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, 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