We call ([[PI].sub.(v,d)], [H.sub.(v,d)) a principal series representation of G.
Theorem 6.2 in [JL] tells that any irreducible admissible infinite-dimensional representation of G is isomorphic to some irreducible principal series representation of G as ([g.sub.C], K)-modules.
Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) be an irreducible principal series representation of G with v = ([v.sub.1], [v.sub.2]) [member of] [C.sup.2], d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA].
Principal series representations. For v = ([v.sub.1], [v.sub.2]) [member of] [C.sup.2] and d = ([d.sub.1], [d.sub.2]) [member of] [Z.sup.2], let [H.sup.[infinity].sub.(v,d)] be the space of smooth functions f on G such that, for [t.sub.1], [t.sub.2] [member of] [C.sup.x], x [member of] C and g [member of] G,
Let ([[PI].sub.(v,d)], [H.sub.(v,d)]) and ([[PI].sub.(v',d')], [H.sub.(v',d')]) be irreducible principal series representations of G with v =([v.sub.1], [v.sub.2]) [member of] [C.sup.2], d = ([d.sub.1], [d.sub.2]) [member of] [LAMBDA], v' = ([v'.sub.1], [v'.sub.2]) [member of] [C.sup.2] and d' = ([d'.sub.1], [d'.sub.2]) [member of] [LAMBDA].
Authors Bruggeman, Lewis, and Zagier present students, academics, researchers, and professional mathematicians working in a wide variety of contexts with an examination of mixed parabolic cohomology gropus and semi-analytic vectors in
principal series representation. The authors have organized the main body of their text in six chapters devoted to Eigenfunctions of the hyperbolic Laplace operator, Maas forms and analytic cohomology for cocompact groups, cohomology of infinite cycle subgroups, Maass formas and semi-analytic cohomolgy of groups with cusps, and a wide variety of other related subjects.