Analytic and Entire Vectors for Representations of Lie Groups

Complex Fourier analysis on a nilpotent Lie group

“…Corollary 3.1 of [5]), which we state and prove in §1. This reduces questions about analytic vectors for G to consideration of analytic vectors for the one-parameter subgroups {exp tX,}.…”

“…Under the stated assumptions on v, we conclude from Proposition 4.1 of [5] that v e Jf %ir)(^A> lay azi. Hence by Theorem 1.1 we have v e 3^f(-rr).…”

“…We omit the details, and instead, to establish the special nature of the nilpotent case, we show in §5 that the space of entire vectors for the left regular representation is dense in L2(G), when G is a simply-connected nilpotent group. By the results of [5] this fails to hold for large classes of noncompact, nonnilpotent Lie groups.…”

“…If this holds for all r, then v will be an entire vector. Conversely, iff is an entire vector, then by Corollary 2.1 of [5] the function X->(7r(exp X)v, v) on g extends holomorphically to gc. Q.E.D.…”

“…Let{JQ, 1 újúd, be an ordered Jordan-Holder basis for g, i.e. if If 77 is a unitary representation of G on a Hilbert space ^(77), let <?f "(77) denote the space of analytic vectors for w, and let the subspaces 2^"i(ir)'^M''a(rr), f>0, be defined as in [5] relative to the basis {X,) (cf. §1).…”