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Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ denote the Lie algebra of the maximal torus. Let $W=N_G(T)/T$ be the Weyl group. The Chevalley Restriction Theorem states that the the natural map $\mathbb{C}[\mathfrak{g}]\rightarrow\mathbb{C}[\mathfrak{t}]$ restricts to a $\mathbb{C}$-algebra isomorphism $$\mathbb{C}[\mathfrak{g}]^G\xrightarrow{\cong}\mathbb{C}[\mathfrak{t}]^W.$$ The latter is generated by $r=rk(G)$ algebraically independent polynomials $f_1,\ldots,f_r$.

Now, one has the root-space decomposition $$\mathfrak{g}=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}.$$ For each root $\alpha\in\Delta$, choose a non-zero root vector $e_{\alpha}\in\mathfrak{g}_{\alpha}$. If $\alpha\in\Pi$ is a (positive) simple root, set $h_{\alpha}:=[e_{\alpha},e_{-\alpha}]\in\mathfrak{t}$. Note that $$\{e_{\alpha}\}_{\alpha\in\Delta}\cup\{h_{\alpha}\}_{\alpha\in\Pi}$$ is then a basis of $\mathfrak{g}$. Let $$\{e_{\alpha}^*\}_{\alpha\in\Delta}\cup\{h_{\alpha}^*\}_{\alpha\in\Pi}$$ be the associated dual basis of $\mathfrak{g}^*$.

$\textbf{Question}:$ Is there a way to choose the generators $f_1,\ldots,f_r$ so that each is nicely expressible as a polynomial in the indeterminates $\{e_{\alpha}^*\}_{\alpha\in\Delta}\cup\{h_{\alpha}^*\}_{\alpha\in\Pi}$?

This question is rather vague, but I'm really just looking for explicit expressions of the $f_1,\ldots,f_r$ in terms of the $\{e_{\alpha}^*\}_{\alpha\in\Delta}\cup\{h_{\alpha}^*\}_{\alpha\in\Pi}$.

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  • $\begingroup$ I'm not optimistic about finding explicit expressions here (or in the closely related case of generators for the center of the universal enveloping algebra). There are quite a few related questions on MO such as mathoverflow.net/questions/182871 (Note that homogeneous generators of the invariants on your right side are already tricky to make explicit.) In any case you may want to use the Killing form to identify $\mathfrak{g}$ with its dual, and may also want to use a Chevalley basis. Everything here works uniformly over any alg closed field of char 0. $\endgroup$ Oct 26, 2014 at 15:56

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