parameter p -> x -> involution(x) = w\delta (w\delta)^2=1 w\delta(w)=1 (twisted involution) reported by KGB Def: support(x)={simple reflections in reduced expression of w} (expression not !, but braid relations support(x) well defined} support(p)=support(p.x) support(p) is proper => p real cohomologically induced from proper parabolic Note: J(p)=coh. induced. from J(P_L) Vogan dual parameter of p: w -> w_0w support(p-dual) proper => p real parabolically induced proper parabolic I(p)=ind (I(p_L)) but J(p) \ne ind J(p_L) Question: cell -> find p support(p) proper OR support(p-dual) proper ------------------------------------------------------------------- Sp(4)xSp(4)xSp(4) Z=Z_2^3 How many subgroups does (Z_2)^n have? How many subspaces does the vectors space (Z_2)^n have? q-binomial coefficient (n\choose m)_q = # of m-dimensional subspaces of F_q^n q-analogues of n/choose m... run over all subgroups A of Z_2^3: G/A