The double affine Hecke algebra

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

Last update: 17 September 2012

The double affine Hecke algebra

With $T_0^{\prime }$ as in (2.4), the double affine Hecke algebra ${\stackrel{\sim }{H}}_n$ is the quotient of $ℂ\stackrel{\sim }{ℬ}$ by

$$\begin{array}{c}(T_0^{\prime }-u_0^{\frac{1}{2}})(T_0^{\prime }+u_0^{-\frac{1}{2}})=0,(T_0^{\vee }-u_n^{\frac{1}{2}})(T_0^{\vee }+u_n^{-\frac{1}{2}})=0,\\ (T_0-t_0^{\frac{1}{2}})(T_0+t_0^{-\frac{1}{2}})=0,(T_n-t_n^{\frac{1}{2}})(T_n-t_n^{-\frac{1}{2}})=0,\text{and}& \text{(3.17)}\\ (T_i-t^{\frac{1}{2}})(T_i+t^{-\frac{1}{2}})=0,\text{for}i=1,\dots ,n-1\text{.}\end{array}$$

Let $X^{{\epsilon }_1},\dots ,X^{{\epsilon }_n}$ and $Y^{{\epsilon }_1^{\vee }},\dots ,Y^{{\epsilon }_n^{\vee }}$ be the elements of $\stackrel{\sim }{ℬ}$ defined in (2.3). The following theorem establishes the "Bernstein presentation" of the double affine Hecke algebra. Often this presentation is used as the definition of the double affine Hecke algebra. The affine Weyl group

$$\begin{array}{cc}{X}^{w\mu }=w{X}^{\mu }{w}^{-1},\text{for}\mu \in {𝔥}_{\mathbb{Z}}^{*}\text{and}w\in W\text{.}& \text{(3.18)}\end{array}$$

Similarly, thhe affine Weyl group

$$W^{\vee }=\{X^{\mu }v\mid \mu \in {𝔥}_{\mathbb{Z}}^{*},v\in W_0\}\text{acts on}\stackrel{\sim }{Y}=\{q^{k/2}{Y}^{{\lambda }^{\vee }}\mid k\in \mathbb{Z},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\},\text{by conjugation.}$$

Write

$$\begin{array}{cc}{Y}^{w{\lambda }^{\vee }}=w{Y}^{{\lambda }^{\vee }}{w}^{-1},\text{for}w\in W^{\vee }\text{and}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}& \text{(3.19)}\end{array}$$

In particular,

$$\begin{array}{cc}{Y}^{s_0^{\vee }{\epsilon }_1^{\vee }}=s_0^{\vee }{Y}^{{\epsilon }_1^{\vee }}{s}_0^{\vee }=q^{-1}{Y}^{-{\epsilon }_1^{\vee }}\text{and}{Y}^{s_0^{\vee }{\epsilon }_j^{\vee }}{s}_0^{\vee }{Y}^{{\epsilon }_j^{\vee }}{s}_0^{\vee }=Y^{{\epsilon }_j^{\vee }},\text{for}j=2,\dots ,n\text{.}& \text{(3.20)}\end{array}$$

1. Let $t_i^{\frac{1}{2}}=u_i^{\frac{1}{2}}=t^{\frac{1}{2}}$ for $i=1,\dots ,n-1\text{.}$ The algebra ${\stackrel{\sim }{H}}_n$ is the quotient of $ℂ{\stackrel{\sim }{ℬ}}_n$ by the relations

   $$T_i-T_i^{-1}=t_i^{\frac{1}{2}}-t_i^{-\frac{1}{2}},\text{for}i=0,1,\dots ,n.$$

   and

   $$T_i{X}^{\mu }=X^{s_i\mu }{T}_i+\frac{((t_i^{\frac{1}{2}}-t_i^{-\frac{1}{2}})+(u_i^{\frac{1}{2}}-u_i^{-\frac{1}{2}})X^{{\alpha }_i})(X^{\mu }-X^{s_i\mu })}{1-X^{2{\alpha }_i}},$$

   for $i=0,\dots ,n,$ where $X^{s_i\mu }$ is defined in (3.18),

   $$X^{{\alpha }_0}=q^{\frac{1}{2}}{X}^{-{\epsilon }_1},{\alpha }_n={\epsilon }_n,\text{and}{\alpha }_i={\epsilon }_i-{\epsilon }_{i+1},\text{for}i=1,\dots ,n-1\text{.}$$

2. The algebra ${\stackrel{\sim }{H}}_n$ is the quotient of $ℂ{\stackrel{\sim }{ℬ}}_n$ by the relations

   $$T_0^{\vee }-{\left(T_0^{\vee }\right)}^{-1}=u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}},T_n-T_n^{-1}=t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}},T_i-T_i^{-1}=t^{\frac{1}{2}}-t^{-\frac{1}{2}},\text{for}i=1,\dots ,n-1,$$

   and

   $$\begin{array}{ccc}{T}_i{Y}^{{\lambda }^{\vee }}& =& Y^{s_i{\lambda }^{\vee }}{T}_i+(t^{\frac{1}{2}}-t^{-\frac{1}{2}})\frac{Y^{{\lambda }^{\vee }}-Y^{s_i{\lambda }^{\vee }}}{1-Y^{-{\alpha }_i^{\vee }}},\text{for}i=1,\dots ,n-1,\\ T_n{Y}^{{\lambda }^{\vee }}& =& Y^{s_n{\lambda }^{\vee }}{T}_n+\frac{((t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }})}{1-Y^{-2{\epsilon }_n^{\vee }}}(Y^{{\lambda }^{\vee }}-Y^{s_n{\lambda }^{\vee }}),\text{and}\\ T_0^{\vee }{Y}^{{\lambda }^{\vee }}& =& Y^{s_0^{\vee }{\lambda }^{\vee }}{T}_0^{\vee }+\frac{((u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}})+(u_0^{\frac{1}{2}}-u_0^{-\frac{1}{2}})Y^{-{\alpha }_0^{\vee }})}{1-Y^{-2{\alpha }_0^{\vee }}}(Y^{{\lambda }^{\vee }}-Y^{s_0^{\vee }{\lambda }^{\vee }})\text{.}\end{array}$$

   where $Y^{s_i{\lambda }^{\vee }}$ is as defined in (3.19),

   $$Y^{{\alpha }_0^{\vee }}=q^{-\frac{1}{2}}{Y}^{-{\epsilon }_1^{\vee }},{\alpha }_n^{\vee }={\epsilon }_n^{\vee },\text{and}{\alpha }_i^{\vee }={\epsilon }_i^{\vee }-{\epsilon }_{i+1}^{\vee },\text{for}i=1,\dots ,n-1\text{.}$$

		Proof.
	Following [M03, p. 81]: Since $T_0^{\prime }=q^{-\frac{1}{2}}{X}^{{\epsilon }_1}{T}_0^{-1},$ $$\begin{array}{ccc}{T}_0{X}^{{\alpha }_0}-X^{-{\alpha }_0}{T}_0& =& T_0{q}^{\frac{1}{2}}{X}^{-{\epsilon }_1}-q^{-\frac{1}{2}}{X}^{{\epsilon }_1}{T}_0\\ & =& T_0{q}^{\frac{1}{2}}{X}^{-{\epsilon }_1}-q^{-\frac{1}{2}}{X}^{{\epsilon }_1}(T_0^{-1}+t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})\\ & =& {\left(T_0^{\prime }\right)}^{-1}-T_0^{\prime }-(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})q^{-\frac{1}{2}}{X}^{{\epsilon }_1}\\ & =& -(u_0^{\frac{1}{2}}-u_0^{-\frac{1}{2}})-(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})q^{-\frac{1}{2}}{X}^{{\epsilon }_1}\\ & =& ((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(u_0^{\frac{1}{2}}-u_0^{-\frac{1}{2}})q^{\frac{1}{2}}{X}^{-{\epsilon }_1})\frac{q^{\frac{1}{2}}{X}^{-{\epsilon }_1}-q^{-\frac{1}{2}}{X}^{{\epsilon }_1}}{1-q{X}^{-2{\epsilon }_1}}\\ & =& ((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(u_0^{\frac{1}{2}}-u_0^{-\frac{1}{2}})X^{{\alpha }_0})\frac{X^{{\alpha }_0}-X^{-{\alpha }_0}}{1-X^{2{\alpha }_0}},\end{array}$$ and, since $X^{-{\epsilon }_n}{T}_n^{-1}=T_n\dots T_1{T}_0^{\vee }{T}_1^{-1}\dots T_n^{-1}$ is conjugate to $T_0^{\vee },$ $$\begin{array}{ccc}{T}_n{X}^{{\alpha }_n}-X^{-{\alpha }_n}{T}_n& =& T_n{X}^{{\epsilon }_n}-X^{-{\epsilon }_n}{T}_n\\ & =& T_n{X}^{{\epsilon }_n}-X^{-{\epsilon }_n}(T_n^{-1}+t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})\\ & =& -(u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}})-(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})X^{-{\epsilon }_n}\\ & =& ((t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}})X^{{\epsilon }_n})\frac{X^{{\epsilon }_n}-X^{-{\epsilon }_n}}{1-X^{2{\epsilon }_n}}\text{.}\end{array}$$ Since $Y^{{\alpha }_0^{\vee }}=q^{-\frac{1}{2}}{Y}^{-{\epsilon }_1^{\vee }}$ and $T_0^{\prime }={\left(T_0^{\vee }\right)}^{-1}{q}^{-\frac{1}{2}}{Y}^{-{\epsilon }_1^{\vee }},$ $$\begin{array}{ccc}{T}_0^{\vee }{Y}^{{\alpha }_0^{\vee }}-Y^{-{\alpha }_0^{\vee }}{T}_0^{\vee }& =& ({\left(T_0^{\vee }\right)}^{-1}+(u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}}))q^{-\frac{1}{2}}{Y}^{-{\epsilon }_1^{\vee }}-q^{\frac{1}{2}}{Y}^{{\epsilon }_1^{\vee }}{T}_0^{\vee }\\ & =& {\left(T_0^{\vee }\right)}^{-1}{q}^{-\frac{1}{2}}{Y}^{-{\epsilon }_1^{\vee }}-q^{\frac{1}{2}}{Y}^{{\epsilon }_1^{\vee }}{T}_0^{\vee }+(u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}})Y^{{\alpha }_0^{\vee }}\\ & =& (u_0^{\frac{1}{2}}-u_0^{-\frac{1}{2}})+(u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}})Y^{{\alpha }_0^{\vee }}\\ & =& ((u_0^{\frac{1}{2}}-u_0^{-\frac{1}{2}})Y^{-{\alpha }_0^{\vee }}+(u_n^{\frac{1}{2}}-u_n^{-\frac{1}{2}}))\frac{Y^{{\alpha }_0^{\vee }}-Y^{-{\alpha }_0^{\vee }}}{1-Y^{-2{\alpha }_0^{\vee }}},\end{array}$$ and, since $T_n^{-1}{Y}^{{\epsilon }_n^{\vee }}=T_n^{-1}{T}_{n-1}^{-1}\dots T_1^{-1}{T}_0{T}_1\dots T_n$ conjugate to $T_0,$ $$\begin{array}{ccc}{T}_n{Y}^{{\alpha }_n^{\vee }}-Y^{-{\alpha }_n^{\vee }}{T}_n& =& T_n{Y}^{{\epsilon }_n^{\vee }}-Y^{-{\epsilon }_n^{\vee }}{T}_n\\ & =& (T_n^{-1}+(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}}))Y^{{\epsilon }_n^{\vee }}-Y^{-{\epsilon }_n^{\vee }}{T}_n\\ & =& \left(T_n^{-1}{Y}^{{\epsilon }_n^{\vee }}\right)-{\left(T_n^{-1}{Y}^{{\epsilon }_n^{\vee }}\right)}^{-1}+(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }}\\ & =& ((t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}+(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}}))\frac{Y^{{\epsilon }_n^{\vee }}-Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n}}\text{.}\end{array}$$ $\square$

SOME REMARK ABOUT LINEARITY.

(Duality) [M03, (4.7.6)] The involution $\iota :{\stackrel{\sim }{ℬ}}_n\to {ℬ}_n$ in Proposition 2.4 descends to an involution $\iota :\stackrel{\sim }{H}\to \stackrel{\sim }{H}$ given by

$$\begin{array}{c}\iota \left(q^{\frac{1}{2}}\right)=q^{-\frac{1}{2}},\iota \left(t^{\frac{1}{2}}\right)=t^{-\frac{1}{2}},\iota \left(t_n^{\frac{1}{2}}\right)=t_n^{-\frac{1}{2}},\iota \left(u_0^{\frac{1}{2}}\right)=u_0^{-\frac{1}{2}},\iota \left(u_n^{\frac{1}{2}}\right)=t_0^{-\frac{1}{2}},\\ \iota \left(T_0^{\vee }\right)=T_0^{-1},\iota \left(T_0^{\prime }\right)={\left(T_0^{\prime }\right)}^{-1},\iota \left(T_i\right)=T_i^{-1},\text{for}i=1,\dots ,n\text{.}\end{array}$$

		Proof.
	A straightforward check shows that the relations in (3.17) are preserved. $\square$

In an attempt to relation the notations in [M03, §4.7], [S99, §3] and [C03, Def. 2.1] let

$${\tau }_0^{\prime }=u_0^{\frac{1}{2}},{\tau }_n^{\prime }=u_n^{\frac{1}{2}},{\tau }_0=t_0^{\frac{1}{2}},{\tau }_n=t_n^{\frac{1}{2}},\text{and}{\tau }_i={\tau }_i^{\prime }=t^{\frac{1}{2}}\text{for}i=1,\dots ,n-1\text{.}$$

The summary of (1.5.1), (4.4.1), (4.4.2), (4.4.3), and (5.1.4) in [M03] is that

$${\tau }_a={\left(t_a{t}_{2a}\right)}^{\frac{1}{2}}=q^{\frac{1}{2}{\kappa }_a}=q^{\frac{1}{2}(k\left(a\right)+k\left(2a\right))},\text{and}{\tau }_a^{\prime }=t_a^{\frac{1}{2}}=q^{\frac{1}{2}{\kappa }_a^{\prime }}=q^{\frac{1}{2}(k\left(a\right)+k\left(2a\right))}\text{.}$$

In our situation

$$\begin{array}{ccccccccccc}{\tau }_n& =& t_n^{\frac{1}{2}}& =& q^{\frac{1}{2}{\kappa }_n}& =& t_{{\epsilon }_n}^{\frac{1}{2}}{t}_{2{\epsilon }_n}^{\frac{1}{2}}& =& q^{\frac{1}{2}k\left({\epsilon }_n\right)+\frac{1}{2}k\left(2{\epsilon }_n\right)}& =& q^{\frac{1}{2}{k}_1+\frac{1}{2}{k}_2},\\ {\tau }_n^{\prime }& =& u_n^{\frac{1}{2}}& =& q^{\frac{1}{2}{\kappa }_n^{\prime }}& =& t_{{\epsilon }_n}^{\frac{1}{2}}& =& q^{\frac{1}{2}k\left({\epsilon }_n\right)-\frac{1}{2}k\left(2{\epsilon }_n\right)}& =& q^{\frac{1}{2}{k}_1-\frac{1}{2}{k}_2},\\ {\tau }_0& =& t_0^{\frac{1}{2}}& =& q^{\frac{1}{2}{\kappa }_0}& =& t_{-{\epsilon }_1+\frac{1}{2}\delta }^{\frac{1}{2}}{t}_{-2{\epsilon }_1+\delta }^{\frac{1}{2}}& =& q^{\frac{1}{2}k({\epsilon }_1+\frac{1}{2}\delta )+k(-2{\epsilon }_1+\delta )}& =& q^{\frac{1}{2}{k}_3+\frac{1}{2}{k}_4},\\ {\tau }_0^{\prime }& =& u_0^{\frac{1}{2}}& =& q^{\frac{1}{2}{\kappa }_0^{\prime }}& =& t_{-{\epsilon }_1+\frac{1}{2}\delta }^{\frac{1}{2}}& =& q^{\frac{1}{2}k({\epsilon }_1+\frac{1}{2}\delta )-k(-2{\epsilon }_1+\delta )}& =& q^{\frac{1}{2}{k}_3-\frac{1}{2}{k}_4},\\ {\tau }_i& =& t^{\frac{1}{2}}& =& q^{\frac{1}{2}\kappa }& =& t_{{\epsilon }_i-{\epsilon }_{i+1}}^{\frac{1}{2}}& =& q^{\frac{1}{2}{k}_5},\end{array}$$

for $i=1,\dots ,n-1,$ and the formulas in [M03, (1.5.1)] correspond to interchanging ${\kappa }_0$ and ${\kappa }_n^{\prime }\text{.}$

The other classical affine root systems are obtained from this one by setting some of the $t_a$ equal to 1. EXPAND ON THIS STATEMENT!

Intertwiners

REPLACE THE $Y^{{\alpha }_0^{\vee }}$s IN THIS SUBSECTION.

From Theorem 3.3(b) it follows that

$${\tau }_i^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{s_i{\lambda }^{\vee }}{\tau }_i^{\vee },\text{for}i=0,1,2,\dots ,n\text{.}$$

if

$$\begin{array}{ccc}{\tau }_i^{\vee }& =& T_i+\frac{t^{-\frac{1}{2}}(1-t)}{1-Y^{-{\alpha }_i^{\vee }}}=T_i^{-1}+\frac{t^{-\frac{1}{2}}(1-t)1-Y^{-{\alpha }_i^{\vee }}}{1-Y^{-{\alpha }_i^{\vee }}},\text{for}i=1,\dots ,n-1,\text{and}\\ {\tau }_n^{\vee }& =& T_n+\frac{t_n^{-\frac{1}{2}}(1-t_n)+t_0^{-\frac{1}{2}}(1-t_0)Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}}\\ & =& T_n^{-1}+\frac{(t_n^{-\frac{1}{2}}(1-t_n)+t_0^{-\frac{1}{2}}(1-t_0)Y^{{\epsilon }_n^{\vee }})Y^{-2{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}},\\ {\tau }_0^{\vee }& =& T_0^{\vee }+\frac{u_n^{-\frac{1}{2}}(1-u_n)+u_0^{-\frac{1}{2}}(1-u_0)Y^{-{\alpha }_0^{\vee }}}{1-Y^{-2{\alpha }_0^{\vee }}}\\ & =& {\left(T_0^{\vee }\right)}^{-1}\frac{(u_n^{-\frac{1}{2}}(1-u_n)+u_0^{-\frac{1}{2}}(1-u_0)Y^{{\alpha }_0^{\vee }})Y^{-2{\alpha }_0^{\vee }}}{1-Y^{-2{\alpha }_0^{\vee }}},\end{array}$$

The operators ${\tau }_0^{\vee },\dots ,{\tau }_n^{\vee }$ satisfy the relations

and

$$\begin{array}{ccc}{\left({\tau }_0^{\vee }\right)}^2& =& \frac{u_n^{-1}(u_0^{\frac{1}{2}}{u}_n^{\frac{1}{2}}-Y^{{\alpha }_0^{\vee }})(u_0^{-\frac{1}{2}}{u}_n^{\frac{1}{2}}+Y^{{\alpha }_0^{\vee }})(u_0^{\frac{1}{2}}{u}_n^{\frac{1}{2}}-Y^{-{\alpha }_0^{\vee }})(u_0^{-\frac{1}{2}}{u}_n^{\frac{1}{2}}+Y^{-{\alpha }_0^{\vee }})}{(1-Y^{{\alpha }_0^{\vee }})(1+Y^{{\alpha }_0^{\vee }})(1-Y^{-{\alpha }_0^{\vee }})(1+Y^{-{\alpha }_0^{\vee }})},\\ {\left({\tau }_n^{\vee }\right)}^2& =& \frac{t_n^{-1}(t_0^{\frac{1}{2}}{t}_n^{\frac{1}{2}}-Y^{{\epsilon }_n^{\vee }})(t_0^{-\frac{1}{2}}{t}_n^{\frac{1}{2}}+Y^{{\epsilon }_n^{\vee }})(t_0^{\frac{1}{2}}{t}_n^{\frac{1}{2}}-Y^{-{\epsilon }_n^{\vee }})(t_0^{-\frac{1}{2}}{t}_n^{\frac{1}{2}}+Y^{-{\epsilon }_n^{\vee }})}{(1-Y^{{\epsilon }_n^{\vee }})(1+Y^{{\epsilon }_n^{\vee }})(1-Y^{-{\epsilon }_n^{\vee }})(1+Y^{-{\epsilon }_n^{\vee }})},\\ {\left({\tau }_i^{\vee }\right)}^2& =& \frac{(t^{\frac{1}{2}}-t^{-\frac{1}{2}}{Y}^{-{\alpha }_i^{\vee }})(t^{\frac{1}{2}}-t^{-\frac{1}{2}}{Y}^{{\alpha }_i^{\vee }})}{(1-Y^{-{\alpha }_i^{\vee }})(1-Y^{{\alpha }_i^{\vee }})},\text{for}i=1,\dots ,n-1\text{.}\end{array}$$

		Proof.
	$$\begin{array}{ccc}{\left({\tau }_n^{\vee }\right)}^2& =& {\tau }_n^{\vee }(T_n-\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}})\\ & =& {\tau }_n^{\vee }{T}_n-\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }}}{1-Y^{2{\epsilon }_n^{\vee }}}{\tau }_n^{\vee }\\ & =& T_n^2-\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}}{T}_n-\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }}}{1-Y^{2{\epsilon }_n^{\vee }}}{T}_n\\ & & +\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }}}{1-Y^{2{\epsilon }_n^{\vee }}}+\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}}\\ & =& (t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})T_n+\frac{(1-Y^{2{\epsilon }_n^{\vee }})(1-Y^{-2{\epsilon }_n^{\vee }})}{(1-Y^{2{\epsilon }_n^{\vee }})(1-Y^{-2{\epsilon }_n^{\vee }})}\\ & & +\frac{((t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }})}{(1-Y^{2{\epsilon }_n^{\vee }})}\frac{((t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }})}{(1-Y^{-2{\epsilon }_n^{\vee }})}\\ & & +(-\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}}+\frac{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})Y^{-2{\epsilon }_n^{\vee }}+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}})T_n\\ & =& \frac{\left(\genfrac{}{}{0ex}{}{1-Y^{2{\epsilon }_n^{\vee }}-Y^{-2{\epsilon }_n^{\vee }}+1+{(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})}^2+{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})}^2}{+(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}+(t_n^{\frac{1}{2}}-t_n^{-\frac{1}{2}})(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }}}\right)}{(1-Y^{2{\epsilon }_n^{\vee }})(1-Y^{-2{\epsilon }_n^{\vee }})}\\ & =& \frac{\left(\begin{array}{c}{t}_n+t_n^{-1}+t_0+t_0^{-1}-1-1-Y^{2{\epsilon }_n^{\vee }}-Y^{-2{\epsilon }_n^{\vee }}\\ +(t_0^{\frac{1}{2}}{t}_n^{\frac{1}{2}}-t_0^{\frac{1}{2}}{t}_n^{-\frac{1}{2}}-t_0^{-\frac{1}{2}}{t}_n^{\frac{1}{2}}+t_0^{-\frac{1}{2}}{t}_n^{-\frac{1}{2}})Y^{-{\epsilon }_n^{\vee }}\\ +(t_0^{\frac{1}{2}}{t}_n^{\frac{1}{2}}-t_0^{\frac{1}{2}}{t}_n^{-\frac{1}{2}}-t_0^{-\frac{1}{2}}{t}_n^{\frac{1}{2}}+t_0^{-\frac{1}{2}}{t}_n^{-\frac{1}{2}})Y^{{\epsilon }_n^{\vee }}\end{array}\right)}{(1-Y^{{\epsilon }_n^{\vee }})(1+Y^{{\epsilon }_n^{\vee }})(1-Y^{-{\epsilon }_n^{\vee }})(1+Y^{-{\epsilon }_n^{\vee }})}\\ & =& \frac{t_n^{-1}(t_0^{\frac{1}{2}}{t}_n^{\frac{1}{2}}-Y^{{\epsilon }_n^{\vee }})(t_0^{-\frac{1}{2}}{t}_n^{\frac{1}{2}}+Y^{{\epsilon }_n^{\vee }})(t_0^{\frac{1}{2}}{t}_n^{\frac{1}{2}}-Y^{-{\epsilon }_n^{\vee }})(t_0^{-\frac{1}{2}}{t}_n^{\frac{1}{2}}+Y^{-{\epsilon }_n^{\vee }})}{(1-Y^{{\epsilon }_n^{\vee }})(1+Y^{{\epsilon }_n^{\vee }})(1-Y^{-{\epsilon }_n^{\vee }})(1+Y^{-{\epsilon }_n^{\vee }})}\end{array}$$ The formula for ${\left({\tau }_i^{\vee }\right)}^2$ is established by exactly the same computation, except with $t_0^{\frac{1}{2}}$ and $t_n^{\frac{1}{2}}$ both replaced by $t^{\frac{1}{2}}\text{.}$ $\square$

Question: Is the affine Hecke algebra a quotient of DAHA? This seems to be true on the braid group. NO?!? The obvious morphism would take $T_0^{\vee }\mapsto 1$ which would violate $(T_0^{\vee }-u_n^{\frac{1}{2}})(T_0^{\vee }+u_n^{\frac{1}{2}})-0\text{.}$ Not clear that reps of affine Hecke would lift to DAHA. The 1 dimensional representations lift or do not lift?

Notes and References

This page is taken from a paper entitled Double affine braid groups and Hecke algebras of classical type by Arun Ram, March 5, 2009.

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