Polynomial invariants of the exceptional Weyl groups Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}^*)$ be the ring of polynomial functions on $\mathfrak{h}$. The Weyl group $W$ acts on $\mathfrak{h}$, and this action extends to an action of $W$ on $S(\mathfrak{h}^*)$. It is a well-known fact that the space of Weyl group invariants $S(\mathfrak{h}^*)^W$ is generated by $r$ algebraically independent homogeneous generators, where $r$ is the dimension of $\mathfrak{h}$ (equivalently, the rank of $\mathfrak{g}$). The degrees of the generators are uniquely determined, though the actual generators themselves are not.
The degrees of the generators for $S(\mathfrak{h}^*)^W$ are well-known and can be found, for example, in Humphreys' book "Reflection groups and Coxeter groups" (Section 3.7). When $\mathfrak{g}$ is of classical type (ABCD), it is also not hard to find explicit examples of generators for $S(\mathfrak{h}^*)^W$ (loc. cit. Section 3.12).

Where, if anywhere, can I find explicit examples of generators for $S(\mathfrak{h}^*)^W$ when $\mathfrak{g}$ is of exceptional type, specifically, for types $E_7$ and/or $E_8$?

I have found explicit examples for types $E_6$ and $F_4$ in a paper by Masaru Takeuchi (On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I  9  1962 313--328 (1962)). I have probably also come across examples for type $G_2$, though I don't recall where at this moment. But I have been unable to find anything for types $E_7$ or $E_8$.
 A: Like Jim, I've had some trouble deciphering some of the older references in the physics literature. Instead, I've recently come across a paper that I probably should have found in my initial reference search. The paper "Invariant polynomials of Weyl groups and applications to the centres of universal enveloping algebras" by C.Y. Lee (Canad. J. Math., Vol. XXVI, No. 3, 1974, pp.583-592) gives for each Lie type an explicit formula for computing generators for the ring $S(\mathfrak{h})^W$. One can then apply the $W$-invariant isomorphism $S(\mathfrak{h}) \rightarrow S(\mathfrak{h}^*)$ coming from the Killing form to obtain an explicit set of generators for $S(\mathfrak{h}^*)^W$.
A: In the paper “Flat Bases of Invariant Polynomials and P-matrices of E7 and E8”, V. Talamini, Journal of Mathematical Physics 51, 023520-1-023520-20 (2010 AIP), I reported explicitly the bases of invariant polynomials for the weyl groups E7 and E8, written in terms of the 7 or 8 variables, respectively (not in the article itself but in a file at FTP directory /epaps/journ_math_phys/E-JMAPAQ-51-060912/ at ftp.aip.org). 
Of the many bases of invariant polynomials that one can write some are called flat and in the article are reported the flat bases, but frome these one may recover any other basis. The basis transformation connecting the flat bases reported in the article with the bases suggested by Mehta in Commun. in Algebra (1988) are also given in the article.
V. Talamini
A: Keeping in mind that a generating set of invariant polynomials having the required degrees is not unique, various computations have been recorded in the literature.   Those I was aware of before 1990 are listed in the references to section 3.12 of my book, but there may have been others I overlooked.   A later paper of interest discusses "canonical" choices of generators, with details for the classical cases as well as dihedral groups (including $G_2$):   
MR1469638 (98j:13007) 13A50 (20F55),
Iwasaki, Katsunori (J-KYUS),
Basic invariants of finite reflection groups. 
J. Algebra 195 (1997), no. 2, 538–547.
Physicists usually look for very explicit expressions, though their notation and approach may be hard for mathematicians to decipher.   Their interest comes from the direction of Casimir operators as Jose points out in his literature citations.   But those operators live in the center of the universal enveloping algebra, which by Harish-Chandra is isomorphic to the Weyl group invariants asked about here.   The complication is that expressions for Casimir operators get much more elaborate-looking in terms of the Lie algebra notation.   (Also, the reflection group theory shows that polynomial invariants and degrees play a uniform role even in 
non-crystallographic cases like $H_3$ and $H_4$ as well as dihedral groups which are not Weyl groups.)
ADDED:  I'd emphasize that writing suitable generators (Casimir operators) for the center of $U(\mathfrak{g})$ should involve a choice of PBW or other basis, though the initial approach might not start with such a basis but rather with the Killing form.  However this is done (non-uniquely), it takes some care to realize from these operators a set of basic polynomial invariants for the Weyl group.    The latter calculation by itself can be done much more straightforwardly, though hardly anyone has taken the trouble to write down (for example) a basic invariant polynomial in 8 variables of degree 30 for $E_8$.  The 1988 paper by M.L. Mehta in Communications in Algebra seems to be a good attempt at giving a comprehensive treatment.   Unfortunately, the journal itself is not so easy to access, and my own copy of the paper photographed from typescript by the journal is barely readable.   
I have had less success in deciphering the physics literature, which may or may not all be mathematically reliable.  In particular, I haven't yet reached any conclusions about what is in the JMP paper cited by Jose.   (That journal is sometimes quite useful but can also be quite frustrating to extract information from for mathematical purposes.)   My only experience has been with the literature on finite (mostly real) reflection groups and their invariants, where the degrees themselves are most important for most applications.   One concrete source I should mention is the added Chapter 7 in the second edition of Grove-Benson Finite Reflection Groups (GTM 99, Springer, 1985).   Their book was first developed as an advanced undergraduate text, then expanded somewhat, and gives more details than my book --- where for instance I left the computation of basic invariants for dihedral groups as an exercise.
A: In Physics these are called the "Casimir operators" and googling this gives the following paper: F. Berdjis and E. Beslmüller Casimir operators for $F_4$, $E_6$, $E_7$ and $E_8$.  For the case of $G_2$, see the paper in my comment above: Casimir operators for the exceptional group $G_2$ by A.M. Bincer and K. Riesselmann.
A: For the record, these invariants (or rather, the ideals of positive-degree invariants) also come up in the Borel presentation of the cohomology ring of the flag manifold G/B, so one can find generators whenever people have computed these rings.  For instance, the preprint "The integral cohomology ring of $E_8/T^1{\cdot}E_7$" by Masaki Nakagawa (2009) gives completely explicit polynomials for E7 in Proposition 2.1 and for E8 in Lemma 2.3.  I can't vouch for their correctness -- a nontrivial computational matter, as Jim remarked -- but the calculations are written out in some detail.
A: Appendices 1 and 2 to the preprint http://arxiv.org/pdf/alg-geom/9202002v1.pdf by Katz-Morrison, now published in J. Alg. Geom., give explicit generating sets in type $E_6$ and $E_7$. There is also some information on basic invariants in type $E_8$ in Appendix 0, but no explicit formulae (for the obvious reason). 
