Double affine Hecke algebras of General Type

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

Last update: 20 September 2012

Double affine Hecke algebras

In the following, for simplicity of exposition we shall assume that we are not in the special case of [15, (2.16)] where the root system is type $C_n$ and ${𝔥}_{\mathbb{Z}}$ is the (co)root lattice. All our results and proofs are valid in this special case but the definition of the double affine Hecke algebra and the formulas in (2.32) and (2.34) may need some slight modification. See Remark 2.3 for details.

Let $R^{\vee }={\left(R^{\vee }\right)}^{+}\cup (-{\left(R^{\vee }\right)}^{+})$ be the set of coroots and dix parameters $c_{{\beta }^{\vee }},$ indexed by ${\beta }^{\vee }\in R^{\vee }+\mathbb{Z}d,$ such that for all $w\in W$ and ${\beta }^{\vee }\in R^{\vee }+\mathbb{Z}d,$

$$\begin{array}{cc}{c}_{{\beta }^{\vee }}=c_{w{\beta }^{\vee }}\text{.}\text{Set}{t}_{{\beta }^{\vee }}=q^{c_{{\beta }^{\vee }}}\text{and}{t}_i=t_{{\alpha }_i^{\vee }}\text{.}& \text{(2.30)}\end{array}$$

The double affine Hecke algebra $\stackrel{\sim }{H}$ is the group algebra $ℂ\stackrel{\sim }{ℬ}$ of the double braid group with the additional relations

$$\begin{array}{cc}{T}_i^2=(t_i^{\frac{1}{2}}-t_i^{-\frac{1}{2}})T_i+1,\text{for}i=0,1,\dots ,n\text{.}& \text{(2.31)}\end{array}$$

The double affine Hecke algebra $\stackrel{\sim }{H}$ has bases

$$\{T_w{X}^{\mu }\mid w\in W,\mu \in {𝔥}_{\mathbb{Z}}^{\vee }\oplus \mathbb{Z}\delta \},\{Y^{{\lambda }^{\vee }}{T}_w^{\vee }\mid w\in W^{\vee },{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\oplus \mathbb{Z}d\},$$

and

$$\{q^k{X}^{\mu }{T}_w{Y}^{{\lambda }^{\vee }}\mid w\in W_0,{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},\mu \in {𝔥}_{\mathbb{Z}}^{*},k\in \frac{1}{e}\mathbb{Z}\}$$

(see [6, Prop. 5.4 and Cor. 5.8]).

In the presence of (2.31) the relations (2.29) are equivalent to

$$\begin{array}{cc}{T}_i^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{s_i{\lambda }^{\vee }}{T}_i^{\vee }+(t_i^{\frac{1}{2}}-t_i^{-\frac{1}{2}})\frac{Y^{{\lambda }^{\vee }}-Y^{s_i{\lambda }^{\vee }}}{1-Y^{-{\alpha }_i^{\vee }}},\text{for}i=0,1,\dots ,n\text{.}& \text{(2.32)}\end{array}$$

where

$$\begin{array}{cc}{\tau }_i^{\vee }=T_i^{\vee }+\frac{t_i^{-\frac{1}{2}}(1-t_i)}{1-Y^{-{\alpha }_i^{\vee }}}={\left(T_i^{\vee }\right)}^{-1}+\frac{t_i^{-\frac{1}{2}}(1-t_i)Y^{-{\alpha }_i^{\vee }}}{1-Y^{-{\alpha }_i^{\vee }}}\text{.}& \text{(2.34)}\end{array}$$

Using that the ${\tau }_i^{\vee }$ satisfy the braid relations and that

$$g^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{g^{\vee }{\lambda }^{\vee }}{g}^{\vee },\text{write}{\tau }_w^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{w{\lambda }^{\vee }}{\tau }_w^{\vee },\text{for}w\in W\text{.}$$

Let $w\in W$ and let $w=s_{i_1}^{\vee }\dots s_{i_{\ell }}^{\vee }$ be a reduced word for $w.$ For $k=1,\dots ,\ell$ let

$$\begin{array}{cc}{\beta }_k^{\vee }=s_{i_{\ell }}^{\vee }{s}_{i_{\ell -1}}^{\vee }\dots s_{i_{k+1}}^{\vee }{s}_{i_k}^{\vee }\text{and}{t}_{{\beta }_k^{\vee }}=t_{i_k},& \text{(2.35)}\end{array}$$

so that the sequence ${\beta }_{\ell }^{\vee },{\beta }_{\ell -1}^{\vee },\dots ,{\beta }_1^{\vee }$ is the sequence of labels of the hyperplanes crossed by the walk $w^{-1}=s_{i_{\ell }}^{\vee }{s}_{i_{\ell -1}}^{\vee }\dots s_{i_1}^{\vee }\text{.}$ For example, in Type $A_2,$ with $w=s_2^{\vee }{s}_0^{\vee }{s}_1^{\vee }{s}_2^{\vee }{s}_1^{\vee }{s}_0^{\vee }{s}_2^{\vee }{s}_1^{\vee }$ the picture is

$$\[ 1 \] \[ w-1 \] \[ {𝔥}^{{\beta }_7^{\vee }} \] \[ {𝔥}^{{\beta }_5^{\vee }} \] \[ {𝔥}^{{\beta }_8^{\vee }} \] \[ {𝔥}^{{\beta }_6^{\vee }} \] \[ {𝔥}^{{\beta }_4^{\vee }} \] \[ {𝔥}^{{\beta }_2^{\vee }} \] \[ {𝔥}^{{\beta }_3^{\vee }} \] \[ {𝔥}^{{\beta }_1^{\vee }} \]$$

Let $v\in W^{\vee }\text{.}$ An alcove walk of type $i_1,\dots ,i_{\ell }$ beginning at $v$ is a sequence of steps, where a step of type $j$ is

$$\begin{array}{cccc} \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]\\ \text{positive}j\text{–crossing}& \text{negative}j\text{–crossing}& \text{positive}j\text{–fold}& \text{negative}j\text{–fold}\end{array}$$

Let $ℬ(v,\overrightarrow{w})$ be the set of alcove walks of type $\overrightarrow{w}=(i_1,\dots ,i_{\ell })$ beginning at $v.$ For a walk $p\in ℬ(v,\overrightarrow{w})$ let

$$\begin{array}{cc}\begin{array}{ccc}{f}^{+}\left(p\right)& =& \{k\mid \text{the}k\text{th step of}p\text{is a positive fold}\},\\ f^{-}\left(p\right)& =& \{k\mid \text{the}k\text{th step of}p\text{is a negative fold}\},\end{array}& \text{(2.36)}\end{array}$$

and

$$\begin{array}{cc}\text{end}\left(p\right)=\text{endpoint of}p\text{(an element of}W\text{).}& \text{(2.37)}\end{array}$$

Let $v,w\in W,$ let $w=s_{i_1}^{\vee }\dots w=s_{i_{\ell }}^{\vee }$ be a reduced word for $w$ and let ${\beta }_{\ell }^{\vee },\dots ,{\beta }_1^{\vee }$ be as defined in (2.35). Then, in $\stackrel{\sim }{H},$

$$X^v{\tau }_w^{\vee }=\sum _{p\in ℬ(v,\overrightarrow{w})}{X}^{\text{end}\left(p\right)}\left(\prod _{k\in f^{+}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-1/2}(1-t_{{\beta }_k^{\vee }})}{1-Y^{-{\beta }_k^{\vee }}}\right)\left(\prod _{k\in f^{-}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-1/2}(1-t_{{\beta }_k^{\vee }})Y^{-{\beta }_k^{\vee }}}{1-Y^{-{\beta }_k^{\vee }}}\right),$$

where the sum is over all alcove walks of type $\overrightarrow{w}=(i_1,\dots ,i_{\ell })$ beginning at $v\text{.}$

		Proof.
	The proof is by induction on the length of $w,$ the base case being the formulas in (2.34). To do the induction step let $p\in ℬ(v,\overrightarrow{w}),$ $$F^{+}\left(p\right)=\left(\prod _{k\in f^{+}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-1/2}(1-t_{{\beta }_k^{\vee }})}{1-Y^{-{\beta }_k^{\vee }}}\right),F^{-}\left(p\right)=\left(\prod _{k\in f^{-}\left(p\right)}\frac{t_{{\beta }_k^{\vee }}^{-1/2}(1-t_{{\beta }_k^{\vee }})Y^{-{\beta }_k^{\vee }}}{1-Y^{-{\beta }_k^{\vee }}}\right)$$ and let $$p_1,p_2\in ℬ(v,\overrightarrow{w}{s}_j)\text{be the two extensions of}p\text{by a step of type}j$$ (by a crossing and a fold, respectively). Let $z=\text{end}\left(p\right)\text{.}$ By induction, a term in $X^v{\tau }_w^{\vee }{\tau }_j^{\vee }$ is $$\begin{array}{c}{X}^z{F}^{+}\left(p\right)F^{-}\left(p\right){\tau }_j^{\vee }=X^z{\tau }_j^{\vee }\left(s_j{F}^{+}\left(p\right)\right)\left(s_j{F}^{-}\left(p\right)\right)\\ =\{\begin{array}{ccc}{X}^z(T_j^{\vee }+\frac{t_j^{-1/2}(1-t_j)}{1-Y^{-{\alpha }_j^{\vee }}})\left(s_j{F}^{+}\left(p\right)\right)\left(s_j{F}^{-}\left(p\right)\right),& & \text{if}{X}^{z{s}_j}=X^z{T}_j^{\vee },\\ X^z({\left(T_j^{\vee }\right)}^{-1}+\frac{t_j^{-1/2}(1-t_j)Y^{-{\alpha }_j^{\vee }}}{1-Y^{-{\alpha }_j^{\vee }}})\left(s_j{F}^{+}\left(p\right)\right)\left(s_j{F}^{-}\left(p\right)\right),& & \text{if}{X}^{z{s}_j}=X^z{\left(T_j^{\vee }\right)}^{-1},\end{array}\\ =X^{\text{end}\left(p_1\right)}{F}^{+}\left(p_1\right)F^{-}\left(p_1\right)+X^{\text{end}\left(p_2\right)}{F}^{+}\left(p_2\right)F^{-}\left(p_2\right)\text{.}\end{array}$$ The last step of $p_2$ is $$\begin{array}{ccccc} \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \text{if}{X}^{z{s}_j}=X^z{T}_j^{\vee },& \text{and}& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \text{if}{X}^{z{s}_j}=X^z{\left(T_j^{\vee }\right)}^{-1},\end{array}$$ $\square$

In some special cases when the affine root system is nonreduced (see [15, (2.1.6)]) the formulas in (2.32) and (2.34) need modification and the definition of the double affine Hecke algebra may need an additional relation. The most involved of these cases is type $(C_n^{\vee },C_n)$ (see [15, (1.4.3)]) where the double affine Hecke algebra needs additional parameters $u_0^{1/2}$ and $u_n^{1/2}$ (in the notation of [15], $u_0^{1/2}={\tau }_0^{\prime }$ and $u_n^{1/2}={\tau }_n^{\prime }$) and additional relations

$$\begin{array}{cc}(T_0^{\prime }-u_0^{\frac{1}{2}})(T_0^{\prime }+u_0^{-\frac{1}{2}})=0\text{and}(T_0^{\vee }-u_n^{\frac{1}{2}})(T_0^{\vee }+u_n^{-\frac{1}{2}}),\text{where}{T}_0^{\prime }=q^{1-\frac{1}{2}}{X}^{{\epsilon }_1}{T}_0^{-1},& \text{(2.38)}\end{array}$$

and the formulas for ${\tau }_n^{\vee }$ and ${\tau }_0^{\vee }$ need to be changed to

$$\begin{array}{cc}{\tau }_n^{\vee }=T_n+\frac{t_n^{-\frac{1}{2}}(1-t_n)+t_0^{-\frac{1}{2}}(1-t_0)Y^{-{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}}=T_n^{-1}+\frac{(t_n^{-\frac{1}{2}}(1-t_n)+t_0^{-\frac{1}{2}}(1-t_0)Y^{{\epsilon }_n^{\vee }})Y^{-2{\epsilon }_n^{\vee }}}{1-Y^{-2{\epsilon }_n^{\vee }}}& \text{(2.39)}\end{array}$$

and

$$\begin{array}{cccc}{\tau }_0^{\vee }& =& T_0^{\vee }+\frac{u_n^{-\frac{1}{2}}(1-u_n)+u_{-}^{-\frac{1}{2}}(1-u_0)q^{\frac{1}{2}}{Y}^{{\epsilon }_1^{\vee }}}{1-q^{-1}{Y}^{-2{\epsilon }_1^{\vee }}}& \text{(2.40)}\\ & =& {\left(T_0^{\vee }\right)}^{-1}+\frac{(u_n^{-\frac{1}{2}}(1-u_n)+u_{-}^{-\frac{1}{2}}(1-u_0)q^{\frac{1}{2}}{Y}^{-{\epsilon }_1^{\vee }})q^{-1}{Y}^{-2{\epsilon }_1^{\vee }}}{1-q^{-1}{Y}^{-2{\epsilon }_1^{\vee }}}\text{.}& \text{(2.41)}\end{array}$$

The statement and proof of the analogue of Theorem 2.2 for this case is the same, except with the factors associated to the 0-folds and $n$-folds replaced by the rational functions in $Y$ which appear in the expressions of ${\tau }_0^{\vee }$ and ${\tau }_n^{\vee }$ in Equations (2.40), (2.41), and (2.39).

Notes and References

This page is taken from a paper entitled A combinatorial formula for Macdonald Polynomials by Arun Ram and Martha Yip.

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