Question: special orbit, distinguished zero orbit: sign representation is NOT obtained by truncated induction from any proper levi sign -> sign*sign=trivial -> principal orbit, which is special -> sign is special... Note: sigma_s = TFInd(sigmaL_s) = TGInd(sigmaL_s) this is true if you mean keep pieces of the induced with the same degree (fake or generic) as the one on L [ false if you mean keep pieces of the induced with minimal (fake or generic) degree... ] DV DISAGREES WITH THE "false" part; and see Jeff's addenda below... ------------------------------------------------------------------- set truncate_degree_induce_character \ ( WeylClassTable Wct_L , CharacterTable ct_G , [int] pi_L , (int->int) degree_function ) = (int,[int]): let embed(WeylElt w) = WeylElt: convert_to(ct_G.root_datum,w) in truncate_induce_character(Wct_L,ct_G,embed,degree_function,pi_L) for degree function: use ct_G.degree or ct_G.generic_degree example: set truncate_by_generic_degree_induce_character \ ( CharacterTable ct_L, CharacterTable ct_G, [int] pi_L) = [int]: ( let (,t) = truncate_degree_induce_character(ct_L.class_table,ct_G,pi_L,ct_G.generic_degree) in {assert(#t=1,"not a unique character in truncated induction");}{allow multiple values} t ) ------------------------------------------------------------------- Note added in proof: Here is an explanation of that "false" assertion, which itself is, I believe, false. In the algorithm I sketched to compute sigma->sigma_s each step involves induction, truncated by generic degree, OR the same thing followed by tensoring with sign. (Here truncated means: keep the lowest degree (fake or generic) terms, or equivalently the same (fake or generic) degree.) If none of the sign operations arise, the final result is that sigma_s is obtained by truncated induction (generic or fake) from sigma_s. However any sign terms that come up along the way (of the recursive algorithm) change fake and generic degrees. I *think* the following statements are true. 1) Given (L,sigma_L) consider ind=ind(sigma_L) (induced from W(L) to W(G)). The terms of ind with minimal generic degree are the same as those with generic degree = generic_degree(sigma_L). The same hold with "generic_degree" in place of degree. 2) Given sigma, the algorithm produces a pair (L,sigma_s_L). Then sigma_s is either: the unique summand of Ind_L^G(sigma_s_L) with the same generic and fake degrees as sigma, or [the unique summand of Ind_L^G(sigma_s_L) with the same generic and fake degrees as sigma\otimes sign]\otimes sign BTW I'm ignoring the 3 (?) exceptional orbits in E7/E8. Probably this isn't quite right but hopefully close.