The structure of local regions

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

Last update: 13 November 2012

The structure of local regions

Recall that the Weyl group acts on

$$T=\text{Hom}(X,{ℂ}^{*})=\{\text{group homomorphisms}t:X\to {ℂ}^{*}\}\text{by}\left(wt\right)\left(X^{\lambda }\right)=t\left(X^{w^{-1}\lambda }\right)\text{.}$$

Any element $t\in T$ is determined by the values $t\left(X^{{\omega }_1}\right),t\left(X^{{\omega }_2}\right),\dots ,t\left(X^{{\omega }_n}\right)\text{.}$ For $t\in T$ define the polar decomposition

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

for all $X^{\lambda }\in X\text{.}$ There is a unique $\gamma \in {ℝ}^n$ and a unique $v\in {ℝ}^n/P$ such that

$$\begin{array}{cc}{t}_r\left(X^{\lambda }\right)=e^{⟨\gamma ,\lambda ⟩}\text{and}{t}_c\left(X^{\lambda }\right)=e^{2\pi i⟨v,\lambda ⟩}\text{for all}\lambda \in P\text{.}& \text{(4.1)}\end{array}$$

In this way we identify the sets $T_r=\{t\in T\mid t=t_r\}$ and $T_c=\{t\in T\mid t=t_c\}$ with ${𝔥}_{ℝ}^{*}$ and ${𝔥}_{ℝ}^{*}/P,$ respectively.

For this paragraph (our goal here is (4.3) below) assume that $q$ is not a root of unity (we will treat the type $A,$ root of unity case in detail in Section 7). The representation theory of $\stackrel{\sim }{H}$ is “the same” for any $q$ which is not a root of unity, i.e. provided $q$ is not a root of unity, the classification and construction of simple $\stackrel{\sim }{H}\text{-modules}$ can be stated uniformly in terms of the parameter $q\text{.}$ Suppose $t\in T$ is such that $t=t_r$ and $\gamma \in {𝔥}_{ℝ}^{*}$ is such that

$$t=e^{\gamma },\text{in the sense that}t\left(X^{\lambda }\right)=e^{⟨\gamma ,\lambda ⟩}\text{for all}{X}^{\lambda }\in X\text{.}$$

For the purposes of representation theory (as in Theorem 3.5) $t$ indexes a central character and so we should assume that $\gamma$ is chosen nicely in its $W\text{-orbit.}$ When

$\begin{array}{cc}q=e\text{and}\gamma \text{is dominant,}\text{i.e.,}⟨\gamma ,\alpha ⟩\ge 0\text{for all}\alpha \in R^{+},& \text{(4.2)}\end{array}$

then

$$Z\left(t\right)=Z\left(\gamma \right),P\left(t\right)=P\left(\gamma \right),\text{and}{ℱ}^{(t,J)}={ℱ}^{(\gamma ,J)}\text{for a subset}J\subseteq P\left(t\right),$$

where

$$Z\left(\gamma \right)=\{\alpha \in R^{+}\mid ⟨\gamma ,\alpha ⟩=0\},P\left(\gamma \right)=\{\alpha \in R^{+}\mid ⟨\gamma ,\alpha ⟩=1\},$$ $$\begin{array}{cc}{ℱ}^{(\gamma ,J)}=\{w\in W\mid R\left(w\right)\cap Z\left(\gamma \right)=\varnothing ,R\left(w\right)\cap P\left(\gamma \right)=J\}\text{.}& \text{(4.3)}\end{array}$$

In this case the combinatorics of local regions is a new chapter in the combinatorics of the Shi arrangement defined in (1.16). Other aspects of the combinatorics of the Shi arrangement can be found in [ALi1999,33–35,37–39], and there are several additional places in the literature [Shi1987], [Xi1994, 1.11, 2.6], [Kos2000,KOP2000] which indicate that there is a deep (and not yet completely understood) connection between the structure and representation theory of the affine Hecke algebra and the combinatorics of the Shi arrangement.

4.4. Intervals in Bruhat order

Using the formulation in (4.3), Theorem 4.6 will give a complete description of the structure of ${ℱ}^{(\gamma ,J)}$ as a subset of the Weyl group when $q$ is not a root of unity. We will treat the type $A,$ root of unity cases in Section 7.

The weak Bruhat order is the partial order on $W$ given by

$$\begin{array}{cc}v\le w\text{if}R\left(v\right)\subseteq R\left(w\right),& \text{(4.5)}\end{array}$$

where $R\left(w\right)$ denotes the inversion set of $w\in W$ as defined in (1.5). This definition of the weak Bruhat order is not the usual definition but is equivalent to the usual one by [Bjo1994, Proposition 2]. A set of positive roots $K$ is closed if $\alpha ,\beta \in K,\alpha +\beta \in R^{+}$ implies that $\alpha +\beta \in K\text{.}$ The closure $\bar{K}$ of a subset $K\subseteq R^{+}$ is the smallest closed subset of $R^{+}$ containing $K\text{.}$ A set of positive roots $K\subseteq R^{+}$ is the inversion set of some permutation $w\in W$ if and only if $K$ is closed and $K^c=R^{+}\backslash K$ is closed (see [Bjo1984, Proposition 2] or [KRa2002, Theorem 5.1]).

The following theorem is proved in [KRa2002, Section 5]. The proof of part (b) of the theorem relies crucially on a theorem of J. Losonczy [Los1999].

Theorem 4.6. Let $\gamma \in {𝔥}_{ℝ}^{*}$ be dominant (i.e., $⟨\gamma ,\alpha ⟩\ge 0$ for all $\alpha \in R^{+}\text{)}$ and let $J\subseteq P\left(\gamma \right)\text{.}$ Let ${ℱ}^{(\gamma ,J)}$ be as given in (4.3).

1. Then ${ℱ}^{(\gamma ,J)}$ is nonempty if and only if $J$ satisfies the condition $$\text{if}\beta \in J,\alpha \in Z\left(\gamma \right)\text{and}\beta -\alpha \in R^{+}\text{then}\beta -\alpha \in J\text{.}$$
2. The sub-root system $R_{\left[\gamma \right]}=\{\alpha \in R\mid ⟨\gamma ,\alpha ⟩\in \mathbb{Z}\},$ has Weyl group $$W_{\left[\gamma \right]}=⟨s_{\alpha }\mid \alpha \in R_{\left[\gamma \right]}⟩$$ and if $W^{\left[\gamma \right]}=\{\sigma \in W\mid R\left(\sigma \right)\cap R_{\left[\gamma \right]}=\varnothing \}$ then $${ℱ}^{(\gamma ,J)}=W^{\left[\gamma \right]}·[{\tau }_{\text{max}},{\tau }_{\text{min}}],$$ where ${\tau }_{\text{max}},{\tau }_{\text{min}}\in W_{\left[\gamma \right]}$ are determined by $$R\left({\tau }_{\text{max}}\right)\cap R_{\left[\gamma \right]}=\bar{J}\text{and}R\left({\tau }_{\text{min}}\right)\cap R_{\left[\gamma \right]}=\overline{(P\left(\gamma \right)\backslash J)\cup Z{\left(\gamma \right)}^c},$$ the complement is taken in the set of positive roots of $R_{\left[\gamma \right]},$ and $[{\tau }_{\text{min}},{\tau }_{\text{max}}]$ denotes the interval between $w_{\text{min}}$ and $w_{\text{max}}$ in the weak Bruhat order in $W_{\left[\gamma \right]}\text{.}$

4.7. Conjugation

Assume that $\gamma$ is dominant (i.e., $⟨\gamma ,\alpha ⟩\ge 0$ for all $\alpha \in R^{+}\text{)}$ and $J\subseteq P\left(\gamma \right)\text{.}$ Let ${ℱ}^{(\gamma ,J)}$ be as given in (4.3). The conjugate of $(\gamma ,J)$ and of $w\in {ℱ}^{(\gamma ,J)}$ are defined by

$$\begin{array}{cc}{(\gamma ,J)}^{\prime }=(-u\gamma ,-u(P\left(\gamma \right)\backslash J))\text{and}\begin{array}{ccc}{ℱ}^{(t,J)}& \stackrel{1-1}{\leftrightarrow }& {ℱ}^{{(t,J)}^{\prime }}\\ w& \leftrightarrow & w^{\prime }=w{u}^{-1},\end{array}& \text{(4.8)}\end{array}$$

where $u$ is the minimal length coset representative of $w_0{W}_{\gamma }\in W/W_{\gamma }$ and $w_0$ is the longest element of $W\text{.}$ In Section 6.7 we shall show that these maps are generalizations of the classical conjugation operation on partitions.

Theorem 4.9. The conjugation maps defined in (4.8) are well defined involutions.

		Proof.
	(a) Since $\gamma$ is dominant, $-u\gamma =-w_0\gamma$ is dominant and thus $⟨-u\gamma ,-u\alpha ⟩=1$ only if $-u\alpha >0\text{.}$ Thus the equation $⟨-u\gamma ,-u\alpha ⟩=1\iff ⟨\gamma ,\alpha ⟩=1$ gives that $P(-u\gamma )=-uP\left(\gamma \right)\text{.}$ (b) Let $v\in W_{\gamma }$ such that $w_0=uv\text{.}$ (By [Bou1968, IV Section 1 Exercise 3], $v$ is unique.) Then $R^{+}\supseteq -w_{0}Z\left(\gamma \right)=-uvZ\left(\gamma \right)=uZ\left(\gamma \right),$ and it follows that $$Z(-u\gamma )=R^{+}\cap \{\alpha \in R\mid ⟨u\gamma ,\alpha ⟩=0\}=R^{+}\cap (uZ\left(\gamma \right)\cup -uZ\left(\gamma \right))=uZ\left(\gamma \right)\text{.}$$ (c) Let $R^{-}=-R^{+}$ be the set of negative roots in $R\text{.}$ Let $v\in W_{\gamma }$ such that $w_0=uv\text{.}$ Then $v$ is the longest element of $W_{\gamma }$ and $R\left(v\right)=Z\left(\gamma \right)\text{.}$ Thus, since $w_0{R}^{-}=R^{+},$ $$\begin{array}{ccc}R\left(u\right)& =& \{\alpha \in R\mid \alpha \in R^{+},w_{0}v\alpha \in R^{-}\}=\{\alpha \in R\mid \alpha \in R^{+},v\alpha \in R^{+}\},\\ & =& R^{+}\backslash R\left(v\right)=R^{+}\backslash Z\left(\gamma \right)\text{.}\end{array}$$ (d) The weight $-u\gamma =-uv\gamma =-w_0\gamma$ is dominant and $-u(P\left(\gamma \right)\backslash J)\subseteq P(-u\gamma )$ since $-uP\left(\gamma \right)=P(-u\gamma )\text{.}$ This shows that ${(\gamma ,J)}^{\prime }$ is well defined. (e) Write $w_0=uv$ where $v$ is the longest element of $W_{\gamma }\text{.}$ Similarly, write $w_0=u^{\prime }{v}^{\prime }$ where $u^{\prime }$ is the minimal length coset representative of $w_0{W}_{w_0\gamma }$ and v′ is the longest element in $W_{w_0\gamma }\text{.}$ Conjugation by $w_0$ is an involution on $W$ which takes simple reflections to simple reflections and $W_{w_0\gamma }=w_0{W}_{\gamma }{w}_0\text{.}$ It follows that $v^{\prime }=w_{0}v{w}_0\text{.}$ This gives $$u^{\prime }u=\left(w_0{v}^{\prime }\right)\left(w_{0}v\right)=w_0{w}_{0}v{w}_0{w}_{0}v=1,$$ and so the second map in (4.8) is an involution. (f) Using (e) and (a), $$\begin{array}{ccc}-u^{\prime }(P(-u\gamma )\backslash (-u(P\left(\gamma \right)\backslash J)))& =& -u^{\prime }(-uP\left(\gamma \right)\backslash (-u(P\left(\gamma \right)\backslash J)))\\ & =& P\left(\gamma \right)\backslash (P\left(\gamma \right)\backslash J)=J,\end{array}$$ and so the first map in (4.8) is an invoution. (g) Let $w\in {𝒢}^{(\gamma ,J)}$ and let $w^{\prime }=w{u}^{-1}\text{.}$ Since $R\left(w\right)\cap Z\left(\gamma \right)=\varnothing ,$ $$\begin{array}{ccc}{u}^{-1}R\left(w{u}^{-1}\right)\cap Z\left(\gamma \right)& =& \{\beta \in R\mid u\beta \in R\left(w{u}^{-1}\right),\beta \in Z\left(\gamma \right)\}\\ & =& \{\beta \in R\mid u\beta \in R^{+},w{u}^{-1}u\beta \in R^{-},\beta \in Z\left(\gamma \right)\}\\ & =& \{\beta \in R\mid \beta \in u^{-1}{R}^{+},w\beta \in R^{-},\beta \in Z\left(\gamma \right)\}\\ & =& \{\beta \in R\mid \beta \in u^{-1}{R}^{+},\beta \in R\left(w\right),\beta \in Z\left(\gamma \right)\}\text{(since}Z\left(\gamma \right)\subseteq R^{+}\text{)}\\ & =& \{\beta \in R\mid \beta \in u^{-1}{R}^{+},\beta \in R\left(w\right)\cap Z\left(\gamma \right)\}\\ & =& \varnothing ,\end{array}$$ and thus, by (b), $$R\left(w^{\prime }\right)\cap Z(-u\gamma )=R\left(w{u}^{-1}\right)\cap uZ\left(\gamma \right)=u(u^{-1}R\left(w{u}^{-1}\right)\cap Z\left(\gamma \right))=\varnothing \text{.}$$ Since $R\left(w\right)\cap P\left(\gamma \right)=J,$ $$\begin{array}{ccc}-u^{-1}R\left(w{u}^{-1}\right)\cap P\left(\gamma \right)& =& \{\beta \in R\mid -u\beta \in R\left(w{u}^{-1}\right),\beta \in P\left(\gamma \right)\}\\ & =& \{\beta \in R\mid -u\beta \in R^{+},-w{u}^{-1}u\beta \in R^{-},\beta \in P\left(\gamma \right)\}\\ & =& \{\beta \in R\mid u\beta \in R^{-},w\beta \in R^{+},\beta \in P\left(\gamma \right)\}\\ & =& \{\beta \in R\mid \beta \in R\left(u\right),\beta \in R^{+}\backslash R\left(w\right),\beta \in P\left(\gamma \right)\}\text{(since}P\left(\gamma \right)\subseteq R^{+}\text{)}\\ & =& \{\beta \in R\mid \beta \in R^{+}\backslash Z\left(\gamma \right),\beta \in R^{+}\backslash R\left(w\right),\beta \in P\left(\gamma \right)\}\\ & =& \{\beta \in R\mid \beta \in R^{+}\backslash Z\left(\gamma \right),\beta \in P\left(\gamma \right)\backslash J\}\text{(since}R\left(w\right)\cap P\left(\gamma \right)=J\text{)}\\ & =& P\left(\gamma \right)\backslash J,\text{since}Z\left(\gamma \right)\text{and}P\left(\gamma \right)\text{are disjoint.}\end{array}$$ Thus, by (a), $$\begin{array}{ccc}R\left(w^{\prime }\right)\cap P(-u\gamma )& =& R\left(w{u}^{-1}\right)\cap -uP\left(\gamma \right)=-u(-u^{-1}R\left(w{u}^{-1}\right)\cap P\left(\gamma \right))\\ & =& -u(P\left(\gamma \right)\backslash J),\end{array}$$ and so the second map in (4.8) is well defined. $\square$

Remark 4.10. In type $A,$ the conjugation involution coincides with the duality operation for representations of $𝔭\text{-adic}$ $GL\left(n\right)$ defined by Zelevinsky [Zel1980]. Zelevinsky’s involution has been studied further in [KZe1996,LTV1999,MWa1986] and extended to general Lie type by Kato [Kat1993] and Aubert [Aub1995]. For $\stackrel{\sim }{H}\text{-modules}$ in type $A,$ this is the involution on modules induced by the Iwahori–Matsumoto involution of $\stackrel{\sim }{H}$ and is detected on the level of characters: it sends an irreducible $\stackrel{\sim }{H}\text{-module}$ $L$ to the unique irreducible $L^{*}$ with $\text{dim}\left({\left(L^{*}\right)}_t^{\text{gen}}\right)=\text{dim}\left({\left(L^{*}\right)}_{t^{-1}}^{\text{gen}}\right)$ for each $t\in T\text{.}$ I would like to thank J. Brundan for clarifying this remark and making it precise.

$$\begin{array}{c} Hα1 Hα2 Hα1+2α2 Hα1+α2+δ Hα1+α2 Hα1+2α2+δ Hα1+2α2-δ Hα2+δ Hα2-δ Hα1-δ Hα1+δ Hα1+α2-δ γ s2s1s2γ s1s2γ s2γ \\ \text{Fig. 2.}\end{array}$$

Examples 4.11. (a) If $\gamma$ is dominant and is generic (as an element of $C\text{)}$ then $Z\left(\gamma \right)=P\left(\gamma \right)=\varnothing$ and ${ℱ}^{(\gamma ,\varnothing )}=W\text{.}$

(b) Let $\rho$ be defined by $⟨\rho ,{\alpha }_i⟩=1,$ for all $1\le i\le n\text{.}$ Then

$$Z\left(\rho \right)=\varnothing ,P\left(\rho \right)=\{{\alpha }_1,\dots ,{\alpha }_n\},\text{and}{ℱ}^{(\rho ,J)}=\{w\in W\mid D\left(w\right)=J\},$$

where $D\left(w\right)=\{{\alpha }_i\mid w{s}_i<w\}$ is the right descent set of $w\in W\text{.}$ The sets ${ℱ}^{(\gamma ,J)}$ which arise here are fundamental to the theory of descent algebras [GRe1989,Reu1993,Sol1976].

(c) This example is a generalization of (b). Suppose that $(\gamma ,J)$ is a local region such that $\gamma$ is regular and integral (i.e., $⟨\gamma ,\alpha ⟩\in {\mathbb{Z}}_{>0}$ for all $\alpha \in R^{+}\text{).}$ Then

$$Z\left(\gamma \right)=\varnothing ,P\left(\gamma \right)\subseteq \{{\alpha }_1,\dots ,{\alpha }_n\},\text{and}{ℱ}^{(\gamma ,J)}=\{w\in W\mid D\left(w\right)\cap P\left(\gamma \right)=J\}\text{.}$$

(d) Let $R$ be the root system of type $C_2$ with simple roots ${\alpha }_1={\epsilon }_1{\alpha }_2={\epsilon }_2-{\epsilon }_1,$ where $\{{\epsilon }_1,{\epsilon }_2\}$ is an orthonormal basis of ${𝔥}_{ℝ}^{*}={ℝ}^2\text{.}$ The positive roots are $R^{+}=\{{\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2,{\alpha }_1+2{\alpha }_2\}\text{.}$ Let $\gamma \in {ℝ}^2$ be given by $⟨\gamma ,{\alpha }_1⟩=0$ and $⟨\gamma ,{\alpha }_2⟩=1\text{.}$ Then $\gamma$ is dominant (i.e., in $C\text{)}$ and integral and

$$Z\left(\gamma \right)=\left\{{\alpha }_1\right\}\text{and}P\left(\gamma \right)=\{{\alpha }_2,{\alpha }_1+{\alpha }_2\}\text{.}$$ $$\begin{array}{c} Hα1 Hα2 Hα1+2α2 Hα1+α2 J={α2,α1+α2} J={α2} J=∅ C s2s1s2C s2s1C s2C \\ \text{Fig. 3.}\end{array}$$

Figure 3 displays the local regions ${ℱ}^{(\gamma ,J)}$ as regions in ${𝔥}_{ℝ}^{*},$ see the remarks after (2.18).

The solid line is the hyperplane corresponding to the root in $Z\left(\gamma \right)$ and the dashed lines are the hyperplanes corresponding to the roots in $P\left(\gamma \right)\text{.}$

(e) Let $R$ be the root system of type $C_2$ as in (d). Let $\gamma \in {ℝ}^2$ be defined by

$$⟨\gamma ,{\alpha }_1⟩=0,⟨\gamma ,{\alpha }_2⟩=\frac{1}{2}\text{.}$$

Then

$$Z\left(\gamma \right)=\left\{{\alpha }_1\right\},P\left(\gamma \right)=\{{\alpha }_1+2{\alpha }_2\}\text{.}$$

If $J=P\left(\gamma \right)$ then the unique minimal element $w_{\text{min}}$ of ${ℱ}^{(\gamma ,J)}$ has $R\left(w_{\text{min}}\right)=\{{\alpha }_2,{\alpha }_1+2{\alpha }_2\}\ne \bar{J}=J\text{.}$

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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