CHAPTER 1
Background and Notation
Let C denote the complex numbers, Q denote the rational numbers, Z denote
the integers, R denote the real numbers, N denote the nonnegative integers, and q
denote an indeterminate. Let
{qr\
r G Q} denote the multiplicative group isomor-
phic to Q under addition. Write C(Q) for the quotient field of the group algebra
of
{qr\
r G Q} over C. Similarly, let R(Q) denote the quotient field of the group
algebra of
{qr\
r G Q} over R. Note that R(Q) can be made into an ordered field
in a manner similar to the rational function field H(q) (see [Ja, Section 5.1]) Write
C for the algebraic closure of C(Q) and let IZ denote the real algebraic closure of
R(Q).
Suppose that $ is a root system. Write Q(&) for the root lattice of $ and let
Q+(I) be the subset of Q($) equal to the N span of the positive roots in $. Let
P(3) denote the weight lattice of $ and let P + ( $ ) be the subset of P(3) consisting
of dominant integral weights. (Sometimes we replace I with the symbol used to
represent the positive simple roots in the notation for the weight and root lattices
and their subsets.)
Let g be a complex semisimple Lie algebra with triangular decomposition g =
n~0l)®n
+
. Let A denote the root system of g and let ( , ) denote the corresponding
Cartan inner product on A. Here we assume that the positive roots of A correspond
to the root vectors in n + . Write TT = {c*i,..., an} for the positive simple roots of
A. We use "" to denote the usual partial order on ()*. In particular, given a and
/3 in ()*, we say that a (3 if and only if (3 a G
Q+{IT).
Let {e^, hi, fi\l i n}
be a standard set of generators for g. Here, e* is a root vector in n
+
corresponding
to the simple root c^, fi is a root vector in n~~ corresponding to the root c^, and
hi,..., hn is a basis of coroots for f).
Classical (infinitesimal) symmetric pairs: Let 0 be a Lie algebra involution
on g and write
ge
for the fixed Lie subalgebra of g with respect to 0. The pair
g,
ge
is a classical (infinitesimal) symmetric pair. We assume throughout the paper
that g,
ge
is an irreducible symmetric pair. The results of this paper extend in a
straightforward manner to the general case. More precisely, a symmetric pair g,
is called irreducible provided that g cannot be written as a direct sum of semisimple
Lie algebras which both admit 0 as an involution. The classification of involutions
and classical irreducible symmetric pairs up to isomorphism can be found in [A]
(see also [L4, Section 7].)
Recall that we have already specified a particular Cartan subalgebra f) of g.
We assume that 0 is a maximally split involution with respect to \) in the sense of
[L3, Section 7]. In particular, 9 is maximally split with respect to \) provided the
following three conditions hold.
(1.1) 9{\}) = h
7
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