Fourier analysis for compact groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 April 2010

Fourier analysis for compact groups

A function $f:G\to ℂ$ is

1. representative if there is a finite dimensional representation $V$ of $G$ and vectors $v,w\in V$ such that $f\left(g\right)=⟨v,gw⟩$ for all $g\in G.$
2. square integrable if $${\parallel f\parallel }_2^2={\int }_{G}f\left(g\right)$$ f g dμ g <∞.
3. smooth if all derivatives exist.
4. real analytic if $f$ has a power series expansion at each point. $$\begin{array}{rcl}C{\left(G\right)}^{\mathrm{rep}}& =& \left\{\text{representative functions }f:G\to ℂ\right\},\\ L^2\left(G\right)& =& \left\{\text{square integrable functions }f:G\to ℂ\right\},\\ C^{\infty }\left(G\right)& =& \left\{\text{smooth functions }f:G\to ℂ\right\},\\ C^{\omega }\left(G\right)& =& \left\{\text{real analytic functions }f:G\to ℂ\right\}.\end{array}$$

We have a map $$\prod _{\lambda \in \hat{G}}{M}_{d_{\lambda }}\left(ℂ\right)\to \text{functions }f:G\to ℂ.$$

The set $\hat{G}$ has a norm $\parallel .\parallel :\hat{G}\to {ℝ}_{\ge 0}.$ For $\left({\hat{f}}^{\lambda }\right)\in \prod _{\lambda \in \hat{G}}{M}_{d_{\lambda }}\left(ℂ\right)$ define

1. $\left({\hat{f}}^{\lambda }\right)$ is finite if all but a finite number of the blocks ${\hat{f}}^{\lambda }$ in $\left({\hat{f}}^{\lambda }\right)$ are 0,
2. $\left({\hat{f}}^{\lambda }\right)$ is square summable if $$\sum _{\lambda \in \hat{G}}\frac{1}{d_{\lambda }}{\parallel f^{\lambda }\parallel }^2<\infty .$$
3. $\left({\hat{f}}^{\lambda }\right)$ is rapidly decreasing if, for all $k\in {\mathbb{Z}}_{>0},\left\{{\parallel \lambda \parallel }^k\parallel {\hat{f}}^{\lambda }\parallel |\lambda \in \hat{G}\right\}$ is bounded,
4. $\left({\hat{f}}^{\lambda }\right)$ is exponentially decreasing if, for some $K\in {ℝ}_{>1},\left\{K^{\parallel \lambda \parallel }\parallel {\hat{f}}^{\lambda }\parallel |\lambda \in \hat{G}\right\}$ is bounded.

Under the map $$\begin{array}{rcl}\left\{\text{functions }f:G\to ℂ\right\}& \to & \prod _{\lambda \in \hat{G}}\\ C{\left(G\right)}^{\mathrm{rep}}& \mapsto & \left\{\text{finite }\left({\hat{f}}^{\lambda }\right)\right\}\\ L^2\left(G,\mu \right)& \mapsto & \left\{\text{square summable }\left({\hat{f}}^{\lambda }\right)\right\}\\ C^{\infty }\left(G\right)& \mapsto & \left\{\text{rapidly decreasing }\left({\hat{f}}^{\lambda }\right)\right\}\\ C^{\omega }\left(G\right)& \mapsto & \left\{\text{exponentially decreasing }\left({\hat{f}}^{\lambda }\right)\right\}\end{array}$$

The space $C{\left(G\right)}^{\mathrm{rep}}$ is dense in $C\left(G\right)$ and $C\left(G\right)\subseteq L^2\left(G\right).$ In fact the sup norm on $C\left(G\right)$ is related to the $L^2$ norm on $L^2\left(G\right)$ and $C\left(G\right)$ is dense in $L^2\left(G\right).$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

page history