Theorem. Compact character expansions and surface integrals [fdip-007C]
AGENTDRAFTED
Let \(G\) be a compact Lie group with normalized Haar
measure. Choose unitary representatives of its complex
irreducibles, indexed by \(\widehat G\), with dimensions \(d_\rho \).
The Peter–Weyl theorem says that the functions
\(\sqrt {d_\rho }\rho (\cdot )_{ij}\) form an orthonormal basis of
\(L^2(G)\). Its central subspace has orthonormal basis
\(\chi _\rho \). Thus a central \(f\in L^2(G)\) has the expansion
\[f=\sum _{\rho \in \widehat G}c_\rho \chi _\rho ,
\qquad c_\rho =\int _G f(h)\overline {\chi _\rho (h)}\,d\mu _G(h),\]
with convergence in \(L^2\). This is a cited statement, namely
Woit's Section 5, Theorem 1 and Corollary 1
[woit2012peterweyl, p. 6]. It does not assert pointwise or
absolute convergence for an arbitrary \(L^2\) function.
For an inequivalent exhaustive family of finite-dimensional unitary
continuous models, TauCeti's
TauCeti.peterWeylBasis constructs the normalized
matrix-coefficient Hilbert basis, with its inner-product convention
for matrix coefficients. Over \(\mathbb C\), its coefficient
\(\langle \rho (g)e_a,e_b\rangle \) is the conjugate of the matrix
entry \(\rho (g)_{ba}\); conjugation, or reindexing by dual
representations, gives the equivalent entry-coordinate basis.
The separate TauCeti.characterBasis constructs
the actual character basis of the central \(L^2\) subspace.
Compact Hausdorff groups over \(\mathbb C\), hence the compact Lie
groups here, satisfy the stated hypotheses. These correspondences
support the two \(L^2\) basis statements, not the uniform absolute
convergence or surface integral established next.
Suppose now that \(f\) is continuous and central and that
\[\sum _{\rho \in \widehat G}|c_\rho |d_\rho <\infty .\]
Then its character expansion converges uniformly and absolutely.
For the closed oriented polygonal surface, with
\(P\) faces and genus \(g\), define the edge integral
\[Z_\Sigma (f)=\int _{G^E}\prod _p f(U_p)\,d\mu (U).\]
Under the displayed summability condition,
\[Z_\Sigma (f)=\sum _{\rho \in \widehat G}
d_\rho ^{2-2g}\left (\frac {c_\rho }{d_\rho }\right )^P,\]
and this numerical series is absolutely convergent.