Free group representations from vector-valued multiplicative functions, I

Free group representations from vector-valued multiplicative functions. III

“…If the two systems are inequivalent (Q a +Q * a −1 ) must be zero by Remark 3.4 in [13]. If the two systems are equivalent one can replace Q a by Qa = 1 2 (Q a − Q * a −1 ).…”

“…In [13] it is proved that any irreducible matrix system admits, up to a normalization, a unique (up to scalar multiples) tuple of strictly positive definite forms (B a ) satisfying (3).…”

“…In [13] it is proved that two multiplicative representations arising from irreducible matrix systems are equivalent as Γ ⋉ C(Ω) representations if and only if the two systems are equivalent; hence for irreducible Γ ⋉ C(Ω) representations the two definitions 2.3 and 2.4 are equivalent. Obviously equivalent systems give raise to equivalent Γ-representations, but the converse is no longer true as we shall see in Section 4.…”

“…The principal series representations considered by Figà-Talamanca and Picardello and Figà-Talamanca and Steger [5], [6] satisfy duplicity, except for the two at the endpoints, which satisfy monotony. Examples of irreducible representations satisfying oddity can be found in the papers of Paschke (see [14], [15]) and Example 6.5 of [13].…”

“…In [13] the first and third authors constructed a class of multiplicative representations. This class is large enough to contain more or less all previously constructed tempered irreducible representations whose construction uses the action of Γ on its Cayley graph.…”