Where does the homomorphism $\Phi$ come from?

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

Last update: 28 February 2013

Where does the homomorphism $\Phi$ come from?

The homomorphism $\Phi :H_{\infty ,1,n}\to H_{r,1,n}$ of Proposition 2.5 is a powerful tool for transporting results about the affine Hecke algebra of type $A$ to the cyclotomic Hecke algebras. In this section we show how this homomorphism arises naturally, from a folding of the Dynkin diagram of ${\stackrel{\sim }{ℬ}}_n,$ and we give some generalizations of the homomorphism $\Phi$ to other types.

Example 1. Type $C_n\text{.}$ The root system $R$ of type $C_n$ can be realized by

$$R=\{\pm 2{\epsilon }_i,\pm ({\epsilon }_j-{\epsilon }_i)\hspace{0.17em}\mid \hspace{0.17em}1\le i,j\le n\},$$

where ${\epsilon }_i$ are an orthonormal basis of ${ℝ}^n\text{.}$ The simple roots and the fundamental weights are given by

$$\begin{array}{c}{\alpha }_1=2{\epsilon }_1,{\alpha }_i={\epsilon }_i-{\epsilon }_{i-1},2\le i\le n,\\ {\omega }_i={\epsilon }_n+{\epsilon }_{n-1}+\dots +{\epsilon }_i,1\le i\le n\text{.}\end{array}$$

If ${\varphi }^{\vee }$ is the highest root of $R^{\vee }$ then $\varphi ={\epsilon }_n+{\epsilon }_{n-1},$ and

$$\begin{array}{cc}{s}_{{\varphi }^{\vee }}=\left(s_{n-1}\dots s_2{s}_1{s}_2\dots s_{n-1}\right)s_n\left(s_{n-1}\dots s_2{s}_1{s}_2\dots s_{n-1}\right)\text{.}& \text{(5.1)}\end{array}$$

Then ${\omega }_n={\epsilon }_n$ is the only miniscule weight,

$$\begin{array}{cc}\begin{array}{ccc}{w}_0& =& (1,-1)(2,-2)\dots (n,-n),\\ w_{0,n}& =& (1,-1)(2,-2)\dots (n-1,-(n-1)),\text{and}\\ w_0{w}_{0,n}& =& (n,-n)=s_n\dots s_2{s}_1{s}_2\dots s_n\text{.}\end{array}& \text{(5.2)}\end{array}$$

Thus, from (4.2), (4.3) and (4.6), $\Omega =\{1,g_n\}\cong \mathbb{Z}/2\mathbb{Z},$

$$\begin{array}{cc}\begin{array}{ccc}{g}_n& =& X^{{\epsilon }_n}{T}_n^{-1}\dots T_2^{-1}{T}_1^{-1}{T}_2^{-1}\dots T_n^{-1},\\ T_0& =& X^{{\epsilon }_n+{\epsilon }_{n-1}}\left(T_n^{-1}\dots T_2^{-1}{T}_1^{-1}{T}_2^{-1}\dots T_n^{-1}\right)T_n^{-1}\left(T_n^{-1}\dots T_2^{-1}{T}_1^{-1}{T}_2^{-1}\dots T_n^{-1}\right),\end{array}& \text{(5.3)}\end{array}$$

and

$$\begin{array}{cc}{g}_n{T}_0{g}_n^{-1}=T_n,\text{and}{g}_n{T}_n{g}_n^{-1}=T_0\text{.}& \text{(5.4)}\end{array}$$

The braid group ${\stackrel{\sim }{ℬ}}_P\left(C_n\right)$ is generated by $T_0,T_1,\dots ,T_n$ and $g_n$ which satisfy relations in (4.6), where the $m_{ij}$ are given by the extended Dynkin diagram ${\stackrel{\sim }{ℬ}}_n,$ see Figure 3. The braid group $ℬ\left(C_n\right)$ is the subgroup generated by $T_1,\dots ,T_n\text{.}$ These elements satisfy the relations in (4.6), where the $m_{ij}$ are given by the Dynkin diagram $C_n\text{.}$ A straightforward check verifies that the map defined by

$$\begin{array}{ccc} 1 2 3 n-2 n-1 n 0 & ⟶& 1 2 3 n-1 n \end{array}$$ $$\begin{array}{cc}\begin{array}{cccc}{\Phi }_{\stackrel{\sim }{C}C}:& {\stackrel{\sim }{ℬ}}_P\left(B_n\right)& ⟶& ℬ\left(C_n\right)\\ & g_n& ⟼& 1,\\ & T_0& ⟼& T_n,\\ & T_i& ⟼& T_i,& 1\le i\le n\text{.}\end{array}& \text{(5.5)}\end{array}$$

extends to a well defined surjective group homomorphism. From the identity (4.3),

$$\begin{array}{ccc}{\Phi }_{\stackrel{\sim }{C}C}\left(X^{{\epsilon }_n}\right)& =& {\Phi }_{\stackrel{\sim }{C}C}\left(X^{{\omega }_n}\right)\\ & =& {\Phi }_{\stackrel{\sim }{C}C}\left(g_n{T}_n{T}_{n-1}\dots T_2{T}_1{T}_2\dots T_{n-1}{T}_n\right)\\ & =& T_n{T}_{n-1}\dots T_2{T}_1{T}_2\dots T_{n-1}{T}_n\text{.}\end{array}$$

By inductively applying the relation $X^{{\epsilon }_i}=T_i{X}^{{\epsilon }_{i-1}}{T}_i$ we get

$${\Phi }_{\stackrel{\sim }{C}C}\left(X^{{\epsilon }_i}\right)=T_i{T}_{i-1}\dots T_2{T}_1{T}_2\dots T_{n-1}{T}_i,\text{for all}\hspace{0.17em}1\le i\le n\text{.}$$

Example 2. Type $A_{n-1}\text{.}$ Since the weight lattice $P$ for the root system of type $C_n$ is the same as the lattice $L$ defined in (1.5) we have an injective homomorphism

$$\begin{array}{cccc}{\Phi }_{\stackrel{\sim }{A}\stackrel{\sim }{C}}:& {ℬ}_{\infty ,1,n}& ⟶& {\stackrel{\sim }{ℬ}}_P\left(C_n\right)\\ & T_i& ⟼& T_i,& 2\le i\le n,\\ & X^{{\epsilon }_i}& ⟼& X^{{\epsilon }_i}\text{.}\end{array}$$

The composition of ${\Phi }_{\stackrel{\sim }{A}\stackrel{\sim }{C}}$ and the map ${\Phi }_{\stackrel{\sim }{C}C}$ from (5.5) is the surjective homomorphism defined by

$$\begin{array}{cccc}\Phi :& {ℬ}_{\infty ,1,n}& ⟶& ℬ\left(C_n\right)\\ & T_i& ⟼& T_i,& 2\le i\le n,\\ & X^{{\epsilon }_i}& ⟼& T_i{T}_{i-1}\dots T_2{T}_1{T}_2\dots T_{i-1}{T}_i,& 1\le i\le n\text{.}\end{array}$$

In fact, it follows from the defining relations of ${ℬ}_{\infty ,1,n}$ and $ℬ\left(C_n\right)$ that the map $\Phi$ is an isomorphism!

The cyclotomic Hecke algebras $H_{r,1,n}(u_1,\dots ,u_r;q)$ are quotients of $ℂℬ\left(C_n\right)$ and in this way the group homomorphism $\Phi$ is the source of the algebra homomorphism

$$\Phi :H_{\infty ,1,n}⟶H_{r,1,n}(u_1,\dots ,u_r;q)$$

which was used extensively in Section 3 to relate the representation theory of the cyclotomic Hecke algebras $H_{r,1,n}(u_1,\dots ,u_r;q)$ to the affine Hecke algebra of type A.

Example 3. Type $D_n\text{.}$ Let $R$ be the root system of type $D_n\text{.}$ Then $R^{\vee }$ is also of type $D_n$ and inspection of the Dynkin diagrams of types ${\stackrel{\sim }{D}}_n$ and $D_n$ yields a surjective algebra homomorphism defined by

$$\begin{array}{ccc} 2 1 3 4 n-2 n-1 n 0 & ⟶& 2 1 3 4 n-1 n \end{array}$$ $$\begin{array}{cccc}{\Phi }_{\stackrel{\sim }{D}D}:& {\stackrel{\sim }{ℬ}}_Q\left(D_n\right)& ⟶& ℬ\left(D_n\right)\\ & g_n& ⟼& 1,\\ & T_0& ⟼& T_n,\\ & T_i& ⟼& T_i,& 1\le i\le n\text{.}\end{array}$$

Examples 1, 2, and 3 show that, for types $A,$ $B$ and $D,$ there exist surjective homomorphisms from the affine Hecke algebra to the corresponding Iwahori-Hecke subalgebra. The following example shows that this is not a general phenomenon: there does not exist a surjective algebra homomorphism from the affine Hecke algebra of type $G_2$ to the corresponding Iwahori-Hecke subalgebra of type $G_2\text{.}$

Example 4. Type $G_2\text{.}$ If $R$ is the root system of type $G_2$ then $P=Q$ and $\Omega =\left\{1\right\}\text{.}$

$$\begin{array}{ccc} 1 2 0 & \nrightarrow & 1 2 \end{array}$$

Proposition 5.6. Let $\stackrel{\sim }{H}\left(G_2\right)$ be the affine Hecke algebra of type $G_2$ as given by (4.6) and (4.10) and let $H\left(G_2\right)$ be the Iwahori-Hecke subalgebra of type $G_2$ generated by $T_1$ and $T_2\text{.}$ There does not exist an algebra homomorphism $\Phi :\stackrel{\sim }{H}\left(G_2\right)\to H\left(G_2\right)$ such that $\Phi \left(T_i\right)=T_i,$ for $1\le i\le 2\text{.}$

		Proof.
	There is an irreducible representation of $\stackrel{\sim }{H}\left(G_2\right)$ given by $$\rho \left(T_1\right)=\left(\begin{array}{cc}q& 0\\ 0& -q^{-1}\end{array}\right),\text{and}\rho \left(T_2\right)=\frac{1}{q+q^{-1}}\left(\begin{array}{cc}2-q^{-2}& q^2-1+q^{-2}\\ 3& q^2-2\end{array}\right),$$ (see [Ram1997, Theorem 6.11]). We show that there does not exist a $2\times 2$ matrix $N$ which satisfies $$N^2=(q-q^{-1})N+1,N\rho \left(T_2\right)N=\rho \left(T_2\right)N\rho \left(T_2\right)\text{and}N\rho \left(T_1\right)=\rho \left(T_1\right)N\text{.}$$ If $N$ exists then $N$ must be diagonal since $N$ commutes with $\rho \left(T_1\right)$ and $\rho \left(T_1\right)$ is a diagonal matrix with distinct eigenvalues. The first equation shows that $N$ is invertible and the second equation shows that $N$ is conjugate to $\rho \left(T_2\right)\text{.}$ It follows that $N$ must have one eigenvalue $q$ and one eigenvalue $-q^{-1}\text{.}$ Thus, either $$N=\left(\begin{array}{cc}q& 0\\ 0& -q^{-1}\end{array}\right)\text{or}N=\left(\begin{array}{cc}-q^{-1}& 0\\ 0& q\end{array}\right)\text{.}$$ However, neither of these matrices satisfies the relation $N\rho \left(T_2\right)N=\rho \left(T_2\right)N\rho \left(T_2\right)\text{.}$ This contradiction shows that the representation $\rho$ cannot be extended to be a representation of $\stackrel{\sim }{H}\left(G_2\right)\text{.}$ $\square$

In spite of the fact, demonstrated by the previous example, that there does not always exist a surjective algebra homomorphism from the affine Hecke algebra onto its Iwahori-Hecke subalgebra, there are interesting surjective homomorphisms from affine Hecke algebras of exceptional type.

Example 5. Type $E_6\text{.}$ For the root system of type $E_6,$ $P/Q\cong \mathbb{Z}/3\mathbb{Z}\text{.}$ Let $\Omega =\{1,g,g^2\}$ where $g$ is as given by (4.3) for the minuscule weight ${\omega }_1$ (see [Bou1968, p. 261]). There are surjective algebra homomorphisms

$$\begin{array}{ccc} 1 2 3 4 5 6 0 & ⟶& 1 2 3 4 5 \end{array}$$ $$\begin{array}{cccc}\Phi :& {\stackrel{\sim }{ℬ}}_Q\left(E_6\right)& ⟶& ℬ\left(A_5\right)\\ & T_0& ⟼& T_5,\\ & T_6& ⟼& T_4,\\ & T_i& ⟼& T_i,& 1\le i\le 5,\end{array}$$

and

$$\begin{array}{ccc} 1 2 3 4 5 6 0 & ⟶& 1 2 3 \end{array}$$ $$\begin{array}{cccc}\Phi \prime :& {\stackrel{\sim }{ℬ}}_P\left(E_6\right)& ⟶& ℬ\left(A_3\right)\\ & g& ⟼& 1,\\ & T_0& ⟼& T_1,\\ & T_6& ⟼& T_2,\\ & T_5& ⟼& T_1,\\ & T_4& ⟼& T_2,\\ & T_i& ⟼& T_i,& 1\le i\le 3\text{.}\end{array}$$

Example 6. Type $F_4\text{.}$ For the root system $R$ of type $F_4$ we have $P=Q$ and $\Omega =\left\{1\right\}\text{.}$ There is a surjective homomorphism

$$\begin{array}{ccc} 1 2 3 4 0 & ⟶& 1 2 3 4 0 \end{array}$$ $$\begin{array}{cccc}\Phi :& \stackrel{\sim }{ℬ}\left(F_4\right)& ⟶& ℬ(A_2\times A_3)\\ & T_i& ⟼& T_i,& 0\le i\le 4\text{.}\end{array}$$

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

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

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