The Harish-Chandra homomorphism

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

Last updates: 5 October 2010

The Harish-Chandra homomorphism

The Weyl group $W_0$ act on $𝔥\mathbb{Z}$ (the lattice of weights of $𝔤$) and ${𝔥}_{\mathbb{Z}}^{*}$ (the lattice of coweights) and the evaluation pairing $$⟨,⟩:{𝔥}_{\mathbb{Z}}^{*}\times {𝔥}_{\mathbb{Z}}\to \mathbb{Z}\text{given by}⟨\mu ,{\lambda }^{\vee }⟩=\mu \left({\lambda }^{\vee }\right),$$ is $W_0$-invariant, $$⟨w\mu ,w{\lambda }^{\vee }⟩=⟨\mu ,{\lambda }^{\vee }⟩\text{for} w\in W_0,\mu \in {𝔥}_{\mathbb{Z}}^{*} \text{and} \lambda \in {𝔥}_{\mathbb{Z}}.$$

The envelping algebra $U𝔤$ has a triangular decomposition $U𝔤=U{𝔫}^{-}U𝔥U{𝔫}^{+}$ coming from the triangular decomposition $𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$ and $$U𝔥=S(𝔥)=ℂ\left[h_1,\dots ,h_r\right],$$ where $h_1,\dots ,h_{\mathrm{r}}$ is a basis of $𝔥$. Thus $U𝔥$ is, conveneniently, a polynomial ring.

For $\mu \in {𝔥}^{*}$, the ring homomorphisms defined by

$$\begin{array}{rccc}{\sigma }_{\mu }:& U𝔥& \to & U𝔥\\ & h& \mapsto & h+⟨\mu ,h⟩\end{array}\text{and}\begin{array}{rccc}{ev}_{\mathrm{<mo>\mapsto </mo>}}:& U𝔥& \to & ℂ\\ & h& \mapsto & ⟨\mu ,h⟩\end{array}$$

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are automorphisms and character of $U𝔥$ respectively. Additionally, the action of $W_0$ on $𝔤$ give automorphisms of $U𝔤$ and

$${\sigma }_{\mu }{\sigma }_{\nu }={\sigma }_{\mu +\nu },{ev}_{\mu }={ev}_0{\sigma }_{\mu },{ev}_{w\mu }={ev}_{\mu }\circ w^{-1},{\sigma }_{w\mu }=w{\sigma }_{\mu }{w}^{-1},$$

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for $\mu \in {𝔥}^{*}$ and $w\in W_0$.

Define a vector space homomorphism by

$${\pi }_0:U𝔤\to U𝔥\text{by}{\pi }_0={\epsilon }^{-}\otimes id\otimes {\epsilon }^{+}:U{𝔫}^{-}\otimes U𝔥\otimes U{𝔫}^{+}\to U𝔥,$$

pi0

where ${\epsilon }^{-}:U{𝔫}^{-}\to ℂ$ and ${\epsilon }^{+}:U{𝔫}^{+}\to ℂ$ are the algebra homomorphisms determined by

$${\epsilon }^{-}\left(y\right)=0\text{and}{\epsilon }^{+}\left(x\right)=0\text{for} x\in {𝔫}^{+} \text{and} y\in {𝔫}^{-}.$$

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The following important theorem says that the center of $U$ is isomorphic to the ring of symmetric functions.

(Harish-Chandra/Chevalley isomorphism) [Bou, VII §8 no. 5 Thm. 2] Let ${\pi }_0:U\to U𝔥$ and $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$ be as in (pi0), and (5.18) respectively. The restriction of ${\pi }_0$ to the center of $U𝔤$, $$\begin{array}{rrcl}{\pi }_0:& Z(U𝔤)& \to & {\sigma }_{\rho }\left(ℂ{[h_1,\dots ,h_r]}^{W0}\right)\\ & z& \mapsto & {\sigma }_{\rho }\left(s\right)\end{array}$$ is an algebra isomorphism, where $s\in ℂ{[h_1,\dots ,h_r]}^{W0}$ is the symmetric funcion determined by $$z \text{acts on} L(\mu ) \text{by} {ev}_{\mu }\left({\sigma }_{\rho }\right(s\left)\right)={ev}_{\mu +\rho }\left(s\right)$$ for all $\mu$.

The quantum group $U=U_q𝔤$ has a triangular decomposition $U=U^{-}{U}^0{U}^{+}$ coming from the triangular decomposition $𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$ and $$U^0={span}_{ℂ}\left\{K^{{\mu }^{\vee }}\mid {\mu }^{\vee }\in {𝔥}_{\mathbb{Z}}\right\},\text{with}{K}^{{\mu }^{\vee }}{K}^{{\nu }^{\vee }}=K^{{\mu }^{\vee }+{\nu }^{\vee }}$$ is the group algebra of ${𝔥}_{\mathbb{Z}}$. If ${\omega }_1^{\vee },\dots ,{\omega }_r^{\vee }$is a $\mathbb{Z}$-basis of ${𝔥}_{\mathbb{Z}}$ and $$L_i=K^{{\omega }_i^{\vee }},\text{then}{U}^0=ℂ\left[L_1^{\pm 1},\dots ,L_r^{\pm 1}\right]$$ so $U^0$ is, conveniently, a Laurent polynomial ring.

For $\mu \in {𝔥}_{\mathbb{Z}}^{*}$, the ring homomorphisms defined by

$$\begin{array}{cccc}{\sigma }_{\mu }:& U^0& \to & U^0\\ & K^{{\lambda }^{\vee }}& \mapsto & K^{w{\lambda }^{\vee }}\end{array}\text{and}\begin{array}{cccc}{ev}_{\mu }:& U^0& \to & ℂ\\ & K^{{\lambda }^{\vee }}& \mapsto & q^{⟨\mu ,{\lambda }^{\vee }⟩}\end{array}$$

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are automorphisms and characters of $U^0$, respectively. Additionally, there are automorphisms,

$${\sigma }_{\mu }{\sigma }_{\nu }={\sigma }_{\mu +\nu },{ev}_{\mu }={ev}_0{\sigma }_{\mu },{ev}_{w\mu }={ev}_{\mu }\circ w^{-1},{\sigma }_{w\mu }=w{\sigma }_{\mu }{w}^{-1},$$

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for $\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and $w\in W_0$.

Define a vector space homomorphism by

$${\pi }_0:U\to U^0\text{by}{\pi }_0={\epsilon }^{-}\otimes id\otimes {\epsilon }^{+}:U^{-}\otimes U^0\otimes U^{+}\to U^0,$$

pihom

where ${\epsilon }^{-}:U^{-}\to ℂ$ and ${\epsilon }^{+}:U^{+}\to ℂ$ are the algebra homomorphisms determined by

$${\epsilon }^{-}\left(F_i\right)=0\text{and}{\epsilon }^{+}\left(E_i\right)=0,\text{for} i=1,\dots ,n.$$

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The following important theorem is the quantum group analogue of [Bou, VII §8 no. 5 Thm. 2]. It says that the center of $U$ is isomorphic to the ring of symmetric functions

$$ℂ{\left[L_1^{\pm 1},\dots ,L_r^{\pm 1}\right]}^{W0}=\left\{f\in ℂ\left[L_1^{\pm 1},\dots ,L_r^{\pm 1}\right]\mid wf=w \text{for} w\in W_0\right\}.$$

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REFER ALSO TO Joseph?, Jantzen?, Chari-Pressley?

(Harish-Chandra/Chevalley isomorphism) ${\pi }_0:U\to U^0$ and $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$ be as in (pihom), and (5.18) respectively. The restriction of ${\pi }_0$ to the center of $U$, $$\begin{array}{rrcl}{\pi }_0:& Z\left(U\right)& \to & {\sigma }_{\rho }\left(ℂ{\left[L_1^{\pm 1},\dots ,L_r^{\pm 1}\right]}^{W0}\right),\\ & z& \mapsto & {\sigma }_{\rho }\left(s\right)\end{array}$$ is an algebra isomorphism, where $s\in ℂ{\left[L_1^{\pm 1},\dots ,L_r^{\pm 1}\right]}^{W0}$ is the symmetric funcion determined by $$z \text{acts on} L(\mu ) \text{by} {ev}_{\mu }\left({\sigma }_{\rho }\right(s\left)\right)={ev}_{\mu +\rho }\left(s\right)$$ for all $\mu$.

References

[Bou] N. Bourbaki Groupes et Algèbres de Lie, Masson, Paris, 1990.

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)

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