Remarks on Gelfand Pairs Associated to Non-Type-I Solvable Lie Groups

✍ Scribed by Katsuhiko Kikuchi

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 323 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3350

✦ Synopsis

dedicated to professor takeshi hirai on his 60th birthday Let S be a connected and simply connected unimodular solvable Lie group and K a connected compact Lie group acting on S as automorphisms. We call the pair (K ; S) a Gelfand pair if the Banach V-algebra L 1 K (S) of all K-invariant integrable functions on S is a commutative algebra. In this paper we give a necessary and sufficient condition for the pair (K ; S) to be a Gelfand pair using the representation theory of non-type-I solvable Lie groups. For a Gelfand pair (K ; S) we realize all irreducible K-spherical representations of K _ S from irreducible unitary representations of S.

1998 Academic Press of S in K is closed. For k # K ? there exists a unitary operator W ? (k) on the representation space H ? of ? by ?(k } x)=W ? (k)?(x)W ? (k) &1 for any x # S. In general W ? is a projective representation of K ? not an ordinary representation. We call W ? the intertwining representation. If K ? is article no.