Bounded operators on direct limit of direct sums of spaces of cusp forms

Question

Consider $S_{2k} (\Gamma_0 (N))$ and let $S(N)$ denote the direct limit of the finite direct sums of the $S_{2k}$. Since each $S_{2k} (\Gamma_0 (N))$ is also a Hilbert space w.r.t. the Petersson inner product, $S(N)$ is as well, and we can consider the C*-algebra $S^*(N)$ of bounded operators on $S(N)$.

So has anyone actually considered this thing yet? If not, can someone comment on the possible relevance of using the Riemann-Hurwitz formula (giving the dimensions of the $S_{2k}$) to describe $S^*(N)$ as the limit of the Bratteli diagrams of the aforementioned finite direct sums? (Since $S^*(N)$ is approximately finite-dimensional, this seems like a natural thing to do...)

(NB. I asked this question on sci.math.research back in the nineties and never got an answer, and figured I'd give it another shot here given the MO userbase. Anything ridiculous here should be blamed on my younger self.)

Answer 1

That colimit of finite-dimensional spaces won't actually be a Hilbert space, but it will nevertheless be quasi-complete. Still, it won't be a representation space for $GL(2,R)$ or $GL(2,Q_p)$, which is what has taken people in other directions, specifically, to look at (for example) $L^2$ completions of spaces of automorphic forms of all levels. It is true that in most applications one restricts attention to "K-finite" vectors in that space, which, at finite places, returns us to a fixed level.

The Atkin-Lehner "newform" idea, recast in representation theory by Casselman, and extended to GL(n) by Jacquet-PiatetskiShapiro-Shalika, strikes me as a singularly noteworthy way to distinguish special vectors in the resulting representations.

Answer 2

Hi, Steve. Just ran across this blog while updating arxiv.org/abs/math/0511361.
While it's not quite related to your question, Gelfand suggested in 1950's that the automorphic forms are nothing but irreducible (infinite dimensional) representations of the Lie groups, like the upper half-plane H or adele GL(n). This is a well established industry and the Langlands program is a part of it. -- Igor