Subject: Re: question abt Lie gps From: Tom Goodwillie Date: Fri, 18 Mar 2005 22:10:12 -0500 To: Don Davis > > > Let G be a compact Lie group and N a closed normal subgroup. > Assume that V is a given finite dimensional real N-representation. > Can one find a finite dimensional real G-representation W and an > N-linear embedding V->W, such that the N-fixed points of W equals > the N-fixed points of V? I believe so. You can immediately reduce to the case in which V has no N-fixed points except 0, by splitting off the fixed part and dealing with it separately. Now if G is a finite group then you can take W to be the induced representation and use characters and Frobenius reciprocity to compute dimension of fixed points. Let X be the character of (N,V). Induce to G and restrict to N. If (dimension of N-fixed part of V) = = 0, then (dimension of N-fixed part of W) = = = = = 0. I would guess that something like this works for compact groups (producing an infinite-dimensional W that will contain a finite-dimensional one that does the job), but I'm a little hazy about induced representations.