Definitions and preliminary results

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

Last update: 31 March 2013

Definitions and preliminary results

The Weyl group. Let $R$ be a reduced irreducible root system in ${ℝ}^n,$ fix a set of positive roots $R^{+}$ and let $\{{\alpha }_1,\dots ,{\alpha }_n\}$ be the corresponding simple roots in $R\text{.}$ Let $W$ be the Weyl group corresponding to $R\text{.}$ Let $s_i$ denote the simple reflection in W corresponding to the simple root ${\alpha }_i$ and recall that $W$ can be presented by generators $s_1,s_2,\dots ,s_n$ and relations

$$\begin{array}{}\begin{array}{cccc}{s}_i^2& =& 1,& \text{for}\hspace{0.17em}1\le i\le n,\\ \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_i{s}_j{s}_i\dots }& =& \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_j{s}_i{s}_j\dots },& \text{for}\hspace{0.17em}i\ne j,\end{array}\end{array}$$

where $m_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩⟨{\alpha }_j,{\alpha }_i^{\vee }⟩,$ and ${\alpha }_i^{\vee }=2{\alpha }_i/⟨{\alpha }_i,{\alpha }_i⟩\text{.}$

The Iwahori-Hecke algebra. Fix $q\in {ℂ}^{*}$ such that $q$ is not a root of unity. The Iwahori-Hecke algebra $H$ is the associative algebra over $ℂ$ defined by generators $T_1,T_2,\dots ,T_n$ and relations

$$\begin{array}{cc}\begin{array}{cccc}{T}_i^2& =& (q-q^{-1})T_i+1,& \text{for}\hspace{0.17em}1\le i\le n,\\ \underset{\underset{m_{ij}\text{factors}}{⏟}}{T_i{T}_j{T}_i\dots }& =& \underset{\underset{m_{ij}\text{factors}}{⏟}}{T_j{T}_i{T}_j\dots },& \text{for}\hspace{0.17em}i\ne j,\end{array}& \text{(1.1)}\end{array}$$

where $m_{ij}$ are the same as in the presentation of $W\text{.}$ For $w\in W$ define $T_w=T_{i_1}\dots T_{i_p}$ where $s_{i_1}\dots s_{i_p}=w$ is a reduced expression for $w\text{.}$ By [Bou1968, Ch. IV §2 Ex. 23], the element $T_w$ does not depend on the choice of the reduced expression. The algebra $H$ has dimension $\mid W\mid$ and the set ${\left\{T_w\right\}}_{w\in W}$ is a basis of $H\text{.}$

The group $X\text{.}$ The fundamental weights are the elements ${\omega }_1,\dots ,{\omega }_n$ of ${ℝ}^n$ given by $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}\text{.}$ The weight lattice is the $W\text{-invariant}$ lattice in ${ℝ}^n$ given by

$$P=\sum _{i=1}^n\mathbb{Z}{\omega }_i\text{.}$$

Let $X$ be the abelian group $P$ except written multiplicatively. In other words,

$$X=\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in P\},\text{and}{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu }=X^{\mu }{X}^{\lambda },\text{for}\hspace{0.17em}\lambda ,\mu \in P\text{.}$$

Let $ℂ\left[X\right]$ denote the group algebra of $X\text{.}$ There is a $W\text{-action}$ on $X$ given by $w{X}^{\lambda }=X^{w\lambda }$ for $w\in W,$ $X^{\lambda }\in X,$ which we extend linearly to a $W\text{-action}$ on $ℂ\left[X\right]\text{.}$

The affine Hecke algebra. The affine Hecke algebra $\stackrel{\sim }{H}$ associated to $R$ and $P$ is the algebra given by

$$\stackrel{\sim }{H}=ℂ\text{-span}\{T_w{X}^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}w\in W,X^{\lambda }\in X\}$$

where the multiplication of the $T_w$ is as in the Iwahori-Hecke algebra $H,$ the multiplication of the $X^{\lambda }$ is as in $ℂ\left[X\right]$ and we impose the relation

$$\begin{array}{cc}{X}^{\lambda }{T}_i=T_i{X}^{s_i\lambda }+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},\text{for}\hspace{0.17em}1\le i\le n\hspace{0.17em}\text{and}\hspace{0.17em}{X}^{\lambda }\in X\text{.}& \text{(1.2)}\end{array}$$

This formulation of the definition of $\stackrel{\sim }{H}$ is due to Lusztig [Lus1989] following work of Bernstein and Zelevinsky. The elements $T_w{X}^{\lambda },$ $w\in W,$ $X^{\lambda }\in X,$ form a basis of $\stackrel{\sim }{H}\text{.}$

Weights. Let

$$T=\{\text{group homomorphisms}\hspace{0.17em}t:X\to {ℂ}^{*}\}\text{.}$$

The torus $T$ is an abelian group with a $W\text{-action}$ given by $\left(wt\right)\left(X^{\lambda }\right)=t\left(X^{w^{-1}\lambda }\right)\text{.}$ For any element $t\in T$ define the polar decomposition

$$t=t_r{t}_c,t_r{t}_c\in T\hspace{0.17em}\text{such that}\hspace{0.17em}{t}_r\left(X^{\lambda }\right)\in {ℝ}_{>0,}\hspace{0.17em}\text{and}\hspace{0.17em}\mid t_c\left(X^{\lambda }\right)\mid =1,$$

for all $X^{\lambda }\in X\text{.}$ Let $Q^{\vee }={\sum }_i\mathbb{Z}{\alpha }_i^{\vee }\text{.}$ There is a unique $\mu \in {ℝ}^n$ and a unique $\nu \in {ℝ}^n/Q^{\vee }$ such that

$$\begin{array}{cc}{t}_r\left(X^{\lambda }\right)=e^{⟨\mu ,\lambda ⟩}\text{and}{t}_c\left(X^{\lambda }\right)=e^{2\pi i⟨\nu ,\lambda ⟩},\text{for all}\hspace{0.17em}\lambda \in P\text{.}& \text{(1.3)}\end{array}$$

In this way we identify the sets $T_r=\{t\in T\hspace{0.17em}\mid \hspace{0.17em}t=t_r\}$ and $T_c=\{t\in T\hspace{0.17em}\mid \hspace{0.17em}t=t_c\}$ with ${ℝ}^n$ and ${ℝ}^n/Q^{\vee },$ respectively.

Central characters.

Theorem 1.4. (Bernstein, Zelevinsky, Lusztig [Lus1983, 8.1]) The center of $\stackrel{\sim }{H}$ is $ℂ{\left[X\right]}^W=\{f\in ℂ\left[X\right]\hspace{0.17em}\mid \hspace{0.17em}wf=f\}\text{.}$

Since $\stackrel{\sim }{H}$ has countable dimension, Dixmier’s version of Schur’s lemma implies that $Z\left(\stackrel{\sim }{H}\right)$ acts on an irreducible $\stackrel{\sim }{H}\text{-module}$ $M$ by scalars. Let $t\in T$ be such that

$$pM=t\left(p\right)M,\text{for all}\hspace{0.17em}p\in Z\left(\stackrel{\sim }{H}\right)\text{.}$$

Since $Z\left(\stackrel{\sim }{H}\right)=ℂ{\left[X\left(T\right)\right]}^W$ it follows that $t\left(p\left(X\right)\right)=\left(wt\right)\left(p\left(X\right)\right)$ for all $w\in W\text{.}$ The $W\text{-orbit}$ $Wt$ of $t$ is the central character of $M\text{.}$ We shall often abuse notation and refer to any weight $s\in Wt$ as “the central character” of $M\text{.}$

Weight spaces. Let $M$ be a finite dimensional $\stackrel{\sim }{H}\text{-module.}$ For each $t\in T$ the $t\text{-weight}$ space of $M$ and the generalized $t\text{-weight}$ space are the subspaces

$$\begin{array}{ccc}{M}_t& =& \{m\in M\hspace{0.17em}\mid \hspace{0.17em}{X}^{\lambda }m=t\left(X^{\lambda }\right)m\hspace{0.17em}\text{for all}\hspace{0.17em}{X}^{\lambda }\in X\}\text{and}\\ M_t^{\text{gen}}& =& \{m\in M\hspace{0.17em}\mid \hspace{0.17em}\text{for each}\hspace{0.17em}{X}^{\lambda }\in X,{(X^{\lambda }-t\left(X^{\lambda }\right))}^{k}m=0\hspace{0.17em}\text{for some}\hspace{0.17em}k\in {\mathbb{Z}}_{>0}\},\end{array}$$

respectively. If $M_t^{\text{gen}}\ne 0$ then $M_t\ne 0\text{.}$ In general $M\ne {⨁}_{t\in T}{M}_t,$ but we do have

$$M=\underset{t\in T}{⨁}{M}_t^{\text{gen}}\text{.}$$

This is a decomposition of $M$ into Jordan blocks for the action of $ℂ\left[X\right]\text{.}$ The set of weights of $M$ is the set

$$\begin{array}{cc}\text{supp}\left(M\right)=\{t\in T\hspace{0.17em}\mid \hspace{0.17em}{M}_t^{\text{gen}}\ne 0\}\text{.}& \text{(1.5)}\end{array}$$

The calibration graph. Let $t\in T\text{.}$ Define a graph $\Gamma \left(t\right)$ with

$$\begin{array}{cc}\text{Vertices:}& Wt,\\ \text{Edges:}& wt⟷s_{i}wt,& \text{if}\left(wt\right)\left(X^{{\alpha }_i}\right)\ne q^{\pm 2}\text{.}\end{array}$$

Proposition 1.6. ([Ram1998] Proposition 2.12) Let $M$ be a finite dimensional irreducible $\stackrel{\sim }{H}\text{-module}$ with central character $t\text{.}$ Then

$$\text{dim}\left(M_s^{\text{gen}}\right)=\text{dim}\left(M_{s\prime }^{\text{gen}}\right)$$

if $s$ and $s\prime$ are in the same connected component of the calibration graph $\Gamma \left(t\right)\text{.}$

If $t\in T$ define

$$\begin{array}{cc}P\left(t\right)=\{\alpha >0\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=q^{\pm 2}\}\text{and}Z\left(t\right)=\{\alpha >0\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=1\}\text{.}& \text{(1.7)}\end{array}$$

For each subset $J\subseteq P\left(t\right)$ define

$$\begin{array}{cc}{ℱ}^{(t,J)}=\{w\in W\hspace{0.17em}\mid \hspace{0.17em}R\left(w\right)\cap Z\left(t\right)=\varnothing ,R\left(w\right)\cap P\left(t\right)=J\},& \text{(1.8)}\end{array}$$

where $R\left(w\right)=\{\alpha >0\hspace{0.17em}\mid \hspace{0.17em}w\alpha <0\}$ is the inversion set of $w\text{.}$ Define a placed shape to be a pair $(t,J)$ such that $t\in T,$ $J\subseteq P\left(t\right)$ and ${ℱ}^{(t,J)}\ne \varnothing \text{.}$ The elements of the set ${ℱ}^{(t,J)}$ are called standard tableaux of shape $(t,J)\text{.}$

Proposition 1.9. ([Ram1998] Theorem 2.14) Let $t\in T\text{.}$ The connected components of the calibration graph $\Gamma \left(t\right)$ are the sets

$$\{wt\hspace{0.17em}\mid \hspace{0.17em}w\in {ℱ}^{(t,J)}\},J\subseteq P\left(t\right),\text{such that}\hspace{0.17em}{ℱ}^{(t,J)}\ne \varnothing \text{.}$$

Calibrated representations. A finite dimensional $\stackrel{\sim }{H}\text{-module}$ $M$ is calibrated if $M_t^{\text{gen}}=M_t,$ for all $t\in T\text{.}$

Proposition 1.10. ([Ram1998] Proposition 4.2)

1. An irreducible $\stackrel{\sim }{H}\text{-module}$ $M$ is calibrated if and only if $\text{dim}\left(M_t^{\text{gen}}\right)=1$ for all weights $t$ of $M\text{.}$
2. If $M$ is an irreducible $\stackrel{\sim }{H}\text{-module}$ with regular central character $t$ (i.e. $Z\left(t\right)=\varnothing \text{)}$ then $M$ is calibrated.

Let ${\alpha }_i$ and ${\alpha }_j$ be simple roots in $R$ and let $R_{ij}$ be the rank two root subsystem of $R$ which is generated by ${\alpha }_i$ and ${\alpha }_j\text{.}$ Let $W_{ij}$ be the Weyl group of $R_{ij},$ the subgroup of $W$ generated by the simple reflections $s_i$ and $s_j\text{.}$ A weight $t\in T$ is calibratable for $R_{ij}$ if one of the following two conditions holds:

1. $t\left(X^{\alpha }\right)\ne 1$ for all $\alpha \in R_{ij},$
2. $R_{ij}$ is of type $C_2$ or $G_2$ (assume that ${\alpha }_i$ is the long root and ${\alpha }_j$ is the short root), $ut\left(X^{{\alpha }_i}\right)=q^2$ and $ut\left(X^{{\alpha }_j}\right)=1$ for some $u\in W_{ij},$ and $t\left(X^{{\alpha }_i}\right)\ne 1$ and $t\left(X^{{\alpha }_j}\right)\ne 1\text{.}$

A placed skew shape is a placed shape $(t,J)$ such that for all $w\in {ℱ}^{(t,J)}$ and all pairs ${\alpha }_i,{\alpha }_j$ of simple roots in $R$ the weight $wt$ is calibratable for $R_{ij}\text{.}$

Theorem 1.11. ([Ram1998] Theorem 3.1 and Proposition 4.1)

1. Let $(t,J)$ be a placed skew shape and let ${ℱ}^{(t,J)}$ be the set of standard tableaux of shape $(t,J)\text{.}$ Define $${\stackrel{\sim }{H}}^{(t,J)}=ℂ\text{-span}\{v_w\hspace{0.17em}\mid \hspace{0.17em}w\in {ℱ}^{(t,J)}\},$$ so that the symbols $v_w$ are a labeled basis of the vector space ${\stackrel{\sim }{H}}^{(t,J)}\text{.}$ Then the following formulas make ${\stackrel{\sim }{H}}^{(t,J)}$ into an irreducible $\stackrel{\sim }{H}\text{-module:}$ For each $w\in {ℱ}^{(t,J)},$ $$\begin{array}{cccc}{X}^{\lambda }{v}_w& =& \left(wt\right)\left(X^{\lambda }\right)v_w,& \text{for}\hspace{0.17em}{X}^{\lambda }\in X\text{, and}\\ T_i{v}_w& =& {\left(T_i\right)}_{ww}{v}_w+(q^{-1}+{\left(T_i\right)}_{ww})v_{s_{i}w},& \text{for}\hspace{0.17em}1\le i\le n,\end{array}$$ where ${\left(T_i\right)}_{ww}=\frac{q-q^{-1}}{1-\left(wt\right)\left(X^{-{\alpha }_i}\right)},$ and we set $v_{s_{i}w}=0$ if $s_{i}w\notin {ℱ}^{(t,J)}\text{.}$
2. If $M$ is an irreducible calibrated representation such that $\text{supp}\left(M\right)=\{wt\hspace{0.17em}\mid \hspace{0.17em}w\in {ℱ}^{(t,J)}\}$ for some placed skew shape $(t,J)$ then $M$ is isomorphic to the module ${\stackrel{\sim }{H}}^{(t,J)}$ constructed in (a).

Remark 1.12. It follows from the results of Rodier [Rod1981] that if $M$ is an irreducible $\stackrel{\sim }{H}\text{-module}$ with regular central character (i.e. $Z\left(t\right)=\varnothing \text{)}$ then $M$ satisfies the hypothesis of the statement of Theorem 1.11 (b).

Langlands classification. The following discussion follows the work of Evens [Eve1996] and [KLu0862716, §8]. For this subsection it is convenient to assume that $q\in {ℝ}_{>0}$ and $q\ne 1\text{.}$ For the general case see [KLu0862716, §8]. Let $t\in T$ and let $t=t_r{t}_c$ be the polar decomposition of $t\text{.}$ Define

$$\nu \left(t\right)\in \sum _{i=1}^nℝ{\alpha }_i^{\vee }\text{by requiring}{t}_r\left(X^{\lambda }\right)=q^{2⟨\lambda ,\nu \left(t\right)⟩},\text{for all}\hspace{0.17em}\lambda \in P\text{.}$$

A finite dimensional $\stackrel{\sim }{H}\text{-module}$ $M$ is tempered if for all weights $t$ of $M$ (as defined in (1.5)) we have

$$⟨{\omega }_i,\nu \left(t\right)⟩\le 0,\text{for all}\hspace{0.17em}1\le i\le n\text{.}$$

The module $M$ is square integrable if $⟨{\omega }_i,\nu \left(t\right)⟩<0$ for all $1\le i\le n\text{.}$ and all weights $t$ of $M\text{.}$

Let $I$ be a subset of the simple roots and let ${\stackrel{\sim }{H}}_I$ be the subalgebra of $\stackrel{\sim }{H}$ generated by $T_i,$ $i\in I,$ and all $X^{\lambda }\in X\text{.}$ We shall say that a finite dimensional ${\stackrel{\sim }{H}}_I\text{-module}$ is tempered if $I$ is the maximal set such that for all weights $t$ of $M,$

$$⟨{\omega }_i,\nu \left(t\right)⟩\le 0,\text{for all}\hspace{0.17em}i\in I\text{.}$$

Theorem 1.13. (see [Eve1996]) Let $I\subseteq \{1,2,\dots ,n\}$ and let $𝒯$ be an irreducible tempered representation of ${\stackrel{\sim }{H}}_I\text{.}$

1. $M_{𝒯,I}={\text{Ind}}_{{\stackrel{\sim }{H}}_I}^{\stackrel{\sim }{H}}\left(𝒯\right)$ has a unique irreducible quotient $L_{𝒯,I}\text{.}$
2. Every irreducible $\stackrel{\sim }{H}\text{-module}$ is isomorphic to $L_{𝒯,I}$ for some pair $(𝒯,I)\text{.}$
3. If $L_{𝒯,I}\cong L_{𝒯\prime ,I\prime }$ then $I=I\prime$ and $𝒯\cong 𝒯\prime$ as ${\stackrel{\sim }{H}}_I\text{-modules.}$

The Langlands parameters of an irreducible $\stackrel{\sim }{H}\text{-module}$ $M$ are given by the pair $(𝒯,I)$ specified by Theorem 1.13 (b).

Classification by indexing triples. Kazhdan and Lusztig [KLu0862716] (see also the important work of Ginzburg [CGi1433132]) gave a refinement of the Langlands classification. Let $G$ be the simple complex algebraic group with root system $R$ and weight lattice $P\text{.}$ An indexing triple $(s,u,\rho )$ consists of

$$\begin{array}{c}\text{a semisimple element}\hspace{0.17em}s\in G,\\ \text{a unipotent element}\hspace{0.17em}u\in G,\end{array}\text{such that}su{s}^{-1}=u^{q^2},$$

and an irreducible representation $\rho$ of the component group $A(s,u)=Z_G(s,u)/Z_G{(s,u)}^{\circ },$ where $Z_G(s,u)=Z_G\left(s\right)\cap Z_G\left(u\right)\text{.}$ Let $K\left({ℬ}_{s,u}\right)$ be the $K\text{-theory}$ of the variety

$${ℬ}_{s,u}=\left\{\text{Borel subgroups of}\hspace{0.17em}G\hspace{0.17em}\text{containing both}\hspace{0.17em}s\hspace{0.17em}\text{and}\hspace{0.17em}u\right\}\text{.}$$

By a theorem of Lusztig [Lus1985] $K\left({ℬ}_{s,u}\right)$ is an $\stackrel{\sim }{H}\text{-module.}$ The group $A(s,u)$ also acts on $K\left({ℬ}_{s,u}\right)$ and this action commutes with the action of $\stackrel{\sim }{H}\text{.}$ The standard modules $M_{s,u,\rho }$ are the $\stackrel{\sim }{H}\text{-modules}$ given by the decomposition

$$K\left({ℬ}_{s,u}\right)=\underset{\rho }{⨁}{M}_{s,u,\rho }\otimes \rho ,$$

where the sum is over all irreducible representations of $A(s,u)\text{.}$

Theorem 1.14. [KLu0862716]

1. If $M_{s,u,\rho }\ne 0$ then it has a unique simple quotient $L_{s,u,\rho }\text{.}$
2. Every simple $\stackrel{\sim }{H}\text{-module}$ isomorphic to some $L_{s,u,\rho }\text{.}$
3. If $L_{s,u,\rho }\cong L_{s\prime ,u\prime ,\rho \prime }$ then there is a $g\in G$ such that $s\prime =gs{g}^{-1},$ $u\prime =gu{g}^{-1},$ and $\rho \prime =\rho \text{.}$

In this way each irreducible $\stackrel{\sim }{H}\text{-module}$ corresponds to a unique (up to conjugation) indexing triple. One can replace $u$ by $n=\text{ln}\hspace{0.17em}u$ in the Lie algebra $𝔤=\text{Lie}\left(G\right)$ (see [CGi1433132, Ch. 8]) so that an indexing triple is

$$\begin{array}{c}\text{a semisimple element}\hspace{0.17em}s\in G,\\ \text{a nilpotent element}\hspace{0.17em}n\in 𝔤,\end{array}\text{such that}\text{Ad}\left(s\right)n=q^{2}n,$$

and an irreducible representation $\rho$ of the component group $A(s,n)=Z_G(s,n)/Z_G{(s,n)}^{\circ },$ where $Z_G(s,n)=Z_G\left(s\right)\cap Z_G\left(n\right)$ and $Z_G\left(n\right)$ is taken with respect to the adjoint action of $G$ on $𝔤\text{.}$ We will use this form of the indexing triples in the examples in later sections.

Principal series modules. Let $t\in T$ and let $ℂv_t$ be the one dimensional $ℂ\left[X\right]\text{-module}$ corresponding to the character $t:X\to {ℂ}^{*}\text{.}$ Specifically, $ℂv_t$ is the one dimensional vector space with basis $\left\{v_t\right\}$ and $ℂ\left[X\right]\text{-action}$ given by

$$X^{\lambda }{v}_t=t\left(X^{\lambda }\right)v_t,\text{for all}\hspace{0.17em}{X}^{\lambda }\in X\text{.}$$

The principal series module corresponding to $t$ is the $\stackrel{\sim }{H}\text{-module}$

$$M\left(t\right)={\text{Ind}}_{ℂX}^{\stackrel{\sim }{H}}\left(ℂv_t\right)\text{.}$$

Theorem 1.15. [Mat1977]

1. Every irreducible $\stackrel{\sim }{H}\text{-module}$ $M$ with central character $t$ is a composition factor of the principal series module $M\left(t\right)\text{.}$
2. If $w\in W$ and $t\in T$ then $M\left(t\right)$ and $M\left(wt\right)$ have the same composition factors.

Theorem 1.16. Kato’s irreducibility criterion [Kat1981]) Let $t\in T$ and let $P\left(t\right)=\{\alpha >0\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=q^{\pm 2}\}\text{.}$ The principal series module $M\left(t\right)$ is irreducible if and only if $P\left(t\right)=\varnothing \text{.}$

Remark. Kato actually proves a more general result and thus needs a further condition for irreducibility. We have simplified matters by specifying the weight lattice $P$ in our construction of the affine Hecke algebra. One can use any $W\text{-invariant}$ lattice in ${ℝ}^n$ and Kato works in this more general situation. When the one uses the weight lattice $P,$ a result of Steinberg [Ste1968-2, 4.2, 5.3] says that the stabilizer $W_t$ of a point $t\in T$ under the action of $W$ is always a reflection group. Because of this Kato’s criterion takes a simpler form.

Weights of induced modules. If $I\subseteq \{1,\dots ,n\}$ define ${\stackrel{\sim }{H}}_I$ to be the subalgebra of $\stackrel{\sim }{H}$ generated by $T_i,$ $i\in I,$ and all $X^{\lambda }\in X\text{.}$

Lemma 1.17. Let $t\in T$ such that $t\left(X^{{\alpha }_i}\right)=q^2$ for all $i\in I$ and let $ℂv_t$ be the one dimensional ${\stackrel{\sim }{H}}_I\text{-module}$ with basis $\left\{v_t\right\}$ and ${\stackrel{\sim }{H}}_I\text{-action}$ given by

$$T_i{v}_t=q{v}_t,\text{for}\hspace{0.17em}i\in I,\text{and}{X}^{\lambda }{v}_t=t\left(X^{\lambda }\right)v_t,\text{for all}\hspace{0.17em}{X}^{\lambda }\in X\text{.}$$

Let $W/W_I$ be the set of minimal length coset representatives of cosets of $W_I$ in $W\text{.}$ Then the weights of the $\stackrel{\sim }{H}\text{-module}$ $M={\text{Ind}}_{{\stackrel{\sim }{H}}_I}^{\stackrel{\sim }{H}}\left(ℂv_t\right)$ are $wt,$ $w\in W/W_I,$ and

$$\text{dim}\left(M_{wt}^{\text{gen}}\right)=\text{(\# of}\hspace{0.17em}u\in W/W_I\hspace{0.17em}\text{such that}\hspace{0.17em}ut=wt\text{).}$$

		Proof.
	The module $M$ has basis $\{T_w\otimes v_t\hspace{0.17em}\mid \hspace{0.17em}w\in W/W_I\}\text{.}$ By writing $T_w=T_{i_1}\dots T_{i_p}$ for a reduced word $w=s_{i_1}\dots s_{i_p}$ and inductively using the defining relation (1.2) we get $$\begin{array}{ccc}{X}^{\lambda }(T_w\otimes v_t)& =& t\left(X^{w^{-1}\lambda }\right)(T_w\otimes v_t)+\sum _{v<w}{a}_v\left(t\right)(T_v\otimes v_t)\\ & =& \left(wt\right)\left(X^{\lambda }\right)(T_w\otimes v_t)+\sum _{v<w}{a}_v\left(t\right)(T_v\otimes v_t),\end{array}$$ where the sum is over $v\in W$ which are less than $w$ in Bruhat order and $a_v\left(t\right)\in ℂ\text{.}$ This shows that the eigenvalues of $X^{\lambda }$ on $M$ are $\left(wt\right)\left(X^{\lambda }\right)\text{.}$ The result follows by counting the multiplicity of each eigenvalue. $\square$

The $\tau$ operators. The maps ${\tau }_i:M_t^{\text{gen}}\to M_{s_{i}t}^{\text{gen}}$ defined below are local operators on $M$ in the sense that they act on each weight space $M_t^{\text{gen}}$ of $M$ separately. The operator ${\tau }_i$ is only defined on weight spaces $M_t^{\text{gen}}$ such that $t\left(X^{{\alpha }_i}\right)\ne 1\text{.}$

Proposition 1.18. ([Ram1998] Proposition 2.7) Let $t\in T$ such that $t\left(X^{{\alpha }_i}\right)\ne 1$ and let $M$ be a finite dimensional $\stackrel{\sim }{H}\text{-module.}$ Define

$$\begin{array}{cccc}{\tau }_i:& M_t^{\text{gen}}& ⟶& M_{s_{i}t}^{\text{gen}}\\ & m& ⟼& (T_i-\frac{q-q^{-1}}{1-X^{-{\alpha }_i}})m\text{.}\end{array}$$

1. The map ${\tau }_i:M_t^{\text{gen}}⟶M_{s_{i}t}^{\text{gen}}$ is well defined.
2. As operators on $M_t^{\text{gen}},$ $X^{\lambda }{\tau }_i={\tau }_i{X}^{s_i\lambda },$ for all $X^{\lambda }\in X\text{.}$
3. As operators on $M_t^{\text{gen}},$ ${\tau }_i{\tau }_i=\frac{(q-q^{-1}{X}^{{\alpha }_i})(q-q^{-1}{X}^{-{\alpha }_i})}{(1-X^{{\alpha }_i})(1-X^{-{\alpha }_i})}\text{.}$
4. Let $1\le i\ne j\le n$ and let $m_{ij}$ be as in (1.1). Then $\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\tau }_i{\tau }_j{\tau }_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\tau }_j{\tau }_i{\tau }_j\dots },$ whenever both sides are well defined operators on $M_t^{\text{gen}}\text{.}$

Lemma 1.19. Let $t\in T$ such that $t\left(X^{{\alpha }_i}\right)=1$ and suppose that $M$ is an $\stackrel{\sim }{H}\text{-module}$ such that $M_t^{\text{gen}}\ne 0\text{.}$ Let $W_t$ be the stabilizer of $t$ under the action of $W$ on $T\text{.}$ Assume that $\bar{w}\in W/W_t$ is such that $t$ and $\bar{w}t$ are in the same connected component of $\Gamma \left(t\right)\text{.}$ Let $w$ be a minimal length coset representative of $\bar{w}\text{.}$ Then

1. $\text{dim}\left(M_{wt}^{\text{gen}}\right)\ge 2,$ and
2. If $M_{s_{j}wt}^{\text{gen}}=0$ then $\left(\bar{w}t\right)\left(X^{{\alpha }_j}\right)=q^{\pm 2}$ and $⟨w^{-1}{\alpha }_j,{\alpha }_i^{\vee }⟩=0\text{.}$

		Proof.
	Let $M\left(t\right)$ be the two dimensional principal series module for the affine Hecke algebra $\stackrel{\sim }{H}{A}_1$ of type $A_1$ (see §2 central character $t_o\text{).}$ Then $M\left(t\right)=M{\left(t\right)}_t^{\text{gen}}$ and has basis $\{v_t,T_1{v}_t\}\text{.}$ Let $n_t$ be a nonzero weight vector in $M_t\text{.}$ There is a unique $\stackrel{\sim }{H}{A}_1\text{-module}$ homomorphism $$\begin{array}{ccc}M\left(t\right)& ⟶& M\\ v_t& ⟼& n_t\end{array}$$ where we view $M$ as an $\stackrel{\sim }{H}{A}_1\text{-module}$ by restriction to the parabolic subalgebra ${\stackrel{\sim }{H}}_{\left\{i\right\}}\subseteq \stackrel{\sim }{H}\text{.}$ This homomorphism must be an injection since $M\left(t\right)$ is irreducible. Thus the vectors $n_t,T_i{n}_t$ span a two dimensional subspace of $M_t^{\text{gen}}$ and $X^{\lambda }\in X$ acts on this subspace by the matrix $${\varphi }_t\left(X^{\lambda }\right)=t\left(X^{\lambda }\right)\left(\begin{array}{cc}1& (q-q^{-1})⟨\lambda ,{\alpha }_i^{\vee }⟩\\ 0& 1\end{array}\right)\text{.}$$ Let $w=s_{i_1}\dots s_{i_p}$ be a reduced expression of $w\text{.}$ Since $t$ and $wt$ are in the same connected component of $\Gamma \left(t\right)$ we can use Proposition 1.18 (c) to show that the map $${\tau }_w={\tau }_{i_1}\dots {\tau }_{i_p}:M_t^{\text{gen}}⟶M_{wt}^{\text{gen}}$$ is well defined and bijective. Thus the vectors ${\tau }_w{n}_t,$ ${\tau }_w{T}_i{n}_t$ span a two dimensional subspace of $M_{wt}^{\text{gen}}$ and, by Proposition 1.18 (b) $X^{\lambda }\in X$ acts on this subspace by the matrix wt $${\varphi }_{wt}\left(X^{\lambda }\right)=t\left(X^{w^{-1}\lambda }\right)\left(\begin{array}{cc}1& (q-q^{-1})⟨w^{-1}\lambda ,{\alpha }_i^{\vee }⟩\\ 0& 1\end{array}\right)\text{.}$$ This proves (a). Then $${\varphi }_{wt}(1-X^{-{\alpha }_j})=(1-t\left(X^{-w^{-1}{\alpha }_j}\right))\left(\begin{array}{cc}1& \frac{(q-q^{-1})t\left(X^{-w^{-1}{\alpha }_j}\right)}{1-t\left(X^{-w^{-1}{\alpha }_j}\right)}⟨-w^{-1},{\alpha }_j,{\alpha }_i^{\vee }⟩\\ 0& 1\end{array}\right)\text{.}$$ Since $M_{s_{j}wt}^{\text{gen}}=0,$ ${\tau }_j:M_{wt}^{\text{gen}}\to M_{s_{j}wt}^{\text{gen}}$ is the zero map and so $${\varphi }_{wt}\left(T_j\right)={\varphi }_{wt}\left(\frac{q-q^{-1}}{1-X^{-{\alpha }_j}}\right)=\frac{q-q^{-1}}{1-t\left(X^{-w^{-1}{\alpha }_j}\right)}\left(\begin{array}{cc}1& \frac{(q-q^{-1})t\left(X^{-w^{-1}{\alpha }_j}\right)}{1-t\left(X^{-w^{-1}{\alpha }_j}\right)}⟨-w^{-1},{\alpha }_j,{\alpha }_i^{\vee }⟩\\ 0& 1\end{array}\right)\text{.}$$ The relation $T_j^2=(q-q^{-1})T_j+1$ is the same as $(T_j-q)(T_j+q^{-1})=0\text{.}$ This relation forces ${\varphi }_{wt}\left(T_j\right)$ to have Jordan blocks of size 1 and eigenvalues $\pm q^{\pm 1}\text{.}$ It follows that $t\left(X^{w^{-1}{\alpha }_j}\right)=q^{\pm 2}$ and $⟨w^{-1}{\alpha }_j,{\alpha }_i^{\vee }⟩=0\text{.}$ $\square$

Notes and References

This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.

page history