Proposition 11

Let $(M, \pi )$ be a finite-dimensional linear representation of G.

a) There exists on M a Hermitian form that is positive, separating, and invariant under G.

b) Suppose that the representation $\pi$ is simple. If Φ and Ψ are nonzero Hermitian forms on M that are invariant under G, then there exists a real number $a$ such that $\Psi =a\Phi$.

Choose a separating positive Hermitian form on the vector space M, and denote it by $\Phi_0$. We define a separating positive Hermitian form Φ that is invariant under G by setting

for $x, x'\in M$.

Let Ψ be a Hermitian form on M; there exists a unique endomorphism A of M such that $\Psi (x, x') = \Phi (x,Ax')$ for $x, x'$ in M. If, moreover, Ψ is invariant under G, then the endomorphism A commutes with the automorphism $\pi (g)$ for $g\in G$. If the representation $\pi$ is simple, then by Schur’s lemma (VIII, p. 47, Theorem 1), A is a homothety and there consequently exists a complex number $a$ such that $\Psi =a\Phi$. Since Φ and Ψ are Hermitian and Φ is nonzero, $a$ is a real number. The proposition follows.

We endow the vector space $\mathbf{C}[G]$ of complex functions on G with the Hilbert space structure whose inner product is given by

For any function $f\in \mathbf{C}[G]$, we denote by $f^*$ the function defined by

for $g\in G$; the mapping $f\mapsto f^*$ is a semilinear involution of $\mathbf{C}[G]$. We also have

for $f, f'\in \mathbf{C}[G]$, with the notation of formula (28) of VIII, p. 410. We therefore have

Let $(M, \pi )$ be a finite-dimensional linear representation of G. We endow the vector space M with the structure of a Hilbert space for which the endomorphisms $\pi (g)$ are unitary (Proposition 11). If we denote by $A^*$ the adjoint of a endomorphism A of M for this structure, then we have Tr($A^*$) $=$ Tr(A). For every $g\in G$, we have $\pi (g^{-1}) =\pi (g)^*$, and therefore $\chi_{\pi}(g^{-1}) =\chi_{\pi}(g)$; in other words, we have $\chi_{\pi}=\chi^*_{\pi}$. The orthogonality relation for characters (VIII, p. 410, Proposition 4) then has the form

for $\lambda , \mu\in \widehat{G}$. It expresses the fact that the family of the characters $(\chi_{\lambda})_{\lambda\in\widehat{G}}$ of the simple representations of G is an orthonormal basis of the Hilbert space $Z(\mathbf{C}[G])$ of central functions.

Let $\pi$ and $\pi '$ be finite-dimensional linear representations of G. We have the relation $\langle \chi_{\pi}|\chi_{\pi'}\rangle_G=$ dim$_{\mathbf{C}}$ Hom$_G(\pi , \pi ')$ (VIII, p. 410, Corollary). The representation $\pi$ is irreducible if and only if $\langle \chi_{\pi}|\chi_{\pi}\rangle_G= 1$.

For every element $\lambda$ of $\widehat{G}$, we endow the vector space $V_{\lambda}$ with the structure of a Hilbert space for which the automorphisms $\pi_{\lambda}(g)$ are unitary. We denote by $\langle v|v'\rangle_{\lambda}$ the inner product of two elements $v, v'$ of $V_{\lambda}$ and by $u^*$ the adjoint of an endomorphism $u$ of $V_{\lambda}$. Let $A = (A_{\lambda})_{\lambda\in\widehat{G}}$ and $A'= (A'_{\lambda})_{\lambda\in\widehat{G}}$ be elements of $F(\widehat{G})$. We write $A^*= (A^*_{\lambda})_{\lambda\in\widehat{G}}$. We have $\mathscr{F}(a^*) = (\mathscr{F}(a))^*$ for every element $a$ of $\mathbf{C}[G]$. Set (55) $\langle A|A'\rangle_{\widehat{G}}=|G|^{-2}\widehat{\tau}(A^*A')$. By formula (10) of VIII, p. 407, we have

(56) $\langle A|A'\rangle_{\widehat{G}}=\frac{1}{|G|^2}\sum_{\lambda\in\widehat{G}}d_{\lambda}$ Tr(A$^*_{\lambda}$A$'_{\lambda}$).

Since $\widehat{\tau}\circ \mathscr{F}=\tau$, formulas (53) and (55) imply that the mapping $\mathscr{F}$ is an isomorphism of Hilbert spaces from $\mathbf{C}[G]$ to $F(\widehat{G})$.

The Schur orthogonality relations (VIII, p. 410) can be reformulated using Hilbertian inner products. Relations (21) and (24) then give the following assertions. For $\lambda \in \widehat{G}$ and $x, x', y, y'$ in $V_{\lambda}$, we have (57) $|G|^{-1}\sum_{g\in G}\overline{\langle x|\pi_{\lambda}(g)x'\rangle}_{\lambda}\langle y|\pi_{\lambda}(g)y'\rangle_{\lambda}=d^{-1}_{\lambda}\overline{\langle x|y\rangle}_{\lambda}\langle x'|y'\rangle_{\lambda}$.

If $\lambda$ and $\mu$ are two distinct elements of $\widehat{G}$, then for $x, x'$ in $V_{\lambda}$ and $y, y'$ in $V_\mu$, we have (58) $\sum_{g\in G}\overline{\langle x|\pi_{\lambda}(g)x'\rangle}_{\lambda}\langle y|\pi_\mu(g)y'\rangle_\mu= 0$.

For every $\lambda \in \widehat{G}$, we choose an orthonormal basis $(e_{\lambda ,i})_{1\leqslant i\leqslant d_{\lambda}}$ of $V_{\lambda}$. For any $g\in G$, we denote by $(\pi^{\lambda}_{ij}(g))$ the matrix of the endomorphism $\pi_{\lambda}(g)$ of $V_{\lambda}$ with respect to this basis; we have (59) $\pi^{\lambda}_{ij}(g) =\langle e_{\lambda ,i}|\pi_{\lambda}(g)e_{\lambda ,j}\rangle_{\lambda}$.

Since the endomorphism $\pi_{\lambda}(g)$ is unitary, its inverse is equal to $\pi_{\lambda}(g)^*$, so that (60) $\overline{\pi^{\lambda}_{ij}(g)}=\pi^{\lambda}_{ji}(g^{-1})$. It then follows from formulas (22) of VIII, p. 409 and (25), p. 409 that the functions $(d_{\lambda})^{1/2}\pi_{ij}^{\lambda}$, for $\lambda \in \widehat{G},1\leqslant i\leqslant d_{\lambda},1\leqslant j\leqslant d_{\lambda}$, form an orthonormal basis of the Hilbert space $\mathbf{C}[G].*$