Classification of calibrated representations

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

Last update: 13 November 2012

Classification of calibrated representations

For simple roots ${\alpha }_i$ and ${\alpha }_j$ in $R$ and let $R_{ij}$ be the rank-two root subsystem of $R$ generated by ${\alpha }_i$ and ${\alpha }_j\text{.}$ A weight $t\in T$ is calibratable if, for every pair $i,j,i\ne j,t$ is a weight of a calibrated representation of the rank-two affine Hecke (sub)algebra generated by $T_i,$ $T_j$ and $ℂ\left[X\right]\text{.}$ A local region

$$(t,J)\text{is}\text{skew}\text{if}wt\text{is calibratable for all}w\in {ℱ}^{(t,J)}\text{.}$$

The classification of irreducible representations of rank-two affine Hecke algebras given in [Ram2002] can be used to state this condition combinatorially. Specifically, a weight $t\in T$ is calibratable if

1. for all simple ${\alpha }_i,$ $1\le i\le n,$ $t\left(X^{{\alpha }_i}\right)\ne 1,$ and
2. for all pairs of simple roots ${\alpha }_i$ and ${\alpha }_j$ such that $\{\alpha \in R_{ij}\mid t\left(X^{\alpha }\right)=1\}\ne \varnothing ,$ the set $\{\alpha \in R_{ij}\mid t\left(X^{\alpha }\right)=q^{\pm 2}\}$ contains more than two elements.

Condition (a) says that $t$ is regular for all rank-1 subsystems of $R$ generated by simple roots. This condition guarantees that the weight is “calibratable” (i.e., appears as a weight of some calibrated representation) for all rank-1 affine Hecke subalgebras of $\stackrel{\sim }{H}\text{.}$ Condition (b) is an “almost regular” condition on t with respect to rank-2 subsystems generated by simple roots.

Remark. The conversion between the definition of calibratable weight and the combinatorial condition given in (a) and (b) is as follows. Consider a rank-two affine Hecke algebra $\stackrel{\sim }{H}\text{.}$

1. By Theorem 2.12, (a) and (d), local regions $(t,J)$ with t regular satisfy (a) and (b) and always contribute calibrated representations of $\stackrel{\sim }{H}\text{.}$
2. Using the notation of [Ram2002], the local regions $(t,J)$ with $t$ nonregular and which satisfy both conditions (a) and (b) are:
   type $A_2:$ none,
   type $C_2:$ $(t_b,\left\{{\alpha }_1\right\})$ and $(t_b,\{{\alpha }_1,{\alpha }_1+{\alpha }_2\})$ (for each of these $P\left(t\right)$ contains 3 elements),
   type $G_2:$ $(t_e,J)$ with $J\ne \varnothing$ and $J\ne P\left(t_e\right)$ (for each of these $P\left(t_e\right)$ contains 4 elements).

From (A) and (B) it follows that the local regions which satisfy (a) and (b) do contribute calibrated weights. The following shows that the other local regions do not contribute calibratable weights.

3. By Lemma 3.1(a) local regions $(t,J)$ with a weight $\xi =wt,$ $w\in F^{(t,J)}$ such that $\xi \left(X^{{\alpha }_i}\right)=1$ do not satisfy (a) and, by inspection of the tables in [Ram2002], they never contribute a calibrated representation.
4. Using the notation of [Ram2002], the local regions which satisfy condition (a) but not condition (b) are
   type $A_2:$ $(t_c,\left\{{\alpha }_2\right\})$ and $(t_d,\left\{{\alpha }_1\right\}),$
   type $B_2:$ $(t_d,\left\{{\alpha }_2\right\}),$
   type $G_2:$ $(t_i,\left\{{\alpha }_2\right\}),$ $(t_f,\left\{{\alpha }_1\right\})\text{.}$

(Note that to satisfy (b) $Z\left(t\right)$ must be nonempty.) From the tables in [Ram2002] we see that none of these local regions supports a calibrated representation.

Remark. The paper [Ram2002] does not treat roots of unity. However, it is interesting to note that, provided $q^2\ne \pm 1,$ the methods of [Ram2002] go through without change to classify all representations of rank-two affine Hecke algebras even when $q^2$ is a root of unity. This classification can be used (as in the previous remark) to show that (a) and (b) above still characterize calibratable weights when $q^2$ is a root of unity such that $q^2\ne \pm 1\text{.}$ The key point is that Lemma 1.19 of [Ram2002] still holds. If $q^2=-1$ then Lemma 1.19 of [Ram2002] breaks down at the next to last line of the proof in the statement “... forces $\varphi \left(wt\left(T_j\right)\right),$ to have Jordan blocks of size 1 ....” When $q^2=-1$ it is possible that $\varphi \left(wt\left(T_j\right)\right)$ has a Jordan block of size 2. If $q^2=1$ then one can change the definition of the $\tau \text{-operators}$ and use similar methods to produce a complete analysis of simple $\stackrel{\sim }{H}\text{-modules,}$ but we shall not do this here, choosing instead to exclude the case $q^2=1$ for simplicity of exposition.

The following lemma provides fundamental results about the structure of irreducible calibrated $\stackrel{\sim }{H}\text{-modules.}$ We omit the proof since it is accomplished in exactly the same way as in [KRa2002, Lemmas 4.1 and 4.2].

Lemma 3.1 Let $M$ be an irreducible calibrated module. Then, for all $t\in T$ such that $M_t\ne 0,$

1. If $t\in T$ such that $M_t\ne 0$ then $t\left(X^{{\alpha }_i}\right)\ne 1$ for all $1\le i\le n\text{.}$
2. If $t\in T$ such that $M_t\ne 0$ then $\text{dim}\left(M_t\right)=1\text{.}$
3. If $t\in T$ such that $M_t$ and $M_{s_{i}t}$ are both nonzero then the map ${\tau }_i:M_t\to M_{s_{i}t}$ is a bijection.

This lemma together with the classification of irreducible modules for rank-two affine Hecke algebras gives the following fundamental structural result for irreducible calibrated $\stackrel{\sim }{H}\text{-modules.}$ The proof is essentially the same as the proof of Proposition 4.3 in [KRa2002]. We repeat the proof here for continuity.

Theorem 3.2 If $M$ is an irreducible calibrated $\stackrel{\sim }{H}\text{-module}$ with central character $t\in T$ then there is a unique skew local region $(t,J)$ such that

$$\text{dim}\left(M_{wt}\right)=\{\begin{array}{cc}1& \text{for all}w\in F^{(t,J)},\\ 0& \text{otherwise.}\end{array}$$

		Proof.
	By Lemma 3.1(b) all nonzero generalized weight spaces of $M$ have dimension 1 and by Lemma 3.1(c) all $\tau \text{-operators}$ between these weight spaces are bijections. This already guarantees that there is a unique local region $(t,J)$ which satisfies the condition. It only remains to show that this local region is skew. Let ${\stackrel{\sim }{H}}_{ij}$ be the subalgebra generated by $T_i,$ $T_j$ and $ℂ\left[X\right]\text{.}$ Since $M$ is calibrated as an $\stackrel{\sim }{H}\text{-module}$ it is calibrated as a ${\stackrel{\sim }{H}}_{ij}\text{-module}$ and so all factors of a composition series of $M$ as an ${\stackrel{\sim }{H}}_{ij}\text{-module}$ are calibrated. Thus the weights of $M$ are calibratable. So $(t,J)$ is a skew local region. $\square$

The following proposition shows that the weight space structure of calibrated representations, as determined in Theorem 3.2, essentially forces the $\stackrel{\sim }{H}\text{-action}$ on a weight basis. The proof is quite similar to the proof of Proposition 4.4 in [KRa2002]. However, we include the details since there is a technicality here; to make the conclusion in (3.4) we use the fact that the group $X$ corresponds to the weight lattice $L=P\text{.}$

Proposition 3.3. Let $M$ be a calibrated $\stackrel{\sim }{H}\text{-module}$ and assume that for all $t\in T$ such that $M_t\ne 0,$

$$\text{(A1)}t\left(X_i^{\alpha }\right)\ne 1\text{for all}1\le i\le n,\text{and}\text{(A2)}\text{dim}\left(M_t\right)=1\text{.}$$

For each $b\in T$ such that $M_b\ne 0$ let $v_b$ be a nonzero vector in $M_b\text{.}$ The vectors $\left\{v_b\right\}$ form a basis of $M\text{.}$ Let ${\left(T_i\right)}_{cb}\in ℂ$ and $b\left(X^{\lambda }\right)\in ℂ$ given by

$$T_i{v}_b=\sum _c{\left(T_i\right)}_{cb}{v}_c\text{and}{X}^{\lambda }{v}_b=b\left(X^{\lambda }\right)v_b\text{.}$$

Then

1. ${\left(T_i\right)}_{bb}=(q-q^{-1})/(1-b\left(X^{-{\alpha }_i}\right)),$ for all $v_b$ in the basis,
2. if ${\left(T_i\right)}_{cb}\ne 0$ then $c=s_{i}b,$
3. ${\left(T_i\right)}_{b,s_{i}b}{\left(T_i\right)}_{s_{i}b,b}=(q^{-1}+{\left(T_i\right)}_{bb})(q^{-1}+{\left(T_i\right)}_{s_{i}b,s_{i}b})\text{.}$

		Proof.
	The definition equation for $\stackrel{\sim }{H},$ $$X^{\lambda }{T}_i-T_i{X}^{s_i\lambda }=(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},$$ forces $$\sum _c(c\left(X^{\lambda }\right){\left(T_i\right)}_{cb}-{\left(T_i\right)}_{cb}b\left(X^{s_i\lambda }\right))v_c=(q-q^{-1})\frac{b\left(X^{\lambda }\right)-b\left(X^{s_i\lambda }\right)}{1-b\left(X^{-{\alpha }_i}\right)}{v}_b\text{.}$$ Comparing coefficients gives $$\begin{array}{c}c\left(X^{\lambda }\right){\left(T_i\right)}_{cb}-{\left(T_i\right)}_{cb}b\left(X^{s_i\lambda }\right)=0,\text{if}b\ne c,\text{and}\\ b\left(X^{\lambda }\right){\left(T_i\right)}_{bb}-{\left(T_i\right)}_{bb}b\left(X^{s_i\lambda }\right)=(q-q^{-1})\frac{b\left(X^{\lambda }\right)-b\left(X^{s_i\lambda }\right)}{1-b\left(X^{-{\alpha }_i}\right)}\text{.}\end{array}$$ These relations give: $$\begin{array}{c}\text{if}{\left(T_i\right)}_{cb}\ne 0\text{then}b\left(X^{s_i\lambda }\right)=c\left(X^{\lambda }\right)\text{for all}{X}^{\lambda }\in X,\text{and}\\ {\left(T_i\right)}_{bb}=\frac{q-q^{-1}}{1-b\left(X^{-{\alpha }_i}\right)}\text{if}b\left(X^{-{\alpha }_i}\right)\ne 1\text{and}b\left(X^{\lambda }\right)\ne b\left(X^{s_i\lambda }\right)\text{for some}{X}^{\lambda }\in X\text{.}\end{array}$$ By assumption (A1), $b\left(X^{{\alpha }_i}\right)\ne 1$ for all $i\text{.}$ For each fundamental weight ${\omega }_i,$ $X^{{\omega }_i}\in X$ and $b\left(X^{s_i{\omega }_i}\right)=b\left(X^{{\omega }_i-{\alpha }_i}\right)\ne b\left(X^{{\omega }_i}\right)$ since $b\left(X^{{\alpha }_i}\right)\ne 1\text{.}$ Thus we conclude that $\begin{array}{cc}{T}_i{v}_b={\left(T_i\right)}_{bb}{v}_b+{\left(T_i\right)}_{s_{i}b,b}{v}_{s_{i}b}\text{with}{\left(T_i\right)}_{bb}=\frac{q-q^{-1}}{1-b\left(X^{-{\alpha }_i}\right)}\text{.}& \text{(3.4)}\end{array}$ This completes the proof of (a) and (b). By the definition of $\stackrel{\sim }{H},$ the vector $$T_i^2{v}_b=({\left(T_i\right)}_{bb}^2+{\left(T_i\right)}_{b,s_{i}b}{\left(T_i\right)}_{s_{i}b,b})v_b+({\left(T_i\right)}_{bb}+{\left(T_i\right)}_{s_{i}b,s_{i}b}){\left(T_i\right)}_{s_{i}b,b}{v}_{s_{i}b}$$ must equal $$((q-q^{-1})T_i+1)v_b=((q-q^{-1}){\left(T_i\right)}_{bb}+1)v_b+(q-q^{-1}){\left(T_i\right)}_{s_{i}b,b}{v}_{s_{i}b}\text{.}$$ Using the formula for ${\left(T_i\right)}_{bb}$ and ${\left(T_i\right)}_{s_{i}b,s_{i}b},$ we find ${\left(T_i\right)}_{bb}+{\left(T_i\right)}_{s_{i}b,s_{i}b}=(q-q^{-1})\text{.}$ So, by comparing coefficients of $v_b,$ we obtain the equation $$\begin{array}{ccc}{\left(T_i\right)}_{b,s_{i}b}{\left(T_i\right)}_{s_{i}b,b}& =& (q-{\left(T_i\right)}_{bb})({\left(T_i\right)}_{bb}+q^{-1})\\ & =& (q^{-1}+{\left(T_i\right)}_{bb})(q^{-1}+{\left(T_i\right)}_{s_{i}b,s_{i}b})\text{.}\end{array}$$ $\square$

Theorem 3.5. Let $(t,J)$ be a skew local region and let ${ℱ}^{(t,J)}$ index the chambers in the local region $(t,J)\text{.}$ Define

$${\stackrel{\sim }{H}}^{(t,J)}=ℂ\text{-span}\{v_w\mid w\in {ℱ}^{(t,J)}\},$$

so that the symbols $v_w$ are a labeled basis of the vector space ${\stackrel{\sim }{H}}^{(t,J)}$ Then the following formulas make ${\stackrel{\sim }{H}}^{(t,J)}$ into an irreducible $\stackrel{\sim }{H}\text{-module:}$ For each $w\in {ℱ}^{(t,J)},$

$$\begin{array}{cc}\begin{array}{ccc}{X}^{\lambda }{v}_w& =& \left(wt\right)\left(X^{\lambda }\right)v_w,\end{array}& \text{for}{X}^{\lambda }\in X,\text{and}\\ \begin{array}{ccc}{T}_i{v}_w& =& {\left(T_i\right)}_{ww}{v}_w+(q^{-1}+{\left(T_i\right)}_{ww})v_{s_{i}w},\end{array}& \text{for}1\le i\le n,\end{array}$$

where ${\left(T_i\right)}_{ww}=(q-q^{-1})/(1-\left(wt\right)\left(X^{-{\alpha }_i}\right)),$ and we set $v_{s_{i}w}=0$ if $s_{i}w\notin {ℱ}^{(t,J)}$.

		Proof.
	Since $(t,J)$ is a skew local region $\left(wt\right)\left(X^{-{\alpha }_i}\right)\ne 1$ for all $w\in {ℱ}^{(t,J)}$ and all simple roots ${\alpha }_i\text{.}$ This implies that the coefficient ${\left(T_i\right)}_{ww}$ is well defined for all $i$ and $w\in {ℱ}^{(t,J)}\text{.}$ By construction, the nonzero weight spaces of ${\stackrel{\sim }{H}}^{(t,J)}$ are ${\left({\stackrel{\sim }{H}}^{(t,J)}\right)}_{wt}^{\text{gen}}={\left({\stackrel{\sim }{H}}^{(t,J)}\right)}_{wt}$ where $w\in {ℱ}^{(t,J)}\text{.}$ Since $\text{dim}\left({\stackrel{\sim }{H}}^{(t,J)}\right)=1$ for $u\in {ℱ}^{(t,J)}$ any proper submodule $N$ of ${\stackrel{\sim }{H}}^{(t,J)}$ must have $N_{wt}\ne 0$ and $N_{w^{\prime }t}=0$ for some $w\ne w^{\prime }$ with $w,w^{\prime }\in {ℱ}^{(t,J)}\text{.}$ This is a contradiction to Corollary 2.19. So ${\stackrel{\sim }{H}}^{(t,J)}$ is irreducible if it is an $\stackrel{\sim }{H}\text{-module.}$ It remains to show that the defining relations for $\stackrel{\sim }{H}$ are satisfied. This is accomplished as in the proof of [KRa2002, Theorem 4.5]. The only relation which is tricky to check is the braid relation. This can be verified as in [KRa2002] or it can be checked by case by case arguments (as in [Ram1998]). $\square$

We summarize the results of this section with the following corollary of Theorem 3.2 and the construction in Theorem 3.5.

Theorem 3.6. Let $M$ be an irreducible calibrated $\stackrel{\sim }{H}\text{-module.}$ Let $t\in T$ be (a fixed choice of) the central character of $M$ and let $J=R\left(w\right)\cap P\left(t\right)$ for any $w\in W$ such that $M_{wt}\ne 0\text{.}$ Then $(t,J)$ is a skew local region and $M\cong {\stackrel{\sim }{H}}^{(t,J)}$ where ${\stackrel{\sim }{H}}^{(t,J)}$ is the module defined in Theorem 3.5.

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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