Multiplicity formulas for representations

Abstract

Our next goal is Theorem 1.24 of the introduction, relating the geometric invariants discussed in Chapters 7 and 8 to representation theory. We begin by discussing a little more carefully the definition of the representation-theoretic multiplicity and character matrices (cf. (1.21)). For the same reasons as in Chapter 11, we work at first in the setting of (11.1). Recall from Definition 11.13 the set L ^z(G_ℝ) of equivalence classes of final limit characters of type z . For Λ ∈ L ^z (G _ℝ), write

$$ M\left( \wedge \right) = {\text{standard limit representation attached}}\;{\text{to}}\; \wedge $$

((15.1)(a))

(cf. (11.2)), and

$$ \pi \left( \wedge \right) = {\text{Langlands quotient of}}\;M\left( \wedge \right). $$

((15.1)(b))

(We leave open the question of which form of the representation to use — Harish-Chandra module or some topological version. Several reasonable possibilities are discussed below.) By Theorem 11.14, π(Λ) is irreducible, and this correspondence establishes a bijection

$$ {L^z}\left( {{G_{\mathbb{R}}}} \right) \leftrightarrow {\Pi^z}\left( {{G_{\mathbb{R}}}} \right),\;\; \wedge \leftrightarrow \pi \left( \wedge \right). $$

((15.1)(c))