Embeddings

•A given representation
(p,V)
may have several different models
of representations

•Different models may reveal different information.

•Main example: all representations of G=GL(n,R) embed into principal series
representations (pl,d,Vl,d):

–V = { f : G! C j f(gb) = f(g) c-1(b) } , [p(h)f](g) = f(h-1g)

–Here b 2 B = lower triangular Borel subgroup,

c(b)
= cl,d(b) = Õ |bj|(n+1)/2
- j - lj sgn(bj)dj ,

and bj are the diagonal elements of the matrix b.

•(__Casselman-Wallach Theorem__) Embedding extends equivariantly to distribution vectors:

V-1 embeds into Vl,d-1 = {s 2 C-1(G) j s(gb) = s(g)c-1(b)}

as a closed subspace.

V-1 embeds into Vl,d-1 = {s 2 C-1(G) j s(gb) = s(g)c-1(b)}

as a closed subspace.