I know that the deligne kusztig varieties corresponding to Suzuki group, Ree group and PGU-_2'(q) are explictly computed. Are there any result for the group G(2). Here 'result' means equation of these varieties

It would help to include a reference for the special cases you quote. Aside from that, is it clear that explicit descriptions of these varieties by polynomials will make it easier to apply them to character theory or such? For instance, describing symplectic or othogonal groups as zero sets of polynomials reveals very little about their structure or representations. Polynomials defining an affine variety are far from unique in any case.
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Jim HumphreysFeb 29 '12 at 0:29

the first three case the corresponding varieties are all maximal curves, and to compute genus of them, the equations are helpful. Sorry the equation mean here is in the function field of deligne -lusztig curve X(w) where w is an element of order 2
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gaussFeb 29 '12 at 5:50

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The clarification helps, but a reference to the known results would also be helpful.
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Jim HumphreysMar 1 '12 at 0:03

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I disagree with Jim's comments on the usefulness of having explicit polynomials. Nowadays you can feed these to software systems and get cohomologies, local cohomologies associated to stratifications and such, all directly applicable to representation theory.
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Reimundo HeluaniMar 11 '12 at 11:08