Strongly stable characters and Theorem 1.29

Abstract

We begin by recalling Langlands’ notion of stable characters. This seems to make sense only for linear groups, so we will not try to define it in the setting of (11.1). Suppose therefore that we are in the setting of Definition 10.3, and that η is a finite-length canonical projective representation of type z of a strong real form δ of G ^Γ. Thus η is in particular a representation of G(ℝ, δ)^can. The character of η is a generalized function Θ(η) on G(ℝ, δ)^can, defined as follows. Suppose f is a compactly supported smooth density on G(ℝ, δ)^can. Then η(f) is a well-defined operator on the space of η. (If we write f as a compactly supported smooth function times a Haar measure, f(g)dg, then η(f) is given by the familiar formula

$$ \eta (f) = \int_{{G{{\left( {\mathbb{R},\delta } \right)}^{{can}}}}} {f(g)\eta (g)dg.)} $$

The operator η(f) is trace class, and the value of the generalized function Θ(η) on the test density f is by definition the trace of this operator:

$$ \Theta \left( \eta \right)(f) = tr\left( {\eta (f)} \right) $$

((18.1)(a))

Of course we can immediately define the trace of any virtual canonical projective representation (cf. (15.5)):

$$ \Theta :K{\Pi^z}\left( {G\left( {\mathbb{R},\delta } \right)} \right) \to \left( {{\text{generalized functions on G}}{{\left( {\mathbb{R},\delta } \right)}^{{can}}}} \right). $$

((18.1)(b))

. This map is injective (since characters of inequivalent irreducible representations are linearly independent).