special_rep.at

W-rep sigma -> special W-rep sigma_s

1) sigma =Springer(O,chi) -> Springer(O_s) where O is in the special piece of O_s

2) different algorithm (that doesn't require Springer(O,chi), which we don't currently have)

Theorem (Lusztig).

1a) Suppose L is a Levi in G, and CL is a left cell of W(L). Then TGInd(CL) is a left cell for W.

b) Suppose C is a left cell in W. Then C\otimes sgn is also a left cell in W.

c) If W \ne 1, then for any left cell C in W, there is a proper L and a left cell CL in L so that
  EITHER TGInd(CL) = C, OR TGIND(CL\otimes sgn) = C\otimes sgn.

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So here is the algorithm for finding the special W-rep sigma_s attached to sigma.

-1) if sigma is special return sigma (this step isn't necessary but speeds up the algorithm)
0)  If W is trivial, then sigma_s = sigma = trivial. Assume W not trivial.
1)  Try to find a proper Levi L and a W(L)-rep sigmaL so that sigma is contained in TGInd(sigmaL);
    if you succeed, then sigma_s = TFInd(sigmaL_s) = TGInd(sigmaL_s).
2)  If you fail, replace sigma by sigmaSgn = sigma\otimes sgn. Step 1) is guaranteed to succeed for sigmaSgn, giving sigmaSgn_s.
3)  The dual special orbit to sigmaSgn_s (found by the duality algorithm) has Springer rep sigma_s.

{note on 3:  sigma -> sigmaSgn -> sigmaSgn_s -> orbit_sgn -> dual(orbit_sgn) -> sigma_s
except in exceptional cases in E7/E8: sigma_s=sigmaSgn_s\otimes sgn}

given G, tables=all_levi_Springer_tables(G), int char_G
the algorithm produces
[(L_n,table_n,char_n),...(L_1,table_1,char_1),(G,0,char_G)
where each table_j,char_j is an integer
start with tables[table_n].characters[char_n], this is special
j-induction takes you