§1. Notation In this section we set up notation. See [28] and [32] for general references. Let H be a connected real linear reductive Lie group, with real Lie algebra f)o and complexified Lie algebra f). Since H is contained in a complex Lie group, we denote by HQ the connected complex Lie subgroup with Lie algebra rj. The center of the universal enveloping algebra U(t)) is written as 3(f))- In what follows analogous notation will be applied to Lie groups denoted by other Roman upper case letters without comment. We will use the standard notation N, Z, R, C and H. Here N means the set of non-negative integers and HI means the M-algebra of quarternionic numbers. We denote by N_|_ the set of positive integers. For x 6 K, we write [x] := sup{n £ Z n #}, the Gaussian integer of x. 1. 0-stable parabolic subalgebra Let G be a connected real linear reductive Lie group. Let K C G be a maximal compact subgroup and fix a Cartan involution 0 so that go £Q + Po 1S a Cartan decomposition of 0o Fix a nondegenerate bilinear form (, ) on g invariant under G and 0, which is positive definite on po and negative definite on £0- This form will be restricted to subspaces and transferred to dual vector spaces without change of notation. If the restriction of (, ) to each subspace a, b with a C b is non-degenerate, we look upon (1.1) a* C b* through this bilinear form. 13
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