finite Hecke algebra is the subalgebra of the affine Hecke algebra $\tilde{H}$ given by $$H=\mathrm{span}\left\{T_{w^{-1}}^{-1} \right| w\in W\}.$$ Let ${\mathbf{1}}_0$ be the element of $H$ given by $$\begin{array}{ll}{{\mathbf{1}}_0}^2={\mathbf{1}}_{0}and{T}_{w^{-1}}^{-1}{\mathbf{1}}_0=q^{-l\left(w\right)}{\mathbf{1}}_0,& (Psf\; 1)\end{array}$$ for $w\in W.$ Two explicit formulas for ${\mathbf{1}}_0$ are $${\mathbf{1}}_0=\frac{1}{W_0\left(q^{-2}\right)}\sum _{w\in W}{q}^{-l\left(w\right)}{T}_{w^{-1}}^{-1}=\frac{1}{W_0\left(q^2\right)}\sum _{w\in W}{q}^{l\left(w\right)}{T}_w,$$ where $W_0\left(t\right)=\sum _{w\in W}{t}^{l\left(w\right)}$ is the Poincaré polynomial of $W$.

As observed in (3.6), $𝕂\left[P\right]=\mathrm{span}\left\{X^{\lambda } \right| \lambda \in P\}$ is a subalgebra of $\tilde{H}.$ The vector space $𝕂\left[P\right]$ also sits inside $\tilde{H}$ in a different way. Since $\left\{X^{\lambda }{\mathbf{1}}_0 \right| \lambda \in P\}$ is a basis of $\tilde{H}{\mathbf{1}}_0$ there is $$\begin{array}{ll}a\; vector\; space\; isomorphism\begin{array}{ccc}𝕂\left[P\right]& \to & \tilde{H}{\mathbf{1}}_0\\ f& \mapsto & f{\mathbf{1}}_0.\end{array}& (Psf\; 2)\end{array}$$

The ring of symmetric functions is $$\begin{array}{ll}𝕂{\left[P\right]}^W=\{f\in 𝕂\left[P\right] | wf=f \; for\; all\; w\in W\}.& (Psf\; 3)\end{array}$$ By a theorem of Bernstein (see [NR, Theorem 1.4]) this subalgebra of $\tilde{H}$ is the center of $\tilde{H},$ $$\begin{array}{ll}𝕂{\left[P\right]}^W=Z\left(\tilde{H}\right).& (Psf\; 4)\end{array}$$ The spherical Hecke algebra is the ring ${\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0$ and the restriction of the map (Psf 2) to $Z\left(\tilde{H}\right)$ is the Satake isomorphism of the following theorem.

Let ${\mathbf{1}}_0$ and $𝕂{\left[P\right]}^W$ be as in (Psf 1) and (Psf 3), respectively. Then $$\begin{array}{clcl}𝕂{\left[P\right]}^W& =Z\left(\tilde{H}\right)& \to & {\mathbf{1}}_0\tilde{H}{\mathbf{1}}_0\\ f& & \mapsto & f{\mathbf{1}}_0\end{array}is\; a\; 𝕂-algebra\; isomorphism.$$

		Proof.
	The map is a well defined homomorphism since, if $f,f_1,f_2 \in Z\left(\tilde{H}\right)$ then $f{\mathbf{1}}_0=f{{\mathbf{1}}_0}^2={\mathbf{1}}_{0}f{\mathbf{1}}_0$ and $f_1{f}_2{\mathbf{1}}_0=f_1{f}_2{{\mathbf{1}}_0}^2=f_1{\mathbf{1}}_0{f}_2{\mathbf{1}}_0.$ Suppose that $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ so that $\lambda$ is on the positive side of $H_{{\alpha }_i}.$ Then, by Proposition 3.2e, $$\begin{array}{rcl}{\mathbf{1}}_0{X}^{s_i\lambda }{\mathbf{1}}_0& =& q^{-1}{\mathbf{1}}_0{T}_{s_i}{X}^{\lambda }{\mathbf{1}}_0\\ & =& q^{-1}{\mathbf{1}}_0{X}^{\lambda }{T}_{s_i}{\mathbf{1}}_0-q^{-1}(q-q^{-1}){\mathbf{1}}_0(Z^{s_i\lambda +{\alpha }_i}+\cdots +X^{\lambda -{\alpha }_i}+X^{\lambda }){\mathbf{1}}_0\\ & =& {\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0-(1-q^{-2}){\mathbf{1}}_0(X^{s_i\lambda +{\alpha }_i}+\cdots +X^{\lambda -{\alpha }_i}+X^{\lambda }){\mathbf{1}}_0.\end{array}$$ so that $$\begin{array}{ll}{\mathbf{1}}_0{X}^{s_i\lambda }{\mathbf{1}}_0=q^{-2}{\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0-(1-q^{-2}){\mathbf{1}}_0(X^{s_i\lambda +{\alpha }_i}+\cdots +X^{\lambda -{\alpha }_i}){\mathbf{1}}_0& (Psf\; 5)\end{array}$$ or, equivalently, $$\begin{array}{ll}{\mathbf{1}}_0(X^{s_i\lambda }+\cdots +X^{\lambda -{\alpha }_i}){\mathbf{1}}_0=q^{-2}{\mathbf{1}}_0(X^{s_i\lambda +{\alpha }_i}+\cdots +X^{\lambda }){\mathbf{1}}_0.& (Psf\; 6)\end{array}$$ For the relation in (Psf 5), $$\begin{array}{rcl}{\mathbf{1}}_0{X}^{s_i}{\mathbf{1}}_0-{\mathbf{1}}_0{X}^{s_i\lambda +{\alpha }_i}{\mathbf{1}}_0& =& q^{-2}{\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0-(1-q^{-2}){\mathbf{1}}_0(X^{s_i\lambda +{\alpha }_i}+\cdots +X^{\lambda -{\alpha }_i}){\mathbf{1}}_0\\ & & -q^{-2}{\mathbf{1}}_0{X}^{\lambda -{\alpha }_i}{\mathbf{1}}_0+(1-q^{-2}){\mathbf{1}}_0(X^{s_i\lambda +2{\alpha }_i}+\cdots +X^{\lambda -2{\alpha }_i}){\mathbf{1}}_0,\end{array}$$ so that $$\begin{array}{ll}{\mathbf{1}}_0{X}^{s_i\lambda }{\mathbf{1}}_0=q^{-2}{\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0+q^{-2}{\mathbf{1}}_0{X}^{s_i\lambda +{\alpha }_i}{\mathbf{1}}_0-{\mathbf{1}}_0{X}^{\lambda -{\alpha }_i}{\mathbf{1}}_0.& (Psf\; 7)\end{array}$$ It follows from these relations that any element of ${\mathbf{1}}_0\tilde{H}{\mathbf{1}}_0$ can, inductively, be written as a linear combination of the elements ${\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0, \lambda \in P^{+}.$ Using Theorem 3.3 to expand ${\mathbf{1}}_0{X}^{\lambda }$ in terms of the basis $\left\{X^{\mu }{T}_{v^{-1}}^{-1} \right| \mu \in P, v\in W\}$ produces $${\mathbf{1}}_0{X}^{\lambda }=X^{w_0\lambda }{T}_{w_0}^{-1}+\sum _{\mu >w_0\lambda }{d}_{\mu ,v}{X}^{\mu }{T}_{v^{-1}}^{-1},$$ and, since these leading terms are all different (as $\lambda$ runs over $P^{+}$), it follows that $$\begin{array}{ll}{\mathbf{1}}_0\tilde{H}{\mathbf{1}}_{0}has\; basis\left\{{\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0 \right| \lambda \in P^{+}\}.& (Psf\; 8)\end{array}$$ The orbit sums $$\begin{array}{ll}{m}_{\lambda }=\sum _{\gamma \in W\lambda }{X}^{\gamma },\lambda \in P^{+},& (Psf\; 9)\end{array}$$ for a basis of $𝕂{\left[P\right]}^W.$ The relation in Proposition 3.2e implies that, if $f\in 𝕂{\left[P\right]}^W$ then $T_{w}f=f{T}_w$ for all $w\in W,$ and so the $m_{\lambda }{\mathbf{1}}_0={\mathbf{1}}_0{m}_{\lambda }{\mathbf{1}}_0$ are in ${\mathbf{1}}_0\tilde{H}{\mathbf{1}}_0.$ Viewing these in terms of the basis $\left\{X^{\mu }{T}_{v^{-1}}^{-1} \right| \mu \in P, v\in W\}$ of $\tilde{H}$ one sees that the $m_{\lambda }{\mathbf{1}}_0, \lambda \in P^{+},$ are linearly independent and so $$\begin{array}{ll}{\mathbf{1}}_0\tilde{H}{\mathbf{1}}_{0}has\; basis\left\{m_{\lambda }{\mathbf{1}}_0 \right| \lambda \in P^{+}\}.& (Psf\; 10)\end{array}$$ The point is that the transition matrix between the basis in (Psf 8) and the basis in (Psf 10) is triangular. $\square$

Hall-Littlewood polynomials

For $\mu \in P$ let $W_{\mu }=\mathrm{Stab}\left(\mu \right)$ be the stabilizer of $\mu .$ The Poincaré polynomial of $W_{\mu }$ is $$\begin{array}{ll}{W}_{\mu }\left(t\right)=\sum _{w\in W_{\mu }}{t}^{l\left(w\right)}.& (Psf\; 11)\end{array}$$ For $\mu \in P,$ the Hall-Littlewood polynomial or Macdonald spherical function $P_{\mu }(X;t)$ is the element of $𝕂{\left[P\right]}^W$ defined by $$\begin{array}{ll}{P}_{\mu }(X;q^{-2}){\mathbf{1}}_0=\left(\sum _{w\in W^{\mu }}{q}^{-l\left(w\right)}{T}_{w^{-1}}^{-1}\right)X^{\mu }{\mathbf{1}}_0,& (Psf\; 12)\end{array}$$ where $W^{\mu }$ is a set of minimal length coset representatives for the cosets in $W/W_{\mu }.$ Since every element $w\in W$ has a unique expression $w=uv$ with $u\in W^{\mu }$ and $v\in W_{\mu },$ $$\begin{array}{rcl}\sum _{w\in W^{\mu }}{q}^{-l\left(w\right)}{T}_{w^{-1}}^{-1}{X}^{\mu }{\mathbf{1}}_0& =& \frac{1}{W_{\mu }\left(q^{-2}\right)}\left(\sum _{u\in W^{\mu }}{q}^{-l\left(u\right)}{T}_{u^{-1}}^{-1}\right)X^{\mu }\left(\sum _{v\in W_{\mu }}{q}^{-l\left(v\right)}{T}_{v^{-1}}^{-1}\right){\mathbf{1}}_0\\ & =& \frac{1}{W_{\mu }\left(q^{-2}\right)}\left(\sum _{u\in W^{\mu }}{q}^{-l\left(u\right)}{T}_{u^{-1}}^{-1}\right)\left(\sum _{v\in W_{\mu }}{q}^{-l\left(v\right)}{T}_{v^{-1}}^{-1}\right)X^{\mu }{\mathbf{1}}_0\\ & =& \frac{W_0\left(q^{-2}\right)}{W_{\mu }\left(q^{-2}\right)}{\mathbf{1}}_0{X}^{\mu }{\mathbf{1}}_0,\end{array}$$ and hence $$P_{\mu }(X,q^{-2}){\mathbf{1}}_{0}is\; exactly{\mathbf{1}}_0{X}^{\mu }{\mathbf{1}}_{0}except\; normalised$$ so that the coefficient of $X^{\mu }{\mathbf{1}}_0$ is 1.

Macdonald's formula for the spherical functions on a $p-$adic group [Mac1, Theorem 4.1.2] is $$\begin{array}{ll}{P}_{\mu }(X;q^{-2})=\frac{1}{W_{\mu }\left(q^{-2}\right)}\sum _{w\in W}w\left(X^{\mu }\prod _{\alpha \in R^{+}}\frac{1-q^{-2}{X}^{-\alpha }}{1-X^{-\alpha }}\right).& (Psf\; 13)\end{array}$$ See [NR, Theorem 2.9a] for a proof in this context.

The following theorem gives additional formulas for $P_{\lambda }(X;q^{-2}).$ $$\begin{array}{ll}A\mathit{positively\; folded}alcove\; walk\; is\; an\; alcove\; walk\; with\; no\; negative\; folds.& (Psf\; 14)\end{array}$$ In the following theorem we shall consider alcove walks which do not necessarily begin at $A.$ This is the natural way to account for the sum over $W^{\lambda }$ which appears in the definition of $P_{\lambda }$ is (Psf 12). The type of $p$ is the sequence of labels of the folds and the wall crossings of $p.$

For $\lambda \in P^{+}$ let $t_{\lambda }\in \tilde{W}$ be the translation in $\lambda$ and let $n_{\lambda }$ be the maximal length element in the double coset $W{t}_{\lambda }W.$

1. [Sc, Theorem 1.1] Let $\lambda \in P^{+}$ and fix a minimal length walk $p_{\lambda }=c_{i_1}^{+}\cdots c_{i_l}^{+}$ from $A$ to $\lambda +A.$ Let $$B_q\left(p_{\lambda }\right)=\{\phantom{|}\begin{array}{l}positively\; folded\; alcove\; walks\; of\\ type\; (i_1,...,i_l) \; which\; begin\; at\; wA\end{array}| w\in W^{\lambda }\}.$$ Then $$P_{\lambda }(X;q^{-2})=\sum _{p\in B_q\left(p_{\lambda }\right)}{q}^{-(l\left(\iota \left(p\right)\right)+l\left(\phi \left(p\right)\right)-f\left(p\right))}{(1-q^{-2})}^{f\left(p\right)}{X}^{\mathrm{wt}\left(p\right)},$$ where $\iota \left(p\right)$ is the alcove where $p$ begins, $\mathrm{wt}\left(p\right)+\phi \left(p\right)A$ is the alcove where $p$ ends, and $f\left(p\right)$ is the number of folds in $p.$
2. Let $\lambda \in P^{+}.$ Then $$q^{l\left(w_0\right)}{W}_0\left(q^{-2}\right)P_{\lambda }{\mathbf{1}}_0=\sum _{x\in W{t}_{\lambda }W}{q}^{l\left(x\right)-l\left(n_{\lambda }\right)}{T}_x.$$

		Proof.
	The proof is accomplished by using Theorem 3.3 to expand the sum in (Psf 12). Since all crossings in the walk $p_{\lambda }$ are positive crossings Theorem 3.3 gives $$\left(\sum _{w\in W^{\lambda }}{q}^{-l\left(w\right)}{T}_{w^{-1}}^{-1}\right)X^{\lambda }=\sum _{w\in W^{\lambda }}{q}^{-l\left(w\right)}\sum _{\begin{array}{}p\in B_q\left(p_{\lambda }\right)\\ \iota \left(p\right)=w\end{array}}{(q-q^{-1})}^{f\left(p\right)}{X}^{\mathrm{wt}\left(p\right)}{T}_{\phi \left(p\right)}^{-1}.$$ Hence $$P_{\lambda }{\mathbf{1}}_0=\sum _{p\in B_q\left(\lambda \right)}{q}^{-l\left(\iota \left(p\right)\right)}{(q-q^{-1})}^{f\left(p\right)}{X}^{\mathrm{wt}\left(p\right)}{q}^{-l\left(\phi \left(p\right)\right)}{\mathbf{1}}_0.$$ Let $\lambda \in P^{+}.$ Let $W_{\lambda }=\mathrm{Stab}\left(\lambda \right)$ and let $w_0$ and $w_{\lambda }$ be the maximal length elements in $W$ and $W_{\lambda },$ respectively. Let $m_{\lambda }$ and $n_{\lambda }$ be the minimal and maximal length elements respectively in the double coset $W{t}_{\lambda }W.$ If $\lambda =2{\omega }_2$ in type $C_2,$ then $W_{\lambda }=\{1,s_1\}, w_{\lambda }=s_1, w_0=s_1{s}_2{s}_1{s}_2, l\left(t_{\lambda }\right)=6, l\left(m_{\lambda }\right)=3,$ and $l\left(n_{\lambda }\right)=10.$ Labeling the alcove $wA$ by the element $w,$ the 32 alcoves $wA$ with $w\in W{t}_{\lambda }W$ make up the four shaded diamonds. Then $$\begin{array}{rcl}{q}^{l\left(w_0\right)}{W}_0\left(q^{-2}\right)P_{\lambda }{\mathbf{1}}_0& =& q^{l\left(w_0\right)}{W}_0\left(q^{-2}\right)q^{-2l\left(w_0\right)+2l\left(w_{\lambda }\right)}\left(\sum _{u\in W^{\lambda }}{q}^{l\left(u\right)}{T}_u\right)X^{\lambda }{\mathbf{1}}_0\\ & =& q^{l\left(w_0\right)}{q}^{-2l\left(w_0\right)}{W}_0\left(q^2\right)q^{-2l\left(w_0\right)+2l\left(w_{\lambda }\right)}\left(\sum _{u\in W^{\lambda }}{q}^{l\left(u\right)}{T}_u\right)T_{m_{\lambda }}{T}_{w_0{w}_{\lambda }}{\mathbf{1}}_0\\ & =& q^{-3l\left(w_0\right)+2l\left(w_{\lambda }\right)}\left(\sum _{u\in W^{\lambda }}{q}^{l\left(u\right)}{T}_u\right)T_{m_{\lambda }}{q}^{l\left(w_0\right)-l\left(w_{\lambda }\right)}\left(\sum _{w\in W}{q}^{l\left(w\right)}{T}_w\right)\\ & =& q^{-2l\left(w_0\right)+l\left(w_{\lambda }\right)}\left(\sum _{\begin{array}{}u\in W^{\lambda }\\ w\in W\end{array}}{q}^{l\left(u\right)+l\left(w\right)}{T}_{u{m}_{\lambda }w}\right)\\ & =& q^{-2l\left(w_0\right)+l\left(w_{\lambda }\right)}\left(\sum _{x\in W{t}_{\lambda }W}{q}^{l\left(x\right)-l\left(m_{\lambda }\right)}{T}_x\right),\end{array}$$ and the result follows from the identity $l\left(n_{\lambda }\right)=l\left(w_0\right)+l\left(t_{\lambda }\right)=l\left(w_0\right)+l\left(w_0\right)-l\left(w_{\lambda }\right)+l\left(m_{\lambda }\right).$ $\square$

The set $B_q\left(p_{\lambda }\right)$ appearing in Theorem 2.1a is heavily dependent on the choice of $p_{\lambda }.$ One way to make this seem less dependent on this choice is as follows. For convenience assume that $\lambda$ is regular (not on any wall). Similar definitions can be made in the general case. Consider the region $$\left[\lambda \right]=\{x=\sum _i{x}_i{\omega }_i | 0\le x_i\le {\lambda }_i\},where\lambda =\sum _i{\lambda }_i{\omega }_i.$$ This region is a union of alcoves and any minimal length walk $p_{\lambda }$ from $A$ to $\lambda +w_{0}A$ lies in $\left[\lambda \right].$ Foldings of the walk $p_{\lambda }$ are then produced by folding the region $\left[\lambda \right]$ along the "creases" formed by the hyperplanes. This process produces a bijection between the paths in $B_q\left(p_{\lambda }\right)$ and the set $B_q\left(\left[\lambda \right]\right)$ of "positively folded foldings" of the region $\left[\lambda \right]$, and the set $B_q\left(\left[\lambda \right]\right)does\; not\; depend\; on\; the\; choice\; of\; the\; initial\; path$ p_{\lambda }$is\; to\; remember\; all\; the\; possible\; initial\; paths\; all\; at\; once.\; This\; translation\; of\; foldings\; was\; explained\; to\; me\; by\; J.\; Ramagge\; in\; Fall\; 2000.$

Let $B_q\left(p_{\lambda }\right)$ be as in Theorem 2.1a and let $p\in B_q\left(p_{\lambda }\right).$ Suppose that $p$ has $f$ folds. For $0\le i\le f,$ let $p^{\left(i\right)}$ be the positively folded walk in $B_q\left(p_{\lambda }\right)$ which coincides with $p$ up to the $i$th fold and is nonfolded thereafter. Then $p^{\left(0\right)},...,p^{\left(f\right)}$ is a sequence of positively folded walks such that $$p^{\left(f\right)}=p,\iota \left(p^{\left(i\right)}\right)=\iota \left(p\right),\phi \left(p^{\left(0\right)}\right)=\iota \left(p\right),and\phi \left(p^{\left(i\right)}\right)=s_{\alpha }\phi \left(p^{(i-1)}\right)$$ if the $i$th fold is on the hyperplane $H_{\alpha ,j}.$ Since $\phi \left(p^{(i-1)}\right)>\phi \left(p^{\left(i\right)}\right)$ and $${(-1)}^{l\left(\phi \left(p^{\left(i\right)}\right)\right)}={(-1)}^{l\left(s_{\alpha }\right)}{(-1)}^{\phi \left(p^{(i-1)}\right)}=(-1){(-1)}^{\phi \left(p^{(i-1)}\right)},$$ $l\left(\phi \left(p^{(i-1)}\right)\right)-l\left(\phi \left(p^{\left(i\right)}\right)\right)-1$ is an even integer $\ge 0.$ It follows that $$\begin{array}{rcl}l\left(\iota \left(p\right)\right)+l\left(\phi \left(p\right)\right)-f\left(p\right)& =& l\left(\iota \left(p\right)\right)-l\phi p^{\left(1\right)}))-1\\ & & +l\phi \left(p^{\left(1\right)}\right))-l\left(\phi \left(p^{\left(2\right)}\right)\right)-1\\ & & +l\left(\phi \left(p^{\left(2\right)}\right)\right)-l\left(\phi \left(p^{\left(3\right)}\right)\right)-1\\ & & +\cdots +l\left(\phi \left(p^{(f-1)}\right)\right)-l\left(\phi \left(p^{\left(f\right)}\right)\right)-1+2l\left(\phi \left(p\right)\right)\end{array}$$ is an even integer $\ge 0.$ This proves that $f\left(p\right)\le l\left(\iota \left(p\right)\right)-l\left(\phi \left(p\right)\right)$ and that $P_{\lambda }(X;q^{-2})$ really is a polynomial in the variable $q^{-2}.$

Demazure operators

The group $W$ acts on $𝕂\left[P\right]=\mathrm{span}\left\{X^{\lambda } \right| \lambda \in P\}$ by $$\begin{array}{ll}w{X}^{\lambda }=X^{w\lambda },for\; w\in W, \lambda \in P.& (Psf\; 15)\end{array}$$ For each $1\le i\le n,$ define Demazure operators $${\Delta }_i:𝕂\left[P\right]\to 𝕂\left[P\right]and{\tilde{\Delta }}_i:𝕂\left[P\right]\to 𝕂\left[P\right]$$ by $$\begin{array}{ll}{\Delta }_{i}f=\frac{1}{1-X^{-{\alpha }_i}}(1-s_i)fand{\tilde{\Delta }}_{i}f=\frac{1}{X^{{\alpha }_i}-1}(X^{{\alpha }_i}-s_i)f,& (Psf\; 16)\end{array}$$ respectively.

Via the isomorphism in (Psf 2), the vector space $𝕂\left[P\right]$ is an $\tilde{H}-$module. Let $$\begin{array}{ll}{C}_i=q^{-2}+q^{-1}{T}_{s_i}=1+q^{-1}{T}_{s_i}^{-1}=(1+q^{-2}){\mathbf{1}}_i,& (Psf\; 17)\end{array}$$ where ${\mathbf{1}}_i$ is the element of $H$ such that ${{\mathbf{1}}_i}^2={\mathbf{1}}_i$ and $T_{s_i}^{-1}{\mathbf{1}}_i=q^{-1}{\mathbf{1}}_i.$ The element ${\mathbf{1}}_i$ is the rank 1 version of the element ${\mathbf{1}}_0$ in (Psf 1).

The following proposition shows that, at $q^{-2}=0,$ the action of $C_i$ on $𝕂\left[P\right]$ is the Demazure operator ${\tilde{\Delta }}_i.$ In geometry, the Demazure operators arise naturally as push-pull operators on the K-theory of the flag variety (see [PR, Proposition]).

Let $\rho ={\omega }_1+\cdots +{\omega }_n$ as in (2.16). As operators on $𝕂\left[P\right],$

1. ${\tilde{\Delta }}_i=X^{-\rho }{\Delta }_i{X}^{\rho }={\Delta }_i+s_i,$
2. $C_i=(1-q^{-2}){\Delta }_i+(s_i+q^{-2})=(1+s_i)\left(\frac{1-q^{-2}{X}^{-{\alpha }_i}}{1-X^{-{\alpha }_i}}\right).$

		Proof.
	Let $\lambda \in P.$ Since $s_i\rho =\rho -⟨\rho ,{\alpha }_i^{\vee }⟩{\alpha }_i=\rho -{\alpha }_i,$ $$\left(X^{-\rho }{\Delta }_i{X}^{\rho }\right)\left(X^{\lambda }\right)=X^{-\rho }\frac{X^{\lambda +\rho }-X^{s_i\lambda +\rho -{\alpha }_i}}{1-X^{-{\alpha }_i}}=\frac{X^{\lambda }-X^{s_i\lambda -{\alpha }_i}}{1-X^{-{\alpha }_i}}={\tilde{\Delta }}_i\left(X^{\lambda }\right),$$ and, as operators, $${\Delta }_i+s_i=\frac{1}{1-X^{-{\alpha }_i}}(1-s_i)+s_i=\frac{1}{1-X^{-{\alpha }_i}}(1-s_i+s_i-X^{-{\alpha }_i}{s}_i)={\tilde{\Delta }}_i.$$ Using Proposition 3.2e, $$\begin{array}{rcl}{q}^{-1}{T}_{s_i}{X}^{\lambda }{\mathbf{1}}_0& =& (q^{-1}{X}^{s_i\lambda }{T}_{s_i}+(1-q^{-2})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}){\mathbf{1}}_0\\ & =& (X^{s_i\lambda }+(1-q^{-2})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}){\mathbf{1}}_0\\ & =& \left(\frac{X^{s_i\lambda }-X^{s_i\lambda -{\alpha }_i}+X^{\lambda }-X^{s_i\lambda }-q^{-2}(X^{\lambda }-X^{s_i\lambda })}{1-X^{-{\alpha }_i}}\right){\mathbf{1}}_0\\ & =& ({\tilde{\Delta }}_i-q^{-2}{\Delta }_i)\left(X^{\lambda }\right){\mathbf{1}}_0,\end{array}$$ and the first formula in (b) now follows from the second formula in (a). Then $$\begin{array}{rcl}{C}_i& =& (q^{-2}+{\tilde{\Delta }}_i-q^{-2}{\Delta }_i)\\ & =& \frac{1}{1-X^{-{\alpha }_i}}(q^{-2}-q^{-2}{X}^{-{\alpha }_i}+1-X^{-{\alpha }_i}{s}_i-q^{-2}+q^{-2}{s}_i)\\ & =& \frac{1-q^{-2}{X}^{-{\alpha }_i}}{1-X^{-{\alpha }_i}}+s_i\frac{q^{-2}-X^{{\alpha }_i}}{1-X^{{\alpha }_i}}=(1+s_i)\left(\frac{1-q^{-2}{X}^{-{\alpha }_i}}{1-X^{-{\alpha }_i}}\right).\end{array}$$ $\square$

Slightly renormalizing the generators of the affine Hecke algebra by setting ${\tilde{T}}_i=q^{-1}{T}_{s_i}$ allows one to let $q^{-1}=0$ so that ${\tilde{T}}_i$ acts on $𝕂\left[P\right]$ by ${\tilde{\Delta }}_i.$ This is the action of the nil affine Hecke algebra on $𝕂\left[P\right].$ Since the ${\tilde{T}}_i$ satisfy the braid relations so do the ${\tilde{\Delta }}_i.$ The first formula in Proposition 1.1 shows that ${\Delta }_i$ is a conjugate of ${\tilde{\Delta }}_i$ and so the ${\Delta }_i$ also satisfy the braid relations. Although $C_i$ equals ${\tilde{\Delta }}_i$ at $q^{-2}=0,$ the operators $C_i$ do not satisfy the braid relations. Furthermore, if $w_0=s_{i_1}\cdots s_{i_l}$ is a reduced word for the longest element then $$C_{i_1}\cdots C_{i_l}=W_0\left(q^{-2}\right){\mathbf{1}}_0+q^{-2}\left(extra\; terms\right).$$ In contrast to the case for Weyl characters (when $q^{-2}=0),$ because of the $q^{-2}$(extra terms) the Hall-Littlewood polynomial cannot be generated by applying the product $C_{i_1}\cdots C_{i_l}$ unless one somehow knows how to throw away the extra terms.

As operators on $𝕂\left[P\right],$ $$q-T_{s_i}=\left(\frac{q^{-1}-q{X}^{-{\alpha }_i}}{1-X^{-{\alpha }_i}}\right)(1-s_i)and$$ $${\mathbf{1}}_0=\frac{1}{W_0\left(q^{-2}\right)}\sum _{w\in W}w\left(\prod _{\alpha \in R^{+}}\frac{1-q^{-2}{X}^{-\alpha }}{1-X^{-\alpha }}\right).$$ The second formula is equivalent to Macdonald's spherical function formula (Psf 13).

Root operators

The idea of root operators is to give an alcove walk interpretation of the action of the operator $C_i$ on $𝕂\left[P\right]$ by considering the projections of the alcove walks onto the line perpendicular to $H_{{\alpha }_i}.$ The main point is the identity (Psf 22) which gives a combinitorial description of the action of $C_i$ on $𝕂\left[P\right].$ The appropriate combinatorics is more or less forced by the Leibnitz rule or tensor product rule for the operator $C_i$ given in (Psf 28).

Let $p$ be a positively folded walk. $$\begin{array}{ll}INSERT\; DIAGRAM& (Psf\; 18)\end{array}$$ Let $1\le i\le n.$ The projection of $p$ onto a line perpendicular to $H_{{\alpha }_i}$ is positively folded alcove walk $\stackrel{\_}{p}$ "with respect to ${\alpha }_i$" (the only important information in the projection is the relative position of the walk to each of the hyperplanes parallel to $H_{{\alpha }_i}$). $$\begin{array}{cc}\begin{array}{c}\\ di+(p) = d+(p_) di-(p) = d-(p_) \end{array}& (Psf\; 19)\end{array}$$ Because $\stackrel{\_}{p}$ is positively folded it is a concatenation of negative-positive sections of the form $c^{-}{c}^{-}\cdots c^{-}f{c}^{+}{c}^{+}\cdots c^{+},$ where $c^{+}$ denotes a positive crossing, $c^{-}$ a negative crossing, and $f$ a (positive) fold. The outer edge (bottom most negative traveling portion and topmost traveling portion) of the walk is a single negative-positive walk $$\begin{array}{ll}{\displaystyle \underset{d^{-}\left(\stackrel{\_}{p}\right) \; factors}{\underbrace{c^{-}{c}^{-}\cdots c^{-}{c}^{-}}}}f{\displaystyle \underset{d^{+}\left(\stackrel{\_}{p}\right) \; factors}{\underbrace{c^{+}{c}^{+}\cdots c^{+}{c}^{+}}}}.\begin{array}{c}\\ \\ \\ \end{array}& (Psf\; 20)\end{array}$$ If $\stackrel{\_}{p}$ is the walk in (Psf 18), the outer edge is the darkened portion of the path, $d^{+}\left(\stackrel{\_}{p}\right)=7$ and $d^{-}\left(\stackrel{\_}{p}\right)=3.$

The root operators $\tilde{e}$ and $\tilde{f}$ change the outer edge of the path $\stackrel{\_}{p}$ and leave all other parts of the walk unchanged. Define $\tilde{e}\stackrel{\_}{p}$ and $\tilde{f}\stackrel{\_}{p}$ to be the positively folded alcove walks which are the same as $\stackrel{\_}{p}$ except that $$\begin{array}{ll}\tilde{f}\stackrel{\_}{p} \; has\; outer\; edge& {\displaystyle \underset{d^{-}\left(\stackrel{\_}{p}\right)+1 \; factors}{\underbrace{c^{-}{c}^{-}\cdots c^{-}{c}^{-}{c}^{-}}}}f{\displaystyle \underset{d^{+}\left(\stackrel{\_}{p}\right)-1 \; factors}{\underbrace{c^{+}{c}^{+}\cdots c^{+}{c}^{+}}}}, \; and\\ \tilde{e}\stackrel{\_}{p} \; has\; outer\; edge& {\displaystyle \underset{d^{-}\left(\stackrel{\_}{p}\right)-1 \; factors}{\underbrace{c^{-}{c}^{-}\cdots c^{-}}}}f{\displaystyle \underset{d^{+}\left(\stackrel{\_}{p}\right)+1 \; factors}{\underbrace{c^{+}{c}^{+}{c}^{+}\cdots c^{+}{c}^{+}}}}.\end{array}$$ If $\stackrel{\_}{p}$ is the walk in (Psf 18) then $$\begin{array}{rcl}\tilde{f}\stackrel{\_}{p}& =& \begin{array}{c}\\ \\ \end{array}\end{array}$$ and $$\begin{array}{rcl}\tilde{e}\stackrel{\_}{p}& =& \begin{array}{c}\\ \\ \end{array}\end{array}$$ The precise rules for the limiting cases, when $d^{+}\left(\stackrel{\_}{p}\right)$ or $d^{-}\left(\stackrel{\_}{p}\right)=0,$ are illustrated by the following example, where the dashed arrows indicate the action of $\tilde{e}$ and $\tilde{f}.$

with $\tilde{f}\left( \right)=0$ and $\tilde{e}\left( \right)=0.$

With notations for $d^{+}\left(\stackrel{\_}{p}\right)$ and $d^{-}\left(\stackrel{\_}{p}\right)$ as in (Psf 19) and (Psf 20), define $$\begin{array}{ll}{d}_i^{+}\left(p\right)=d^{+}\left(\stackrel{\_}{p}\right)and{d}_i^{-}\left(p\right)=d^{-}\left(\stackrel{\_}{p}\right),& (Psf\; 21)\end{array}$$ where $\stackrel{\_}{p}$ is the projection of $p$ onto the line perpendiculat to ${\alpha }_i.$ The walks ${\tilde{e}}_{i}p$ and ${\tilde{f}}_{i}p$ are the walks obtained from $p$ by changing the corresponding edges $p$ (so that the projections of ${\tilde{e}}_{i}p$ and ${\tilde{f}}_{i}p$ are $\tilde{e}\stackrel{\_}{p}$ and $\tilde{f}\stackrel{\_}{p},$ respectively). $$INCLUDE\; 2\; PICTURES\; HERE$$

The i-string of $p$ $S_i\left(p\right)$ is the set of paths generated from $p$ by applying the root operators ${\tilde{e}}_i$ and ${\tilde{f}}_i.$ The head of the $i-$string is the path $h$ in $S_i\left(p\right)$ which has $d_i^{-}\left(h\right)=0.$ If $\mathrm{wt}\left(h\right)=\lambda$ and $\lambda$ is on the positive side of $H_{{\alpha }_i}$ then $$\begin{array}{ll}{C}_i{X}^{\lambda }=q^{-1}{X}^{s_i\lambda }{T}_{s_i}^{-1}+(1-q^{-2})(X^{s_i\lambda }+X^{s_i\lambda +{\alpha }_i}+\cdots +X^{\lambda -{\alpha }_i})+X^{\lambda },& (Psf\; 22)\end{array}$$ and the terms in this sum correspond to the paths in the $i-$string $S_i\left(h\right).$

$q-$Crystals

The $q-$crystals provide a combinatorial model for the spherical Hecke algebra ${\mathbf{1}}_0\tilde{H}{\mathbf{1}}_0$ in the basis of Hall-Littlewood polynomials. Three structural properties motivate the definition of $q-$crystals:

1. The Hall-Littlewood polynomials are normalised versions of the basis ${\mathbf{1}}_0{X}^{\lambda }{\mathbf{1}}_0.$
2. The element ${\mathbf{1}}_0$ is characterised by the property that $C_i{\mathbf{1}}_0=(1+q^{-2}){\mathbf{1}}_0$ for all $1\le i\le n.$
3. The action of $Ci$ on $\tilde{H}{\mathbf{1}}_0\cong 𝕂\left[P\right]$ is captured in the combinatorics of $i-$strings.

These properties indicate that the combinatorics of Hall-Littlewood polynomials can be captured with the root operators.

Let $$\begin{array}{ll}\begin{array}{c}{B}_{\mathrm{univ}} \; be\; the\; set\; of\; positively\; folded\; alcove\; walks\\ which\; begin\; in\; the\; 0-polygon\; WA.\end{array}& (Psf\; 23)\end{array}$$ If $B$ is a finite subset of $B_{\mathrm{univ}}$ the character of $B$ is $$\begin{array}{ll}\mathrm{char}\left(B\right)=\sum _{p\in B}{q}^{-(\iota \left(p\right)+\phi \left(p\right)-f\left(p\right))}{(1-q^{-2})}^{f\left(p\right)-c\left(p\right)}{X}^{\mathrm{wt}\left(p\right)},& (Psf\; 24)\end{array}$$ where $p$ has $f\left(p\right)$ folds, $\iota \left(p\right)A$ is the alcove where $p$ begins, $\mathrm{wt}\left(p\right)+\phi \left(p\right)A$ is the alcove where $p$ ends and $$\begin{array}{ll}\begin{array}{rl}c\left(p\right) \; is& the\; number\; of\; folds\; of\; p\\ & \text{<em>touching one of the hyperplanes</em>} H_{{\alpha }_1},...,H_{{\alpha }_n} .\end{array}& (Psf\; 25)\end{array}$$ A q-crystal is a finite subset $B$ of $B_{\mathrm{univ}}$ which is closed under the action of the root operators.

A positively folded alcove walk is i-dominant if it never touches the hyperplane $H_{{\alpha }_i,-1}.$ The head $h$ of an $i-$string $S_i\left(p\right)$ is $i-$dominant and $S_i\left(h\right)=S_i\left(p\right).$ A positively folded alcove walk $$\begin{array}{ll}p \; is\text{<em>dominant</em>}if\; p\subseteq C-\rho ,& (Psf\; 26)\end{array}$$ where $$C-\rho =\{\mu \in {𝔥}_{ℝ}^{*} | ⟨\mu ,{\alpha }_i^{\vee }⟩>-1 \; for\; 1\le i\le n\}.$$ In other words, a positively folded alcove walk $p$ is dominant if it is $i-$dominant for all $i,$ $1\le i\le n.$

Let $B$ be a $q-$crystal. Then, with notations as in (Psf 24)-(Psf 26), $$\mathrm{char}\left(B\right)=\sum _{\genfrac{}{}{0ex}{}{p\in B}{p\subseteq C-\rho }}{q}^{-(\iota \left(p\right)+\phi \left(p\right)-f\left(p\right))}{(1-q^{-2})}^{f\left(p\right)-c\left(p\right)}{P}_{\mathrm{wt}\left(p\right)}.$$

		Proof.
	If $p$ is a dominant walk let $$B_q\left(p\right) \; be\; the\; q-crystal\; generated\; by\; p$$ under the action of the root operators ${\tilde{e}}_i$ and ${\tilde{f}}_i.$ The point is that the set of all positively folded alcove walks is partitioned into "equivalence classes" given by the sets $B_q\left(p\right)$ such that $p\in B_{\mathrm{univ}}$ is dominant and $p$ is the unique dominant walk in $B_q\left(p\right).$ Because ${\mathbf{1}}_0$ is characterized by the property that $C_i{\mathbf{1}}_0=(1+q^{-2}){\mathbf{1}}_0$ and the action of $C_i$ is modeled by the combinatorics of $i-$strings (Psf 25), this equivalence relation is generated by the relations $p\sim {\tilde{f}}_{i}p$ and $p\sim {\tilde{e}}_{i}p.$ $\square$

Products and restrications

The results in this section are generalizations of the Littlewood-Richardson rules. These are obtained as corollaries of Theorem 5.1.

The combinatorial definition of the root operators given above is essentially a consequence of the Leibniz rule for the Demazure operator, $$\begin{array}{ll}{\Delta }_i\left(fg\right)={\Delta }_i\left(f\right)g+\left(s_{i}f\right)\left({\Delta }_{i}g\right),for\; f,g\in 𝕂\left[P\right].& (Psf\; 27)\end{array}$$ The corresponding rule for the operators $C_i$ is $$\begin{array}{ll}{C}_i\left(fg\right)=\left(C_{i}f\right)g+\left(s_{i}f\right)\left((C_i-(1+q^{-2}))g\right),for\; f,g\in 𝕂\left[P\right].& (Psf\; 28)\end{array}$$ This identity is implicit in the additivity in $\lambda$ of the relation in Proposition 3.2e (the product $T_{s_i}{X}^{\lambda +\mu }=\left(T_{s_i}{X}^{\lambda }\right)X^{\mu }$ can be expanded in two different ways using Proposition 3.2e).

In order to define the product of $p_1\otimes p_2$ of walks $p_1,p_2\in B_{\mathrm{univ}}$ the final direction $\phi \left(p_1\right)$ of $p_1$ and the initial direction $\iota \left(p_2\right)$ of $p_2$ need to be taken into account. (To properly model the multiplication of Hall-Littlewood polynomials we must account for the effect of ${\mathbf{1}}_0$ in the product $P_{\mu }{\mathbf{1}}_0{P}_{\nu }{\mathbf{1}}_0$ and we cannot just concatenate walks as in the alcove walk algebra). Define $p_1\otimes p_2,$ recursively, by $$\begin{array}{lcl}If\; \phi \left(p_1\right)=\iota \left(p_2\right)& then& p_1\otimes p_2=p_1{p}_2, \; the\; concatenation\; of\; p_1 \; and\; p_2, \; and\\ if\; \phi \left(p_1\right)\ne \iota \left(p_2\right)& then& p_1\otimes p_2=p_1^{\text{'}}\otimes p_2,\end{array}$$ where $p_1^{\text{'}}$ is the alcove walk constructed by the following procedure. Let $c_{i_1}^{{\epsilon }_1}\cdots c_{i_r}^{{\epsilon }_r}$ be a minimal length walk from $\phi \left(p_1\right)$ to $\iota \left(p_2\right).$ If ${\epsilon }_1=-$ let $p_1^{\text{'}}=p_1{c}_{i_1}^{-}.$ If ${\epsilon }_1=+$ let $H_{\alpha ,j}$ be the hyperplane crossed by the last step of $p_1{c}_{i_1}^{-}$ and change the last negative crossing of $H_{\alpha ,j}$ in $p_1$ to a fold to obtain a new path $p_1^{\text{'}}$ with $\phi \left(p_1^{\text{'}}\right)=\phi \left(p_1{c}_{i_1}^{+}\right).$

In terms of root operators, the Leibniz rule (Psf 28) translates to the property (Psf 29) $$\begin{array}{rcl}{\tilde{e}}_i(p_1\otimes p_2)& =& \{\begin{array}{ll}\left({\tilde{e}}_i{p}_1\right)\otimes p_2,& if\; d_i^{+}\left(p_1\right)\ge d_i^{-}\left(p_2\right),\\ p_1\otimes \left({\tilde{e}}_i{p}_2\right),& if\; d_i^{+}\left(p_1\right)<d_i^{-}\left(p_2\right),\end{array}\phantom{\}}and\\ {\tilde{f}}_i(p_1\otimes p_2)& =& \{\begin{array}{ll}\left({\tilde{f}}_i{p}_1\right)\otimes p_2,& if\; d_i^{+}\left(p_2\right)>d_i^{-}\left(p_2\right)\left(\stackrel{\_}{p_1\otimes p_2}=\begin{array}{c}\\ \\ \end{array}\right),\\ p\otimes \left({\tilde{f}}_{i}q\right),& if\; d_i^{+}\left(p\right)\le d_i^{-}\left(q\right)\left(\stackrel{\_}{p\otimes q}=\begin{array}{c}\\ \\ \end{array}\right),\end{array}\phantom{\}}\end{array}$$ for $p_1,p_2\in B_{\mathrm{univ}}.$ It follows from this version of the Leibnitz rule that if $B_1$ and $B_2$ are $q-$crystals then the product $$\begin{array}{ll}{B}_1\otimes B_2=\{p_1\otimes p_2 | p_1\in B_1, p_2\in B_2\}is\; also\; aq-crystal,& (Psf\; 30)\end{array}$$ and $$\begin{array}{ll}\mathrm{char}(B_1\otimes B_2)=\mathrm{char}\left(B_1\right)\mathrm{char}\left(B_2\right).& (Psf\; 31)\end{array}$$ This last property is not completely trivial. The general case follows from the rank one case (projecting onto the line perpendicular to $H_{\alpha }$). More importantly, the definition of the product $\otimes$ is forced by (Psf 29)-(Psf 31).

The following theorem is a version of [Sc, Theorem 1.2] and results in [KM] and [Ha].

Recall the notations from Theorem 2.1. If $\lambda \in P^{+}$ let $B_q\left(p_{\lambda }\right)$ be the $q-$crystal generated by $p_{\lambda },$ where $p_{\lambda }$ is a fixed minimal length alcove walk from $A$ to $\lambda +A.$ For $\mu ,\nu \in P^{+},$ $$P_{\mu }{P}_{\nu }=\sum _{\genfrac{}{}{0ex}{}{p\in B_q(p\nu )}{p_{\mu }\otimes p\subseteq C-\rho }}{q}^{-(l\left(\iota \left(p\right)\right)+l\left(\phi \left(p\right)\right)-f\left(p\right))}{(1-q^{-2})}^{f\left(p\right)-c\left(p\right)}{P}_{\mu +\mathrm{wt}\left(p\right)}.$$

		Proof.
	Using (Psf 30) and (Psf 31) and applying Theorem 5.1 to the $q-$crystal $B_q\left(p_{\mu }\right)\otimes B_q\left(p_{\nu }\right)$ gives $$\begin{array}{rcl}{P}_{\mu }{P}_{\nu }{\mathbf{1}}_0& =& \mathrm{char}\left(B_q\left(p_{\mu }\right)\right)\mathrm{char}(B_q{p}_{\nu }\left)\right){\mathbf{1}}_0\\ & =& \mathrm{char}((B_q\left(p_{\mu }\right)\otimes B_q\left(p_{\nu }\right)){\mathbf{1}}_0\\ & =& \sum _{\genfrac{}{}{0ex}{}{p=p_1\otimes p_2\in B_q\left(p_{\mu }\right)\otimes B_q\left(p_{\nu }\right)}{p_1\otimes p_2\subseteq C-\rho }}{q}^{-(l\left(\iota \left(p\right)\right)+l\left(\phi \left(p\right)\right)-f\left(p\right))}{(1-q^{-2})}^{f\left(p\right)-c\left(p\right)}{P}_{\mu +\mathrm{wt}\left(p_2\right)}{\mathbf{1}}_0\end{array}$$ since every path in $p_1\in B_q\left(p_{\mu }\right)$ which is constrained in $C-\rho$ has weight $\mu$ (so that $\mathrm{wt}(p_1\otimes p_2)=\mu +\mathrm{wt}\left(p_2\right).)$ $\square$

Fix $J\subseteq \{1,2,...,n\}.$ The subgroup of $W$ generated by the reflections in the hyperplanes $H_{{\alpha }_j}, j\in J,$ $$W_J=⟨s_j | j\in J⟩,acts\; on\; {𝔥}_{ℝ}^{*},with\; C_J=\{\mu \in {𝔥}_{ℝ}^{*} | ⟨\mu ,{\alpha }_j^{\vee }⟩>0 \; for\; j\in J\}$$ as a fundamental chamber. Let $H_J=\mathrm{span}\left\{T_{w^{-1}}^{-1} \right| w\in W_J\}$ and let ${\mathbf{1}}_J\in H_J$ be given by $${{\mathbf{1}}_J}^2=<mi>J</mi>and{T}_{w^{-1}}^{-1}{\mathbf{1}}_J=q^{-l\left(w\right)}{\mathbf{1}}_J,for\; w\in W_J.$$ For $\mu \in P$ let $W_{\mu }^J$ be the stabilizer of $\mu$ under the $W_J$ action on $P$ and $$\begin{array}{ll}\begin{array}{l}define\; P_{\mu }^J(X;q^{-2})\in 𝕂{\left[P\right]}^{W_J} \; by\\ P_{\mu }^J(X;q^{-2}){\mathbf{1}}_0=\left(\sum _{w\in W_J^{\mu }}{q}^{-l\left(w\right)}{T}_{w^{-1}}^{-1}\right)X^{\mu }{\mathbf{1}}_0,\end{array}& (Psf\; 32)\end{array}$$ where $W_J^{\mu }$ is a set of minimal length coset representatives for the cosets in $W_J/W_{\mu }^J.$ Then $$\begin{array}{ll}up\; to\; normalization\; P_{\lambda }^J(X;q^{-2}){\mathbf{1}}_0 \; equals\; {\mathbf{1}}_J{X}^{\lambda }{\mathbf{1}}_0,& (Psf\; 33)\end{array}$$ and $$\begin{array}{ll}{\mathbf{1}}_J\tilde{H}{\mathbf{1}}_0 \; has\; basis\; \left\{P_{\lambda }^J{\mathbf{1}}_0 \right| \lambda \in P_J^{+}\},where\; P_J^{+}=P\cap \stackrel{\_}{C_J},& (Psf\; 34)\end{array}$$ with $\stackrel{\_}{C_J}=\{\mu \in {𝔥}_{ℝ}^{*} | ⟨\mu ,{\alpha }_j^{\vee }⟩>0 \; for\; j\in J\}.$

Recall the notations from Theorem 2.1. If $\mu \in P^{+}$ let $B_q\left(p_{\mu }\right)$ be the $q-$crystal generated by $p_{\mu },$ where $p_{\mu }$ is a fixed minimal length alcove walk from $A$ to $\mu +A.$ Let $\lambda \in P^{+}$ and let $J\subseteq \{1,2,...,n\}.$ Then $$P_{\lambda }=\sum _{\genfrac{}{}{0ex}{}{p\in B\left(\lambda \right)}{p\subseteq C_J-{\rho }_J}}{q}^{-(l\left(\iota \left(p\right)\right)+l\left(\phi \left(p\right)\right)-f\left(p\right))}{(1-q^{-2})}^{f\left(p\right)-c_J\left(p\right)}{P}_{\mathrm{wt}\left(p\right)}^J,$$ where $C_J-{\rho }_J=\{\mu \in {𝔥}_{ℝ}^{*} | ⟨\mu ,{\alpha }_j^{\vee }⟩>-1 \; for\; j\in J\}$ and $c_J\left(p\right)$ is the number of folds of $p$ which touch a hyperplane $H_{{\alpha }_j}$ with $j\in J.$

		Proof.
	A J-crystal is a set of positively folded alcove walks $B$ which is closed under the operators ${\tilde{e}}_j, {\tilde{f}}_j,$ for $j\in J.$ Since $P_{\lambda }{\mathbf{1}}_0=\mathrm{char}\left(B_q\left(p_{\lambda }\right)\right){\mathbf{1}}_0,$ the statement follows by applying Theorem 5.1 to $B_q\left(p_{\lambda }\right)$ viewed as a $J-$crystal. $\square$

Notes and References

The above notes are taken from section 4 of the paper

[Ram] A. Ram, Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (2006), 963-1013.

References

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[GL] S. Gaussent and P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 no. 1 (2005), 35-88.

[GR] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 no. 8 (2004), 1263-1283.

[Ha] T. Haines, Structure constants for Hecke and representation rings, Int. Math. Res. Not. 39 (2003), 2103-2119.

[IM] N. Iwahori and H. Hatsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $𝔭-$adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5-48.

[Ki] K. Killpatrick, A Combinatorial Proof of a Recursion for the q-Kostka Polynomials, J. Comb. Th. Ser. A 92 (2000), 29-53.

[KM] M. Kapovich and J.J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation theory, arXiv: math.RT/0411182.

[LS] A. Lascoux and M.P. Schützenberger, Le monoïde plaxique, Quad. Ricerce Sci. 109 (1981), 129-156.

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