Orbital Symmetric Spaces and Finite Multiplicity

Abstract

It is proven that any algebraic symmetric space has finite multiplicity. The multiplicity is then shown to be bounded provided that is the case for reductive symmetric spaces (a widely-believed, but as yet unproven fact). The method of proof involves an amalgamation of the Mackey Machine and the Orbit Method. A necessary and sufficient orbital condition is also derived for the quasi-regular representation of a symmetric space to be irreducible.

Finite multiplicity theorems for induction and restriction

“…Let G be real algebraic (not necessarily reductive), σ an involution and H = G σ . R. Lipsman proved that the multiplicity of the abstract Plancherel formula for G/H is uniformly bounded under the hypothesis that this statement is true in the reductive case([23, Theorem 7.3]). Theorem B shows that his hypothesis is true because there always exists an open H c -orbit on G c /B for any complex reductive symmetric pair (G c , H c ).…”

Attributes and applications of the Corwin-Greenleaf multiplicity function

“…Remark. In 30] it is proven that n = n n ( ), whence in this scenario we actually have n = n . Thus the CG multiplicity function describes the bounded multiplicity for algebraic solvable symmetric spaces.…”