We let M _ 2 ( \ mathbb { C } ) denote the 2x2 matrices over \ mathbb { C }, U denote the upper triangular unipotent matrices ( that is, matrices with'1's on the diagonal, a zero below and anything above ), and T denote the matrices of the form \ operatorname { Diag } ( t, t ^ {-1 } ).
42.
If " G " is a split reductive group and " U " is the unipotent radical of a Borel subgroup " B ", then a Whittaker model for a representation is an embedding of it into the induced ( Gelfand Graev ) representation Ind ( ? ), where ? is a non-degenerate character of " U ", such as the sum of the characters corresponding to simple roots.
43.
David Roberts provided a natural combinatorial link between the Artin Hasse exponential and the regular exponential in the spirit of the ergodic perspective ( linking the " p "-adic and regular norms over the rationals ) by showing that the Artin Hasse exponential is also the generating function for the probability that an element of the symmetric group is unipotent in characteristic " p ", whereas the regular exponential is the probability that an element of the same group is unipotent in characteristic zero.
44.
David Roberts provided a natural combinatorial link between the Artin Hasse exponential and the regular exponential in the spirit of the ergodic perspective ( linking the " p "-adic and regular norms over the rationals ) by showing that the Artin Hasse exponential is also the generating function for the probability that an element of the symmetric group is unipotent in characteristic " p ", whereas the regular exponential is the probability that an element of the same group is unipotent in characteristic zero.
45.
The underlying reduced scheme of the geometric kernel is isomorphic to the additive group " G " " a " and is the unipotent radical of the geometric fiber of " G ", but this reduced subgroup scheme of the geometric fiber is not defined over " k " ( i . e ., it does not arise from a closed subscheme of " G " over the ground field ) and the unipotent " k "-radical is trivial.
46.
The underlying reduced scheme of the geometric kernel is isomorphic to the additive group " G " " a " and is the unipotent radical of the geometric fiber of " G ", but this reduced subgroup scheme of the geometric fiber is not defined over " k " ( i . e ., it does not arise from a closed subscheme of " G " over the ground field ) and the unipotent " k "-radical is trivial.
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