Since E is finite-dimensional, there exists a Hilbert space structure on E. The result follows by applying the preceding remark and corollary 2 of V, p. 392.

This follows from the proposition and corollary 3 of V, p. 392.

In fact, the matrix coefficients of an irreducible unitary representation are continuous and bounded, and are therefore square-integrable on G, since G is compact.

Since the representation $\pi$ is square-integrable (lemma 1), its formal degree $c_\mu(\pi )$ relative to $\mu$ is defined (def. 4 of V, p. 423); it is a strictly positive real number. Let $(e_i)_{i\in I}$ be a finite orthonormal family in E. Let $x$ be an element of E of norm 1 (there exists one since E is nonzero). For $i\in I$, one has

(prop. 8 of V, p. 424). Let us sum this formula over $i\in I$. We obtain

Card(I) $=c_\mu(\pi )\int_G\sum_{i\in I}|\langle e_i|\pi (g)x\rangle |^2d\mu(g)$

by Bessel's inequality (EVT, V, p. 21, prop. 4). Hence the cardinal of I is bounded above by $c_\mu(\pi )$. This implies that the dimension of E is finite.

One may then apply the foregoing to an orthonormal basis $(e_i)_{i\in I}$ of E. One obtains, by EVT, V, p. 22, prop. 5,

dim(E) $=c_\mu(\pi )\int_G\sum_{i\in I}|\langle e_i|\pi (g)x\rangle |^2d\mu(g)$

which completes the proof.

In particular, it is therefore permissible to speak of the character $\chi_{\pi}$ of an irreducible unitary representation $\pi$ of G. It is a continuous function on G.

Let $x\in G$. Since G is unimodular, one has $\chi^*_{\pi}(x) =$ Tr($\pi (x^{-1})$), whence $\chi^*_{\pi}(x) =$ Tr($\pi (x)$) since $\pi$ is a unitary representation.

$\int'$ 1 $''$

$_G$ dim(E)

for every $(x, y, x', y')\in E^4$;

b) Let $\pi_1($resp. $\pi_2)$ be an irreducible unitary representation of G in a Hilbert space $E_1($resp. $E_2)$. If $\pi_1$ and $\pi_2$ are not isomorphic, one has

for every $(x, x', y, y')\in E^2_1\times E^2_2$.

This follows at once from Prop. 8 of V, p. 424 and Prop. 9 of V, p. 424, taking into account Lemma 1 and the formula $c_\mu(\pi ) =$ dim(E) (Prop. 2).

This results from Proposition 1 of V, p. 457 and Corollary 2, noting that for every orthonormal basis $(e_i)_{i\in I}$ of the space of a finite-dimensional unitary representation $\pi$ of G, and for every $g\in G$, one has the formula

$\chi_{\varrho}=\sum_{\pi\in\widehat{G}}$ dim(Hom$_G(\pi , \varrho )$)$\chi_{\pi}=\sum_{\pi\in\widehat{G}}$ dim(Hom$_G(\varrho , \pi )$)$\chi_{\pi}$.

b) Let $\varrho_1$ and $\varrho_2$ be finite-dimensional continuous representations of G. One has

$\langle \chi_{\varrho_1}|\chi_{\varrho_2}\rangle =$ dim(Hom$_G(\varrho_1, \varrho_2)$) $=$ dim(Hom$_G(\varrho_2, \varrho_1)$).

c) A finite-dimensional continuous representation $\varrho$ of G is irreducible if and only if $\|\chi_{\varrho}\|^2= 1$.

Since $\varrho$ is semisimple, it is isomorphic to the direct sum of its $\pi$-isotypic components $M_{\pi}(\varrho )$ for $\pi \in \widehat{G}$. Denoting by $m_{\pi}(\varrho )$ the multiplicity of $\pi$ in $\varrho$ (Def. 10 of V, p. 398), one then has

Assertion a) therefore results from the fact that

$m_{\pi}(\varrho ) =$ dim Hom$_G(\pi , \varrho ) =$ dim Hom$_G(\varrho , \pi )$

(formula (1) of V, p. 377 and Corollary 2 of V, p. 387).

By bilinearity and orthonormality of the characters, assertion a) implies that

$\langle \chi_{\varrho_1}|\chi_{\varrho_2}\rangle =\sum_{\pi\in\widehat{G}}$ dim(Hom$_G(\pi , \varrho_1)$) dim(Hom$_G(\pi , \varrho_2)$), and on the other hand one has canonical isomorphisms

Hom$_G(\varrho_1, \varrho_2)\rightarrow \bigoplus_{(\pi_1,\pi_2)\in\widehat{G}\times\widehat{G}}$ Hom$_G(M_{\pi_1}(\varrho_1),M_{\pi_2}(\varrho_2))$

$\rightarrow \bigoplus_{\pi\in\widehat{G}}$ Hom$_G(M_{\pi}(\varrho_1),M_{\pi}(\varrho_2))$

(formula (1) of V, p. 377), whence it follows that

dim(Hom$_G(\varrho_1, \varrho_2)$) $=\sum_{\pi\in\widehat{G}}m_{\pi}(\varrho_1)m_{\pi}(\varrho_2)$,

whence assertion b). Assertion c) also results from a) and Schur’s Lemma (prop. 6 of V, p. 386).

The character of $\pi_2$ is a continuous central function on G, hence the linear mapping $u$ = $\pi_1(\overline{\chi}_{\pi_2})$ is defined and belongs to the space Hom$_G(\pi_1, \pi_1)$. It is a homothety by Schur’s Lemma (prop. 6 of V, p. 386), whose trace is

Tr($u$) $=$ Tr$(\int_G\overline{\chi_{\pi_2}(g)}\pi_1(g)d\mu(g))$

By the orthogonality relations, the trace of $u$ is therefore zero if $\pi_1$ is not isomorphic to $\pi_2$, and equal to 1 otherwise. The assertion follows.

By cor. 6 of V, p. 389, the mapping $b$ is well defined.

Let $\pi_1$ and $\pi_2$ be elements of $\widehat{G}$. Let $\pi \in \widehat{G}$. The $\pi$-isotypical component of the restriction $\varpi$ of $\pi_1\boxtimes \pi_2$ to $G_1\times \{e\}$ is equal to $\varpi$ if $\pi_1=\pi$ and is zero in the contrary case (lemma 8 of V, p. 384). Thus, the unitary representation $\pi_1\boxtimes \pi_2$ determines $\pi_1$ up to isomorphism; analogously, it determines $\pi_2$, which proves that $b$ is injective.

Let us prove that $b$ is surjective. Let $\pi$ be an irreducible unitary representation of $G_1\times G_2$ in a Hilbert space E. It is finite-dimensional (prop. 2). Let $\pi_1$ be an irreducible unitary representation of $G_1$ in a Hilbert space $E_1$ such that the restriction of $\pi$ to $G_1\times \{e\}$ contains a subrepresentation isomorphic to $\pi_1$ (lemma 3 of V, p. 379). Let F = Hom$_{G_1}(\pi_1, \pi )$. It is a nonzero finite-dimensional vector space. For $h\in G_2$ and $u\in F$, let $\varrho (h)(u)$ be the linear mapping from $E_1$ into E defined by $x\mapsto \pi (e, h)(u(x))$. Since $G_1\times \{e\}$ and $\{e\} \times G_2$ commute in G, the mapping $\varrho (h)(u)$ belongs to the space F. The mapping $\varrho$ is a linear representation of $G_2$ in the space F; it is continuous. Since F is not reduced to 0, there exists an irreducible subrepresentation $\pi_2$ of $\varrho$ (lemma 3 of V, p. 379). Let $E_2$ be the space of $\pi_2$. The linear mapping $v: E_1\otimes E_2\rightarrow E$ such that $x\otimes u\mapsto u(x)$ for $x\in E_1$ and $u\in E_2$ is then a nonzero G-morphism of $\pi_1\boxtimes \pi_2$ into $\pi$; since the representations $\pi$ and $\pi_1\boxtimes \pi_2$ are irreducible and finite-dimensional, the morphism $v$ is an isomorphism (lemma 2 of V, p. 378).

This proposition is to be compared with theorem 1 of A, VIII, p. 208.

The unital subalgebra Θ(G) of $\mathscr{C}(G)$ is stable under conjugation. It coincides with the subalgebra Υ(G) by prop. 2 of V, p. 457, hence it separates the points of G (cor. of theorem 4 of V, p. 454). Consequently, it is dense in $\mathscr{C}(G)$ by TG, X, p. 40, cor. 2, and, fortiori, it is dense in the space $L^2(G) ($cf. INT, IV, p. 155, §4, n$^o7$, prop. 13).

Recall that the unitary biregular representation of G is the representation $\boldsymbol{\varrho }_G$ of $G\times G$ in $L^2(G)$ such that $(g_1, g_2)\mapsto \boldsymbol{\gamma }_G(g_1)\boldsymbol{\delta }_G(g_2)$.

By prop. 7 of V, p. 422 (applied with $Z =\{e\}$), the space $\boldsymbol{\varrho }_G(\pi )$ is a subrepresentation of $\boldsymbol{\varrho }_G$ and the linear mapping from $\overline{E}_{\pi}\otimes E_{\pi}$ into $L^2(G)$ which associates to $x\otimes y$ the matrix coefficient $g\mapsto \langle x|\pi (g)y\rangle$ is a $(G\times$ G)-isomorphism of $\overline{\pi}\boxtimes \pi$ onto $\boldsymbol{\varrho }_G(\pi )$. The subrepresentation $\boldsymbol{\varrho }_G(\pi )$ is therefore irreducible (prop. 3 of V, p. 461).

The spaces $\boldsymbol{\varrho }_G(\pi )$ are pairwise orthogonal; they are irreducible subrepresentations of the biregular representation of G (prop. 5) which are pairwise non-isomorphic (prop. 3 of V, p. 461). The theorem then results from the fact that the sum Θ(G) of the spaces $\boldsymbol{\varrho }_G(\pi )$ for $\pi$ ranging over $\widehat{G}$ is dense in $L^2(G)$ (prop. 4).

Let F be the $(\overline{\pi}\boxtimes \pi$)-isotypical component of $\boldsymbol{\varrho }_G$. The space F contains $\boldsymbol{\varrho }_G(\pi )$ (prop. 5). Moreover, for every $\tau \in \widehat{G}$ different from $\pi$, the intersection of F and $\boldsymbol{\varrho }_G(\tau )$ is zero (prop. 3 of V, p. 461). It follows that $F =\boldsymbol{\varrho }_G(\pi )$ by applying the theorem.

$\pi \bigoplus\in \widehat{G}\pi$dim($\pi$ ).

By the theorem and proposition 5, the restriction of the biregular representation of G to the subgroup $H =\{e\} \times G$ is isomorphic to the Hilbert sum of the restrictions to H of the representations $\overline{\pi}\boxtimes \pi$ for $\pi \in \widehat{G}$. These are isomorphic to the direct sum of dim($\pi$ ) copies of $\pi$ (lemma 8 of V, p. 384). This implies the result for the right regular representation, and the case of the left regular representation is similar.

The argument is similar to that of the preceding corollary, considering $\boldsymbol{\varrho }_G(\pi )$ as a subrepresentation of $\boldsymbol{\delta }_G$ (resp. of $\boldsymbol{\gamma }_G$).

a) The space Θ(G) ;

b) The algebraic direct sum of the subspaces $\boldsymbol{\varrho }_G(\pi )$ of $L^2(G)$;

c) The space of G-finite vectors (cf. V, p. 376) of $\boldsymbol{\gamma }_G$;

d) The space of G-finite vectors of $\boldsymbol{\delta }_G$;

e) The space of $(G\times G)$-finite vectors of $\boldsymbol{\varrho }_G$.

In particular, every G-finite vector of $\boldsymbol{\gamma }_G,\boldsymbol{\delta }_G$ or $\boldsymbol{\varrho }_G$ belongs to $\mathscr{C}(G)$.

Let us denote by $F_a$ (resp. $F_b, F_c, F_d, F_e$) the space defined by condition a) (resp. b), c), d), e)). We have already observed that $F_a= F_b$.

We have $F_b\subset F_c$ because $\boldsymbol{\varrho }_G(\pi )$ is a finite-dimensional subrepresentation of $\boldsymbol{\gamma }_G$ for every $\pi \in \widehat{G}$. Conversely, let $f$ be a G-finite vector of $\boldsymbol{\gamma }_G$. The subspace $E_f$ generated by the functions $\boldsymbol{\gamma }_G(g)f$ for $g\in G$ is a finite-dimensional subrepresentation of $\boldsymbol{\gamma }_G$. It is equal to the direct sum of its $\pi$-isotypic components for $\pi \in \widehat{G}$ (Prop. 1 of V, p. 457). By Cor. 4 of V, p. 463, this implies that $E_f$ is contained in the sum of the spaces $\boldsymbol{\varrho }_G(\pi )$, hence $F_c\subset F_b$.

By analogous reasoning, one obtains $F_b= F_d$, and since $\boldsymbol{\varrho }_G(\pi )$ is a subrepresentation of $\boldsymbol{\varrho }_G$, one establishes in the same way that $F_b= F_e$.

In fact, every element of E is a G-finite vector of the representation $\boldsymbol{\gamma }_G$.

Let $x\in E$ and let U be an open neighbourhood of $x$. The set of measures $\nu \in \mathscr{M}(G)$ such that $\varrho (\nu )x\in U$ is open in $\mathscr{M}(G)$ for the topology of compact convergence (cf. n$^o2$ of V, p. 400). It contains $\varepsilon_e$, hence it contains a measure of the form $\nu =f_1\cdot \mu$ where $f_1\in \mathscr{C}(G)$ (INT, VIII, p. 171, § 4, n$^o7$, Prop. 19) and, consequently, it contains a measure of the form $f_2\cdot \mu$ where $f_2$ is a G-finite function (Prop. 6 of V, p. 464 and Prop. 4 of V, p. 462).

The subrepresentation F of $\boldsymbol{\gamma }_G$ generated by $f_2$ is finite-dimensional. Let $\widetilde{F}$ be the image of the linear mapping $f\mapsto \varrho (f)x$ of F into E. The space $\widetilde{F}$ is finite-dimensional, and it contains the element $\varrho (f_2)x$ of U. Since $\varrho (g)\varrho (f) =\varrho (\boldsymbol{\gamma }_G(g)f)$ for every $g\in G$ and every $f\in F$ (formula (1) of V, p. 406), the space $\widetilde{F}$ is a subrepresentation of $\varrho$ which meets U. The proposition is proved.

a) The G-morphism $p_{\pi}=$ dim($\pi$ )$\varrho (\overline{\chi}_{\pi})$ of E into E is a continuous projector of E whose image is the $\pi$-isotypical component of $\varrho$;

b) If $\varrho$ is a unitary representation, then the projector $p_{\pi}$ is the orthoprojector of E with image $M_{\pi}(\varrho )$.

Let us prove a). The linear mapping $p_{\pi}=$ dim($\pi$ )$\varrho (\overline{\chi}_{\pi})$ is well-defined, since E is quasi-complete and G is compact (V, p. 401). It is an element of Hom$_G(\varrho , \varrho )$ since the character of $\pi$ is a continuous central function on G.

Let $\varpi$ be a finite-dimensional subrepresentation of E and F its space. The mapping $p_{\pi}$ induces, by passing to subspaces, the endomorphism (dim $\pi$ )$\varpi (\overline{\chi}_{\pi})$ of F. This endomorphism is therefore zero if $\varpi$ is not isomorphic to $\pi$, and is the identity mapping of F otherwise (in fact, this is the case if $\varpi$ is irreducible by Cor. 5 of V, p. 460, and the general case follows from this since $\varpi$ is semisimple by Prop. 1 of V, p. 457).

Finally, since the sum of the finite-dimensional subrepresentations of E is dense in E (Prop. 7) and $p_{\pi}$ is continuous, it follows that $p_{\pi}$ is a projector whose image is the $\pi$-isotypical component of $\varrho$. If E is Hilbert space, the projector $p_{\pi}$ is hermitian by Cor. 1 of V, p. 458, hence it is an orthoprojector (Lemma 3, (ii) of I, p. 133).

dim(E$^G$) $=\int_G\chi_{\varrho}(g)d\mu(g)$.

The first assertion is the particular case of the preceding corollary when $\pi$ is the trivial representation of dimension 1 of G. The second follows from it since the dimension of $E^G$ is the trace of the orthoprojector onto $E^G$, that is,

dim(E$^G$) $=$ Tr$(\int_G\varrho (g)d\mu(g))=\int_G\chi_{\varrho}(g)d\mu(g)$.

In particular, when E is finite-dimensional, there exists a vector $x\not = 0$ in E such that $\varrho (g)x=x$ for every $g\in G$ if and only if

The isotypical components of $\varrho$ corresponding to non-isomorphic irreducible representations of G are orthogonal (prop. 8 of V, p. 394). Since every finite-dimensional unitary representation of G is semisimple (prop. 1 of V, p. 457), the sum of the isotypical components $M_{\pi}(\varrho )$ for $\pi \in \widehat{G}$ is dense in E (prop. 7). The corollary follows.

(1) $\chi_{\pi}*\chi_{\sigma}= 0$ for all $\pi , \sigma$ belonging to $\widehat{G}, \pi \not =\sigma$,

(2) $\chi_{\pi}*\chi_{\pi}=$ dim($1\pi$ )$\chi_{\pi}$ for every $\pi$ belonging to $\widehat{G}$.

Let $\pi$ and $\sigma$ be irreducible representations of G. One has

dim($\pi$ )$(\chi_{\pi}*\chi_{\sigma}) =$ dim($\pi$ )$\boldsymbol{\gamma }_G(\chi_{\pi})\chi_{\sigma}$

(lemma 4 of V, p. 407). The function dim($\pi$ )$\boldsymbol{\gamma }_G(\chi_{\pi})\chi_{\sigma}$ is the orthogonal projection of $\chi_{\sigma}$ on the $\overline{\pi}$-isotypical component of $\boldsymbol{\gamma }_G$ (cor. 2 of V, p. 465), that is, on $\boldsymbol{\varrho }_G(\pi )$ (cor. 4 of V, p. 463). Since $\chi_{\sigma}$ belongs to $\boldsymbol{\varrho }_G(\sigma )$, the result follows from theorem 1 of V, p. 462.

In fact, this graph is the image of the continuous mapping of the compact space $G\times G$ into itself defined by $(x, y)\mapsto (x, yxy^{-1})$.

We denote by $G^{\sharp}$ the space of conjugacy classes of G endowed with the quotient topology; it is a compact space by lemma 2 and TG, I, p. 78, prop. 8. Let $\varpi : G\rightarrow G^{\sharp}$ be the canonical projection. The mapping of $\mathscr{C}(G^{\sharp})$ into $\mathscr{C}(G)$ defined by $f\mapsto f\circ \varpi$ identifies the involutive algebra $\mathscr{C}(G^{\sharp})$ with the involutive subalgebra of $\mathscr{C}(G)$ formed by the continuous central functions.

The measures on $G^{\sharp}$ are identified with the central measures on G (V, p. 402, def. 1).

The linear form on $\mathscr{C}(G^{\sharp})$ defined by $f\mapsto \int_Gf$ is then a positive measure of mass 1 on $G^{\sharp}$, denoted by $\mu^{\sharp}$. For every $p\in [1,+\infty ]$, we denote by $\mathscr{L}^p(G^{\sharp})$ (resp. $L^p(G^{\sharp})$) the space $\mathscr{L}_{\mathbf{C}}^p(G^{\sharp}, \mu^{\sharp})$ (resp. $L^p_{\mathbf{C}}(G^{\sharp}, \mu^{\sharp})$). We identify $L^p(G^{\sharp})$ with the closure of $\mathscr{C}(G^{\sharp})$ in $L^p(G)$; in particular, it is a closed subspace of $L^p(G)$.

We also denote by $\Theta (G^{\sharp}) = \Theta (G)\cap \mathscr{C}(G^{\sharp})$ the space of central matrix coefficients of G. It is a unital involutive subalgebra of $\mathscr{C}(G^{\sharp})$.

When G is a Lie group, the space $\Theta (G^{\sharp})$ is also denoted by ZΘ(G) in LIE, IX, p. 71.

Consider the unitary representation $\sigma$ of G in the space $\boldsymbol{\varrho }_G(\pi )$ defined by $\sigma (g) =\boldsymbol{\varrho }_G(g, g)$. Since the unitary representation $\boldsymbol{\varrho }_G(\pi )$ of $G\times G$ is isomorphic to $\overline{\pi}\boxtimes \pi$ (prop. 5 of V, p. 462), its character is given by

for every $g\in G$. The space $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$ is the subspace of the elements invariant under this representation, and its dimension is equal to

(cor. 3 of V, p. 466 and cor. 3 of V, p. 459). Since the character of $\pi$ is a nonzero element of $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$, it is a basis of this space.

By the Peter-Weyl theorem (th. 1 of V, p. 462), the closed subspace $L^2(G^{\sharp})$ of $L^2(G)$ formed by the elements invariant under the unitary representation $g\mapsto \boldsymbol{\varrho }_G(g, g)$ of G is the Hilbert sum of the subspaces $L^2(G^{\sharp})\cap \boldsymbol{\varrho }_G(\pi )$ for $\pi \in \widehat{G}$. The proposition therefore follows from the preceding lemma and cor. 3 of V, p. 459.

The second assertion follows from prop. 9. As to the first assertion, consider the linear representation $\varrho$ of G on the Banach space $\mathscr{C}(G)$ defined by

for every $f\in \mathscr{C}(G)$ and every $x\in G$. It is a continuous isometric representation, and $\mathscr{C}(G^{\sharp})$ is the space of the elements invariant under this representation. The continuous projector $p=\varrho (1)$ of $\mathscr{C}(G)$ therefore has image $\mathscr{C}(G^{\sharp})$ (cor. 3 of V, p. 466); it has norm $\leqslant 1$.

Let $f\in \mathscr{C}(G^{\sharp})$ and let $\varepsilon >0$. There exists $\widetilde{f}\in \Theta (G)$ such that $\|f-\widetilde{f}\|\leqslant \varepsilon$ (prop. 4 of V, p. 462), and then $\|f-p(\widetilde{f})\|=\|p(f-\widetilde{f})\|\leqslant \varepsilon$; since $p(\widetilde{f})\in \mathscr{C}(G^{\sharp})$, we conclude that $\Theta (G^{\sharp})$ is dense in $\mathscr{C}(G^{\sharp})$.

Let $R(G)$ denote the Grothendieck ring of continuous representations of $G$ in finite-dimensional separated complex vector spaces (cf. LIE, IX, p. 70 and A, VIII, p. 182, applied to the additive class of continuous representations of $G$ in finite-dimensional separated complex vector spaces, viewed as $\mathbf{C}$[G]-modules). Since every continuous linear representation of $G$ in a finite-dimensional separated topological vector space is semisimple (prop. 1 of V, p. 457), the abelian group $R(G)$ is free and the classes of irreducible unitary representations $\pi \in \widehat{G}$ form a basis of it (A, VIII, p. 186, prop. 7).

b) The mapping $u: R(G)\otimes_{\mathbf{Z}}\mathbf{C}\rightarrow \Theta (G^{\sharp})$ such that $u(\pi \otimes 1) =\chi_{\pi}$ for every $\pi \in \widehat{G}$ is an isomorphism of algebras over $\mathbf{C}$.

The first assertion results from Proposition 9 and Corollary 1.

Since the classes of the representations $\pi \in \widehat{G}$ form a basis of the free $\mathbf{Z}$-module $R(G)$, the mapping $u$ is well-defined. It is a morphism of $\mathbf{C}$-algebras, and it is an isomorphism by a).

Let us denote by $\sigma$ the restriction of $\varrho$ to the compact subgroup $G$ of $H$. Let $\pi$ be an irreducible representation of $G$. The endomorphism $\sigma (\chi_{\pi})\in \mathscr{L}(E)$ is of finite rank, since its image is the $\overline{\pi}$-isotypical component of $\sigma$ (cor. 2 of V, p. 465) and this is finite-dimensional by hypothesis. Consequently, the endomorphism $\sigma (f)$ is of finite rank for every $f\in \Theta (G^{\sharp})$ and is compact for every $f\in \mathscr{C}(G^{\sharp})$ (cor. 1 of V, p. 468 and cor. of prop. 2 of III, p. 4).

Let $j$ denote the canonical injection of $G$ into $H$. It is a continuous mapping which is $\nu$-proper for every measure $\nu$ on $G$ (INT, V, p. 69, § 6, No.$^o1$, Remark 1). Let $\mathfrak{B}$ be the filter of compact neighbourhoods of the element $e$ in $G$. For every $V\in \mathfrak{B}$, there exists a positive continuous central function $f_V$ on $G$ with support contained in $V$ whose integral over $G$ is equal to $1$. Let $\beta_V$ denote the image measure $j(f_V\cdot \mu)$; it is a positive measure with compact support on $H$ such that $\varrho (\beta_V) =\sigma (f_V)$ (INT, V, p. 71, § 6, No.$^o2$, Theorem 1). From the foregoing, the filter basis on $\mathscr{M}_c(H)$ which is the image of $\mathfrak{B}$ under the mapping $V\mapsto \beta_V$ satisfies the conditions of Proposition 2 of V, p. 402, and the assertion follows.

Corollary 4 (Weyl's criterion for equidistribution)

Let $M$ be a set of positive central measures on $G$ such that $\nu (G) = 1$ for every $\nu \in M$. Let $\mathfrak{F}$ be a filter on $M$. In order that the filter $\mathfrak{F}$ converge vaguely to the measure $\mu^{\sharp}$ on $G^{\sharp}$, it is necessary and sufficient that, for every non-trivial irreducible unitary representation $\pi$, one have

lim$_{\nu ,\mathfrak{F}}\int_G\chi_{\pi}(x)d\nu (x) = 0$.

Since $\nu (G^{\sharp}) = 1 =\mu^{\sharp}(G^{\sharp})$ for $\nu \in M$, the hypothesis means that

lim$_{\nu ,\mathfrak{F}}\int_{G^{\sharp}}\chi_{\pi}(x)d\nu (x) =\int_{G^{\sharp}}\chi_{\pi}(x)d\mu^{\sharp}(x)$

for every representation $\pi \in \widehat{G}$, hence it is equivalent to the condition

lim$_{\nu ,\mathfrak{F}}\int_{G^{\sharp}}f(x)d\nu (x) =\int_{G^{\sharp}}f(x)d\mu^{\sharp}(x)$

for every function $f\in \Theta (G^{\sharp})$ by linearity. Since the space $\Theta (G^{\sharp})$ is dense in $\mathscr{C}(G^{\sharp})$ (Corollary 1), the hypothesis is therefore equivalent to the convergence of the filter $\mathfrak{F}$ to $\mu^{\sharp}$ in $\mathscr{M}(G^{\sharp})$ endowed with the topology of simple convergence in $\mathscr{C}$ (G), which coincides with the topology of vague convergence since G is compact (cf. INT, III, p. 59, § 1, No.$^o9$).