Whittaker functions, crystal bases and quantum groups

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 June 2010

Combinatoritcs and spherical functions

The Weyl group $W_0$ acts on ${𝔥}_{\mathbb{Z}}^{*}.$ $W_0$ has generators $s_1,\dots ,s_n$ with relations $$s_i^2=1\text{ and }{s}_i{s}_j{s}_i\dots =\left(m_{ij}\text{ factors}\right)=s_j{s}_i{s}_j\dots =\left(m_{ij}\text{ factors}\right),$$ where $\frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i^{\vee }}\nless {𝔥}^{{\alpha }_j^{\vee }}.$ Let $$u_0=\sum _{w\in W_0}w\text{ and }{e}_0=\sum _{w\in W_0}{\left(-1\right)}^{l\left(w\right)}w\text{ in }ℂW_0,$$ so that $w{u}_0=u_0$ and $e_{0}w={\left(-1\right)}^{l\left(w\right)}w.$

The Hecke algebra $H_0$ has generators $T_1,\dots ,T_n$ with $$T_i^2=\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)T_i+1\text{ and }{T}_i{T}_j{T}_i\dots \left(m_{ij}\text{ factors}\right)=T_j{T}_i{T}_j\dots \left(m_{ij}\text{ factors}\right)$$

Let $T_w=T_{i_1}\dots T_{i_l}$ if $w=s_{i_1}\dots s_{i_l}$ is reduced, $$\left\{T_w|w\in W_0\right\}\text{ is a basis of }{H}_0.$$ Let $$1_0=\sum _{w\in W_0}{\left(t^{\frac{1}{2}}\right)}^{l\left(w\right)}{T}_w\text{ and }{\epsilon }_0=\sum _{w\in W_0}{\left(-t^{-\frac{1}{2}}\right)}^{l\left(w\right)}{T}_w,$$ so that $$T_w{1}_0={\left(t^{\frac{1}{2}}\right)}^{l\left(w\right)}{1}_0\text{ and }{\epsilon }_1{T}_w={\left(-t^{-\frac{1}{2}}\right)}^{l\left(w\right)}{\epsilon }_0.$$

Symmetric functions

$$\left\{\begin{array}{r}{𝔥}_{\mathbb{Z}}\\ {𝔥}_{\mathbb{Z}}^{*}\end{array}\right.\text{ dual vector spaces, }⟨,⟩:{𝔥}_{\mathbb{Z}}^{*}\times {𝔥}_{\mathbb{Z}}\to \frac{1}{e}\mathbb{Z},$$ $W_0$ a finite subgroup of $\mathrm{GL}\left({𝔥}_{\mathbb{Z}}\right)$ generated by reflections. Then $W_0$ acts on the group algebra $$K_{T^{\vee }}\left(\rho t\right)=\mathrm{span}\left\{y^{{\lambda }^{\vee }}|{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\right\}$$ with $y^{{\lambda }^{\vee }}{y}^{{\sigma }^{\vee }}=y^{{\lambda }^{\vee }+{\sigma }^{\vee }}\text{ by }w{y}^{{\lambda }^{\vee }}=y^{w{\lambda }^{\vee }}.$ The algebra of symmetric functions is $$K_{T^{\vee }}{\left(\rho t\right)}^{W_0}=\left\{f\in K_{T^{\vee }}\left(\rho t\right)|wf=f,\text{ for all }w\in W_0\right\}.$$ Then $$K_{T^{\vee }}{\left(\rho t\right)}^{\det }=\left\{f\in K_{T^{\vee }}\left(\rho t\right)|wf=\det \left(w\right)f,\text{ for all }w\in W_0\right\}$$ is a free $K_{T^{\vee }}{\left(\rho t\right)}^{W_0}$-module of rank 1.

$K_{T^{\vee }}{\left(\rho t\right)}^{W_0}$ and $K_{T^{\vee }}{\left(\rho t\right)}^{\det }$ have bases $$m_{{\lambda }^{\vee }}={\pi }_0{y}^{{\lambda }^{\vee }}\text{ and }{\alpha }_{{\lambda }^{\vee }+{\rho }^{\vee }}={\epsilon }_0{y}^{{\lambda }^{\vee }},{\lambda }^{\vee }\in {𝒫}^{+}=\frac{{𝔥}_{\mathbb{Z}}}{W_0}$$ where $${\pi }_0=\sum _{w\in W_0}w\text{ and }{\epsilon }_0=\sum _{w\in W_0}\det \left(w^{-1}\right)w.$$

Weyl character formula

$$\begin{array}{rcl}{K}_{T^{\vee }}{\left(\rho t\right)}^{W_0}& \to & K_{T^{\vee }}{\left(\rho t\right)}^{\det }\\ f& \mapsto & a_{\rho }f\\ m_{{\lambda }^{\vee }}\text{(naive basis)}& & \\ s_{{\lambda }^{\vee }}& \leftarrow & a_{{\lambda }^{\vee }+{\rho }^{\vee }}\text{ (naive basis)}\end{array}$$ $m_{{\lambda }^{\vee }}$ are the monomial symmetric functions.

$s_{{\lambda }^{\vee }}$ are the Weyl characters or Schur functions.

Note: $$K_{T^{\vee }}{\left(\rho t\right)}^{W_0}=K_{G^{\vee }}\left(\rho t\right)=K\left(G^{\vee }\text{-modules}\right)$$ and $$s_{{\lambda }^{\vee }}\text{ are the classes of the simple modules.}$$

Affine Hercke algebra $H$ and Gindikin-Karpelevich

$ℂ\left[X\right]=\mathrm{span}\left\{X^{\lambda }|\lambda \in {𝔥}_{\mathbb{Z}}^{*}\right\}\text{ and }{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu }.$$H$ is actually generated by subalgebras $H_0$ and $ℂ\left[X\right]$ with $T_i{X}^{\lambda }=X^{s_i\lambda }{T}_i+\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-X^{-{\alpha }_i}}\left(X^{\lambda }-X^{s_i\lambda }\right).$

Intertwines ${\tau }_i$ are ${\tau }_i{X}^{\lambda }=X^{s_i\lambda }{\tau }_i,$$${\tau }_i={{\rm T}}_i-\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-X^{-{\alpha }_i}}\text{ and }{{\rm T}}_i={\tau }_i+\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-X^{-{\alpha }_i}}.$$ In this language, Gindikin-Karpelevich is $$1_0=\sum _{w\in W_0}{\tau }_w\prod _{\alpha \in {ℛ}^{+},w\alpha \in {ℛ}^{-}}\left(t^{\frac{1}{2}}\right)\left(\frac{1-t^{-1}{X}^{-\alpha }}{1-X^{-\alpha }}\right)$$ $${\epsilon }_0=\sum _{w\in W_0}{\tau }_w\prod _{\alpha \in {ℛ}^{+},w\alpha \in {ℛ}^{-}}\left(-t^{\frac{1}{2}}\right)\left(\frac{1-t{X}^{-\alpha }}{1-X^{-\alpha }}\right)$$ see Prop 3.15 in the thesis of Martha Yip. $$\left\{T_w{X}^{\lambda }|\lambda \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0\right\}\text{ is a basis of }H.$$

Casselamn-Shalika

Case $q=0,t=0;$ Hermann Weyl; $G\left(ℂ\right)$$$\begin{array}{rcl}{u}_0ℂ\left[X\right]=ℂ{\left[X\right]}^{W_0}& \tilde{\to }& ℂ{\left[X\right]}^{\det }={\epsilon }_0ℂ\left[X\right]\\ m_{\lambda }=u_0{X}^{\lambda }& & \\ s_{\lambda }=\text{Weyl character/Schur function}& \leftarrow & a_{\lambda +\rho }={\epsilon }_0{X}^{\lambda +\rho }\\ h& \mapsto & a_{\rho }h\text{(Weyl denominator/ Vandermonde det)}\end{array}$$

Case $q=0;$ Lusztig; $G\left(ℂ\left(\left(t\right)\right)\right)$ $$\begin{array}{rclll}ℂ{\left[X\right]}^{W_0}=Z\left[H\right]& \to & 1_{0}H{1}_0& \tilde{\to }& {\epsilon }_{0}H{1}_0\\ {𝒫}_{\lambda }\left(0,t\right)& \leftarrow & 1_0{X}^{\lambda }{1}_0& & \\ s_{\lambda }& \leftarrow & C_{\lambda }& \leftarrow & A_{\lambda +\rho }={\epsilon }_0{X}^{\lambda +\rho }{1}_0\\ & & h& \mapsto & A_{\rho }h\end{array}$$$$H{1}_0=ℂ\left[X\right]1_0=\text{polynomial representation}$$$${𝒫}_{\lambda }\left(0,t\right)=\text{Hall-Littlewood polynomial=Macdonald spherical function}$$$$A_{\rho }=\prod _{\alpha \in {ℛ}^{+}}\left(t^{\frac{1}{2}}{X}^{\frac{\alpha }{2}}-t^{-\frac{1}{2}}{X}^{-\frac{\alpha }{2}}\right)={\left(t^{\frac{1}{2}}\right)}^{l\left(w_0\right)}{X}^{\rho }\prod _{\alpha \in R^{+}}\left(1-t^{-1}{X}^{-\alpha }\right)$$ (see arXiv0401298 with Nelson.)

Double affine Hecke algebra $\tilde{H}$

$$\tilde{H}=\mathrm{span}\left\{q^{\frac{k}{e}}{X}^{\mu }{T}_w{Y}^{{\lambda }^{\vee }}|k\in \mathbb{Z},\mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0,{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\right\}\supseteq H^{\vee }=\mathrm{span}\left\{X^{\mu }{T}_w|\mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0\right\}\supseteq H_0=\mathrm{span}\left\{T_w|w\in W_0\right\}$$ with $$q^{\frac{1}{2}}\in Z\left(\tilde{H}\right),X^{\mu }{X}^{\nu }=X^{\mu +\nu },Y^{{\lambda }^{\vee }}{Y}^{{\sigma }^{\vee }}=Y^{{\lambda }^{\vee }+{\sigma }^{\vee }}.$$ $H_0$ is generated by $T_1,\dots ,T_n$ with $$T_i^2=\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)T_i+1\text{ and }{T}_i{T}_{j}Ti\dots \left(m_{ij}\text{ factors}\right)=T_j{T}_{i}Tj\dots \left(m_{ij}\text{ factors}\right)$$ where $\frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i}\nless {𝔥}^{{\alpha }_j}.$

$H^{\vee }$ has a unique 1-dimensional module $\mathrm{span}\left(1\right)$ with $T_{i}1=t^{\frac{1}{2}}1$ for $i=1,\dots ,n.$ The polynomial representation of $\tilde{H}$ is $$\begin{array}{rcl}{\mathrm{Ind}}_{H^{\vee }}^{\tilde{H}}\left(1\right)& =& \tilde{H}1=\mathrm{span}\left\{q^{\frac{k}{e}}{Y}^{{\lambda }^{\vee }}1|k\in \mathbb{Z},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\right\}\\ & =& K_{T^{\vee }}\left(\rho t\right)1.\end{array}$$

Macdonald polynomials

Let $1_0,{{\rm E}}_0\in H_0$ be such that $$1_0{T}_i=t^{\frac{1}{2}}{1}_0\text{ and }{{\rm E}}_0{T}_i=\left(-t^{-\frac{1}{2}}\right){{\rm E}}_0$$ for $1,\dots ,n.$ At $t=1,$$1_0={\pi }_0$ and ${\epsilon }_0={{\rm E}}_0.$ Then $$K_{T^{\vee }}\left(\rho t\right)1=\tilde{H}1\supseteq 1_0\tilde{H}1=K_{T^{\vee }}{\left(\rho t\right)}^{W_0}1.$$ The nonsymmetric Macdonald polynomial $${𝒫}_{{\lambda }^{\vee }}={𝒫}_{{\lambda }^{\vee }}\left(q,t\right)$$ is given by $${𝒫}_{{\lambda }^{\vee }}1=1_0{E}_{{\lambda }^{\vee }}1.$$ Define $A_{{\lambda }^{\vee }+{\rho }^{\vee }}=A_{{\lambda }^{\vee }+{\rho }^{\vee }}\left(q,t\right)$ in $K_{T^{\vee }}\left(\rho t\right)$ by $$A_{{\lambda }^{\vee }+{\rho }^{\vee }}1={{\rm E}}_0{E}_{{\lambda }^{\vee }+{\rho }^{\vee }}1.$$

Case $q,t;$ Cherednik-Opdam-Macdonald; $G\left(ℂ\left(\left(t,s\right)\right)\right)?$

Let $\tilde{H}$ be the double affine Hecke algebra $$\left\{X^{\mu }{T}_w{Y}^{{\lambda }^{\vee }}|\mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0,{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\right\}\text{ is a basis of }\tilde{H}.$$ Let $1$ be such that $T_{w}1={\left(t^{\frac{1}{2}}\right)}^{l\left(w\right)}1$ and $Y^{{\lambda }^{\vee }}1=t^{⟨{\lambda }^{\vee },\rho ⟩}1$$$\begin{array}{rclll}ℂ{\left[X\right]}^{W_0}1=1_0\tilde{H}1& & & & {\epsilon }_1\tilde{H}1\\ {𝒫}_{\lambda }\left(q,t\right)1& \leftarrow & 1_0{E}_{\lambda }1& & \\ {𝒫}_{\lambda }\left(q,qt\right)1& \leftarrow & \leftarrow & \leftarrow & A_{\lambda +\mu }\left(q,,,t\right)={\epsilon }_0{E}_{\lambda +\mu }1\\ & & h& \mapsto & A_{\rho }h\end{array}$$$$\tilde{H}=`ℂ\left[X\right]1=\text{polynomial representation of }\tilde{H}$$$$E_{\lambda }=E_{\lambda }\left(q,t\right)=\text{nonsymmetrical Macdonald polynomial }$$$$A_{\rho }=\prod _{\alpha \in {ℛ}^{+}}\left(t^{\frac{1}{2}}{X}^{\frac{\alpha }{2}}-t^{-\frac{1}{2}}{X}^{-\frac{\alpha }{2}}\right),\text{ see Prop 2.13 in Yip's thesis.}$$ (see Macdonald Seminaire Bourbaki 1995).

Big picture

$$\begin{array}{rcl}{K}_{T^{\vee }}{\left(\rho t\right)}^{W_0}1=1_0\tilde{H}1& \to & {{\rm E}}_0\tilde{H}1\\ f1& \mapsto & A_{{\rho }^{\vee }}\left(q,t\right)f1\\ 1_0{E}_{{\lambda }^{\vee }}1={𝒫}_{{\lambda }^{\vee }}\left(q,t\right)1& & \\ {𝒫}_{{\lambda }^{\vee }}\left(q,qt\right)1& \leftarrow & A_{{\lambda }^{\vee }+{\rho }^{\vee }}\left(q,t\right)1={{\rm E}}_0{E}_{{\lambda }^{\vee }+{\rho }^{\vee }}1\end{array}$$ At $q=0$ this picture becomes $$\begin{array}{rcl}{K}_{T^{\vee }}{\left(\rho t\right)}^{W_0}1=1_{0}H{1}_0& \to & {{\rm E}}_{0}H{1}_0\\ f{1}_0& \mapsto & A_{{\rho }^{\vee }}\left(0,t\right)f{1}_0\\ 1_0{Y}^{{\lambda }^{\vee }}{1}_0={𝒫}_{{\lambda }^{\vee }}\left(0,t\right)1_0& & \\ s_{{\lambda }^{\vee }}{1}_0{𝒫}_{{\lambda }^{\vee }}\left(0,0\right)1_0& \leftarrow & A_{{\lambda }^{\vee }+{\rho }^{\vee }}\left(0,t\right)1={{\rm E}}_0{Y}^{{\lambda }^{\vee }+{\rho }^{\vee }}1\end{array}$$ where $H=\mathrm{span}\left\{T_w{Y}^{{\lambda }^{\vee }}|w\in W_0,{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\right\}$

At $q=0,t=1,$ this becomes $$\begin{array}{rcl}{K}_{T^{\vee }}{\left(\rho t\right)}^{W_0}& \to & K_{T^{\vee }}{\left(\rho t\right)}^{\det }\\ f& \mapsto & a_{{\rho }^{\vee }}f\\ {\pi }_0{Y}^{{\lambda }^{\vee }}=m_{{\lambda }^{\vee }}& & \\ s_{{\lambda }^{\vee }}& \leftarrow & a_{{\lambda }^{\vee }+{\rho }^{\vee }}={\epsilon }_0{Y}^{{\lambda }^{\vee }+{\rho }^{\vee }}\end{array}$$

Remarks

1. At $q\ne 0,$$Z\left(\tilde{H}\right)$ is trivial. ($Z\left(\tilde{H}\right)=ℂ\left[q^{\pm \frac{1}{e}}\right]$).
   At $q=0,Z\left(\tilde{H}\right)$ is big, and contains $$K_{T^{\vee }}{\left(\rho t\right)}^{W_0}=Z\left(H\right)\text{(theorem of Bernstein).}$$
2. The Satake isomorphism is $$\begin{array}{rcl}{1}_{0}H{1}_0& & K_{T^{\vee }}{\left(\rho t\right)}^{W_0}{1}_0\\ 1_0{Y}^{{\lambda }^{\vee }}{1}_0& \mapsto & {𝒫}_{{\lambda }^{\vee }}\left(0,t\right)1_0\end{array}$$ and ${𝒫}_{{\lambda }^{\vee }}\left(0,t\right)$ is the Macdonald spherical function or Hall-Littlewood polynomial.
3. $H\cong$ Grothendieck ring (product is convolution) of $I$ equivariant perverse sheaves on $G/I.$
   $1_{0}H{1}_0=$ Grothendieck ring of $K$ equivariant perverse sheaves on $G/K.$
   $G/I=$ affine flag variety, $G/K=$ Loop Grassmanian
   $\left\{s_{{\lambda }^{\vee }}{1}_0\right\}$ is the Kazhdan-Lusztig basis of $1_{0}H{1}_0,$ ie $s_{{\lambda }^{\vee }}1$ is the image of $IC\left(K{h}_{{\lambda }^{\vee }}\left(t^{-1}\right)K,\stackrel{\_}{\mathbb{Q}}\right)$.
4. $$\left.\begin{array}{r}{K}_{T^{\vee }}{\left(\rho t\right)}^{\det }\\ {{\rm E}}_0\tilde{H}1\\ {{\rm E}}_{0}H1\end{array}\right\}\text{are "Fock spaces"}$$ and $1_0\tilde{H}1\cong {{\rm E}}_0\tilde{H}1$ are 'boson-Fermion correspondences'. The big picture at $q=0$ is a 1981 paper of Lusztig which started "geometric Langlands".
5. In $\tilde{H},$ $$T_i{X}^{\mu }=X^{s_i\mu }+\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)\frac{X^{\mu }-X^{s_i\mu }}{1-X^{{\alpha }_i}}\text{(Bernstein-Lusztig relation)}$$ is equivalent to $${\tau }_i{X}^{\mu }=X^{s_i\mu }{\tau }_i,\text{ where }$$$${\tau }_i=T_i+\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-X^{{\alpha }_i}}=T_i^{-1}+\frac{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)X^{{\alpha }_i}}{1-X^{{\alpha }_i}}\text{(intertiner)}$$

If $Y^{{\lambda }^{\vee }}=s_{i_1}\dots s_{i_l}$ is a minimal length walk to $Y^{{\lambda }^{\vee }}$ in $W,$ then, in $\tilde{H},$ $$Y^{{\lambda }^{\vee }}=T_{i_1}^{{\epsilon }_1}\dots T_{i_l}^{{\epsilon }_l}$$ where $${\epsilon }_k=1$$ if the $k$th step is
and where $${\epsilon }_k=-1$$ if the $k$th step is
and $$E_{{\lambda }^{\vee }}={\tau }_{i_1}\dots {\tau }_{i_l}1.$$

Using folded alcove walks this can be exapnded to give a formula $$E_{{\lambda }^{\vee }}=\sum _{\text{folded alcove paths }p}\left(\text{explicit coefficients}\right)y^{\mathrm{end}\left(p\right)},$$ which has similar coefficients to the Haglund-Haiman-Loehr formula for $E_{{\lambda }^{\vee }}$ in type ${\mathrm{GL}}_n$ and generalises $$s^{{\lambda }^{\vee }}=\sum _{\text{column strict tableaux }p}{y}^{\mathrm{wt}\left(p\right)}=\sum _{\text{Littelmann paths }p}{y}^{\mathrm{end}\left(p\right)}$$ and the positively folded walks labeling points in MV intersections $IwI\cap U_0^{-}vI.$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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