# Decomposition of the Delta Function on a Lie group in terms of irreps

While reading a paper I stumbled over the following formula:

$$\delta(g)=\sum_{\rho\text{ irreps of G}}\mathrm{dim}(\rho)\mathrm{tr}(\rho(g))$$

The specific group in this context was $$G=\mathrm{SU}(2)$$, if this is important. Hence $$\mathrm{dim}(\rho)=2j+1$$. According to the paper, $$\delta$$ is just the function-notation for the detla-function on $$\mathrm{SU}(2)$$, i.e.

$$\int_{\mathrm{SU}(2))}\delta(g)f(g)\,\mathrm{d}g=f(1)$$

for any $$f:\mathrm{SU}(2)\to \mathbb{C}$$ and where $$\mathrm{d}g$$ denotes the Haar measure.

Can anybody tell me how this formula is derived? Furthermore, is it (mathematically speaken) precise? (It is from a physics paper^^). Does this formula have a name?

This result is true for any compact Lie group $$G$$, and is a consequence of the Peter-Weyl theorem. This theorem says that the space $$L^2(G)$$ of square-integrable functions, with inner product $$\langle f,\,k\rangle = \int_G f(g)\overline{k(g)}\,dg$$ ($$dg$$ denoting the normalised Haar metric), has an orthonormal (Schauder) basis $$\{\sqrt{\dim\rho}\,\rho_{ij} \mid \rho \text{ an irrep of }G,\ 1\le i,j\le \dim\rho\},$$ where $$\rho_{ij}$$ denotes the $$ij$$ matrix element of $$\rho$$ (with respect to some bases of the representation space). Orthonormality of these elements says that $$\langle \sqrt{\dim\rho}\,\rho_{ij}, \sqrt{\dim\rho'}\,\rho'_{mn}\rangle = \delta_{\rho\rho'}\delta_{im}\delta_{jn}.$$ Completeness says that any $$f\in L^2(G)$$ can be decomposed into this orthonormal basis $$f = \sum_{\rho \text{ irrep}}\sum_{i,j=1}^{\dim\rho}\langle f,\sqrt{\dim\rho}\rho_{ij}\rangle \sqrt{\dim\rho}\rho_{ij}$$ and this can be written informally as $$\sum_{\rho\text{ irrep}}\dim\rho\sum_{i,j=1}^{\dim\rho}\overline{\rho_{ij}(h)}\rho_{ij}(g) = \delta(h^{-1}g).$$ Taking $$h=e$$ and using the fact that $$\rho_{ij}(e) = [I]_{ij}=\delta_{ij}$$ gives the result you are using.
The statement is precise in the sense that the Peter-Weyl theorem is precise (and, for example, infinite sums are defined as limits relative to the topology in $$L^2(G)$$ derived from the inner product). Writing the sum in terms of a delta function is a bit informal; formally, the sum has to be understood as only making sense when integrated against a function $$f\in L^2(G)$$ (although I'm sure one can make it precise using distributions).