1.1. Preliminaries
LECTURE 1
Basic concepts
Let F be a number field. For each place v of F, let Fv be the completion of Fat v.
For each finite v, let Ov be the ring of integers of Fv and denote by Pv its maximal
ideal. Let Wv be a generator of Pv. Let qv = [Ov: Pv] and fix an absolute valueI Iv such that lwv lv = q;;^1.
Let G b e a split connected reductive algebraic group over F. This simply means
a Zariski closed subgroup of GLN(F) for some N, where Fis the algebraic closure
of F , whose radical (maximal normal connected solvable subgroup) consists of only
semisimple elements. The radical is then equal to the connected component of
the center of G. Being split simply means that G has a maximal abelian subgroup
consisting entirely of diagonalizable elements, a maximal torus, which is isomorphic
over F to some power of p*.
Let B be a Borel subgroup of G (over F), i.e., a maximal connected solvable
subgroup of G. Let T be a maximally split torus of G contained in B. Then
B = TU, where U is the unipotent radical of B. The unipotent subgroup U
determines a set of simple roots ~ and positive roots R+ for T , upon acting on the
Lie algebra g of G.
Let P be a parabolic subgroup of G , i.e., a conjugate of a closed subgroup
of G containing B. We will assume P is standard by P :J B. We let N be the
uni potent radical (maximal connected normal uni potent subgroup) of P. Then
P = MN, where M is a reductive subgroup, called a Levi subgroup. We will fix
M by assuming T c M. Let A be the split component of M, i.e., the connected
component of the center of M (the maximally split subtorus of the center of M, if
the group is not necessarily split over F). The parabolic subgroup P is maximal if
the dimension of A/ A n ZG is one, where Z G is the center of G. Then the adjoint
action of A on the Lie algebra of N h as a unique reduced eigenfunction a, the
simple root of A in N. There exists a unique simple root of T whose restriction
to A is a. We will denote this root of T also by a and always identify them with
each other. Other roots of A in N are simply multiples (considered additively) of
a. Throughout these lectures P is always assumed to be maximal. We refer to
[B2,Sat,Sp] as our main references for algebraic groups and their structure t heory.
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