The affine Hecke algebra

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

Last update: 20 February 2013

The affine Hecke algebra

Though we shall never really use the data $(G,B,T)$ it is conceptually useful to note that there is an affine Hecke algebra associated to each triple $(G\supseteq B\supseteq T)$ where

• $G$ is a connected reductive complex algebraic group,
• $B$ is a Borel subgroup,
• $T$ is a maximal torus.

An example of this data is when $G={GL}_n\left(ℂ\right),$ $B$ is the subgroup of upper triangular invertible matrices, and $T$ is the subgroup of invertible diagonal matrices.

The reason that we can avoid the data $(G\supseteq B\supseteq T)$ is that it is equivalent to different data $(W,C,L)$ where

• $W$ is a finite real reflection group with reflection representation ${𝔥}_{ℝ}^{*},$
• $C$ is a fixed fundamental chamber for the $W\text{-action,}$
• $L$ is a $W\text{-invariant}$ lattice in ${𝔥}_{ℝ}^{*}\text{.}$

This will be our basic data. In the example where $G={GL}_n\left(ℂ\right)$ and $B$ and $T$ are the upper triangular and diagonal matrices, respectively,

$$\begin{array}{cc}W=S_n,{𝔥}_{ℝ}^{*}={ℝ}^n=\sum _{i=1}^nℝ{\epsilon }_i,C=\{\mu =\sum _{i=1}^n{\mu }_i{\epsilon }_i\hspace{0.17em}\mid \hspace{0.17em}{\mu }_1\le \dots \le {\mu }_n\},L=\sum _{i=1}^n\mathbb{Z}{\epsilon }_i,& \text{(1.1)}\end{array}$$

where $W=S_n$ is the symmetric group, acting on ${𝔥}_{ℝ}^{*}={ℝ}^n$ by permuting the orthonormal basis ${\epsilon }_1,\dots ,{\epsilon }_n\text{.}$ This example will be treated in depth in Sections 5–7. We shall show that the labeling sets ${ℱ}^{(t,J)}$ for weight spaces of affine Hecke algebra representations that are introduced in (2.18) and Corollary 2.19 and used for the classification in Theorem 3.6 are generalizations of standard Young tableaux.

The components $W$ and $L$ in the data $(W,C,L)$ are obtained from $(G\supseteq B\supseteq T)$ by

$$W=N\left(T\right)/T,X=\text{Hom}(T,{ℂ}^{*})=\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in L\},$$

where $N\left(T\right)$ is the normalizer of $T$ in $G$ and $\text{Hom}(T,{ℂ}^{*})$ is the set of algebraic group homomorphisms from $T$ to ${ℂ}^{*}\text{.}$ The notation is designed so that the multiplication in the group $X$ is

$$\begin{array}{cc}{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu }=X^{\mu }{X}^{\lambda },\text{for}\hspace{0.17em}\mu ,\lambda \in L,& \text{(1.2)}\end{array}$$

see [Bou1968, III Section 8]. The reflection (or defining) representation of the group $W$ is given by its action on ${𝔥}_{ℝ}^{*}=ℝ{\otimes }_{\mathbb{Z}}L\cong {ℝ}^n$ and with respect to a $W\text{-invariant}$ inner product $⟨,⟩$ on ${𝔥}_{ℝ}^{*}$ the group $W$ is generated by reflections $s_{\alpha }$ in the hyperplanes

$$\begin{array}{cc}{H}_{\alpha }=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}⟨x,\alpha ⟩=0\},\alpha \in R^{+}\text{.}& \text{(1.3)}\end{array}$$

See the picture which appears just before Theorem 1.17. The chambers are the connected components of ${𝔥}_{ℝ}^{*}-\left({\bigcup }_{\alpha \in R^{+}}{H}_{\alpha }\right)$ and these are the fundamental regions for the action of $W$ on ${𝔥}_{ℝ}^{*}\text{.}$ Fixing a choice of a fundamental chamber $C$ corresponds to the choice of the set $R^{+}$ of positive roots, which corresponds to the choice of $B$ in $G\text{.}$

In our formulation we may view the set $R^{+}$ as a labeling set for the reflecting hyperplanes $H_{\alpha }$ in ${𝔥}_{ℝ}^{*}$ and

$$\begin{array}{cc}C=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}⟨x,\alpha ⟩>0\hspace{0.17em}\text{for all}\hspace{0.17em}\alpha \in R^{+}\}\text{.}& \text{(1.4)}\end{array}$$

For a root $\alpha \in R,$ the positive side of the hyperplane $H_{\alpha }$ is the side towards $C,$ i.e., $\{\lambda \in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}⟨\lambda ,\alpha ⟩>0\},$ and the negative side of $H_{\alpha }$ is the side away from $C\text{.}$

For $w\in W,$ the inversion set of $W$ is

$$\begin{array}{cc}R\left(w\right)=\{\alpha \in R^{+}\hspace{0.17em}\mid \hspace{0.17em}w\alpha \in R^{-}\},& \text{(1.5)}\end{array}$$

where $R^{-}=-R^{+}\text{.}$ There is a bijection

$$\begin{array}{cc}\begin{array}{ccc}W& \leftrightarrow & \left\{\text{fundamental chambers for}\hspace{0.17em}W\hspace{0.17em}\text{acting on}\hspace{0.17em}{𝔥}_{ℝ}^{*}\right\},\\ W& \mapsto & w^{-1}C\end{array}& \text{(1.6)}\end{array}$$

and the chamber $w^{-1}C$ is the unique chamber which is on the positive side of $H_{\alpha }$ for $\alpha \ne R\left(w\right)$ and on the negative side of $H_{\alpha }$ for $\alpha \in R\left(w\right).$

The simple roots ${\alpha }_1,\dots ,{\alpha }_n$ in $R^{+}$ index the walls $H_{{\alpha }_i}$ of the fundamental chamber $C$ and the corresponding reflections $s_1,\dots ,s_n$ generate $W\text{.}$ In fact, $W$ can be presented by generators $s_1,s_2,\dots ,s_n$ and relations

$$\begin{array}{cc}\begin{array}{cccc}{s}_i^2& =& 1& \text{for}\hspace{0.17em}1\le i\le n,\\ \underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{s_i{s}_j{s}_i\dots }& =& \underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{s_j{s}_i{s}_j\dots }& \text{for}\hspace{0.17em}i\ne j,\end{array}& \text{(1.7)}\end{array}$$

where the (acute) angle $\pi /m_{ij}$ between the hyperplanes $H_{{\alpha }_i}$ and $H_{{\alpha }_j}$ determines the value $m_{ij}\text{.}$

Fix $q\in {ℂ}^{*}$ with $q^2\ne \pm 1\text{.}$ The Iwahori–Hecke algebra $H$ associated to $(W,C)$ is the associative algebra over $ℂ$ defined by generators $T_1,T_2,\dots ,T_n$ and relations

$$\begin{array}{cc}\begin{array}{cccc}{T}_i^2& =& (q-q^{-1})T_i+1& \text{for}\hspace{0.17em}1\le i\le n,\\ \underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{T_i{T}_j{T}_i\dots }& =& \underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{T_j{T}_i{T}_j\dots }& \text{for}\hspace{0.17em}i\ne j,\end{array}& \text{(1.8)}\end{array}$$

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, Chapter IV, Section 2 Exercise 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 affine Hecke algebra $\stackrel{\sim }{H}$ associated to $(W,C,L)$ algebra given by

$$\begin{array}{cc}H=ℂ\text{-span}\{T_w{X}^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}w\in W,X^{\lambda }\in X\}& \text{(1.9)}\end{array}$$

where the multiplication of the $T_w$ is as in the Iwahori–Hecke algebra $H,$ the multiplication of the $X^{\lambda }$ is as in (1.2) and we impose the relation

$$\begin{array}{cc}{X}^{\lambda }{T}_i=T_i{X}^{s_i\lambda }+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},\text{for}\hspace{0.17em}1\le i\le n\hspace{0.17em}\text{and}\hspace{0.17em}{X}^{\lambda }\in X\text{.}& \text{(1.10)}\end{array}$$

This formulation of the definition of $\stackrel{\sim }{H}$ is due to Lusztig [Lus1983] following work of Bernstein and Zelevinsky. The elements $T_w{X}^{\lambda },$ $w\in W,$ $X^{\lambda }\in X,$ form a basis of $\stackrel{\sim }{H}\text{.}$

The group algebra of $X,$

$$\begin{array}{cc}ℂ\left[X\right]=ℂ\text{-span}\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in L\},& \text{(1.11)}\end{array}$$

is a subalgebra of $\stackrel{\sim }{H}$ with a $W\text{-action}$ obtained by linearly extending the $W\text{-action}$ on $X,$

$$\begin{array}{cc}w{X}^{\lambda }=X^{w\lambda },\text{for}\hspace{0.17em}w\in W,X^{\lambda }\in X\text{.}& \text{(1.12)}\end{array}$$

Theorem 1.13 (Bernstein, Zelevinsky, Lusztig [Lus1983, 8.1]). The center of $\stackrel{\sim }{H}$ is $ℂ{\left[X\right]}^W=\{f\in ℂ\left[X\right]\hspace{0.17em}\mid \hspace{0.17em}wf=f\hspace{0.17em}\text{for all}\hspace{0.17em}w\in W\}\text{.}$

		Proof.
	Assume $$z=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda }{T}_w\in Z\left(\stackrel{\sim }{H}\right)\text{.}$$ Let $m\in W$ be maximal in Bruhat order subject to $c_{\gamma ,m}\ne 0$ for some $\gamma \in L\text{.}$ If $m\ne 1$ there exists a dominant $\mu \in L$ such that $c_{\gamma +\mu -m\mu ,m}=0$ (otherwise $c_{\gamma +\mu -m\mu ,m}\ne 0$ for every dominant $\mu \in L,$ which is impossible since $z$ is a finite linear combination of $X^{\lambda }{T}_w\text{).}$ Since $z\in Z\left(\stackrel{\sim }{H}\right)$ we have $$z=X^{-\mu }z{X}^{\mu }=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda -\mu }{T}_w{X}^{\mu }\text{.}$$ Repeated use of the relation (1.10) yields $$T_w{X}^{\mu }=\sum _{\nu \in L,v\in W}{d}_{\nu ,v}{X}^{\nu }{T}_v$$ where $d_{\nu ,v}$ are constants such that $d_{w\mu ,w}=1,$ $d_{\nu ,w}=0$ for $\nu \ne w\mu ,$ and $d_{\nu ,v}=0$ unless $v\le w\text{.}$ So $$z=\sum _{\lambda \in L,w\in W}{c}_{\lambda ,w}{X}^{\lambda }{T}_w=\sum _{\lambda \in L,w\in W}\sum _{\nu \in L,v\in W}{c}_{\lambda ,w}{d}_{\nu ,v}{X}^{\lambda -\mu +\nu }{T}_v$$ and comparing the coefficients of $X^{\gamma }{T}_m$ gives $c_{\gamma ,m}=c_{\gamma +\mu -m\mu ,m}{d}_{m\mu ,m}\text{.}$ Since $c_{\gamma +\mu -m\mu ,m}=0$ it follows that $c_{\gamma ,m}=0,$ which is a contradiction. Hence $z={\sum }_{\lambda \in L}{c}_{\lambda }{X}^{\lambda }\in ℂ\left[X\right]\text{.}$ The relation (1.10) gives $$z{T}_i=T_{i}z=\left(s_{i}z\right)T_i+(q-q^{-1})z\prime$$ where $z\prime \in ℂ\left[X\right]\text{.}$ Comparing coefficients of $X^{\lambda }$ on both sides yields $z\prime =0\text{.}$ Hence $z{T}_i=\left(s_{i}z\right)T_i,$ and therefore $z=s_{i}z$ for $1\le i\le n\text{.}$ So $z\in ℂ{\left[X\right]}^W\text{.}$ $\square$

It is often convenient to assume that $W$ acts irreducibly on ${𝔥}_{ℝ}^{*}$ and that the lattice $L$ is the weight lattice

$$\begin{array}{cc}P=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}⟨x,\alpha ⟩\in \mathbb{Z}\hspace{0.17em}\text{for all}\hspace{0.17em}\alpha \in R^{+}\}=\sum _{i=1}^n\mathbb{Z}{\omega }_i,& \text{(1.14)}\end{array}$$

where the fundamental weights are the elements ${\omega }_1,\dots ,{\omega }_n$ of ${ℝ}^n$ given by

$$\begin{array}{cc}⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij},\text{where}\hspace{0.17em}{\alpha }_i^{\vee }=\frac{2{\alpha }_i}{⟨{\alpha }_i,{\alpha }_i⟩}& \text{(1.15)}\end{array}$$

and ${\delta }_{ij}$ is the Kronecker delta. Many facts are easier to state in this case and the general case can always be reduced to this one. We will make some further remarks on this reduction at the end of this section.

Consider the connected regions of the negative Shi arrangement ${𝒜}^{-}$ [ALi1999,Shi1997,Shi1994,Shi1987,Sta1996,Sta1998], i.e., the arrangement of (affine) hyperplanes given by

$$\begin{array}{c}\begin{array}{c} \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_1+{\alpha }_2-\delta } \] \[ H_{{\alpha }_1-\delta } \] \[ H_{{\alpha }_1-\delta } \] \[ H_{{\alpha }_2-\delta } \] \[ H_{{\alpha }_1+2{\alpha }_2-\delta } \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ {\lambda }_1 \] \[ {\lambda }_{s_1} \] \[ {\lambda }_{s_2{s}_1} \] \[ {\lambda }_{s_2} \] \[ {\lambda }_{s_1{s}_2} \] \[ {\lambda }_{s_1{s}_2{s}_1} \] \[ {\lambda }_{s_2{s}_1{s}_2} \] \[ {\lambda }_{s_1{s}_2{s}_1{s}_2} \]\end{array}\\ The\; arrangement{𝒜}^{-}\\ \text{Fig. 1.}\end{array}$$

$$\begin{array}{cc}{𝒜}^{-}=\{H_{\alpha },H_{\alpha -\delta }\hspace{0.17em}\mid \hspace{0.17em}\alpha \in R^{+}\}\text{where}{H}_{\alpha }=\{x\in {ℝ}^n\hspace{0.17em}\mid \hspace{0.17em}⟨x,\alpha ⟩=0\},H_{\alpha -\delta }=\{x\in {ℝ}^n\hspace{0.17em}\mid \hspace{0.17em}⟨x,\alpha ⟩=-1\}\text{.}& \text{(1.16)}\end{array}$$

Each chamber $w^{-1}C,$ $w\in W,$ contains a unique region of ${𝒜}^{-}$ which is a cone, and the vertex of this cone is the point ${\lambda }_w$ which appears in the following theorem.

Theorem 1.17 [Ste1975]. Suppose that $W$ acts irreducibly on ${𝔥}_{ℝ}^{*}$ and that $X=\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in P\}$ where $P$ is the weight lattice. The algebra $ℂ\left[X\right]$ is a free $ℂ{\left[X\right]}^W\text{-module}$ with

$$\text{basis}\hspace{0.17em}\{X^{{\lambda }_w}\hspace{0.17em}\mid \hspace{0.17em}w\in W\},\text{where}\hspace{0.17em}{\lambda }_w=w^{-1}\left(\sum _{s_{i}w<w}{\omega }_i\right)\text{.}$$

		Proof.
	The proof is accomplished by establishing three facts: Let $f_y,y\in W,$ be a family of elements of $\mathbb{Z}\left[X\right]\text{.}$ Then $\text{det}\left(z{f}_y\right)$ is divisible by ${\prod }_{\alpha \in R^{+}}{(X^{\alpha }-1)}^{\mid W\mid /2}\text{.}$ $\text{det}\hspace{0.17em}{\left(z{X}^{{\lambda }_y}\right)}_{z,y\in W}={\prod }_{\alpha >0}{(1-X^{\alpha })}^{\mid W\mid /2}\text{.}$ If $f\in \mathbb{Z}\left[X\right]$ then there is a unique solution to the equation $$\sum _{w\in W}{a}_w{X}^{{\lambda }_w}=f,\text{with}\hspace{0.17em}{a}_w\in \mathbb{Z}{\left[X\right]}^W\text{.}$$ (a) For each $\alpha \in R^{+}$ subtract row $z{f}_y$ from row $s_{\alpha }z{f}_y\text{.}$ Then this row is divisible by $(1-X^{-\alpha })\text{.}$ Since there are $\mid W\mid /2$ pairs of rows $(z{f}_y,s_{\alpha }z{f}_y)$ the whole determinant is divisible by ${(1-X^{-\alpha })}^{\mid W\mid /2}\text{.}$ For $\alpha ,\beta \in R^{+}$ the factors $(1-X^{-\alpha })$ and $(1-X^{-\beta })$ are coprime, and so $\text{det}\left(z{f}_y\right)$ is divisible by ${\prod }_{\alpha \in R^{+}}{(1-X^{-\alpha })}^{\mid W\mid /2}\text{.}$ This product and the product in the statement of (a) differ by the unit ${\left(X^{2\rho }\right)}^{\mid W\mid /2}$ in $\mathbb{Z}\left[X\right]\text{.}$ (b) By (a), $\text{det}\left(z{X}^{{\lambda }_y}\right)$ is divisible by ${(X^{\alpha }-1)}^{\mid W\mid /2}\text{.}$ The top coefficient of $\text{det}\left(z{X}^{{\lambda }_y}\right)$ is equal to $$\prod _{z\in W}z{X}^{{\lambda }_z}=\prod _{z\in W}\prod _{\underset{s_{i}z<z}{i}}{X}^{{\omega }_i}=\prod _{i=1}^n{X}^{(\mid W\mid /2){\omega }_i}={\left(X^{\rho }\right)}^{\mid W\mid /2},$$ and the top coefficient of ${(X^{\alpha }-1)}^{\mid W\mid /2}$ is ${\left(X^{2\rho }\right)}^{\mid W\mid /2}\text{.}$ (c) Assume that $a_y\in \mathbb{Z}{\left[X\right]}^W$ are solutions of the equation ${\sum }_{y\in W}{X}^{{\lambda }_y}{a}_y=f\text{.}$ Act on this equation by the elements of $W$ to obtain the system of $\mid W\mid$ equations $$\sum _{y\in W}\left(z{X}^{{\lambda }_y}\right)a_y=zf,z\in W\text{.}$$ By (a) the matrix ${\left(z{X}^{{\lambda }_y}\right)}_{z,y\in W}$ is invertible and so this system has a unique solution with $a_y\in \mathbb{Z}{\left[X\right]}^W\text{.}$ In fact, the $a_y$ can be obtained by Cramer’s rule. Cramer’s rule provides an expression for $a_y$ as a quotient of two determinants. By (a) and (b) the denominator divides the numerator to give an element of $\mathbb{Z}\left[X\right]\text{.}$ Since each determinant is an alternating function, the quotient is an element of $\mathbb{Z}{\left[X\right]}^W\text{.}$ $\square$

Remark. In [Ste1975] Steinberg proves this type of result in full generality without the assumptions that $W$ acts irreducibly on ${𝔥}_{ℝ}^{*}$ and $L=P\text{.}$ Note also that the proof given above is sketchy, particularly in the aspect that the top coefficient of the determinant is what we have claimed it is. See [Ste1975] for a proper treatment of this point.

1.18. Deducing the ${\stackrel{\sim }{H}}_L$ representation theory from ${\stackrel{\sim }{H}}_P$

It is often easier to work with the representation theory of $\stackrel{\sim }{H}$ in the case when $L=P\text{.}$ It is important to be able to convert from this case to the case of a general lattice $L\text{.}$ If $W$ acts irreducibly on ${𝔥}_{ℝ}^{*}$ then the lattice $L$ satisfies

$$Q\subseteq L\subseteq P,\text{where}P=\sum _{i=1}\mathbb{Z}{\omega }_i\text{and}Q=\sum _{i=1}\mathbb{Z}{\alpha }_i$$

are the weight lattice and the root lattice, respectively. The group $\Omega =P/Q$ is a finite group (either cyclic or isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\text{).}$ It corresponds to the center of the corresponding complex algebraic group. Let us denote the corresponding affine Hecke algebras by

$${\stackrel{\sim }{H}}_Q\subseteq {\stackrel{\sim }{H}}_L\subseteq {\stackrel{\sim }{H}}_P,$$

according which lattice is used to make the group $X\text{.}$

Theorem 1.19 [RRa2003]. Then there is an action of the finite group $P/L$ on ${\stackrel{\sim }{H}}_P,$ by ring automorphisms, such that

$${\stackrel{\sim }{H}}_L={\left({\stackrel{\sim }{H}}_P\right)}^{P/L}=\{h\in {\stackrel{\sim }{H}}_P\hspace{0.17em}\mid \hspace{0.17em}gh=h\hspace{0.17em}\text{for all}\hspace{0.17em}g\in P/L\},$$

is the subalgebra of fixed points under the action of the group $P/L\text{.}$

This theorem is exactly what is needed to apply a (not very well known) version of Clifford theory to completely classify the representations of ${\stackrel{\sim }{H}}_L$ in terms of the representations of ${\stackrel{\sim }{H}}_P,$ see [RRa2003].

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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