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→ℂ is

1. representative if there is a finite dimensional representation V of G and vectors v,w∈V such that fg=vgw for all g∈G.
2. square integrable if ∥f∥22=∫Gfgfgdμg<∞.
3. smooth if all derivatives exist.
4. real analytic if f has a power series expansion at each point. CGrep=representative functions  f:G→ℂ,L2G=square integrable functions  f:G→ℂ,C∞G=smooth functions  f:G→ℂ,CωG=real analytic functions  f:G→ℂ.

We have a map ∏λ∈G^Mdλℂ→functions  f:G→ℂ.

The set G^ has a norm ∥.∥:G^→ℝ≥0. For f^λ∈∏λ∈G^Mdλℂ define

1. f^λ is finite if all but a finite number of the blocks f^λ in f^λ are 0,
2. f^λ is square summable if ∑λ∈G^1dλ∥fλ∥2<∞.
3. f^λ is rapidly decreasing if, for all k∈ℤ>0,∥λ∥k∥f^λ∥|λ∈G^ is bounded,
4. f^λ is exponentially decreasing if, for some K∈ℝ>1,K∥λ∥∥f^λ∥|λ∈G^ is bounded.

Under the map functions  f:G→ℂ→∏λ∈G^CGrep finite  f^λL2Gμ square summable  f^λC∞G rapidly decreasing  f^λCωG exponentially decreasing  f^λ

The space CGrep is dense in CG and CG⊆L2G. In fact the sup norm on CG is related to the L2 norm on L2G and CG is dense in L2G.

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)

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