Two boundary Hecke Algebras and the combinatorics of type $C$

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

Last updated: 27 January 2015

Classification of irreducible representations of $H_2$

In this section we do a complete classification of the irreducible representations of the algebra $H_2^{\text{ext}}\text{.}$ An important reason for doing the complete classification of the irreducible representations of $H_2^{\text{ext}}$ is to provide a sound basis for the definition of a skew local region (see the remarks immediately after the definition of skew local region in (3.10)). The classification and construction of calibrated representations of $H_k^{\text{ext}}$ in terms of skew local regions in Theorem 3.3 is important for determining the irreducible representations of $H_k^{\text{ext}}$ which arise in the Schur-Weyl duality framework (see Theorem 5.5). We will do the clasification of irreducible $H_2^{\text{ext}}$ representations under genericity assumptions on the parameters: $t^{\frac{1}{2}}$ is not a root of unity and $$\begin{array}{cc}{\displaystyle t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}},-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}}\notin \{1,-1,t^{\pm \frac{1}{2}},-t^{\pm \frac{1}{2}},t^{\pm 1},-t^{\pm 1}\}\text{and}{t}_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}}\ne {(-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}})}^{\pm 1}\text{.}}& \text{(4.1)}\end{array}$$ More specifically, this condition is used for the (rank 2) computation in equation (4.4). Similar methods apply to the nongeneric cases but the final classification needs to be stated differently and we will not deal with the nongeneric cases here. The nongeneric case $t_0^{\frac{1}{2}}=t_k^{\frac{1}{2}}=t^{\frac{1}{2}}$ is done in [Ram2002, Ram2003] and [Ree1997]; the case where $t_0^{\frac{1}{2}}=t_k^{\frac{1}{2}}\ne t^{\frac{1}{2}}$ appears in [Eno2006] (see also [KRa2002]).

The algebra $H_2$ is generated by $W_1^{\pm 1},W_2^{\pm 1},T_0,$ and $T_1\text{,}$ and the Weyl group ${𝒲}_0$ is generated by $s_0$ and $s_1$ with relations $s_i^2=1$ and $s_0{s}_1{s}_0{s}_1=s_1{s}_0{s}_1{s}_0\text{.}$ By (2.29), $$H_2^{\text{ext}}=ℂ\left[W_0^{\pm }\right]\otimes H_2\text{as algebras,}$$ and therefore it is sufficient to do the classification of irreducible representations of $H_2$ (all irreducible representations of $ℂ\left[W_0^{\pm 1}\right]$ are one dimensional and determined by the image of $W_0\text{,}$ and all irreducible representations of $H_2^{\text{ext}}$ are the tensor product of an irreducible representation of $ℂ\left[W_0^{\pm 1}\right]$ and an irreducible representation of $H_2\text{).}$

The group ${𝒲}_0$ acts on ${\left({ℂ}^{\times }\right)}^2$ by $$\begin{array}{cc}{\displaystyle s_0({\gamma }_1,{\gamma }_2)=({\gamma }_1^{-1},{\gamma }_2)\text{and}{s}_1({\gamma }_1,{\gamma }_2)=({\gamma }_2,{\gamma }_1)\text{.}}& \text{(4.2)}\end{array}$$ By (2.35), the intertwiners are $${\tau }_0=T_0-\frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})W_1^{-1}}{1-W_1^{-2}}\text{and}{\tau }_1=T_1-\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-W_1{W}_2^{-1}}\text{.}$$

Classification of central characters

Following [Ram2002, §5], the classification of irreducible $H_k^{\text{ext}}\text{-modules}$ begins with a classification of possible pairs $(Z\left(\text{c}\right),P\left(\text{c}\right))=(Z\left(\gamma \right),P\left(\gamma \right))$ (where $\gamma$ and $\text{c}$ are related as in (2.32)). It is straightforward (though slightly tedious) to enumerate all the possibilities by taking note of the following:

(0)	Since $(Z\left(w\gamma \right),P\left(w\gamma \right))=(wZ\left(\gamma \right),wP\left(\gamma \right)),$ it is sufficient to do the analysis for a single representative $\gamma$ of each ${𝒲}_0\text{-orbit}$ on ${\left({ℂ}^{\times }\right)}^k\text{.}$
(1)	The ${𝒲}_0\text{-orbits}$ of roots are $\{\pm {\epsilon }_1,\pm {\epsilon }_2\}$ and $\{\pm ({\epsilon }_1\pm {\epsilon }_2)\},$ and our preferred representative of the ${𝒲}_0\text{-orbit}$ will have ${\epsilon }_1$ or ${\epsilon }_1-{\epsilon }_2$ in $Z\left(\gamma \right)$ if $Z\left(\gamma \right)\ne \varnothing \text{.}$
(2)	If $Z\left(\gamma \right)=\varnothing$ and $P\left(\gamma \right)\ne \varnothing$ then our preferred representative of the ${𝒲}_0\text{-orbit}$ will have ${\epsilon }_1$ or ${\epsilon }_1-{\epsilon }_2$ in $Z\left(\gamma \right)\text{.}$

With these preferences, the classification of $(Z\left(\gamma \right),P\left(\gamma \right))$ is accomplished by noting that

(a)	if $\gamma \in \{(1,1),(-1,-1)\}$ then $(Z\left(\gamma \right),P\left(\gamma \right))=(\{{\epsilon }_1,{\epsilon }_2,{\epsilon }_1\pm {\epsilon }_2\},\varnothing )\text{;}$
(b)	if $\gamma \in \{(1,-1),(-1,1)\}$ then $(Z\left(\gamma \right),P\left(\gamma \right))=(\{{\epsilon }_1,{\epsilon }_2\},\varnothing )\text{;}$
(c)	${\epsilon }_1-{\epsilon }_2\in Z\left(\gamma \right)$ if and only if $\gamma =({\gamma }_1,{\gamma }_1)\text{;}$
(d)	${\epsilon }_1+{\epsilon }_2\in Z\left(\gamma \right)$ if and only if $\gamma =({\gamma }_1,{\gamma }_1^{-1})\text{;}$
(e)	${\epsilon }_1\in Z\left(\gamma \right)$ if and only if $\gamma =(1,{\gamma }_2)$ or $\gamma =(-1,{\gamma }_2)\text{;}$
(f)	${\epsilon }_2\in Z\left(\gamma \right)$ if and only if $\gamma =({\gamma }_1,1)$ or $\gamma =({\gamma }_1,-1)\text{;}$
(g)	${\epsilon }_1\in P\left(\gamma \right)$ if and only if $\gamma =({\gamma }_1,{\gamma }_2)$ with ${\gamma }_1\in \{t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}},-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}},-t_0^{\frac{1}{2}}{t}_k^{-\frac{1}{2}},t_0^{-\frac{1}{2}}{t}_k^{-\frac{1}{2}}\}\text{;}$
(h)	${\epsilon }_1-{\epsilon }_2\in P\left(\gamma \right)$ if and only if $\gamma =({\gamma }_1,{\gamma }_2)$ with ${\gamma }_2={\gamma }_1{t}^{\pm 1}\text{;}$
(i)	${\epsilon }_1+{\epsilon }_2\in P\left(\gamma \right)$ if and only if $\gamma =({\gamma }_1,{\gamma }_2)$ with ${\gamma }_1{\gamma }_2=t^{\pm 1}\text{.}$

We shall freely use the conversion between $\gamma =({\gamma }_1,{\gamma }_2)$ and $\text{c}=(c_1,c_2)$ given by (2.32), $${\gamma }_1=-t^{c_1},{\gamma }_2=-t^{c_2},\text{and write}\hspace{0.17em}(Z\left(\text{c}\right),P\left(\text{c}\right))=(Z\left(\gamma \right),P\left(\gamma \right))\text{.}$$ Representatives of the 12 possible $(Z\left(\text{c}\right),P\left(\text{c}\right))$ with $\mathbb{Z}\left(\text{c}\right)=\varnothing$ are displayed in Figure 1. Representatives of the 9 possible $(Z\left(\text{c}\right),P\left(\text{c}\right))$ with $\mathbb{Z}\left(\text{c}\right)\ne \varnothing$ are displayed in Figure 2. It works out that, in each case, the pair $(Z\left(\text{c}\right),P\left(\text{c}\right))$ is attained by an element $\text{c}$ that has real coordinates (the one complex character in the equal parameter case that behaves differently from the real characters, namely the point $t_b$ in [Ram2002, Figure 5.1], does not appear in the generic unequal parameter case assumed in (4.1)).

With notation as at the beginning of Section 3, in Figures 1 and 2, the fundamental region $C$ is the shaded area, the solid lines are the hyperplanes ${𝔥}^{\alpha }$ for $\alpha \in R^{+},$ and the dotted hyperplanes are labeled by the equation which defines them. If $\text{c}=(c_1,c_2)\in C$ so that $0\le c_1\le c_2$ then $$Z\left(\text{c}\right)=\left\{\text{solid hyperplanes through}\hspace{0.17em}\text{c}\right\}\text{and}P\left(\text{c}\right)=\left\{\text{dotted hyperplanes through}\hspace{0.17em}\text{c}\right\}\text{.}$$ The bijection $$\begin{array}{cc}{\displaystyle \begin{array}{ccc}{𝒲}_0& ⟷& \left\{\text{chambers}\right\}\\ w& ⟼& w^{-1}C\end{array}\text{identifies each}\hspace{0.17em}{ℱ}^{(\text{c},J)}\hspace{0.17em}\text{with a set of chambers,}}& \text{(4.3)}\end{array}$$ a local region in ${𝔥}_{ℝ}^{*}\text{.}$ As illustrated by the example at the bottom right of Figures 1 and 2, ${ℱ}^{(\text{c},J)}$ is identified with the set of chambers that are on the negative side of the hyperplanes in $J$ and on the positive side of the hyperplanes in $P\left(\text{c}\right)-J\text{.}$ For each $(\text{c},J)$ the corresponding configuration of boxes $\kappa$ is displayed in the local region of chambers corresponding to the elements of ${ℱ}^{(\text{c},J)}$ by (4.3). In Figure 1, only the boxes on positive diagonals are shown since they determine the entire doubled configuration when $Z\left(\text{c}\right)=\varnothing \text{.}$ The diagram at the bottom right of each figure gives an example of the correspondence between chambers corresponding to ${ℱ}^{(\text{c},J)}\text{,}$ the elements of ${ℱ}^{(\text{c},J)}$ and the standard fillings of the corresponding configuration of boxes $\kappa \text{:}$ the point $\text{c}=(r_1,r_1+1)$ in the bottom right of Figure 1, and the point $\text{c}=(0,1)$ in the bottom right of Figure 2.

In Figure 2, the small graphs nearby each marked $\text{c}=(c_1,c_2)$ indicate the structure (generalized weight spaces and intertwiner maps) of the irreducible modules $M$ of central character $\text{c}\text{.}$ This structure is determined below in Section 4.2. There is a vertex in the chamber $w^{-1}C$ for each element of a basis of $M_{w\text{c}}^{\text{gen}}$ and there is an edge if the matrix of ${\tau }_i$ (or $T_i$ if ${\tau }_i$ is not defined on $M_{w\text{c}}^{\text{gen}}\text{)}$ is nonzero in the entry corresponding to the two vertices that are connected. $$\begin{array}{c}\text{Figure 1: Regular central characters in rank 2. See the description in Section 4.1}\\ \begin{array}{c} c1=0 c1=c2 c2=0 c2=c1+1 c2=-c1+1 c2=r1 c2=r2 c1=r1 c1=r2 1 1 2 -1 -2 s1 2 1 -2 -1 s0s1 2 -1 -2 1 s1s0s1 1 -2 -1 2 s0 -1 2 1 -2 s1s0 -2 1 2 -1 s0s1s0 -2 -1 2 1 s1s0s1s0 -1 -2 1 2 J=∅ J={ε1-ε2} J={ε2} J={ε2,ε1-ε2} \end{array}\end{array}$$ $$\begin{array}{c}\text{Figure 2: Non-regular points}\\ \begin{array}{c} c1=0 c1=c2 c2=0 c2=c1+1 c2=-c1+1 c2=r1 c2=r2 c1=r1 c1=r2 1 s1 s1s0 s1s0s1 -2 -1 2 1 -1 -2 2 1 1 -2 2 -1 2 -1 1 -2 J=∅ J={ε1-ε2} J={ε1±ε2} \end{array}\end{array}$$

Construction of the irreducible $H_2\text{-modules}$

The group ${𝒲}_0$ acts on ${\left({ℂ}^{\times }\right)}^2$ as in (4.2) and the central characters are the ${𝒲}_0\text{-orbits}$ on ${\left({ℂ}^{\times }\right)}^2\text{.}$ The regular central characters are the ${𝒲}_0\text{-orbits}$ of $\gamma =({\gamma }_1,{\gamma }_2)\in {\left({ℂ}^{\times }\right)}^2$ that have $Z\left(\gamma \right)=\varnothing ,$ i.e. where the intertwining operators in (2.38) are defined. Let $ℂ\left[W\right]=ℂ[W_1^{\pm 1},W_2^{\pm 1}]\subseteq H_2\text{.}$ By Kato's criterion (see [Ram2003, Proposition 2.11b]), for central characters $\gamma =({\gamma }_1,{\gamma }_2)$ with $P\left(\gamma \right)=\varnothing$ there is a single irreducible module of dimension eight given by $$L_{({\gamma }_1,{\gamma }_2)}={\text{Ind}}_{ℂ\left[W\right]}^H\left({ℂ}_{{\gamma }_1,{\gamma }_2}\right),\text{where}\hspace{0.17em}{ℂ}_{{\gamma }_1,{\gamma }_2}=ℂv\hspace{0.17em}\text{with}\hspace{0.17em}{W}_{1}v={\gamma }_{1}v\hspace{0.17em}\text{and}\hspace{0.17em}{W}_{2}v={\gamma }_{2}v\text{.}$$ All irreducible modules with $\gamma =({\gamma }_1,{\gamma }_2)$ with $Z\left(\gamma \right)=\varnothing$ are calibrated and can be constructed as in Theorem 3.3.

Representatives of the ${𝒲}_0\text{-orbits}$ of $\gamma =({\gamma }_1,{\gamma }_2)\in {\left({ℂ}^{\times }\right)}^2$ which have $Z\left(\gamma \right)\ne \varnothing$ and $P\left(\gamma \right)\ne \varnothing$ are as follows: $$\begin{array}{cc}{\displaystyle \begin{array}{ccc}\gamma =({\gamma }_1,{\gamma }_2)& Z\left(\gamma \right)& P\left(\gamma \right)\\ (t^{\frac{1}{2}},t^{\frac{1}{2}}),(-t^{\frac{1}{2}},-t^{\frac{1}{2}})& \{{\epsilon }_1-{\epsilon }_2\}& \{{\epsilon }_1+{\epsilon }_2\}\\ (t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}},t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}}),(-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}},-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}})& \{{\epsilon }_1-{\epsilon }_2\}& \{{\epsilon }_1,{\epsilon }_2\}\\ (1,t),(-1,-t)& \left\{{\epsilon }_1\right\}& \{{\epsilon }_1-{\epsilon }_2,{\epsilon }_1+{\epsilon }_2\}\\ (\pm 1,t_0^{\frac{1}{2}},t_k^{\frac{1}{2}}),(\pm 1,-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}}),& \left\{{\epsilon }_1\right\}& \left\{{\epsilon }_2\right\}\end{array}}& \text{(4.4)}\end{array}$$ This classification is valid under the genericity assumption on the parameters (4.1), which guarantees that none of these representatives are in the ${𝒲}_0\text{-orbit}$ of another.

The following analysis of modules of central character $\gamma =({\gamma }_1,{\gamma }_2)$ in (4.4) shows that no irreducible calibrated $H_2\text{-modules}$ appear at these central characters. As in (3.5), the values $r_1$ and $r_2$ are defined by $$-t^{r_1}=-t_k^{\frac{1}{2}}{t}_0^{-\frac{1}{2}}\text{and}-t^{r_2}=t_k^{\frac{1}{2}}{t}_0^{\frac{1}{2}}\text{.}$$ Case $({\gamma }_1,{\gamma }_2)=(-1,-t^{r_i})$ for $i=1$ or $2\text{:}$ Let $H_{\left\{0\right\}}$ be the subalgebra of $H_2$ generated by $T_0,W_1^{\pm 1},W_2^{\pm 1}\text{.}$ For each of $i=1$ and $i=2,$ there are two irreducible modules of central character $\text{c}=(0,r_i)\text{:}$ $$L_{(0,r_i)}^{+}={\text{Ind}}_{H_{\left\{0\right\}}}^{H_2}\left({ℂ}_{(r_i,0)}\right),\text{where}\hspace{0.17em}{ℂ}_{(r_i,0)}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -t^{r_i}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -v,}\\ {\displaystyle T_{0}v}& =& {\displaystyle t_0^{\frac{1}{2}}v,}\end{array}$$ and $$L_{(0,r_i)}^{-}={\text{Ind}}_{H_{\left\{0\right\}}}^{H_2}\left({ℂ}_{(-r_i,0)}\right),\text{where}\hspace{0.17em}{ℂ}_{(-r_i,0)}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle t^{-r_i}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -v,}\\ {\displaystyle T_{0}v}& =& {\displaystyle -t_0^{-\frac{1}{2}}v\text{.}}\end{array}$$ With $M=L_{(0,r_i)}^{+},$ the generalized weight space decomposition is $$\begin{array}{cc}{\displaystyle M=M_{(r_i,0)}^{\text{gen}}\oplus M_{(0,r_i)}^{\text{gen}},\text{with}\text{dim}\left(M_{(r_i,0)}^{\text{gen}}\right)=\text{dim}\left(M_{(0,r_i)}^{\text{gen}}\right)=2\text{.}}& \text{(4.5)}\end{array}$$ The element $W_1{W}_2^{-1}$ acts on $M_{(r_i,0)}^{\text{gen}}$ with eigenvalues $t^{r_i}\text{.}$ Since the parameters are generic (see (4.1)), $t^{r_i}\ne t^{\pm 1}$ and thus, by (2.43), ${\tau }_1^2$ has no kernel. Thus the intertwiner ${\tau }_1:M_{(r_i,0)}^{\text{gen}}\to M_{(0,r_i)}^{\text{gen}}$ is invertible and $M=L_{(0,r_i)}^{+}$ is irreducible. Replacing $r_i$ with $-r_i$ in (4.5) yields the decomposition of $M=L_{(0,r_i)}^{-}$ analogously.

Case $({\gamma }_1,{\gamma }_2)=(-t^{\frac{1}{2}},-t^{\frac{1}{2}})\text{:}$ Let $H_{\left\{1\right\}}$ be the subalgebra of $H_2$ generated by $T_1,W_1^{\pm 1},W_2^{\pm 1}\text{.}$ There are two irreducible modules of central character $\text{c}=(\frac{1}{2},\frac{1}{2})\text{:}$ $$L_{(\frac{1}{2},\frac{1}{2})}^{+}={\text{Ind}}_{H_{\left\{1\right\}}}^{H_2}\left(C_{(-\frac{1}{2},\frac{1}{2})}\right),\text{where}\hspace{0.17em}{ℂ}_{(-\frac{1}{2},\frac{1}{2})}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -t^{-\frac{1}{2}}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -t^{\frac{1}{2}}v,}\\ {\displaystyle T_{1}v}& =& {\displaystyle t^{\frac{1}{2}}v,}\end{array}$$ and $$L_{(\frac{1}{2},\frac{1}{2})}^{-}={\text{Ind}}_{H_{\left\{1\right\}}}^{H_2}\left(C_{(\frac{1}{2},-\frac{1}{2})}\right),\text{where}\hspace{0.17em}{ℂ}_{(\frac{1}{2},-\frac{1}{2})}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -t^{\frac{1}{2}}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -t^{-\frac{1}{2}}v,}\\ {\displaystyle T_{1}v}& =& {\displaystyle -t^{-\frac{1}{2}}v\text{.}}\end{array}$$ With $M=L_{(\frac{1}{2},\frac{1}{2})}^{+},$ the generalized weight space decomposition is $$\begin{array}{cc}{\displaystyle M=M_{(\frac{1}{2},\frac{1}{2})}^{\text{gen}}\oplus M_{(-\frac{1}{2},\frac{1}{2})}^{\text{gen}},with\text{dim}\left(M_{(\frac{1}{2},\frac{1}{2})}^{\text{gen}}\right)=\text{dim}\left(M_{(-\frac{1}{2},\frac{1}{2})}^{\text{gen}}\right)=2\text{.}}& \text{(4.6)}\end{array}$$ The element $W_1^{-1}$ acts on $M_{(\frac{1}{2},\frac{1}{2})}^{\text{gen}}$ with eigenvalues $-t^{\frac{1}{2}}\text{.}$ Since the parameters are generic (see (4.1)), $-t^{\frac{1}{2}}\notin \{-t^{\pm r_1},-t^{\pm r_2}\}$ and thus, by (2.42), ${\tau }_0^2$ has no kernel. Thus the intertwiner ${\tau }_0:M_{(\frac{1}{2},-\frac{1}{2})}^{\text{gen}}\to M_{(-\frac{1}{2},-\frac{1}{2})}^{\text{gen}}$ is invertible and $M=L_{(\frac{1}{2},\frac{1}{2})}^{+}$ is irreducible. Similarly, the structure of $M=L_{(\frac{1}{2},\frac{1}{2})}^{-}$ is given by swapping $\frac{1}{2}$ and $-\frac{1}{2}$ in (4.6).

Case $({\gamma }_1,{\gamma }_2)=(-t^{r_i},-t^{r_i})$ for $i=1$ or $2\text{:}$ Let $H_{\left\{0\right\}}$ be the subalgebra of $H_2$ generated by $T_0,W_1^{\pm 1},W_2^{\pm 1}\text{.}$ For each of $i=1$ and $i=2,$ there are two irreducible modules of central character $\text{c}=(r_i,r_i)\text{:}$ $$L_{(r_i,r_i)}^{+}={\text{Ind}}_{H_{\left\{0\right\}}}^{H_2}\left({ℂ}_{(r_i,-r_i)}\right),\text{where}\hspace{0.17em}{ℂ}_{(r_i,-r_i)}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -t^{r_i}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -t^{-r_i}v,}\\ {\displaystyle T_{0}v}& =& {\displaystyle t_0^{\frac{1}{2}}v,}\end{array}$$ and $$L_{(r_i,r_i)}^{+}={\text{Ind}}_{H_{\left\{0\right\}}}^{H_2}\left({ℂ}_{(-r_i,r_i)}\right),\text{where}\hspace{0.17em}{ℂ}_{(-r_i,r_i)}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -t^{-r_i}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -t^{r_i}v,}\\ {\displaystyle T_{0}v}& =& {\displaystyle t_0^{-\frac{1}{2}}v\text{.}}\end{array}$$ The irreducibility of $L_{(r_i,r_i)}^{+}$ and $L_{(r_i,r_i)}^{-}$ is not immediate. We will show that $M=L_{(r_i,r_i)}^{+}$ is irreducible; the irreducibility of $L_{(r_i,r_i)}^{-}$ is proved analogously.

The generalized weight space decomposition of $M=L_{(r_i,r_i)}^{+}$ is $$M=M_{(r_i,-r_i)}^{\text{gen}}\oplus M_{(-r_i,r_i)}^{\text{gen}}\oplus M_{(r_i,r_i)}^{\text{gen}}\text{with}\begin{array}{rcl}{\displaystyle \text{dim}\left(M_{(r_i,-r_i)}^{\text{gen}}\right)}& =& {\displaystyle \text{dim}\left(M_{(-r_i,r_i)}^{\text{gen}}\right)=1,}\\ {\displaystyle \text{dim}\left(M_{(r_i,r_i)}^{\text{gen}}\right)}& =& {\displaystyle 2\text{.}}\end{array}$$ The element $W_1{W}_2^{-1}$ acts on $M_{(r_i,-r_i)}^{\text{gen}}$ with eigenvalue $t^{r_i-(-r_i)}\text{.}$ Since the parameters are generic (see (4.1)), $t^{2r_i}\ne t^{\pm 1}$ and thus, by (2.43), ${\tau }_1^2$ has no kernel. Thus the intertwiner ${\tau }_1:M_{(r_i,-r_i)}^{\text{gen}}\to M_{(-r_i,r_i)}^{\text{gen}}$ is invertible. As a $H_{\left\{0\right\}}\text{-module,}$ $M_{(r_i,r_i)}^{\text{gen}}$ is irreducible (2-dimensional). So either $N=M_{(r_i,r_i)}^{\text{gen}}$ is an $H_2\text{-submodule}$ or $M$ is irreducible.

For the purposes of deriving a contradiction, assume that $N=M_{(r_i,r_i)}^{\text{gen}}$ is an $H_2\text{-submodule}$ of $M\text{.}$ The space $N$ has a basis $$\{n_{\gamma },T_1{n}_{\gamma }\}\text{with}{W}_1{n}_{\gamma }=-t^{r_i}{n}_{\gamma },\text{and}{W}_2{n}_{\gamma }=-t^{r_i}{n}_{\gamma }\text{.}$$ By (2.24), $W_1^{-1}{T}_1{n}_{\gamma }=T_1{W}_2^{-1}{n}_{\gamma }+(t^{\frac{1}{2}}-t^{-\frac{1}{2}})W_1^{-1}{n}_{\gamma }=T_1(-t^{-r_i})n_{\gamma }+(t^{\frac{1}{2}}-t^{-\frac{1}{2}})(-t^{-r_i})n_{\gamma }$ and the action of $W_1^{-1}$ and $W_1^{-2}$ on the basis $\{n_{\gamma },T_1{n}_{\gamma }\}$ are given by the matrices $$\rho \left(W_1^{-1}\right)=(-t^{-r_i})\left(\begin{array}{cc}1& (t^{\frac{1}{2}}-t^{-\frac{1}{2}})\\ 0& 1\end{array}\right)\text{and}\rho \left(W_1^{-2}\right)=\rho {\left(W_1^{-1}\right)}^2=t^{-2r_i}\left(\begin{array}{cc}1& 2(t^{\frac{1}{2}}-t^{-\frac{1}{2}})\\ 0& 1\end{array}\right)\text{.}$$ Thus $$\begin{array}{rcl}{\displaystyle \rho (1-W_1^{-2})}& =& {\displaystyle (1-t^{-2r_i})\left(\begin{array}{cc}1& \frac{-2(t^{\frac{1}{2}}-t^{-\frac{1}{2}})t^{-2r_i}}{1-t^{-2r_i}}\\ 0& 1\end{array}\right)\text{and}}\\ {\displaystyle \rho {(1-W_1^{-2})}^{-1}}& =& {\displaystyle \frac{1}{(1-t^{-2r_i})}\left(\begin{array}{cc}1& \frac{2(t^{\frac{1}{2}}-t^{-\frac{1}{2}})t^{-2r_i}}{1-t^{-2r_i}}\\ 0& 1\end{array}\right)\text{.}}\end{array}$$ Since $N$ is a submodule of $M\text{,}$ $0={\tau }_0=T_0-{\displaystyle \frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})W_1^{-1}}{1-W_1^{-2}}}$ (see (2.35) for the formula for ${\tau }_0\text{),}$ and so $$\begin{array}{rcl}{\displaystyle \rho \left(T_0\right)}& =& {\displaystyle ((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})W_1^{-1}){(1-W_1^{-2})}^{-1}}\\ & =& {\displaystyle \frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-t^{-r_i})}{1-t^{-2r_i}}\left(\begin{array}{cc}1& \frac{(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(t^{\frac{1}{2}}-t^{-\frac{1}{2}})(-t^{-r_i})}{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-t^{-r_i})}\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& \frac{2(t^{\frac{1}{2}}-t^{-\frac{1}{2}})t^{-2r_i}}{1-t^{-2r_i}}\\ 0& 1\end{array}\right)}\\ & =& {\displaystyle \frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-t^{-r_i})}{1-t^{-2r_i}}\left(\begin{array}{cc}1& \frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{-t^{r_i}}(\frac{2\left(t^{-r_i}\right)}{1-t^{-2r_i}}+\frac{(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})}{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-t^{-r_i})})\\ 0& 1\end{array}\right)\text{.}}\end{array}$$ Recall, from (3.5), that $-t^{r_i}=\pm t_k^{\pm \frac{1}{2}}{t}_0^{\frac{1}{2}},$ so that $$\frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-t^{-r_i})}{1-t^{-2r_i}}=\frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(\pm t_k^{\mp \frac{1}{2}}{t}_0^{-\frac{1}{2}})}{1-t_k^{\mp 1}{t}_0^{-1}}=t_0^{\frac{1}{2}}\text{.}$$ The eigenvalues of $\rho \left(T_0\right)$ are $t_0^{\frac{1}{2}}$ and, since $(T_0-t_0^{\frac{1}{2}})(T_0+t_0^{-\frac{1}{2}})=0,$ the Jordan blocks of $\rho \left(T_0\right)$ are of size 1, forcing $$\begin{array}{rcl}{\displaystyle 0}& =& {\displaystyle \frac{2(-t^{-r_i})}{1-t^{-2r_i}}+\frac{(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})}{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-t^{-r_i})}=\frac{2(-t^{-r_i})}{1-t^{-2r_i}}+\frac{(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})}{(1-t^{-2r_i})t_0^{\frac{1}{2}}}}\\ & =& {\displaystyle \frac{2(-t^{-r_i})t_0^{\frac{1}{2}}+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})}{(1-t^{-2r_i})t_0^{\frac{1}{2}}}=\frac{2(\pm t_k^{\mp \frac{1}{2}}{t}_0^{-\frac{1}{2}})t_0^{\frac{1}{2}}+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})}{(1-t^{-2r_i})t_0^{\frac{1}{2}}}=\frac{\pm (t_k^{\frac{1}{2}}+t_k^{-\frac{1}{2}})}{(1-t^{-2r_i})t_0^{\frac{1}{2}}}\text{.}}\end{array}$$ This is a contradiction since, by the generic condition on parameters in (4.1), $1\ne (-t^{r_1})(-t^{r_2})=(-t_k^{\frac{1}{2}}{t}_0^{-\frac{1}{2}})\left(t_k^{\frac{1}{2}}{t}_0^{\frac{1}{2}}\right)=-{\left(t_k^{\frac{1}{2}}\right)}^2\text{.}$ Thus $N$ is not a submodule of $M\text{,}$ and so $M$ is irreducible.

Case $({\gamma }_1,{\gamma }_2)=(-1,-t)\text{:}$ Let $H_{\left\{1\right\}}$ be the subalgebra of $H_2$ generated by $T_1,W_1^{\pm 1},W_2^{\pm 1}\text{.}$ There are two irreducible modules of central character $\text{c}=(0,1)\text{:}$ $$L_{(0,1)}^{+}={\text{Ind}}_{H_{\left\{1\right\}}}^{H_2}\left({ℂ}_{(-1,0)}\right),\text{where}\hspace{0.17em}{ℂ}_{(-1,0)}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -t^{-1}v,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -v,}\\ {\displaystyle T_{1}v}& =& {\displaystyle t^{\frac{1}{2}}v,}\end{array}$$ and $$L_{(0,1)}^{-}={\text{Ind}}_{H_{\left\{1\right\}}}^{H_2}\left({ℂ}_{(1,0)}\right),\text{where}\hspace{0.17em}{ℂ}_{(-1,0)}=ℂv\hspace{0.17em}\text{with}\begin{array}{rcl}{\displaystyle W_{1}v}& =& {\displaystyle -tv,}\\ {\displaystyle W_{2}v}& =& {\displaystyle -v,}\\ {\displaystyle T_{1}v}& =& {\displaystyle -t^{-\frac{1}{2}}v\text{.}}\end{array}$$ The irreducibility of $L_{(0,1)}^{+}$ and $L_{(0,1)}^{-}$ is not immediate. We will show that $M=L_{(0,1)}^{+}$ is irreducible; the irreducibility of $L_{(0,1)}^{-}$ is proved analogously.

The generalized weight space decomposition of $M=L_{(0,1)}^{+}$ is $$M=M_{(-1,0)}^{\text{gen}}\oplus M_{(1,0)}^{\text{gen}}\oplus M_{(0,1)}^{\text{gen}}\text{with}\begin{array}{rcl}{\displaystyle \text{dim}\left(M_{(-1,0)}^{\text{gen}}\right)}& =& {\displaystyle \text{dim}\left(M_{(1,0)}^{\text{gen}}\right)=1,}\\ {\displaystyle \text{dim}\left(M_{(0,1)}^{\text{gen}}\right)}& =& {\displaystyle 2\text{.}}\end{array}$$ The element $W_1^{-1}$ acts on $M_{(-1,0)}^{\text{gen}}$ with eigenvalue $-t\text{.}$ Since the parameters are generic (see (4.1)), $-t\notin \{-t^{\pm r_1},-t^{\pm r_2}\}$ and thus, by (2.42), ${\tau }_0^2$ has no kernel. Thus the intertwiner ${\tau }_0:M_{(-1,0)}^{\text{gen}}\to M_{(1,0)}^{\text{gen}}$ is invertible. Since $M_{(0,1)}^{\text{gen}}$ is irreducible as a $H_{\left\{0\right\}}\text{-module,}$ we have either $N=M_{(0,1)}^{\text{gen}}$ is an $H_2\text{-submodule}$ or $M$ is irreducible.

For the purposes of deriving a contradiction, assume that $N=M_{(0,1)}^{\text{gen}}$ is an $H_2\text{-submodule}$ of $M\text{.}$ The space $N$ has a basis $$\{n_{\gamma },T_0{n}_{\gamma }\}\text{with}{W}_1{n}_{\gamma }=-n_{\gamma },\text{and}{W}_2{n}_{\gamma }=-t{n}_{\gamma }\text{.}$$ By (C2) and (B3), $$\begin{array}{rcl}{\displaystyle W_1{W}_2^{-1}{T}_0{n}_{\gamma }}& =& {\displaystyle T_0{W}_1^{-1}{W}_2^{-1}{n}_{\gamma }+((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})W_1^{-1})\frac{W_1-W_1^{-1}}{1-W_1^{-2}}{W}_2^{-1}{n}_{\gamma }}\\ & =& {\displaystyle T_0{t}^{-1}{n}_{\gamma }+((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-1))t^{-1}{n}_{\gamma },}\end{array}$$ and the action of $W_1{W}_2^{-1}$ on the basis $\{n_{\gamma },T_0{n}_{\gamma }\}$ is given by the matrix $$\rho \left(W_1{W}_2^{-1}\right)=\left(\begin{array}{cc}{t}^{-1}& ((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-1))t^{-1}\\ 0& t^{-1}\end{array}\right)\text{.}$$ Thus $$\begin{array}{rcl}{\displaystyle \rho (1-W_1{W}_2^{-1})}& =& {\displaystyle \left(\begin{array}{cc}1-t^{-1}& -((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+\left(t_k^{\frac{1}{2}}0t_k^{-\frac{1}{2}}\right)(-1))t^{-1}\\ 0& 1-t^{-1}\end{array}\right)}\\ & =& {\displaystyle (1-t^{-1})\left(\begin{array}{cc}1& -\frac{((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+\left(t_k^{\frac{1}{2}}0t_k^{-\frac{1}{2}}\right)(-1))t^{-1}}{1-t^{-1}}\\ 0& 1\end{array}\right)}\end{array}$$ and $$\rho {(1-W_1{W}_2^{-1})}^{-1}=\frac{1}{(1-t^{-1})}\left(\begin{array}{cc}1& \frac{((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-1))t^{-1}}{1-t^{-1}}\\ 0& 1\end{array}\right)\text{.}$$ If $N$ is a submodule of $M$ then $0={\tau }_1=T_1-\frac{t^{\frac{1}{2}}+t^{-\frac{1}{2}}}{1-W_1{W}_2^{-1}}$ (see (2.35) for the formula for ${\tau }_1\text{).}$ Thus $$\rho \left(T_1\right)=t^{\frac{1}{2}}\left(\begin{array}{cc}1& \frac{((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})(-1))t^{-1}}{1-t^{-1}}\\ 0& 1\end{array}\right)\text{.}$$ Since $(T_1-t^{\frac{1}{2}})(T_1+t^{-\frac{1}{2}})=0$ the Jordan blocks of $\rho \left(T_1\right)$ are of size 1 forcing $$0=(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})-(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}})=t_0^{-\frac{1}{2}}(t_0^{\frac{1}{2}}+t_k^{-\frac{1}{2}})(t_0^{\frac{1}{2}}-t_k^{\frac{1}{2}})\text{.}$$ This is a quadratic equation in $t_0^{\frac{1}{2}}$ with two solutions, $t_0^{\frac{1}{2}}=t_k^{\frac{1}{2}}$ and $t_0^{\frac{1}{2}}=-t_k^{-\frac{1}{2}}\text{.}$ This is a contradiction since, by the generic condition on parameters in (4.1), $-t^{-r_1}=-t_0^{\frac{1}{2}}{t}_k^{-\frac{1}{2}}\ne -1$ and $-t^{r_2}=t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}}\ne -1\text{.}$ Thus $N$ is not a submodule of $M\text{,}$ and so $M$ is irreducible.

Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type $C$ Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).

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