text:
\section{Introduction}

In our previous paper \cite{LRS} we provided a construction of an ``abstract'' Fock space $\cF_\ell$ in a general Lie type setting.
The construction is given by simple combinatorial ``straightening relations'' which
generalize the Kashiwara-Miwa-Stern \cite{KMS} formulation of the $q$-Fock space from the
type A case. We showed that the abstract Fock space is a combinatorial realization of the
graded Grothendieck group of finite dimensional representations of the quantum group at
a root of unity, where the standard basis elements $\vert\lambda \rangle$
correspond to the Weyl modules $\Delta_q(\lambda)$ and the KL-basis $C_\lambda$
corresponds to the simple modules $L_q(\lambda)$.

In Section 1 we prove a product theorem (Theorem \ref{TenPrdThm})
in abstract Fock space which generalizes the type A
theorem of Leclerc and Thibon \cite[Theorem 6.9]{LT}.  Our proof
follows the same pattern as the proof for Type A given in \cite[Theorem 6.9]{LT}
except that, in order to deal with general Lie type,
we have replaced the use of ribbon tableaux
with the crystal basis and Littelmann paths.
The basic philosophy of our technique is similar to the main idea of a paper of Guilhot \cite{Gu}
but we also make use of the elegant cancellation technique of
Littelmann \cite[proof of Theorem 9.1]{Li} to complete the proof.  This technique provides
a combinatorial control of the Demazure operator used in the proof of \cite[Lemma 4.4]{Knp}.
We have not considered the unequal parameter case in this paper but the close relation
between our context and that of \cite{Knp} cries out for an extension of the tensor
product theorem for abstract Fock space to unequal parameters.

The Casselman-Shalika formula is important in the representation theory of
p-adic groups (see \cite{CS}), in its relation
to the affine Hecke algebra (see for example \cite{BBF}) and in the
geometric Langlands program (see \cite{FGV} and \cite{NP}).
In Section 2 we show that the Casselman-Shalika formula can be
derived as a special case of the Steinberg tensor product theorem for abstract Fock space.
This derivation is done by using the relationship between the abstract Fock space and
the affine Hecke algebra as detailed in \cite[Theorem 4.7]{LRS}.

In Section 3, we review the connection between the abstract Fock space and the
representations of quantum groups at roots of unity (Theorem 3.1) and the Steinberg-Lusztig
tensor product theorem (Theorem 3.2).  The Steinberg-Lusztig tensor product theorem is the
primary motivation for the product theorem in abstract Fock space.  Our approach does provide
an alternative proof of the  Steinberg-Lusztig tensor product theorem for representations of
quantum groups at  roots of unity (though hardly elementary since proving the
Steinberg-Lusztig tensor product theorem this way relies on deep results of Kazhdan-Lusztig
\cite{KL94}
and Kashiwara-Tanisaki \cite{KT95}).

As explained in \cite{LT}, the Steinberg-Lustzig tensor product theorem and
the abstract Fock space are intimately related to the
LLT polynomials defined in type A by Lascoux, Leclerc and Thibon \cite{LLT}.
Fundamentally, the LLT polynomials are taking the role of the characters of the
Frobenius twisted Weyl modules which, by the Steinberg-Lusztig tensor product theorem, are
simple modules for the quantum group at a root of unity.
General Lie type definitions
of LLT polynomials have been given by Grojnowski-Haiman \cite{GH}
and Lecouvey \cite{Lcy}.
In the second half of Section 3, we summarize a 2008 letter
from C.\ Lecouvey to A.\ Ram which explains
that a consequence of the tensor product theorem for
abstract Fock space is that the definition from \cite{GH} and the definition from \cite{Lcy}
coincide up to a  power of $t^{\frac12}$.

Kazhdan and Lusztig \cite{KL94} established an equivalence of categories between an appropriate
category of representations of the affine Lie algebra (of negative level) and the finite dimensional
representations of the quantum group (of the finite dimensional Lie algebra) at a root of unity.
In Section 4 we review this correspondence and make explicit the tensor product theorem
in terms of representations of the affine Lie algebra.  This produces a character formula for certain
negative level irreducible highest weight representations of the affine Lie algebra.  From the point of
view of this paper this character formula is an easy consequence of \cite{KL94} and
\cite{Lu89}.  We find it difficult to believe that this formula has not been noticed before but we
have not yet been able to locate a suitable specific reference.

We thank all the institutions which have supported our work on this paper,
particularly the Univ. of Melbourne,  the
Australian Research Council (grants DP1201001942 and DP130100674)
and  ICERM (Institute for Computational and Experimental Research in Mathematics).
We thank Kari Vilonen and Ting Xue for generous support of a visit of Martina Lanini to
University of Melbourne funded by their Australian Research Council grant DP150103525.

\section{A product theorem in abstract Fock space $\cF_\ell$}\label{sectionFelldefin}

Let $W_0$ be a finite Weyl group, generated by simple reflections
$s_1, \ldots, s_n$, and acting on a lattice of weights $\fa_\ZZ^*$.
For example, this situation arises when $T$ is a maximal torus of a reductive algebraic group $G$,
\begin{equation}
\fa_\ZZ^* = \Hom(T,\CC^\times)
\qquad\hbox{and}\qquad
W_0 = N(T)/T,
\label{wtsWeylgpdefn}
\end{equation}
where $N(T)$ is the normalizer of $T$ in $G$.  The simple reflections
in $W_0$ correspond to a choice of Borel subgroup $B$ of $G$ which contains $T$.
Let $R^+$ denote the positive roots.  Let $\alpha_1, \ldots, \alpha_n$ be the simple roots and let
$\alpha_1^\vee, \ldots, \alpha_n^\vee$ be the simple coroots.
The \emph{dot action} of $W_0$ on $\fa_\ZZ^*$ is given by
\begin{equation}
w\circ\lambda = w(\lambda+\rho)-\rho,
\qquad\hbox{where}\quad \rho = \hbox{$\frac12$}\sum_{\alpha\in R^+} \alpha
\label{dotaction}
\end{equation}
is the half sum of the positive roots for $G$ (with respect to $B$).

Fix $\ell\in \ZZ_{>0}$.  The \emph{abstract Fock space} $\mathcal{F}_\ell$ is
the $\ZZ[t^{\frac12},t^{-\frac12}]$-module generated by $\{ \vert \lambda\rangle \ |\ \lambda\in \fa_{\ZZ}^{*}\}$ with relations
\begin{equation}
\vert s_i\circ\lambda\rangle=\begin{cases}
-\vert \lambda\rangle, &\hbox{if $\langle\lambda+\rho,\alpha_i^\vee\rangle \in \ell\ZZ_{\ge 0}$,} \\
-t^{\frac12}\vert \lambda\rangle, &\hbox{if $0<\langle\lambda+\rho,\alpha_i^\vee\rangle<\ell $,} \\
-t^{\frac12}\vert s_i\circ\lambda^{(1)} \rangle -\vert \lambda^{(1)} \rangle -t^{\frac12}\vert \lambda\rangle,
&\hbox{if $ \langle\lambda+\rho,\alpha_i^\vee\rangle > \ell$ and $\langle\lambda+\rho,\alpha_i^\vee\rangle\not\in \ell\ZZ$,}
\end{cases}
\label{Fstraightening}
\end{equation}
where
$\lambda^{(1)} = \lambda - j\alpha_i$
if $\langle\lambda+\rho, \alpha_i^\vee\rangle = k\ell + j$ with $k\in \ZZ_{>0}$ and $j\in \{1, \ldots, \ell-1\}$.

The following picture illustrates the terms in \eqref{Fstraightening}.  This is the case $G=SL_2$ with $\ell=5$,
$\langle\omega_1,\alpha_1^\vee\rangle=1$ and $\alpha_1=2\omega_1$ and, in the picture,
$\lambda$ corresponds to the third case of \eqref{Fstraightening}, $\mu$ to the first case and $\nu$ to the second case.
$$\setlength{\unitlength}{0.5cm}
\begin{picture}(23,3)
\put(-5,2){\line(1,0){32}}

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\put(24,2){\circle*{0.3}}
\put(25,2){\circle*{0.3}}
\put(26,2){\circle*{0.3}}

\put(-4.8,0.8){$\scriptstyle{-14}$}
\put(-3.8,0.8){$\scriptstyle{-13}$}
\put(-2.8,0.8){$\scriptstyle{-12}$}
\put(-1.8,0.8){$\scriptstyle{-11}$}
\put(-0.8,0.8){$\scriptstyle{-10}$}
\put(0.4,0.8){$\scriptstyle{-9}$}
\put(1.4,0.8){$\scriptstyle{-8}$}
\put(2.4,0.8){$\scriptstyle{-7}$}
\put(3.4,0.8){$\scriptstyle{-6}$}
\put(4.4,0.8){$\scriptstyle{-5}$}
\put(5.4,0.8){$\scriptstyle{-4}$}
\put(6.4,0.8){$\scriptstyle{-3}$}
\put(7.4,0.8){$\scriptstyle{-2}$}
\put(8.4,0.8){$\scriptstyle{-\rho}$}
\put(9.85,0.8){$\scriptstyle{0}$}
\put(10.85,0.8){$\scriptstyle{\omega_1}$}
\put(11.85,0.8){$\scriptstyle{2}$}
\put(12.85,0.8){$\scriptstyle{3}$}
\put(13.85,0.8){$\scriptstyle{4}$}
\put(14.85,0.8){$\scriptstyle{5}$}
\put(15.85,0.8){$\scriptstyle{6}$}
\put(16.85,0.8){$\scriptstyle{7}$}
\put(17.85,0.8){$\scriptstyle{8}$}
\put(18.85,0.8){$\scriptstyle{9}$}
\put(19.75,0.8){$\scriptstyle{10}$}
\put(20.75,0.8){$\scriptstyle{11}$}
\put(21.75,0.8){$\scriptstyle{12}$}
\put(22.75,0.8){$\scriptstyle{13}$}
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\put(25.75,0.8){$\scriptstyle{16}$}

\put(-1,1.2){\line(0,1){1.5}}
\put(4,1.2){\line(0,1){1.5}}
\put(9,1.2){\line(0,1){1.5}}
\put(14,1.2){\line(0,1){1.5}}
\put(19,1.2){\line(0,1){1.5}}
\put(24,1.2){\line(0,1){1.5}}

\put(19.75,2.3){$\scriptstyle{\lambda}$}
\put(17.75,2.3){$\scriptstyle{\lambda^{(1)}}$}
\put(-0.8,2.3){$\scriptstyle{s_1\circ\lambda^{(1)} }$}
\put(-2.8,2.3){$\scriptstyle{s_1\circ\lambda}$}

\put(18.75,3){$\scriptstyle{\mu}$}
\put(-1.8,3){$\scriptstyle{s_1\circ\mu}$}

\put(11.75,2.3){$\scriptstyle{\nu}$}
\put(5.4,2.3){$\scriptstyle{s_1\circ\nu}$}

\end{picture}
$$
Define a $\ZZ$-linear involution
$\overline{\phantom{T}}\colon \mathcal{F}_{\ell}\to \mathcal{F}_{\ell}$ by
\begin{equation}
\overline{t^{\frac12}} = t^{-\frac12}
\qquad\hbox{and}\qquad
\overline{\vert \lambda \rangle} = (-1)^{\ell(w_0)}(t^{-\frac12})^{\ell(w_{0})-N_\lambda}\, \vert w_0\circ \lambda \rangle.
\label{Fellbar}
\end{equation}
where $w_0$ is the longest element of $W_0$, $\ell(w_0) = \Card(R^+)$ is the length of $w_0$, and
$N_\lambda = \Card\{\alpha\in R^+\ |\ \langle \lambda+\rho, \alpha^\vee\rangle \in \ell\ZZ\}.$

The \emph{dominant integral weights}  with the \emph{dominance partial order} $\le$ are the elements of
\begin{equation}
\begin{array}{c}
(\fa_\ZZ^*)^+
= \{\lambda\in \fa_\ZZ^*\ |\ \hbox{$\langle \lambda+\rho, \alpha_i^\vee\rangle> 0$ for $i=1, 2,\ldots, n$} \}
\\
\\
\hbox{with}\qquad
\mu\le \lambda\qquad\hbox{if}\quad
\mu \in \lambda - \sum_{\alpha\in R^+} \ZZ_{\ge 0}\alpha.
\end{array}
\label{domintwtsdefn}
\end{equation}
In \cite[Theorem 1.1 and Proposition 2.1]{LRS} we showed that
$\cF_\ell$ has bases
\begin{equation}
\{ \vert\lambda\rangle\ |\ \lambda\in (\fa_\ZZ^*)^+\}
\qquad\hbox{and}\qquad
\{ C_\lambda\ |\ \lambda\in (\fa_\ZZ^*)^+\}
\label{Fellbases}
\end{equation}
where $C_\lambda$ are determined by
\begin{equation}
\overline{C_\lambda} = C_\lambda
\qquad\hbox{and}\qquad
C_\lambda = \vert \lambda\rangle + \sum_{\mu\ne \lambda} p_{\mu\lambda} \vert\mu\rangle,
\qquad\hbox{with $p_{\mu\lambda}\in t^{\frac12}\ZZ[t^{\frac12}]$.}
\label{Clambdadefn}
\end{equation}

\subsection{The action of $\KK[X]^{W_0}$ on $\cF_\ell$}

Letting $\KK = \ZZ[t^{\frac12}, t^{-\frac12}]$, the group algebra of $\fa_\ZZ^*$ is
\begin{equation}
\KK[X] = \hbox{$\KK$-span}\{ X^\mu\ |\ \mu\in \fa_\ZZ^*\}
\quad\hbox{with}\quad
X^\mu X^\nu = X^{\mu+\nu}.
\label{KXdefn}
\end{equation}
The Weyl group $W_0$ acts $\KK$-linearly on $\KK[X]$ by
\begin{equation}
wX^\mu = X^{w\mu},\ \hbox{for $w\in W_0$ and $\mu\in \fa_\ZZ^*$,}
\quad\hbox{and}\quad
\KK[X]^{W_0} = \{ f\in \KK[X]\ |\ wf = f\}
\end{equation}
is the ring of \emph{symmetric functions}.

Let $V$ be the free $\KK$-module generated
by $\{ \vert \lambda\rangle \ |\ \lambda\in \fa_{\ZZ}^{*}\}$ so that
\begin{equation}
\cF_\ell \cong V/I,
\label{Fasqtt}
\end{equation}
where $I$ is the subspace of $V$
consisting of $\KK$-linear combinations of
the elements
$$
\begin{array}{cl}
a_\lambda = \vert s_i\circ\lambda\rangle+\vert \lambda\rangle,
&\hbox{with $\langle\lambda+\rho,\alpha_i^\vee\rangle \in \ell\ZZ_{\ge 0}$,} \\
b_\lambda = \vert s_i\circ\lambda\rangle+t^{\frac12}\vert \lambda\rangle,
&\hbox{with $0<\langle\lambda+\rho,\alpha_i^\vee\rangle<\ell $, and} \\
c_\lambda = \vert s_i\circ\lambda\rangle+t^{\frac12}\vert s_i\circ\lambda^{(1)} \rangle
+\vert \lambda^{(1)} \rangle + t^{\frac12}\vert \lambda\rangle,\quad
&\hbox{with $ \langle\lambda+\rho,\alpha_i^\vee\rangle > \ell$ and $\langle\lambda+\rho,\alpha_i^\vee\rangle\not\in \ell\ZZ$.}
\end{array}
$$

Let $\mathring{\fg}$ be the Lie algebra of the reductive group $G$ alluded to in \eqref{wtsWeylgpdefn}.
Let $\varphi$ be the highest weight of the adjoint representation and let
$\varphi^\vee\in [\mathring{\fg}_\varphi, \mathring{\fg}_{-\varphi}]$ such that
$\langle \varphi, \varphi^\vee\rangle=2$ (so that $\varphi^\vee$ is an appropriate normalized
highest short coroot of $\mathring{\fg}$).
\begin{equation}
\hbox{The \emph{dual Coxeter number} is} \qquad
h = \langle \rho, \varphi^\vee\rangle + 1.
\label{Coxeternumber}
\end{equation}
The \emph{level $(-\ell-h)$ action} of
$\KK[X]$ on $V$ is the $\KK$-linear extension of
\begin{equation}
X^\mu \cdot\vert \gamma\rangle = \vert -\ell w_0\mu+\gamma\rangle,
\qquad\hbox{for $\mu, \gamma\in \fa_\ZZ^*$.}
\label{Vaction}
\end{equation}
Letting $w_0$ be the longest element of $W_0$, define
$$\mu^* = -w_0\mu
\qquad\hbox{and}\qquad w^*= w_0ww_0,
\qquad\hbox{for $\mu\in \fa_\ZZ^*$ and $w\in W$.}
$$
(This notation is such that if $\mu\in \fa_\ZZ^*$ and $L_{\mathring{\fg}}(\mu)$ denotes the
irreducible $\mathring{\fg}$-module of
highest weight $\mu$ then the dual $L_{\mathring{\fg}}(\mu)^*\cong L_{\mathring{\fg}}(\mu^*)$
and ${}^*\colon W_0\to W_0$ is the involutive automorphism of $W_0$
induced by the automorphism of the Dynkin diagram specified by $s_i^* = s_{i^*}$.)
Then
\begin{equation}
X^\mu \cdot\vert \gamma\rangle = \vert \ell \mu^*+\gamma\rangle
\qquad\hbox{and}\qquad
(w\mu)^*
= w^*\mu^*.
\label{staraction}
\end{equation}

The following proposition establishes an action of the ring of symmetric functions
$\KK[X]^{W_0}$ on the abstract Fock space $\cF_\ell$.  From the point of view of
Theorem \ref{abstracttoHecke} below, this action is coming from an action of the center
of the affine Hecke algebra which, by an important result of Bernstein, is the ring
of symmetric functions (inside the affine Hecke algebra).  Our proof of
Proposition \ref{KXaction} provides an independent proof of the existence of the action
of $\KK[X]^{W_0}$ without referring to the affine Hecke algebra and the characterization of its center.

\begin{prop} \label{KXaction}
The action of $\KK[X]$ on $V$ given in \eqref{Vaction} induces a $\KK$-linear action of
the ring $\KK[X]^{W_0}$ of symmetric functions on $\cF_\ell$  by
$$\left(\sum_{w\in W_0} X^{w\mu}\right) \cdot \vert \gamma\rangle
= \sum_{w\in W_0} \vert \ell(w\mu)^*+\gamma\rangle,
\quad\hbox{for $\mu\in \fa_\ZZ^*$ and $\gamma\in \fa_\ZZ^*$.}
$$
\end{prop}
\begin{proof}
Let $f$ be an element of the subspace $I$ defined in \eqref{Fasqtt}, let $\mu\in \fa_\ZZ^*$
and let $i\in \{1, \ldots, n\}$.  Summing over a set of representatives of the cosets in $\{1, s_i^*\}\backslash W_0$,
$$\left(\sum_{w\in W_0} X^{w\mu}\right) \cdot f
= \left( \sum_{v\in \{1,s_i^*\}\backslash W_0} (X^{v\mu}+X^{s_i^*v\mu})\right)\cdot f,$$
where the representatives $v\in \{1, s_i^*\}\backslash W_0$ are chosen such that
$\langle v\mu, \alpha_{i^*}^\vee\rangle \in \ZZ_{\ge 0}$.

\smallskip\noindent
Case 1:  $f = \vert s_i\circ\lambda\rangle
+ \vert \lambda\rangle$ with $\langle \lambda+\rho, \alpha_i^\vee\rangle \in \ell\ZZ_{\geq 0}$.  Then
\begin{align*}
(X^{s_i^*v\mu}&+X^{v\mu})\cdot (\vert s_i\circ\lambda\rangle
+ \vert \lambda\rangle)\\
&=
\vert \ell (s_i^*v\mu)^* + s_i\circ\lambda\rangle
+\vert \ell (v\mu)^* + s_i\circ\lambda\rangle
+ \vert \ell (s_i^*v\mu)^* + \lambda\rangle
+ \vert \ell (v\mu)^* + \lambda\rangle \\
&=
\vert \ell s_iv^*\mu^* + s_i\circ\lambda\rangle
+\vert \ell v^*\mu^* + s_i\circ\lambda\rangle
+ \vert \ell s_iv^*\mu^* + \lambda\rangle
+ \vert \ell v^*\mu^* + \lambda\rangle \\
&=
\vert s_i\circ( \ell v^*\mu^* + \lambda)\rangle
+\vert s_i\circ( \ell s_iv^*\mu^* + \lambda)\rangle
+ \vert \ell s_iv^*\mu^* + \lambda\rangle
+ \vert \ell v^*\mu^* + \lambda\rangle \\
&=  \begin{cases}
a_{\ell v^*\mu^*+s_i\circ \lambda}+a_{\ell v^*\mu^*+\lambda},
&\hbox{if $\langle \ell v^*\mu^*, \alpha_i^\vee\rangle > \langle \lambda+\rho, \alpha_i^\vee\rangle$,} \\
a_{\ell s_iv^*\mu^* + \lambda}+a_{\ell v^*\mu^*+\lambda},
&\hbox{if $\langle \ell v^*\mu^*, \alpha_i^\vee\rangle \le \langle\lambda+\rho, \alpha_i^\vee\rangle$. }
\end{cases}
\end{align*}
Thus the right hand side is an element of $I$.

\smallskip\noindent
Case 2: $f = \vert s_i\circ\lambda\rangle
+ t^{\frac12}\vert \lambda\rangle$ with $0<\langle \lambda+\rho, \alpha_i^\vee\rangle < \ell$.
Then
$\ell v^*\mu^*+s_i\circ\lambda = (\ell v^*\mu^*+\lambda)^{(1)}$ so that
\begin{align*}
(X^{s_i^*v\mu}&+X^{v\mu})\cdot (\vert s_i\circ\lambda\rangle
+ t^{\frac12}\vert \lambda\rangle)\\
&=
\vert \ell (s_i^* v\mu)^* + s_i\circ\lambda\rangle
+\vert \ell (v\mu)^* + s_i\circ\lambda\rangle
+ t^{\frac12}\vert \ell (s_i^* v\mu)^* + \lambda\rangle
+ t^{\frac12}\vert \ell (v\mu)^* + \lambda\rangle \\
&=
\vert \ell s_i v^*\mu^* + s_i\circ\lambda\rangle
+\vert \ell v^*\mu^* + s_i\circ\lambda\rangle
+ t^{\frac12}\vert \ell s_i v^*\mu^* + \lambda\rangle
+ t^{\frac12}\vert \ell v^*\mu^* + \lambda\rangle \\
&=
\vert s_i\circ ( \ell v^*\mu^* + \lambda) \rangle
+\vert ( \ell v^*\mu^* + \lambda)^{(1)}\rangle
+ t^{\frac12}\vert s_i\circ( \ell v^*\mu^* + s_i\circ \lambda)\rangle
+ t^{\frac12}\vert \ell v^*\mu^* + \lambda\rangle \\
&=
\vert s_i\circ ( \ell v^*\mu^* + \lambda) \rangle
+ t^{\frac12}\vert s_i\circ( \ell v^*\mu^* + \lambda)^{(1)} \rangle
+\vert ( \ell v^*\mu^* + \lambda)^{(1)}\rangle
+ t^{\frac12}\vert \ell v^*\mu^* + \lambda\rangle \\
&= \begin{cases}
c_{\ell v^*\mu^*+\lambda}, &\hbox{if $\langle v^*\mu^*, \alpha_i^\vee\rangle\in \ZZ_{>0}$, } \\
2b_{\ell v^*\mu^* + \lambda}, &\hbox{if $\langle v^*\mu^*, \alpha_i^\vee\rangle =0$,}
\end{cases}
\end{align*}
since if $s_i^*v\mu\ne v\mu$ then $s_iv^*\mu^*\ne v^*\mu^*$ and
$\langle v^*\mu^*, \alpha_i^\vee\rangle\in \ZZ_{>0}$ then
$\langle \ell v^*\mu^*+\lambda+\rho, \alpha_i^\vee\rangle > \ell$ and
$\langle \ell v^*\mu^*+\lambda+\rho, \alpha_i^\vee\rangle \not\in \ell\ZZ$.
Thus the right hand side is an element of $I$.

\smallskip\noindent
Case 3:
Assume $\lambda\in \fa_\ZZ^*$ with $ \langle\lambda+\rho,\alpha_i^\vee\rangle > \ell$ and $\langle\lambda+\rho,\alpha_i^\vee\rangle\not\in \ell\ZZ$.
If $\mu\in \fa_\ZZ^*$ and $\langle \nu, \alpha_i^*\rangle\in \ZZ_{\ge 0}$ then
$$s_i\circ (v^*\mu^*+\nu) = s_i(v^*\mu^*+\nu+\rho)-\rho = s_iv^*\mu^* + s_i\circ\nu
\quad\hbox{and}\quad
(\ell \nu^* + \lambda)^{(1)} = \ell \nu^* + \lambda^{(1)},$$
so that, with $\langle \lambda+\rho, \alpha_i^\vee\rangle = k\ell+j$ with $k>0$ and $0\le j<\ell$,
\begin{align*}
(&X^{s_i^*v\mu}+X^{v\mu})\cdot (\vert s_i\circ\lambda\rangle
+t^{\frac12}\vert s_i\circ \lambda^{(1)}\rangle +\vert \lambda^{(1)}\rangle
+ t^{\frac12}\vert \lambda\rangle)\\
&=
\vert \ell (s_i^* v\mu)^* + s_i\circ\lambda\rangle
+\vert \ell (v\mu)^* + s_i\circ\lambda\rangle
+t^{\frac12}\vert \ell (s_i^*v\mu)^*+ s_i\circ \lambda^{(1)} \rangle
+t^{\frac12}\vert \ell (v\mu)^*+ s_i\circ \lambda^{(1)} \rangle \\
&\qquad +\vert \ell (s_i^*v\mu)^*+ \lambda^{(1)} \rangle
+\vert \ell (v\mu)^* + \lambda^{(1)} \rangle
+ t^{\frac12}\vert \ell (s_i^* v\mu)^* + \lambda\rangle
+ t^{\frac12}\vert \ell(v\mu)^* + \lambda\rangle \\
&=
\vert \ell s_iv^*\mu^* + s_i\circ\lambda\rangle
+\vert \ell v^*\mu^* + s_i\circ\lambda\rangle
+t^{\frac12}\vert \ell s_i v^*\mu^*+ s_i\circ \lambda^{(1)} \rangle
+t^{\frac12}\vert \ell v^*\mu^*+ s_i\circ \lambda^{(1)} \rangle \\
&\qquad +\vert \ell s_i v^*\mu^*+ \lambda^{(1)} \rangle
+\vert \ell v^*\mu^* + \lambda^{(1)} \rangle
+ t^{\frac12}\vert \ell s_i v^*\mu^* + \lambda\rangle
+ t^{\frac12}\vert \ell v^*\mu^* + \lambda\rangle \\
&=
\vert s_i\circ(\ell v^*\mu^* + \lambda) \rangle
+\vert s_i\circ( \ell s_i v^*\mu^* + \lambda)\rangle
+t^{\frac12}\vert s_i\circ (\ell v^*\mu^* +\lambda^{(1)}) \rangle
+t^{\frac12}\vert s_i\circ(\ell s_i v^*\mu^* + \lambda^{(1)}) \rangle \\
&\qquad +\vert \ell s_i v^*\mu^* + \lambda^{(1)} \rangle
+\vert (\ell v^*\mu^* + \lambda)^{(1)} \rangle
+ t^{\frac12}\vert \ell s_i v^*\mu^* + \lambda\rangle
+ t^{\frac12}\vert \ell v^*\mu^* + \lambda\rangle \\
&=\begin{cases}
c_{\lambda+ \ell v^* \mu^*}+c_{s_i\circ \lambda^{(1)}+v^*\mu^*},
&\hbox{if $\langle \ell v^*\mu^*, \alpha_i^\vee\rangle > \ell k>0$,}
\\
c_{\lambda+ \ell v^* \mu^*}+c_{\lambda+ s_i v^*\mu^*},
&\hbox{if $0<\langle \ell v^*\mu^*, \alpha_i^\vee\rangle < \ell k$,}
\\
c_{\lambda+ \ell v^* \mu^*}
+b_{\lambda+ s_i v^*\mu^*}
+b_{s_i\circ \lambda^{(1)}+v^*\mu^*},
&\hbox{if $\langle \ell v^*\mu^*, \alpha_i^\vee\rangle = \ell k$.}
\end{cases}
\end{align*}
Thus the right hand side is an element of $I$.

These computations show that $I$ is stable under the action of $\KK[X]^{W_0}$.
Thus the action of $\KK[X]^{W_0}$ on $\cF_\ell = V/I$ is well defined.
\end{proof}

 \begin{remark}  One might be tempted to try to define an action of
 $\KK[X]$ on $\cF_\ell$ by $X^\mu\cdot \vert \gamma \rangle = \vert \gamma+\ell\mu\rangle$
 for $\mu, \gamma\in \fa_\ZZ^*$ but this action is not well defined.  For example in the
 $G=SL_2$ case with $\ell=5$ pictured after \eqref{Fstraightening}, one would have
 $0 = X^{\omega_1}\cdot \vert -1\rangle = \vert 5-1\rangle = \vert 4\rangle$,
 which is a contradiction to \eqref{Fellbases}.
 On the other hand $0=(X^{-\omega_1}+X^{\omega_1})\cdot \vert -1\rangle
 = \vert -5-1\rangle + \vert 4\rangle = 0$, as it should be.
 \end{remark}

\subsection{The product theorem}

 Let $\mathring{\fg}$ be the Lie algebra of the reductive group $G$ alluded to in \eqref{wtsWeylgpdefn}.
For $\lambda\in (\fa_\ZZ^*)^+$ let $L_{\mathring{\fg}}(\lambda)$ be the irreducible
$U\mathring{\fg}$-module of highest weight $\lambda$ and let $B(\lambda)$ be the
crystal of $L_{\mathring{\fg}}(\lambda)$,
$$B(\lambda) = \{ \hbox{LS paths $p$ of type $\lambda$}\}
\quad\hbox{and}\quad
\hbox{$\mathrm{wt}(p)$ denotes the endpoint of $p$,}
$$
see \cite[\S5]{Ra}. The \emph{Weyl character} corresponding to $\lambda$ is
the element of $\KK[X]^{W_0}$ given by
\begin{equation}
s_{\lambda} = \mathrm{char}(L_{\mathring{\fg}}(\lambda))
=\frac{\displaystyle{\sum_{w\in W_0} \det(w) X^{w\circ \lambda} } }
{\displaystyle{ \sum_{w\in W_0} \det(w) X^{w\circ 0} } }
= \sum_{p \in B(\lambda)} X^{\mathrm{wt}(p)}.
\label{Weylchdefn}
\end{equation}

An \emph{$\ell$-restricted} dominant integral weight is $\lambda_0\in (\fa_\ZZ^*)^+$
such that $\langle \lambda_0,\alpha_i^\vee\rangle < \ell$ for $i\in \{1, \ldots, n\}$.
In other words, if $\omega_1, \ldots, \omega_n$ are the fundamental weights for
$\mathring{\fg}$ then
a weight $\lambda_0\in (\fa_\ZZ^*)^+$ is $\ell$-restricted if $\lambda_0$ is an
element of
\begin{equation}
\Pi_\ell = \{ a_1\omega_1+\cdots+a_n\omega_n\ |\ a_1, \ldots, a_n\in \{0,1,\ldots, \ell-1\} \}.
\label{Pielldefn}
\end{equation}

\begin{thm} \label{TenPrdThm}
Let $\lambda\in (\fa_\ZZ^*)^+$ be a dominant integral weight and write
$$\lambda = \ell \lambda_1 + \lambda_0,
\qquad\hbox{with $\lambda_0\in \Pi_\ell$ and $\lambda_1\in (\fa_\ZZ^*)^+$.}
$$
Then, with $C_\lambda\in \cF_\ell$ as in \eqref{Clambdadefn}
and the $\KK[X]^{W_0}$-action on $\cF_\ell$ as in Proposition \ref{KXaction},
$$C_\lambda = s_{\lambda_1^*} \cdot C_{\lambda_0}.$$
\end{thm}
\begin{proof}
The proof is accomplished in two steps:
\begin{enumerate}
\item[(a)] Show that $s_{\lambda_1^*} \cdot C_{\lambda_0}$ is bar invariant.
\item[(b)] Show that $s_{\lambda_1^*} \cdot C_{\lambda_0 }
= \vert \lambda\rangle + \sum_{\mu\ne \lambda} c_{\mu} \vert\mu\rangle$
with $c_{\mu}\in t^{\frac12}\ZZ[t^{\frac12}]$.
\end{enumerate}
Proof of (a):  The bar involution and $N_\gamma$ are defined in \eqref{Fellbar}.
Since $\langle -\ell w_0\mu, \alpha^\vee\rangle\in \ell\ZZ$ then
\begin{align*}
N_{\gamma-\ell w_0\mu}
= \Card&\{\alpha\in R^+\ |\ \langle \gamma-\ell w_0\mu+\rho, \alpha^\vee\rangle \in \ell\ZZ\} \\
&= \Card\{\alpha\in R^+\ |\ \langle \gamma+\rho, \alpha^\vee\rangle \in \ell\ZZ\}
=N_{\gamma}.
\end{align*}
Thus
\begin{align*}
\overline{X^\mu\cdot \vert \gamma\rangle }
&= \overline{\vert \gamma -\ell w_0\mu \rangle }
= (-1)^{\ell(w_0)}
(t^{-\frac12})^{\ell(w_0)-N_{\gamma-\ell w_0\mu}}
\vert w_0\circ(\gamma-\ell w_0\mu)\rangle \\
&= (-1)^{\ell(w_0)}
(t^{-\frac12})^{\ell(w_0)-N_{\gamma-\ell w_0\mu}}
\vert w_0(\gamma+\rho)-\rho-\ell\mu \rangle \\
&= (-1)^{\ell(w_0)}
(t^{-\frac12})^{\ell(w_0)-N_{\gamma-\ell w_0\mu}}
\vert w_0\circ \gamma-\ell\mu \rangle \\
&= (-1)^{\ell(w_0)}
(t^{-\frac12})^{\ell(w_0)-N_{\gamma}}
\vert w_0\circ \gamma-\ell \mu \rangle \\
&= (-1)^{\ell(w_0)}
(t^{-\frac12})^{\ell(w_0)-N_{\gamma}}
X^{w_0\mu}\cdot \vert w_0\circ \gamma\rangle
= X^{w_0\mu}\cdot \overline{ \vert \gamma\rangle},
\end{align*}
and since $s_{\lambda_1^*}$ is $W_0$-invariant,
$$\overline{ s_{\lambda_1^*} \cdot C_{\lambda_0} }
= (w_0 s_{\lambda_1^*})\cdot \overline{C_{\lambda_0}}
=s_{\lambda_1^*} \cdot C_{\lambda_0}.$$

\smallskip\noindent
(b)
Let $\lambda=\lambda_0+\ell\lambda_1$ as in the statement of the Theorem and let
$a \equiv b$ mean $a=b \bmod t^{\frac12}$.
By the second formula in \eqref{Clambdadefn},
\begin{equation}
s_{\lambda_1^*}\cdot C_{\lambda_0} \equiv s_{\lambda_1^*}\cdot \vert \lambda_0 \rangle
= \sum_{p\in B(\lambda_1^*)} X^{\mathrm{wt}(p)}  \vert \lambda_0 \rangle
=\sum_{p\in B(\lambda_1^*)} \vert \ell\mathrm{wt}(p)^*+\lambda_0 \rangle.
\label{SCexpansion}
\end{equation}
By \eqref{Fstraightening},
if $\lambda\in\mathfrak{a}_{\mathbb{Z}}^*$ and $\langle\lambda+\rho,\alpha_i^\vee\rangle\geq 0$
then
$$
\vert s_i\circ\lambda \rangle
\equiv
\begin{cases}
- \vert \lambda \rangle,
&\hbox{if $\langle\lambda+\rho,\alpha_i^\vee\rangle \in \ell\ZZ_{\geq 0}$,}\\
0, &\hbox{if $0 < \langle \lambda+\rho, \alpha_i^\vee \rangle < \ell$,} \\
- \vert \lambda^{(1)} \rangle, &\hbox{otherwise,}
\end{cases}
$$
where $\lambda^{(1)}=\lambda-j\alpha_i$ if $\langle\lambda+\rho,\alpha_i^\vee\rangle=k\ell+j$ with $j\in\{0,1,\ldots, \ell-1\}$.  Since $\lambda^{(1)} = \lambda$ if $\langle \lambda+\rho, \alpha_i^\vee\rangle\in \ell\ZZ_{\ge 0}$, the first case can be viewed as a special case of the last case
to read
$$
\vert s_i\circ\lambda \rangle
\equiv
\begin{cases}
0, &\hbox{if $0 < \langle \lambda+\rho, \alpha_i^\vee \rangle < \ell$,} \\
- \vert \lambda^{(1)} \rangle, &\hbox{otherwise.}
\end{cases}
$$

Assume $\langle \nu+\rho, \alpha_i^\vee\rangle\in \ZZ_{\le0}$ and let
$\lambda = s_i\circ(\lambda_0 + \ell \nu)$.
Since $\langle \rho, \alpha_i^\vee\rangle = 1$ then
\begin{align*}
\langle \lambda+\rho, \alpha_i^\vee\rangle
&= \langle s_i\circ(\lambda_0+\ell\nu)+\rho, \alpha_i^\vee\rangle
=\langle s_i(\lambda_0+\ell\nu+\rho), \alpha_i^\vee\rangle
=\langle \lambda_0+\ell\nu+\rho, s_i\alpha_i^\vee\rangle \\
&= - \langle \lambda_0+\ell\nu+\rho, \alpha_i^\vee\rangle
= \ell(-\langle \nu+\rho, \alpha_i^\vee\rangle) + (\ell -1 - \langle \lambda_0, \alpha_i^\vee\rangle).
\end{align*}
Since $\lambda_0\in \Pi_\ell$ then $0\le \ell-1-\langle \lambda_0,\alpha_i^\vee\rangle <\ell$
and so
\begin{align*}
\lambda^{(1)}
&= \lambda
- (\ell-1-\langle \lambda_0, \alpha_i^\vee\rangle)\alpha_i
= s_i\circ(\lambda_0 + \ell \nu)
- (\ell-1-\langle \lambda_0, \alpha_i^\vee\rangle)\alpha_i \\
&= s_i\lambda_0 + \ell s_i\nu+s_i\rho-\rho
- (\ell-1-\langle \lambda_0, \alpha_i^\vee\rangle)\alpha_i \\
&= (\lambda_0 - \langle \lambda_0, \alpha_i^\vee\rangle \alpha_i)
+\ell(s_i\nu+s_i\rho-\rho)+(\ell-1)\alpha_i
- (\ell-1-\langle \lambda_0, \alpha_i^\vee\rangle)\alpha_i \\
&= \lambda_0 + \ell(s_i\circ \nu).
\end{align*}
Thus, since $s_i\circ\lambda = \lambda_0+\ell\nu$,
$$
\vert \lambda_0+\ell\nu \rangle
\equiv
\begin{cases}
0, &\hbox{if $\langle \nu+\rho, \alpha_i^\vee\rangle = 0$,} \\
- \vert \lambda_0+\ell(s_i\circ \nu)\rangle,
&\hbox{if $\langle \nu+\rho, \alpha_i^\vee\rangle < 0$.} \\
\end{cases}
$$
Since $s_i\circ \nu = \nu$ when $\langle \nu+\rho, \alpha_i^\vee\rangle=0$, then
$\vert \lambda_0+\ell\nu \rangle
\equiv
- \vert \lambda_0+\ell (s_i\circ \nu)\rangle$ when
$\langle \nu+\rho, \alpha_i^\vee \rangle\in \ZZ_{\le 0}$ and,
replacing $\nu$ by $s_i\circ \nu$, gives
$\vert \lambda_0+\ell\nu \rangle\equiv
- \vert \lambda_0+\ell(s_i\circ \nu)\rangle$
for $\langle \nu+\rho, \alpha_i^\vee \rangle\in \ZZ_{\ge 0}$.
Thus
\begin{equation}
\vert \lambda_0+\ell\nu \rangle
\equiv
- \vert \lambda_0+\ell(s_i\circ \nu)\rangle,
\qquad\hbox{for $\nu\in \fa_\ZZ$.}
\label{modtstraightening}
\end{equation}

With formula \eqref{modtstraightening} established, follow
\cite[proof of Theorem 5.5]{Ra} (see also \cite[proof of Theorem 9.1]{Li})
to define an involution $\iota$ on the set $B(\lambda_1^*)\setminus\{p_{\lambda_1^*}^+\}$,
where $p_{\lambda_1^*}^+$ is the unique highest weight path in $B(\lambda_1^*)$.

\medskip
Let $p\in B(\lambda_1^*)$ and $p\neq p_{\lambda_1^*}^+$.
Since $p\neq p_{\lambda_1^*}^+$ the path $p$ crosses a wall out of the fundamental chamber
at some point during its trajectory.
Let $r$ be such that the first time $p$ leaves the dominant chamber is by crossing the
hyperplane $\{ x\in \fa_\RR^*\ |\ \langle x, \alpha_r^\vee\rangle = 0\}$.
Letting $\tilde e_r$ and $\tilde f_r$ denote the root operators on $B(\lambda_1^*)$,
the $r$-string containing $p$ is
$$S_r(p) =\{q\in B(\lambda_1^*)\ |\
\hbox{$q=\tilde{e}_r^kp$ or $q=\tilde{f}_r^k p$ where $k\in \ZZ_{\geq 0}$}\}.$$
  Let
$$
\hbox{$\iota(p)$ be the element of $S_r(p)$ such that
$\mathrm{wt}(\iota(p)) = s_r\circ \mathrm{wt}(p)$.}
$$
By \eqref{modtstraightening},
$$
\vert \lambda_0+\ell\wt(p)^*\rangle
\equiv
- \vert \lambda_0+\ell (s_r^*\circ \wt(p)^*) \rangle
= - \vert \lambda_0+\ell(s_r\circ \wt(p))^*\rangle
= - \vert \lambda_0+\ell\wt(\iota(p))^*\rangle,
$$
and so the map $\iota$ partitions the set $B(\lambda_1^*)\setminus\{p_{\lambda_1^*}^+\}$
into pairs $\{p, \iota(p)\}$ which cancel each other in the mod $t^{\frac{1}{2}}$
straightening of the terms of
$s_{\lambda_1^*}\cdot C_{\lambda_0}$ in \eqref{SCexpansion}.
Thus
$$s_{\lambda_1^*}\cdot C_{\lambda_0} \equiv
\vert \ell(\lambda_1^*)^*+\lambda_0\rangle
=\vert \lambda_0+ \ell \lambda_1\rangle,
\qquad\hbox{which proves (b).}
$$
\end{proof}

\section{The Casselman-Shalika formula}

In order to establish the Casselman-Shalika formula it is necessary to use the connection
between the abstract Fock space $\cF_\ell$ and the affine Hecke algebra $H$.  Let us recall
this relationship from \cite{LRS}.

\subsection{The affine Hecke algebra $H$}

Keep the notation for the finite Weyl group $W_0$, the simple reflections $s_1, \ldots, s_n$
and the weight lattice $\fa_\ZZ^*$ as in \eqref{wtsWeylgpdefn}.
For $i,j\in \{1, \ldots, n\}$ with $i\ne j$, let
$m_{ij}$ denote the order of $s_is_j$ in $W_0$
so that $s_i^2=1$ and $(s_is_j)^{m_{ij}}=1$ are the relations for the Coxeter presentation of $W_0$.
Let $\KK = \ZZ[t^{\frac12}, t^{-\frac12}]$.  The \emph{affine Hecke algebra} is
\begin{equation}
H = \hbox{$\KK$-span}\{X^\mu T_w\ |\ \mu\in \fa_\ZZ^*, w\in W_0\},
\label{affHeckedefn}
\end{equation}
with $\KK$-basis $\{X^\mu T_w\ |\ \mu\in \fa_\ZZ^*, w\in W_0\}$ and relations
\begin{equation}
(T_{s_i} - t^{\frac{1}{2}})(T_{s_i} + t^{-\frac{1}{2}}) =0, \qquad
\underbrace{T_{s_i}T_{s_j}T_{s_i}\ldots}_{m_{ij}\ \mathrm{factors}}
= \underbrace{T_{s_j}T_{s_i}T_{s_j}\ldots}_{m_{ij}\ \mathrm{factors}},
\end{equation}
\begin{equation}
X^{\lambda + \mu} = X^{\lambda}X^{\mu}, \quad\hbox{and}\quad
T_{s_i}X^{\lambda} - X^{s_i\lambda}T_{s_i} = (t^{\frac{1}{2}} - t^{-\frac{1}{2}})\left(\frac{X^{\lambda} - X^{s_i\lambda}}{1 - X^{-\alpha_i}}\right),
\label{Hrels}
\end{equation}
for $i,j\in \{1,\ldots, n\}$ with $i\ne j$ and $\lambda, \mu\in \fa_\ZZ^*$.
The \emph{bar involution on $H$} is the $\ZZ$-linear automorphism
$\overline{\phantom{T}}\colon H\to H$ given by
\begin{equation}
\overline{t^{\frac12}} = t^{-\frac12}, \qquad \overline{T_{s_i}} = T_{s_i}^{-1},
\qquad\hbox{and}\qquad
\overline{X^\lambda} = T_{w_0}X^{w_0\lambda}T_{w_0}^{-1}.
\label{baronH}
\end{equation}
for $i=1,\ldots, n$ and
$\lambda, \mu\in  \fa_\ZZ^*$.
For $\mu\in \fa_\ZZ^*$ and $w\in W_0$ define
\begin{equation}
X^{t_\mu w} = X^\mu (T_{w^{-1}})^{-1}
\qquad\hbox{and}\qquad
T_{t_\mu w}
=T_x X^{\mu^+}T_{w_{\mu^+}}(T_{w^{-1}xw_{\mu^+}})^{-1},
\label{BernsteinCoxeterconversion}
\end{equation}
where $\mu^+$ is the dominant representative of $W_0\mu$, $x\in W_0$ is minimal length such that
$\mu = x\mu^+$ and $w_{\mu^+}$ is the longest element of the stabilizer $W_{\mu^+} = \mathrm{Stab}_{W_0}(\mu+)$.
Define
$$\varepsilon_0
= (-t^{\frac12})^{\ell(w_0)}\sum_{z\in W_0} (-t^{-\frac12})^{\ell(z)} T_z
\qquad\hbox{and}\qquad
\mathbf{1}_0
= (t^{-\frac12})^{\ell(w_\nu)}\sum_{z\in W_0} (t^{\frac12})^{\ell(z)} T_z,
$$
so that
\begin{equation}
\overline{\varepsilon_0} = \varepsilon_0,
\quad
\overline{\mathbf{1}_0} = \mathbf{1}_0,
\qquad\hbox{and}\qquad
\varepsilon_0 T_{s_i} = -t^{-\frac12}\varepsilon_0,
\quad\hbox{and}\quad
T_{s_i}\mathbf{1}_0  = t^{\frac12}\mathbf{1}_0,
\label{e0defn}
\end{equation}
for $i\in \{1, \ldots, n\}$.
The algebra $\KK[X]$ defined in \eqref{KXdefn} is a subalgebra of $H$ and, by a theorem of
Bernstein (see \cite[Theorem 1.4]{NR}), the center of $H$ is the ring of symmetric functions,
\begin{equation}
Z(H) = \KK[X]^{W_0}.
\label{centerofH}
\end{equation}

\begin{remark}
Formulas \eqref{baronH} and \eqref{BernsteinCoxeterconversion} are just a reformulation of the usual
bar involution
and the conversion
between the Bernstein and Coxeter presentations of the affine Hecke algebra
(see for example \cite[Lemma 2.8 and (1.22)]{NR}).
\end{remark}

\subsection{The relation between $H$ and the abstract Fock space $\cF_\ell$}

In this subsection we follow \cite[\S4.2]{LRS}.
The \emph{affine Weyl group} is
\begin{equation}
W = \{ t_\mu w\ |\ \mu\in \fa_\ZZ^*, w\in W_0\}, \qquad\hbox{with}\qquad
t_\mu t_\nu = t_{\mu+\nu}, \quad\hbox{and}\quad
wt_\mu = t_{w\mu} w,
\label{affWeyldefn}
\end{equation}
for $\mu, \nu\in \fa_\ZZ^*$ and $w\in W_0$.
Let $\varphi^\vee$ and $h$ be as in \eqref{Coxeternumber}.
For $\ell\in \ZZ_{>0}$,
the \emph{level $(-\ell-h)$ dot action of $W$ on $\fa_\ZZ^*$} is given by
\begin{equation}
(t_\mu w)\circ\lambda = (w\circ\lambda) - \ell \mu = w(\lambda+\rho) - \rho - \ell\mu,
\label{levelelldotaction}
\end{equation}
for $\mu\in \fa_\ZZ^*$, $w\in W_0$ and $\lambda\in \fa_\ZZ^*$.  Note that this is
an extension of the dot action of $W_0$ given in \eqref{dotaction}.
Define
\begin{equation}
A_{-\ell-h} = \{ \nu\in \fa^*_\ZZ\ |\
\hbox{$\langle \nu,\varphi^\vee\rangle \ge -\ell-1$ and
$\langle \nu, \alpha_i^\vee\rangle \le -1$ for $i\in \{1, \ldots, n\}$}\}.
\label{ellAlcove}
\end{equation}
and
\begin{equation}
\cP^+_{-\ell-h} = \bigoplus_{\nu\in A_{-\ell-h}} \varepsilon_0 H\mathbf{p}_\nu,
\label{Heckemoduledefn}
\end{equation}
where $\varepsilon_0$ is as in \eqref{e0defn} and $\mathbf{p}_\nu$ are formal symbols
indexed by $\nu\in A_{-\ell-h}$ satisfying
$$\overline{\mathbf{p}_\nu} = \mathbf{p}_\nu
\qquad\hbox{and}\qquad
\hbox{$T_y\mathbf{p}_\nu = (t^{\frac12})^{\ell(y)}\mathbf{p}_\nu$ for $y\in W_\nu$,}
$$
where $W_\nu = \mathrm{Stab}_W(\nu)$ is the stabilizer of $\nu$
under the level $(-\ell-h)$ dot action of $W$ on $\fa_\ZZ^*$.
Define a bar involution
\begin{equation}
\overline{\phantom{T}}\colon \cP_{-\ell-h}^+\to \cP_{-\ell-h}^+
\qquad\hbox{by}\qquad
\overline{\varepsilon_0 f \mathbf{p}_\nu } = \varepsilon_0 \bar f \mathbf{p}_\nu,
\quad\hbox{for $\nu\in A_{-\ell-h}$ and $f\in H$.}
\label{barforP}
\end{equation}
For $\lambda\in \fa^*_\ZZ$ define
\begin{equation}
[X_\lambda] = [X_{w_0v\circ\nu}]
= \varepsilon_0 X^v \mathbf{p}_\nu,
\qquad\hbox{where}\quad
\lambda = w_0v\circ\nu \quad\hbox{with $\nu\in A_{-\ell-h}$,}
\label{bracketTXdefn}
\end{equation}
and $v\in W$ is such that $X^{vu} = X^{v}T_{u}$ for any $u\in W_{\nu}$.  It is helpful
to stress that the $(-\ell-h)$ dot action of \eqref{levelelldotaction} applies here so that, when
$v = t_\mu w$ with $\mu\in \fa_\ZZ^*$ and $w\in W_0$,
then $\lambda = -\ell w_0\mu+(w_0w)\circ \nu$ and
\begin{equation}
[X_\lambda] = [X_{-\ell w_0\mu+(w_0w)\circ \nu}] = \varepsilon_0 X^{t_\mu w}\mathbf{p}_\nu
= \varepsilon_0 X^\mu(T_{w^{-1}})^{-1}\mathbf{p}_\nu.
\label{bracketXexpanded}
\end{equation}

With these notations, a main result of \cite{LRS} is
\begin{thm} (see \cite[Theorem 4.7]{LRS}) \label{abstracttoHecke}
Let $\le$ be the dominance order on the set $(\fa_\ZZ^*)^+$ of
dominant integral weights.
Then the $\KK$-linear map $\Phi\colon \cF_\ell\to \cP^+_{-\ell-h}$ given by
\begin{equation}
\Phi(\ \vert\lambda\rangle\ ) = [X_\lambda],
\qquad \hbox{for $\lambda\in \fa_\ZZ^*$},
\label{Phidefn}
\end{equation}
is a well defined $\KK$-module isomorphism satisfying
$\overline{\Phi(f)} = \Phi(\overline{f})$.
\end{thm}

\noindent
Since elements of $Z(H)=\KK[X]^{W_0}$ commute with $\varepsilon_0$ there
is a $\KK[X]^{W_0}$-action on $\cP_{-\ell-h}$ by left multiplication.  The pullback of
this action by the isomorphism $\Phi$ is the source of the $\KK[X]^{W_0}$
action on $\cF_\ell$ given in Proposition \ref{KXaction},
\begin{equation}
z\Phi(f) = \Phi(zf),
\qquad\hbox{for $z\in Z(H) = \KK[X]^{W_0}$ and $f\in \cF_\ell$.}
\label{Phicommute}
\end{equation}

\subsection{Deducing the Casselman-Shalika formula}

For $\mu\in \fa_\ZZ^*$ define the ``Whittaker function'' $A_\mu\in  \varepsilon_0 H \mathbf{1}_0$ by
\begin{equation}
A_\mu = \varepsilon_0 X^\mu \mathbf{1}_0.
\label{Amudefn}
\end{equation}
See, for example, \cite[\S6]{HKP} for the connection between $p$-adic groups and
the affine Hecke algebra and the
explanation of why $A_\mu$ is equivalent to the data of a (spherical) Whittaker function
for a $p$-adic group.
As proved carefully in \cite[Theorem 2.7]{NR}, it follows from \eqref{e0defn} and \eqref{Hrels} that
$$
\hbox{$\varepsilon_0 H \mathbf{1}_0$ has $\KK$-basis}\quad
\left\{ A_{\lambda+\rho}\ |\ \hbox{
$\langle \lambda+\rho, \alpha_i\rangle\in \ZZ_{\ge 0}$
for $i\in \{1, \ldots, n\}$}\right\}.
$$
Following \cite[Theorem 2.4]{NR}, the Satake isomorphism,
$\KK[X]^{W_0}\cong \mathbf{1}_0H\mathbf{1}_0$, and
the Casselman-Shalika formula, $A_{\lambda+\rho}=s_\lambda A_\rho$,
can be formulated by the following diagram of vector space (free $\KK$-module) isomorphisms:
\begin{equation}
\begin{matrix}
Z(H) = \KK[X]^{W_0} &\stackrel{\sim}{\longrightarrow} &\mathbf{1}_0 H \mathbf{1}_0
&\stackrel{\sim}{\longrightarrow} &\varepsilon_0 H \mathbf{1}_0 \\
f &\longmapsto &f\mathbf{1}_0 &\longmapsto &A_\rho f \mathbf{1}_0 \\
s_{\lambda} &\longmapsto &s_{\lambda}\mathbf{1}_0 &\longmapsto &A_{\lambda+\rho}\end{matrix}
\label{LuszCasShi}
\end{equation}
This diagram has particular importance due to the fact that $\KK[X]^{W_0}$ is an avatar of the
Grothendieck group of the category $\mathrm{Rep}(G)$
of finite dimensional representations of $G$,
the spherical Hecke algebra $\mathbf{1}_0 H \mathbf{1}_0$ is a form of the Grothendieck
group of $K$-equivariant perverse sheaves on the loop Grassmanian $Gr$,
and $\varepsilon_0 H\mathbf{1}_0$
is isomorphic to the Grothendieck group of Whittaker sheaves (appropriately formulated
$N$-equivariant sheaves
on $Gr$), see \cite{FGV}.

Our proof of the Casselman-Shalika formula is accomplished by restricting Theorem \ref{TenPrdThm}
to the summand in \eqref{Heckemoduledefn} corresponding to $-\rho\in A_{-\ell-h}$.
We shall identify this summand with $\varepsilon_0 H \mathbf{1}_0$ via the $Z(H)$-isomorphism
$$
\begin{matrix}
\varepsilon_0 H \mathbf{1}_0 &\stackrel{\sim}{\longrightarrow} &\varepsilon_0 H \mathbf{p}_{-\rho} \\
\varepsilon_0 X^\mu \mathbf{1}_0 &\longmapsto &\varepsilon_0 X^\mu \mathbf{p}_{-\rho}
\end{matrix}
$$
Using the level $(-\ell-h)$ dot action of $W$ from \eqref{levelelldotaction},
the stabilizer of $-\rho$ is $W_0$ and
$$
W\circ (-\rho) = \{ t_{-\lambda}\circ(-\rho)\ |\ \lambda\in \fa_\ZZ^*\}
= \{ \ell\lambda-\rho\ |\ \lambda\in \fa_\ZZ^*\}.
$$
Since
$\langle (\ell\lambda-\rho)+\rho, \alpha^\vee\rangle \in \ell\ZZ$ for $\alpha\in R^+$,
the straightening law \eqref{Fstraightening} for elements of $W\circ (-\rho)$ is
\begin{equation}
\vert s_i\circ (\ell\lambda-\rho)\rangle = - \vert \ell\lambda-\rho\rangle.
\label{negrhostraightening}
\end{equation}

\begin{thm} (Casselman-Shalika)  For $\lambda\in (\fa_\ZZ^*)^+$ and $\mu\in \fa_\ZZ^*$ let $s_\lambda$ be the Weyl character as defined in \eqref{Weylchdefn} and let $A_\mu$ be the Whittaker function as defined
in \eqref{Amudefn}. Then
$$s_\lambda A_\rho = A_{\lambda+\rho}.$$
\end{thm}
\begin{proof}
Using \eqref{negrhostraightening},
$$\overline{\vert \ell\lambda-\rho \rangle}
= (-1)^{\ell(w_0)}(t^{-\frac12})^{\ell(w_0)-\ell(w_0)}
\vert w_0\circ (\ell\lambda-\rho)\rangle
= (-1)^{\ell(w_0)} \vert w_0\circ (\ell \lambda-\rho) \rangle
= \vert \ell\lambda -\rho \rangle
$$
and thus $\vert \ell\lambda-\rho\rangle$ satisfies the conditions of \eqref{Clambdadefn} so that
\begin{equation}
C_{\ell\lambda-\rho} = \vert \ell\lambda-\rho \rangle,
\qquad\hbox{for $\lambda\in (\fa_\ZZ^*)^+$.}
\label{KLisket}
\end{equation}
By \eqref{bracketXexpanded} and \eqref{Amudefn},
$$[X_{-\ell w_0\mu-\rho}]
= \varepsilon_0 X^\mu T_{w_0}^{-1}\mathbf{p}_{-\rho}
= t^{-\ell(w_0)/2} \varepsilon_0 X^\mu \mathbf{p}_{-\rho}
= t^{-\ell(w_0)/2}A_\mu,
\qquad\hbox{for $\mu\in (\fa_\ZZ^*)^+$}.$$
Using \eqref{Phicommute}, \eqref{KLisket}, \eqref{Phidefn}
and that $w_0\rho = -\rho$,
\begin{align*}
t^{-\ell(w_0)/2} s_\lambda &A_{\rho}
= s_\lambda [X_{-\ell w_0\rho - \rho}]
= s_\lambda [X_{(\ell-1)\rho}]
= s_\lambda \Phi(\vert (\ell-1)\rho\rangle)
= \Phi(s_\lambda \, \vert (\ell-1)\rho\rangle\ ) \\
&= \Phi(s_{\lambda} C_{(\ell-1)\rho}) = \Phi(C_{-\ell w_0\lambda+(\ell-1)\rho}),
\qquad\hbox{by Theorem \ref{TenPrdThm},} \\
&= \Phi(\vert (-\ell w_0\lambda)+(\ell-1)\rho \rangle)
= [X_{-\ell w_0\lambda+(\ell-1)\rho}]
= [X_{-\ell w_0(\lambda+\rho)-\rho}] \\

= t^{-\ell(w_0)/2} A_{\lambda+\rho}.
\end{align*}
\end{proof}

\section{Quantum groups and LLT polynomials}

In this section we describe the main motivation for Theorem \ref{TenPrdThm} namely,
the Steinberg-Lusztig tensor product theorem for representations of quantum groups
at roots of unity.  Then we explain the connection between these results and the theory of
LLT polynomials.

\subsection{Representations of quantum groups at a root of unity}

Let $\mathring{\fg}$ be the Lie algebra of the group $G$ alluded to in \eqref{wtsWeylgpdefn}.
Let $q\in \CC^\times$ and let $U_q(\mathring{\fg})$ be the Drinfel'd-Jimbo quantum group corresponding to $\mathring{\fg}$.  Let
\begin{align*}
&\Delta_q(\lambda) &\hbox{the Weyl module for $U_q(\mathring{\fg})$ of highest weight $\lambda$,} \\
&L_q(\lambda) &\hbox{the simple module for $U_q(\mathring{\fg})$ of highest weight $\lambda$,}
\end{align*}
Let
$$\hbox{$K(\hbox{fd$U_q(\mathring{\fg})$-mod})$
be the free $\ZZ[t^{\frac12},t^{-\frac12}]$-module generated by symbols
$[\Delta_q(\lambda)]$,}
$$
for $\lambda\in \fa_\ZZ^*$. For $\mu\in\fa_\ZZ^*$, denote by $W^\mu$, resp. ${}^\mu W$, the set of minimal length coset representatives for $W/W_\mu$, resp. $W_\mu\backslash W$.
Define elements $[L_q(w_0 y\circ\nu)]$,
for $\nu\in A_{-\ell-h}$ and $y\in {}^0 W$ such that $w_0 y\in W^\nu$, by the equation
$$[\Delta_q(w_0 x\circ\nu)] =
\sum_{y\le x}
\left(\sum_{i\in \ZZ_{\ge 0}} \left[ \frac{\Delta_q(w_0 x\circ\nu)^{(i)}}
{\Delta_q(w_0 x\circ\nu)^{(i+1)} } :
L_q(w_0 y\circ\nu)\right] (t^{\frac12})^i\right) [L_q(w_0 y\circ\nu)],$$
where $[M:L_q(\mu)]$ denotes the multiplicity of the simple $\fg$-module
$L_q(\mu)$ of highest weight
$\mu$ in a composition series of $M$ and
$$\Delta_q(\lambda)=\Delta_q(\lambda)^{(0)}
\supseteq \Delta_q(\lambda)^{(1)}\supseteq
\cdots
\qquad\hbox{is the Jantzen filtration of $\Delta_q(\lambda)$}
$$
(see, for example, \cite[\S1.4, \S2.3 and \S2.10 and Cor.\ 2.14]{Sh}  and \cite[\S4]{JM} for the Jantzen filtration in this context).

The combination of \cite[(3.20)]{LRS}
and \cite[Theorem 4.7]{LRS}) is the following connection between the representation theory of
the quantum group at a root of unity and the abstract Fock space.

\begin{thm} \label{qgrouptoFockspace}
Let $\ell\in \ZZ_{>0}$ and let $q\in \CC^\times$ such that $q^{2\ell}=1$.\hfil\break
Let
$\KK = \ZZ[t^{\frac12}, t^{-\frac12}]$.  Then the $\KK$-linear map given by
$$\begin{matrix}
K(\hbox{\emph{fd$U_q(\mathring{\fg})$-mod}}) &\stackrel{\Psi_2}\longrightarrow & \cF_\ell \\
[\Delta_q(\lambda)] &\longmapsto &\vert \lambda \rangle \\
[L_q(\lambda)] &\longmapsto &C_\lambda
\end{matrix}
\qquad\hbox{is a well defined isomorphism of $\ZZ[t^{\frac12}, t^{-\frac12}]$-modules.}
$$
\end{thm}

The enveloping algebra $U\mathring{\fg}$ has a presentation by generators
$e_1,\ldots, e_n$, $f_1, \ldots, f_n$ and $h_1, \ldots, h_n$
and Serre relations and the quantum group $U_q\mathring{\fg}$
has a presentation by generators $E_1, \ldots, E_n$, $F_1, \ldots, F_n$, and $K_1, \ldots, K_n$
and quantum Serre relations such that, at $q=1$,
$E_i$ becomes $e_i$ and $F_i$ becomes $f_i$.
Following \cite{Lu89} and \cite[Theorem 9.3.12]{CP},
with appropriate restrictions on $\ell$ as in \cite[just before Proposition 9.3.5 and Theorem 9.3.12]{CP},
the \emph{Frobenius map} is the Hopf algebra homomorphism
\begin{equation}
\begin{matrix} Fr \colon &U_q\mathring{\fg} &\longrightarrow &\ \ U\mathring{\fg}\hfill \\
&E_i^{(r)} &\mapsto
&\begin{cases} e_i^{(r/\ell)}, &\hbox{if $\ell$ divides $r$,} \\
0, &\hbox{otherwise,} \\
\end{cases} \\
&F_i^{(r)} &\mapsto
&\begin{cases} f_i^{(r/\ell)}, &\hbox{if $\ell$ divides $r$,} \\
0, &\hbox{otherwise,} \\
\end{cases} \\
&K_i &\mapsto &\ \ 1. \hfill
\end{matrix}
\end{equation}
The \emph{Frobenius twist} of a $U\mathring{\fg}$-module $M$
is the $U_q\mathring{\fg}$-module $M^{Fr}$
with underlying vector space $M$ and $U_q\mathring{\fg}$-action given by
$$um = Fr(u)m, \qquad\hbox{for $u\in U_q\mathring{\fg}$ and $m\in M$.} $$

\begin{thm} \label{SLThm}  (\cite[Theorem 7.4]{Lu89}, see also \cite[11.2.9]{CP})
Let $\ell\in \ZZ_{>0}$ and let $\Pi_\ell$ be as defined in \eqref{Pielldefn}.
Let $\lambda\in (\fa_\ZZ^*)^+$ and write
$$\lambda = \ell\lambda_1+\lambda_0,
\qquad\hbox{with $\lambda_0\in \Pi_\ell$ and $\lambda_1\in (\fa_\ZZ^*)^+$.}
$$
Let $q\in \CC^\times$ be such that $q^{2\ell}=1$ and let $L_q(\lambda)$ denote the
simple $U_q\mathring{\fg}$-module of highest weight $\lambda$.
Then
$$
L_q(\lambda) \cong \Delta(\lambda_1)^{Fr}\otimes L_q(\lambda_0),$$
where $\Delta(\mu)$ denotes the irreducible $U\mathring{\fg}$-module of highest weight $\mu$.
\end{thm}

\noindent
Accepting Theorem \ref{qgrouptoFockspace}, Theorem \ref{SLThm} is equivalent to the product theorem for abstract Fock space,
Theorem \ref{TenPrdThm}.

\subsection{LLT polynomials for general Lie type}

In \cite{LLT} and \cite[(43)]{LT} and \cite[Definition 6.6]{GH},
the LLT polynomials for type A are defined by
\begin{equation}
G^{(\ell)}_{\mu/\nu}(x, t^{-1})
\sum_{T\in SSRT_\ell(\mu/\nu)} t^{-\mathrm{spin}(T)}x^T,
\label{LLTdefn}
\end{equation}
where $SSRT_\ell(\mu/\nu)$ is the set of semistandard $\ell$ ribbon tableaux of
shape $\mu/\nu$, $\mathrm{spin}(T)$ is the spin of the tableaux $T$ and
$X^T$ is the weight of the tableaux $T$ (see \cite[\S6.5]{GH} for an efficient review of the
combinatorial definitions of semistandard ribbon tableaux, spin and weight).

In Lecouvey \cite{Lcy}, there is a definition of LLT polynomials for general Lie type
generalizing the definition of \cite{LLT} from type A which proceeds as follows.
Define a $\KK$-algebra homomorphism
$$
\begin{matrix}
\psi_\ell\colon &\KK[X] &\longrightarrow &\KK[X] \\
&X^\mu &\longmapsto &X^{\ell\mu}
\end{matrix}
\quad\hbox{so that}\quad
\psi_\ell(s_\lambda) = \mathrm{char}(\Delta_q(\lambda)^{Fr}),
$$
in the framework of Theorem \ref{SLThm}.  Then \cite[(57)]{Lcy} defines
\begin{equation}
G_\mu^\ell
= \sum_{\lambda\in (\fa_\ZZ^*)^+} p_{\ell\lambda, \mu}  s_\lambda,
\label{Lecouveydefn}
\end{equation}
where $p_{\ell\lambda,\mu}\in \ZZ[t^{\frac12}]$ are as in \eqref{Clambdadefn}.
As pointed out in \cite[Cor.\ 5.1.3]{Lcy},  Theorem \ref{qgrouptoFockspace} gives
\begin{equation*}
\psi_\ell(s_\lambda)
= \mathrm{char}(\Delta(\lambda)^{Fr})
= \mathrm{char}(L_q(\ell\lambda))
= \sum_{\mu\in (\fa_\ZZ^*)^+} p_{\ell\lambda,\mu}(1) \mathrm{char}(\Delta_q(\mu))
= \sum_{\mu\in (\fa_\ZZ^*)^+} p_{\ell\lambda,\mu}(1) s_\mu.
\end{equation*}
As explained carefully in
\cite[Theorem 4.8(b)]{LRS}, the polynomials $p_{\ell\lambda,\mu}$ are
parabolic singular Kazhdan-Lusztig polynomials.

In \cite[Definition 5.12 and Corollary 6.4]{GH} there is another definition of LLT polynomials for general
Lie type:
\begin{equation}
\cL^G_{L, \beta,\gamma}
= t^{l_{\beta-\gamma}+\ell(w)-\ell(v)}
\sum_{\lambda\in (\fa_\ZZ^*)^+} Q_{\mu\nu}^\lambda s_\lambda,
\quad\hbox{where
}\quad
s_{\lambda^*} \cdot \vert\nu\rangle = \sum_{\mu} Q_{\mu\nu}^\lambda \vert\mu\rangle
\label{GHdefn}
\end{equation}
determine the polynomials $Q_{\nu\mu}^\lambda$.  Here $G$ is the
reductive algebraic group alluded to in \eqref{wtsWeylgpdefn},
$L$ is a Levi subgroup of $G$ with Weyl group $W_\nu$,
$l_{\beta-\gamma}$ is the nonnegative integer defined in \cite[Remark 5.10]{GH},
and
$
\mu=v\circ (\eta+\ell\beta) \quad\hbox{and}\quad
\nu = w\circ (\eta+k\gamma),
\quad\hbox{where}\quad
v\in W_0 t_{\beta} W_\eta
\quad\hbox{and}\quad
w\in W_0 t_\gamma W_\eta
$$
are minimal representatives.  At this point, the reader's discomfort
occurring from the transitions between $\beta$ and $\gamma$ and $v$ and $w$
and $\mu$ and $\nu$ is mitigated by recognizing that the relation between these two
definitions occurs in the special case $\nu=0$: Theorem \ref{TenPrdThm}
and \eqref{KLisket} and the definition of $Q_{\mu\nu}^\lambda$ in \eqref{GHdefn} give
$$C_{\ell\lambda} = s_{\lambda^*}\cdot C_0
= s_{\lambda^*}\cdot \vert 0\rangle = \sum_\mu Q_{\mu0}^\lambda \vert \mu\rangle,
\quad\hbox{and comparing with \eqref{Clambdadefn} gives}\quad
p_{\ell\lambda,\mu} = Q_{\mu,0}^\lambda$$
and specifies the close relationship between $G_\mu^\ell$ and $\cL_{L, \beta, \gamma}^G$
which occurs at $\nu=0$.  They are the same up to a power of $t$.

\section{Tensor product theorem on affine Lie algebra representations}

Let $\mathring{\fg}$ be the Lie algebra of $G$ and let
$\fg = \mathring{\fg}\otimes_\CC \CC[\epsilon, \epsilon^{-1}] + \CC K +\CC d$
be the corresponding affine Kac-Moody Lie algebra (see \cite[\S6.2]{Kac} -- we follow the notation
of \cite[(3.17)]{LRS}).
Let $\ell\in \ZZ_{>0}$ and let $h$ be the dual Coxeter number.
As explained in \cite[Theorem 3.2]{LRS}, an important result of Kazhdan-Lusztig establishes a
relation between level $(-\ell-h)$-representations in parabolic
category $\cO_{\mathring{\fg}}^\fg$ for the affine Lie algebra
and the finite dimensional representations
of the quantum group $U_q\mathring{\fg}$ with $q^{2\ell}=1$.

Let
$$\fg' = [\fg, \fg] = \mathring{\fg}\otimes_\CC \CC[\epsilon,\epsilon^{-1}] + \CC K.$$
By restriction, the modules in $\cO_{\mathring{\fg}}^\fg$ are $\fg'$-modules.  Let $\Lambda_0$ be
the fundamental weight of the affine Lie algebra so that $L(c\Lambda_0+\lambda)$ is an
irreducible highest weight $\fg$-module of level $c$ (i.e. $K$ acts by the constant $c$).

\begin{thm} \label{negleveltoquantum}
\cite[Theorem 38.1]{KL94}
There is an equivalence of categories
\begin{equation*}
\begin{matrix}
\left\{
\begin{matrix}
\hbox{finite length $\fg'$-modules} \\
\hbox{of level $-\ell-h$ in $\cO_{\mathring{\fg}}^\fg$}
\end{matrix}
\right\}
&\stackrel{\Psi_1}{\longrightarrow}
&\left\{
\begin{matrix}
\hbox{finite dimensional $U_q(\mathring{\fg})$-modules}\\
\hbox{with $q^{2\ell}=1$}
\end{matrix}
\right\}
\\
\Delta_{\mathring{\fg}}^\fg((-\ell-h)\Lambda_0+\lambda)
&\longmapsto
&\Delta_q(\lambda) \\
L((-\ell-h)\Lambda_0+\lambda)
&\longmapsto
&L_q(\lambda)
\end{matrix}
\end{equation*}
\end{thm}

\noindent
This statement of Theorem \ref{negleveltoquantum} is for the simply-laced (symmetric) case.  With the proper
modifications to this statement the result holds for non-simply laced
cases as well, see \cite[\S 8.4]{Lu94} and \cite{Lu95}.

Let $\lambda\in \fa_\ZZ^*$.
Under the composition of the map $\Psi_1$ in Theorem \ref{negleveltoquantum} and
the map $\Psi_2$ from Theorem  \ref{qgrouptoFockspace},
$$\Psi_2(\Psi_1([L((-\ell-h)\Lambda_0+\ell\lambda)])) = \Psi_2([L_q(\ell\lambda)])
= C_{\ell\lambda} = \vert \ell\lambda \rangle.$$
Thus it follows from Theorem \ref{negleveltoquantum}, Theorem \ref{qgrouptoFockspace} and \eqref{KLisket} that
\begin{equation}
L((-\ell-h)\Lambda_0+\ell\lambda) = \Delta_{\mathring{\fg}}^\fg((-\ell-h)\Lambda_0 +\ell\lambda)
= \Ind_{\mathring{\fg_0}+\fb}^{\fg}(L_{\mathring{\fg}}(\ell\lambda))
\cong U\fg \otimes_{U\fk} L_{\mathring{\fg}}(\ell\lambda),
\label{affineFtwistmodule}
\end{equation}
where
$$
\fk = \bigoplus_{k\in \ZZ_{\ge 0}} \epsilon^k
\Big(\fa\oplus
\bigoplus_{\alpha\in R^+} \mathring{\fg}_\alpha+\mathring{\fg}_{-\alpha}\Big)
\quad\hbox{with $R^+$ the set of positive roots of $\mathring{\fg}$.}
$$
As given in \eqref{Weylchdefn}, the Weyl character formula for the
$\mathring{\fg}$-module $L_{\mathring{\fg}}(\ell\lambda)$ is
\begin{equation}
\mathrm{char}(L_{\mathring{\fg}}(\ell\lambda)) = s_{\ell\lambda}
= \left(\prod_{\alpha\in R^+}\frac{1}{1-X^{-\alpha}}\right)
\cdot
\sum_{w\in W_0} \det(w)X^{w\circ \ell\lambda}.
\end{equation}
Letting $q = e^\delta$ and using the Poincar\'e-Birkhoff-Witt theorem,
the character of the $\fg$-module in \eqref{affineFtwistmodule} is
\begin{align}
&\mathrm{char}(L((-\ell-h)\Lambda_0+\ell\lambda))
=\mathrm{char}(\Delta_{\mathring{\fg}}^\fg((-\ell-h)\Lambda_0 +\ell\lambda))
= \mathrm{char}(U\fg \otimes_{U\fk} L_{\mathring{\fg}}(\lambda))
\nonumber \\
&=
s_{\ell\lambda}
\prod_{k\in \ZZ_{>0}} \left( \frac{1}{(1-q^{-k})^n}
\prod_{\alpha\in R^+} \frac{1}{1-q^{-k}X^\alpha}
\cdot \frac{1}{1-q^{-k}X^{-\alpha}}\right)
 \\
&=
\left(\prod_{k\in \ZZ_{>0}} \frac{1}{(1-q^{-k})^n}\right)
\left(\prod_{k\in \ZZ_{>0}} \prod_{\alpha\in R^+} \frac{1}{1-q^{-k}X^\alpha}\right)
\left(\prod_{k\in \ZZ_{\ge 0}} \prod_{\alpha\in R^+} \frac{1}{1-q^{-k}X^{-\alpha}}\right)
\left(\sum_{w\in W_0} \det(w) X^{w\circ \ell\lambda}\right).
\nonumber
\end{align}
This formula is reminiscent of Weyl-Kac character formula for integrable representations,
but we have not yet found a reference for it in the literature.
As we have explained in \eqref{affineFtwistmodule}, this formula is an easy consequence of \cite{KL94} and \cite{Lu89}.

The equivalence in Theorem \ref{negleveltoquantum} is an equivalence of monoidal categories where
the product on the left hand side is the fusion tensor product $\hat\otimes$
and the product on right hand side is the tensor product coming from the Hopf algebra
structure of $U_q\mathring{\fg}$.
Thus, in terms of affine Lie algebra representations, the Lusztig-Steinberg tensor product theorem
says that
$$\hbox{if $\lambda\in (\fa_\ZZ^*)^+$ \quad and \quad
$\lambda = \lambda_0+\ell\lambda_1$ with $\lambda_0\in \Pi_\ell$}$$
where $\Pi_\ell$ is as in \eqref{Pielldefn}, then
\begin{align}
L((-\ell-h)\Lambda_0+\lambda)
&\cong L((-\ell-h)\Lambda_0+\lambda_0)\hat\otimes L((-\ell-h)\Lambda_0+\ell\lambda_1)
\nonumber \\
&\cong L((-\ell-h)\Lambda_0+\lambda_0)\hat\otimes
\Delta_{\mathring{\fg}}^{\fg}((-\ell-h)\lambda_0+\ell\lambda_1).
\end{align}

redpajama_set_name: RedPajamaArXiv


Spank shops specialize in selling commodities at such inflated prices that they are effectively worthless to the customer. The sales usually include an 'exit date' at which point the investor can expect to receive his return – but before which the spank shop closes down and moves on. Access to personal data on the target 'victim' allows the trader to 'get inside the head' of the victim and offer what is likely to be a tempting deal.
The scandal came to light when an anonymous former commodity broker handed the Mail on Sunday a memory stick containing 2000 files on customer details, and claimed to have a database of 27,000 similar files. The Mail is working with both the Information Commissioners Office (ICO, able to impose fines of up to £500,000 for serious breaches of the Data Protection Act) and the Financial Conduct Authority (FCA, able to impose unlimited fines).
Barclays issued a statement Sunday. "Our initial investigations suggest this is isolated to customers linked to our Barclays Financial Planning business which we ceased operating as a service in 2011. Based on what we have seen, this appears to be data from 2008 or earlier." This certainly fits with the information provided by the whistleblower – the files contain very detailed and personal information that would help a legitimate financial planner advise on the most appropriate investments.
Barclays Financial Planning had a troubled history. The files disclosed to the Mail "are believed to have been customers of the now defunct Barclays Financial Planning business, which was fined £7.7m in 2011 and ordered to pay up to £59m in compensation for mis-selling investment funds to more than 12,000 customers," notes the Guardian.
Colin Tankard, managing director of Digital Pathways, believes that the solution is a combination of activity monitoring/auditing and data encryption. The former controls staff who have a legitimate right to access the data by "limiting the number of records they can access and watching, and stopping… any unusual activity such as the copying large numbers of records," he said.


text:
ITU names new sponsor
Oakley joins the ITU World Championship Series.
Rogerhospedales March 1, 2011
The International Triathlon Union today announced Oakley as the Official Global Eyewear Sponsor of the Dextro Energy Triathlon ITU World Championship Series.
Arguably the most recognisable eyewear brand in the world, Oakley will be the seventh global partner of the world's premier triathlon series, which has created major sponsorship and broadcast opportunities since launching in 2009.
Current Oakley athletes include Olympic gold medallists Emma Snowsill and Jan Frodeno, World Champions Javier Gomez and Emma Moffatt, and a bevy of World Cup winners and Olympians such as Brad Kahlefeldt, Lisa Norden, Nicola Spirig and Kris Gemmell.
Last year 21,000 athletes competed in mass-participation races in front of 700,000 spectators and global television audiences in 149 countries. It is expected to attract higher numbers in 2011, helping Series sponsors and partners extend their reach to wider audiences.
"With its product quality, Oakley has been favoured by many triathletes from elite to amateur competitors," said Marisol Casado, ITU President and IOC Member. "The new partnership will earn big cheers from existing fans and attract new audiences through the exciting sport of triathlon."
"Oakley is a global high profile brand that ITU is excited to develop successful partnerships with. Attracting Oakley as global partners of the Dextro Energy Triathlon ITU World Championships Series is another clear statement that we are successfully establishing the series as a major sports platform," added Paul Coleman, Managing Director of ITU World Championship Series.
Entering its third year, the Dextro Energy Triathlon ITU World Championship Series is the premier triathlon series in the world and the most competitive. Athletes compete in seven Series events vying for the title of ITU Triathlon World Champion. In 2011, the Series will be staged in seven world-class sites. Starting off in Sydney in April and touching down in Yokohama, Madrid, Kitzbühel, Hamburg, London and finishing in Beijing for the Grand Final in September.

redpajama_set_name: RedPajamaCommonCrawl


Hi Blogger fam! Here to share a moment I never thought would happen, yet I always dreamed of it! I'm releasing a new poetry collection!
Pain is Fertile Ground focuses on how facing pain opens our hearts to receive the love we've always wanted and live the lives ordained for us since the beginning of time!
For updates, connect with me on Facebook, Twitter, and LinkedIn!
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Whats similar about all the countries in list A?
They are all democracies settled and developed with western ideals. This is just a propaganda exercise stating that "small government" is the reason for their success. Its not. Its only one factor of many.


A marriage certificate is an official record certifying that two people have undertaken a marriage ceremony. In the majority of jurisdictions, a marriage certificate is issued by a government official following the civil registration of the marriage. A marriage certificate may be required for several reasons: among others, because of issues of legitimacy of a child, divorce proceedings, change of a party's name, etc.
The official area where the marriage was registered (district, sub-district and county).
The bride's residence at the time of marriage.
The bride's rank or profession.
The full name and occupation of the bride's father.
The groom's residence at the time of marriage.
The groom's rank or profession.
The full name and occupation of the groom's father.
Details of the marital status of both the bride and the groom (bachelor/spinster if either party had never been married before, widow/widower if either party had been married and their previous partner died, or previously dissolved if either party had previously been divorced).
The marriage witnesses' full names.
The registrar's full name and signature.
The date when the certificate was issued.
Re-issued marriage certificates include expressions such as "Certified to be a true copy of an entry in the marriage register". This is a standard expression included in every official marriage certificate since it refers to the fact that the certificate is a copy of the details contained in the marriage register. It does not mean that it is a photocopy of an original record and therefore can be used for all legal purposes.


"Sarah Parsons has never seen Amber House, the grand Maryland estate that's been in her family for three centuries. She's never walked its hedge maze nor found its secret chambers; she's never glimpsed the shades that haunt it, nor hunted for lost diamonds in its walls.
But all of that is about to change. After her grandmother passes away, Sarah and her friend Jackson decide to search for the diamonds--and the house comes alive. She discovers that she can see visions of the house's past, like the eighteenth-century sea captain who hid the jewels, or the glamorous great-grandmother driven mad by grief. She grows closer to both Jackson and a young man named Richard Hathaway, whose family histories are each deeply entwined with her own. But when the visions start to threaten the person she holds most dear, Sarah must do everything she can to get to the bottom of the house's secrets, and stop the course of history before it is cemented forever."
The whole premise behind this book - an old house full of treasure & memories, maybe even ghosts sounds fantastic and is right up my alley. After reading this though, it wasn't what I was expecting. It started quite slow and I was expecting a spooky, stay up all night kind of book but it wasn't. I'm not saying this was terrible, it just turned into something different.
Sarah, her mum & her autistic brother are staying in her recently deceased grandmother's house (Amber House) because Sarah's mother is getting it ready to be sold. Sarah's mum used to live in Amber House when she was a little girl but something happened and she never really wanted to come back. I didn't pick the reason why the mother wanted to stay away but other things were quite easy to pick up on. I'm pretty bad at picking out what is going to happen next in movies and in books, so for someone like me to get clued in means others will know the ending quite easily too.
Don't get me wrong I really enjoyed reading it once I got halfway through mainly because the mysterious back story to who this little girl in white is and why can she see & interact with Sarah if she isn't a ghost? Of course with any teenage fiction these days there has to be a love triangle, luckily I was really glad with who Sarah picked in the end.
The ending was really good and I don't think there needs to be a trilogy written, I feel this book was great on it's own. I've seen the blurb for the next book and I don't think I'll rush out to read it when it's released mainly because this ended really well.
My final thoughts are that this would make a fantastic teen movie. It has everything in there. Suspense, mystery, romance, rich guy on a boat and a ball party where the main character can dress up in a ball gown.
Thanks to NetGalley & Scholastic Inc. for my kindle edition.
You couldn't get a more perfect cover that tells you what happens in the book. A spooky house, a maze and a ball. But as this was a kindle edition I didn't get to close the book to look at the front cover, which I like to do often.


Parents, please take time to read this important letter detailing our comprehensive plan for achieving student growth and success for this school year.
Don't forget! This is your last chance to put your best smile on!
See Ms. Halverson or your homeroom teacher for an entry form today!
Dr. Mike Weddington is our new Assistant Principal. This is his 13th year in Wilkes County Schools. We are delighted that 13 is lucky for CWMS!
Mr. Foster is completing his degree in administration at the University of North Carolina - Wilmington. He is doing the first half of his Administrative Internship with us during first semester.
Mrs. Privette has taught seventh and eighth grade language arts and social studies in Wilkes County for more than twenty years. Her favorite aspect of Central Wilkes Middle School is its diversity, and her favorite attribute in a student is curiosity. She is an avid reader and runner and is active in a local church. Her two sons, Hank and Jack both graduated from Wilkes Central, and her husband, Chris is a local wood worker.
Kendall R. and Alea M. competed in the 8th Annual Gravity Games in Lenoir. Read all about it in an article from the Blowing Rock News at the link. Click the headline or READ MORE ABOUT DAILY ANNOUNCMENTS for the whole story and an awesome picture of Alea.
Mrs. Eggett has a passion for her students and had a dream for them to have their own playground. Through numerous fund raisers, her passion, and her persistence, the playground has FINALLY become a reality. Click on the headline or READ MORE ABOUT DAILY ANNOUNCEMENTS to enjoy the photo album.


text:
'Fantastical 2' for Mac Updated With Native Exchange Support, Availability, Printing and More
Popular calendar app Fantastical 2 for Mac was today updated with a long list of new features, introducing a multitude of small but significant changes. The new version of the app includes Exchange support with native account syncing so Exchange users can respond to invitations and use availability lookup, categories, and people lookup.
With the new availability feature, enabled through a calendar service like Exchange, Google Apps, or OS X Server, Fantastical 2 users can quickly check their colleagues' availability and schedules to expedite the process of setting up meetings and appointments. For Google Apps users, there's an option to load Google Hangouts links for group meetings and there's support for push updates and Google Contact search when adding invitees.
Printing is now available and there are multiple layout options available for printing calendar schedules. Users can choose a specific calendar to print and choose whether to show all-day events, timed events, or both. There are also options for calendar keys, black and white printing, and a mini calendar at the upper right of the page. Daily, monthly, weekly and yearly calendars are available to print, letting users share a hard copy of their schedules with others.
Other important new features include iCloud shared calendar notifications, an extra time zone line to let users view other time zones when scheduling an event, multiple selection to move or delete multiple items at one time, and a new option to start week or month views on the current day or week.
"We launched Fantastical 2 for Mac one year ago, and our goal was to reinvent Fantastical itself," said Michael Simmons, Co-founder of Flexibits. "Today's update is the biggest we've ever shipped and we think our users are going to love it."
There are several other feature additions, bug fixes, and improvements to the Fantastical 2 app, which should improve performance for all users.
Fantastical 2 can be downloaded from the Mac App Store for $49.99. [Direct Link]
Article Link: 'Fantastical 2' for Mac Updated With Native Exchange Support, Availability, Printing and More
https://www.macrumors.com/2016/03/30/fantastical-2-major-update/
smizzle
$50 for a calendar app seems a bit excessive... It's a shame Microsoft dropped development on Sunrise. That was an awesome free calendar app. If paying this much is the alternative I'll probably just go back to the stock calendar if they ever completely pull the plug on Sunrise.
Reactions: tyasjs and Aston441
ArtOfWarfare
smizzle said:
I bought the original Fantastical as part of a bundle with 9 other apps for $20. I thought I got a good deal for my money.
Buying just Fantastical for $50 is a ripoff.
Also, printing? Really? Having a printer is like having a landline or DVD player - plenty of people still have old ones laying around, but the only old people are still buying them. It's a dying industry - it'll be gone within a few years. It seems really silly to invest any time or effort in adding this feature.
Reactions: jocamero and DIRSGT
nefan65
I trialed this one, and BusyCal. Not impressed with either really. I have O365 from work, and still use Mail and Cal for Exchange. The only issue I have is that new mail is usually a few seconds behind my iPhone [or Outlook if I used it], and iCal gets flaky at times. If I put a new event/appointment in, it will add it and a few seconds later it'll just say New Event. If I change it, and check my phone or Outlook, it shows 2 - 3 events; the one that should be there, and a couple that just say "New Event"...odd.
Other than that, ya $50 for a calendar app is nuts. It's worth maybe $9.99.
Reactions: Nuvi, tardegrade and Aston441
It's expensive but I use it everyday, it's well supported with updates and it's the best calendar app out there. Natural language calendar entry is incredible.
Reactions: Easttime, CarlJ and dwfaust
tgara
nefan65 said:
I'm in the same boat.... I tried both Fantastical 2 and BusyCal and a the time, neither had Exchange support that was on par with Apple's Calendar. Since that feature is critical for me, both were a no go.
That said, I am willing to give this update of Fantastical a try to see how Exchange works. I like the menubar feature, and the iOS apps are great. But, $50 is a lot to spend on a calendar app. It better work really good if I'm going to drop that much.
nefan65, I had the same issue you did with the New Event bit. Have you tried turning the Exchange calendar off, then on again? If that doesn't work, you may need to delete your Exchange account and re-add it. That fixed the problem for me.
AndyK
It's a good app but the pricing just doesn't justify the feature list. Especially considering they still sell the iPhone / iPad apps separately from each other & haven't gone universal. That and the inclusion of printing (I find it hard to believe they get as many requests for it as they claim on their blog).
That being said I do own it and I like it. I purchased it at launch when it was on discount with iTunes credit I won in a giveaway. Not sure even after having it so long I'd shell out from my own pocket in the future though.
All in all, they just seem a little out of touch with a slow and low development curve, which doesn't fuel any excitement or enthusiasm for what future releases could bring.
zkmusa
I have been a long-time user of Fantastical 1, the menu bar app. This app is indispensable to me. When Fantastical 2 came out, I could not justify spending $50 for the app, as I am not in need of a full calendar application.
I am still hoping the developer will create a spinoff menubar app that doesn't have the features of the full calendar app, but updates the original Fantastical. I would gladly pay $20 for this.
tgara said:
Thanks! I did delete and re-add. I'll turn off and wait a bit and see if maybe there was a weird sync thing going on, and re-enable. Worth a shot...
Futurix
ArtOfWarfare said:
Clearly you are not working in an office.
Reactions: Nuvi, Easttime, thefourthpope and 6 others
liven2
This upgrade is duplicating all my exchange calendar events over and over and over. I now have hundreds of duplicate entries related to my exchange account. NOT happy!! User beware!
philipmv
MacRumors really should include a short disclaimer at the beginning of advertisement posts like this saying this was a "paid article submission." I miss the MacRumors of old, when articles were rumors, not ads.
Reactions: Nuvi and thisisnotmyname
Sleepyshibe
Aika said:
That was my one argument for using Fantastical too but when Fantastical 2 came out at $50 and then Apple added language processing to the native Calendar app, I saw no reason to upgrade and went back to using the built in calendar.
AppleInLVX
Waiting for it to be part of a bundle so I can get more value out of it. That 50 bucks is 70 CAD, and there is no way in hell I'm dropping 70 bucks for something that replicates the features of what comes with the OS. 20 bucks CAD is the limit for standalone. With a bundle of other stuff it might be worth it.
Weaselboy
philipmv said:
If this were a sponsored review/article it would be disclosed in the article. See this.
Reactions: CoastalOR, MaximizedAction and MacDawg
BornAgainMac
Florida Resident
Needs Travel support. I really like that Apple feature. One time it was a Saturday and I got an alert to leave about 15 minutes earlier. It was due to delays on the freeway that just happened.
Futurix said:
What am I sitting in right now if not a cubical in an office?
You must be working for a fairly unique company then. In my experience paper scheduling is still king in most places, even in most 'new media' companies. Unfortunately.
Our conference rooms each have iPads on the door indicating when the next meeting is. Tap on it if you want to reserve it for something quick.
I personally feel the iPads are overkill... You could get a $100-$200 Android Tablet to do the same thing, but oh well, I'm not the one calling the shots here.
The iPads all synchronize with the calendars on our Exchange servers.
That's pretty slick. Although in practice, what does it gain? I assume that the meeting rooms are all booked through Exchange anyway, and if you wanted a room, its bookings would show up there. I guess I wonder how often someone shows up randomly at a room wanting to book it for later.
C DM
macrumors Sandy Bridge
I would wager that there are far fewer places with anything close to that compared to those that still print things out (especially those outside the tech sector, and there are sure plenty of those).
jeffg819
Print out my calendar every morning and use it to record various things about my day. Was using Calendar for this and Fantastical for all other activities. One less step in the morning routine!
ScottNWDW
I've been using Fantastical 2 since it came out. It is well worth the price. At first I was taking aback by the price, but like other apps I use, if I USE the app on a regular basis it is well worth the price. I use OmniFocus and that was a huge investment, but it is the best damned task app I could find. With this update, Fantastical is the best calendar app out there and in my opinion well worth the price.
This update brought some great new features to the app and one of the best new features is the native support for Microsoft Exchange. My workplace uses Exchange and I have my own personal calendars. I like that the category colors that I have chosen in Exchange will come through to Fantastical. All other calendar apps that I have used or tried had the exchange calendars in just one color. Now they are all in the same colors as my computers at work. I also like the printing capabilities and the fact that you can have the various calendar sets.
I get annoyed and tired of people who think that app developers are supposed to just give their apps away for free all the time. If you want to wait until the app is included in some bundle to make it worth the price then you really don't need the app that much.
Reactions: Aika and Weaselboy
nnoble
[doublepost=1459506977][/doublepost]Congratulations on living and working in an advanced IT environment with good connectivity. I'm aware such environments exist. Like you, Flexibits were also working under the illusion that the rest of the world is like that. I work in a part of the world that can't fully guarantee electricity 24 hours a day. Internet connectivity is even less reliable. There is a lot to be said for getting out and about and seeing how the rest of the world operates. Printer manufacturers will be in business for a very long time to come.
nnoble said:
This is so true. People always have to have the paper. I work as a concierge and every day I get people wanting me to print their airline boarding passes even though they have the airline's own app with the digital boarding pass right there that they can use. I don't think printer manufacturers have to worry about anything. From what I see we are all printing more things than we did in the past.


text:
Q: How can I watch a folder on an sftp server for new files? I log in to a sftp server, and to see the files I type ls. Is there any way to recursively watch a folder? For now I just keep typing ls -tlh. Is there a way for that to happen without me typing it? Like a -watch flag or something? Any help would be greatly appreciated.

A: Its unclear what you mean by "recursively watch" - your command does not appear to be doing any recursion or looking in any subdirectories.
Under linux the "watch" command can be used to repeatedly issue a command periodically - so I suspect you want
  watch -n1 'ls -tlh'

Which is equivalent to typing ls -tlh every second.   The default frequency is every 2 seconds (ie watch 'ls -tlh')
I don't think its what you want, but to answer the question as you asked it - if you want to see a tree of all the files and subdirectories in a subdirectory repeatedly, there is a utility "tree" (apt install tree under Ubuntu). You could then use
 watch 'tree /path/to/monitor'

redpajama_set_name: RedPajamaStackExchange


1996, 225 Four Winns Sundowner. Cuddy Cabin. 5.8FSI/SX Volvo Inboard/Outboard 265 HP. Aluminum Prop. Boat weight 3590 lbs. 11 Passenger 1900 lbs max. capacity. 50 gal. fuel tank. Porta Potty. Carpet on Floor. Binimi Top. Three tone colors on hull and upholstery Tan, White and Green. Battery switch. Two new batteries. Mooring cover. Marine Radio.
2017 Venture Custom Aluminum Tandem Axle Trailer. Disk brakes. Aluminum wheels. Radial Tires. Led lights. Extra bunks. Fulton Winch. Swing tongue. Carpet Side Guides. Keel Roller. Tongue Jack. Spare tire. Retracting buckle straps. No bottom paint.
Lowrance 7" touch screen GPS/Fish Finder GEN2. New (2) Big Jon Triple Rod holders, removable. Custom made Sunbrella Fabric Full Cover. New trolling plate. anchor, rope, dock lines, and a few lifejackets included.
I have owned the boat since 2011. Serviced and Winterized at Lindy's Sales in Quanicassee MI. Stored indoors during the winter months. This is a great family boat for fishing or tubing. Top speed about 41 MPH. Motor/Outdrive all original. No hour meter. This boat is in very good shape for its age. Handles very well in the Saginaw Bay's rough water.
Call, Email or Text Martin. 989-737-48 three four.


Cheerleading Competition - 6:00 p.m.
Rapides Parish Coliseum ($7.00) - Children 6 & under get in free!
Each night during the carnival the Rapides Parish Fair offers visitors the opportunity to exciting entertainment along with the carnival rides. We will offer karaoke, bands and more. Come to the fair.


French police officers patrol in Longpont, north of Paris, France, Thursday, Jan. 8, 2015.
Friends and classmates of a suspect in the deadly attack on the satirical magazine Charlie Hebdo in Paris say he has an alibi. The 18-year-old, who turned himself in to police on Wednesday, was reportedly in school at the time and consequently couldn't have taken part in the shooting.
On Wednesday, police identified three suspects in the attack: two brothers — Said Kouachi, 32, Cherif Kouachi, 34, — and 18-year-old Mourad Hamyd. The attack claimed the lives of 12 people.
By Wednesday evening, several people who identified themselves as Hamyd's classmates went on Twitter, saying they saw Hamyd in class at the time of the shooting. It's unclear what role, if any, Hamyd is suspected to have played in the attack as police haven't released any information beyond his name.
Using the hashtag #MouradHamydInnocent, friends protested his innocence.
The tweet below roughly translates as, "He is in my high school. I saw that smile this morning as usual."
Despite some early reports that three people were involved in the attack, just two shooters are seen in the pictures and videos taken in the immediate aftermath of the attack.
A police officer pleads for mercy on the pavement before being shot in the head by masked gunmen during an attack on the headquarters of the French satirical newspaper Charlie Hebdo in Paris, France on Jan. 7.
Hamyd's name initially began circulating on social media Wednesday along with the names of the Kouachi brothers, who were still on the loose by Thursday afternoon.
French Interior Minister Cazeneuve said Said Kouachi had been "formally identified" in a picture of the attack on the Charlie Hebdo headquarters.
According to France 24, Hamyd's name was included in the first search warrant issued early on Wednesday evening. He turned himself into to police "after seeing his name circulated on social networks," a source told France 24. Le Monde reported that Hamyd turned himself over to police in the city of Charleville-Mezieres.
Mashable contacted the Charleville-Mezieres police who said they weren't able to release information.
Hamyd attended the Lycee Monge in Charleville-Mezières, according to French press. Several students, claiming to be from the same high school, were the driving force behind the #MouradHamydInnocent hashtag.
Malachy Browne, a journalist with reported.ly, spoke with one of the students, who insisted that Hamyd couldn't have taken part in the shooting.
French media outlets quoted other friends who also said Hamyd was in school at the time of the attack.
On Thursday, Interior Minister Cazenueve announced that nine people had been taken into custody as the investigation into the shooting continues. It is unclear if Hamyd is among them or if he is still being held by authorities.


34 S E P T E M B E R / O C TO B E R 2 0 1 8 S olar customer data is increasingly being used for quantifiable savings by solar installers, devel- opers, energy arbitrators and utilities. The soft- ware companies special- ized in gathering and aggregating this data on residential, commercial and industrial levels are also building more functional systems that will help to standardize energy efficiency and asset management platforms, benefi- cial for all levels of data use. Solar data usage is contingent on cus- tomers' willingness to share their data with the local utility, which is no small hurdle for the industry, suggests Matt Kuo, the vice president of product at Atlanta-based Urjanet. "Initiatives like Green Button are great, but adoption has not been as rapid as many people expect- ed. That's why we exist," he says. The Green Button initiative is "an industry-led effort that responds to a White House call-to-action to provide utility customers with easy and secure access to their energy usage information in a consumer-friendly and computer- friendly format," according to the U.S. Department of Energy (DOE). While such standards are slow to emerge, companies like Urjanet and UtillityAPI independently contact users for data sharing permission, aggregate and clean the data, and often manage the utility-to-customer-to-developer inter- face. Urjanet, which launched its solar-spe- cific service, Utility Data for Solar, in May, has access to residential and com- mercial energy usage, cost and location data from more than 750 electric utilities in over 15 countries, Kuo says. By offer- ing on-demand, accurate and complete energy usage, cost and location data from more than 900 utilities, Utility Data for Solar aims to bring an automated, streamlined process to existing and emerging solar markets worldwide. Solar installers typically spend a lot of time analyzing utility data for a potential residential customer, drawing on historic usage, rate thresholds and grid connec- tion scenarios. With access to a solar data service, this information can be delivered rapidly and at a low cost, says Daniel Roesler, the co-founder, CEO and CTO of UtilityAPI. His company provides comprehensive data on a single meter for a mere $15 one-time charge, with ongo- ing meter reads at $2 each. "We only charge if we can get the data back within 24 hours," he says. "With utility data on the backend, a solar installer can access a prospect's actual address, electric usage, costs and tariff," Kuo says. "This means that, before even sending someone to the property for a site visit, the installer can assess system size, project customer ROI and determine whether they're a good candidate for solar. With this information in hand, installers can expect more efficient sales cycles and, ultimately, higher revenue." On the commercial side, solar develop- ers not only need to present a cost/bene- fit analysis to potential customers, but also to analyze the feasibility of a low- cost grid connection, dependent on data about local substations and other utility infrastructure. "We've done about 200 commercial applications over the last four years, yet the number of projects that are actually getting through utility review is only about five percent," says Tim McDuffie, the director of engineering at CalCom Solar, based in Visalia, Calif. Merging residential and C&I custom- ers into a community solar development can be even more challenging without detailed solar data and grid availability knowledge. Since community solar proj- ects often include energy storage — if not grid service arbitrage — historic information is needed to launch the proj- ect, and ongoing data is needed to man- age it. "We can pull in ongoing utility data for a customer, match it with performance data and calculate true savings — not some savings estimate. This is especially necessary for battery systems," Roesler says. "Then for the asset management side, our data reporting can be fed into their performance analytics for real-time report- ing and future action recommendations." Working with regulators As distributed energy resources prolif- erate, utilities increasingly must balance decisions about future infrastructure investments as they transition from elec- tricity generators to wires-and-poles companies. Both regulations and soft- ware advances will help in the broader use and analysis of solar data. California's Public Utilities Commission, for example, is now study- ing a ground-breaking set of regulations for demand response providers that even- "WE CAN PULL IN ONGOING UTILITY DATA FOR A CUSTOMER, MATCH IT WITH PERFORMANCE DATA AND CALCULATE TRUE SAVINGS — NOT SOME SAVINGS ESTIMATE."


text:
Mind Over Masters: The Question of Free Will
The Whispering Mind: The Enduring Conundrum of Consciousness
Engineering the Brain: Deploying a New Neural Toolkit
Notes and Neurons: In Search of the Common Chorus
Time The Familiar Stranger
Brilliant Breakthroughs: Neuroscience
The Illusion of Certainty: Risk, Probability, and Chance
Consciousness: Explored and Explained
My Neurons, My Self
What It Means to be Human
Program Media
Do we make conscious decisions? Or are all of our actions predetermined? And if we don't have free will, are we responsible for what we do? Modern neurotechnology is now allowing scientists to study brain activity neuron by neuron to try to determine how and when our brains decide to act. In this program, experts probe the latest research and explore the question of just how much agency we have in the world, and how the answer impacts our ethics, our behavior, and our society.
TOPICS: Mind & Brain, Science in Society
TAGS: 2016, Alfred Mele, Azim Shariff, christof koch, do we have free will, Free Will (Quotation Subject), Guilt, Metaphysics, Mind Over Masters, New York City, NYC, Philosophy (Field Of Study), physics, praise, predetermined, responsibility, sin, string theory, Tamar Kushnir, the free will debate, The Libet Experiment, The Question of Free Will, what is free will
PLAYLISTS: Big Ideas
Original Program Date: Saturday, May 30, 2015
Moderator: Emily Senay
Participants: Christof Koch, Tamar Kushnir, Alfred Mele, Azim Shariff
Benjamin Libet
The Libet Experiment 01
The Libet Veto Experiment
Free Will Gas Station
Newton's Second Law of Motion
Quantum Mechanical Wave Function
Making a Choice
Brain Map
Fighting in British Parliament
Bar Fight
Neuroscience Classroom
Drowning witch
Cracking Witches Back
Criminal Headlines
4 Year old explain physical limitations
TAGS: 2016, Alfred Mele, Azim Shariff, christof koch, do we have free will, Free Will (Quotation Subject), Guilt, Metaphysics, Mind Over Masters, New York City, NYC, Philosophy (Field Of Study), physics, praise, predetermined, responsibility, sin, string theory, Tamar Kushnir, the free will debate, The Libet Experiment, The Question of Free Will, what is free will, 2016, Alfred Mele, Azim Shariff, christof koch, do we have free will, Free Will (Quotation Subject), Guilt, Metaphysics, Mind Over Masters, New York City, NYC, Philosophy (Field Of Study), physics, praise, predetermined, responsibility, sin, string theory, Tamar Kushnir, the free will debate, The Libet Experiment, The Question of Free Will, what is free will
Emily SenayPhysician Journalist
Emily Senay is a physician, medical and public health educator, broadcast journalist, and author. She is an assistant professor of Medicine in the Department of Preventive Medicine at the Icahn School of Medicine at Mount Sinai and a clinician in the World Trade Center Health Program in New York City.
Tamar KushnirDevelopmental Psychologist
Tamar Kushnir is an associate professor at Cornell University and the director of the Early Childhood Cognition Laboratory. Her research examines the origins of causal and social knowledge in early childhood, and how children acquire this knowledge through play, observation, and social interaction.
Azim ShariffPsychologist
Azim Shariff is an assistant professor of psychology at the University of Oregon and the director of the Culture and Morality Lab. He graduated with his Ph.D. in Social Psychology from the University of British Columbia in 2010, before joining the UO faculty.
Christof KochNeuroscientist
Born in the American Midwest, Christof Koch grew up in Holland, Germany, Canada, and Morocco. He studied Physics and Philosophy and was awarded his Ph.D. in Biophysics. In 1987, Koch joined the California Institute of Technology as a Professor in Biology and Engineering.
Alfred MelePhilosopher
Alfred Mele is the William H. and Lucyle T. Werkmeister Professor of Philosophy at Florida State University. He is the author of 10 books, including Free Will and Luck, Effective Intentions, A Dialogue on Free Will and Science, and Free: Why Science Hasn't Disproved Free Will.
What were the experiments that prove that free will doesn't exist.
Three types of Free Will
What does a post free will world look like?
Free Will and Punishment
Does Free Will exist?
If there is no free will how can you test it?


> affected this for example?
anything would have a param with spaces.
> KOPT_/boot/datacom=yes: not found". This a bug or feature?
my computer too. Just not had time to actually fix it.
Yeah. this sounds like bug.
> like # acpi_osi="!Windows 2006"
Thanks for your feedback, and sorry for the breakages due to syslinux-5.
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Last 5 Pages
A random public picture relating to the lorien trust LARP system, taken from Flickr . Click on a photo to visit the full size version.
Jackals/Tripoli
The city is located on the edge of the Unforgiving desert and is built upon on a rocky outcrop of land. Tripoli was founded by the Persians who named it Oea. The climate is typical of Arabia, with hot, dry summers, cool winters and some modest rainfall.
Having been driven back east by various circumstances the Persians built Oea on the east side of the Unforgiving desert, hoping the combination of a manmade barrier backing up the physical barrier of the desert would protect them from further territorial loss.
Long after the Persians had lost the city to the Ottoman the Arab tribes came to power, surging North from the Gardens of Babylon. Many of the weakened states they encountered submitted with little fight, the great city Baghdad supplying a rare example of prolonged defiance. When the Arabs finally took Baghdad, Tripoli became cut off from its support base. The Arabs surrounded the city, and rather than fight the fortress they waited for the inevitable surrender, not willing to spend more of their own lives when the passage of time could do their work for them. Cut off from support with no hope of supplies the isolated defenders gave up Tripoli.
Tripoli is mainly used by the Arabs as a place of rest for trade caravans along the old empire road. It acts as the last rest stop before the Unforgiving desert to the West. The City of Abbabis lies on the other side of the desert and the two cities work closely together in various matters.
As a structure Tripoli is incredibly defensible, as long as it is not cut off from the vital supply lines that maintain it. Tripoli is built on a rocky outcrop, with commanding views over the surrounding land for many miles. This land is hard to work and offers little reward for a lot of effort, hence the city's reliance on imported ford.
Tripoli is often used as a military training facility for Arabian men at arms. Ores are brought in by trade caravans so the men can learn to smith and keep their equipment in a good state of repair.
A number of years ago Tripoli was subjected to a terrible famine, because of its close proximity to one of the Wasir pillars of salt. Following inaction from the Jackals, Tripoli was saved from disorder by a group of knights from the Heartlands. After the knights settled the situation in Tripoli they started to look after their own agenda. Whilst attempting to impose cultural change much death and destruction was caused.
The Faction mustered its great warriors, and after a number of battles managed to rout the orders of knights from Tripoli, bringing it once more under the aegis of the Empire, and allowing it to begin to rebuild itself to match its former glories.
Arabian Cities


Pretty simple really. Put together by Pascalle Burton. I'm in issue #8 right opposite Emily xyz. Brisbane misses her.
Go to The Lavender Room to order a copy or back issues.
To get your copy coffee stained like mine, see Vanessa Page in a silly mood.
I'm stoked to announce the Queensland Poetry Festival 2012 is launched and rolling. It is looking to be a cracker event and among the incredible line up of international and local poets performing, this Piedhillprawn will be a part of the event for the first time.
On Sunday 26th August 11am, I'll be performing with Tiggy Johnson in the 'Storm and Honey' session in the Shopfront Space. This is a collaborative project we've been working on over the past year. Don't miss it. The session also includes the set 'MC Lady Lazarus vs. DJ Thought Fox' which is certain to be explosive. Yep, page poets and stage poets for the full experience in the shopfront space.
A Million Bright Things is the showcase on Saturday night 8pm where Queensland Poet in Residence a.rawlings launches the Queensland focussed component of her series Sound Poetry and Visual Poetry. This will be followed by a short set from every performer on the program. Not to be missed.
Also performing at QPF this year is Robert Adamson, who 'has fished the Hawkesbury River for poems for more than four decades'. I love Adamson's work and am super excited to be able to see several performances from him. There are so many more so go and check out the program for yourself. See you there.