(i) $\lambda -w(\lambda )\geq 0$ (resp. $>0$) for all $w\in W$ such that $w\not= 1$;

(ii) for all $w\in W$ such that $w\not= 1,\lambda -w(\lambda )$ is a linear combination with positive coefficients (resp. positive coefficients not all zero) of simple roots;

(iii) $\langle \lambda , K_{\alpha}\rangle \geq 0$ (resp. $>0$) for every positive root $\alpha$;

(iv) $\langle \lambda , K_{\alpha}\rangle \geq 0$ (resp. $>0$) for every simple root $\alpha$.

The equivalence if (iii) and (iv) is immediate. Since the set of the $K_{\alpha}$ can be identified with the inverse root system of $R(G,T) ($§4, no. 5), the equivalence of (i) and (iii) follows from Chap. VI, §1, no. 6, Prop. 18 and Cor. The implication (ii) $\Rightarrow$ (i) is trivial, and the opposite implication follows from loc. cit.

Denote by $X_{++}$ the set of elements of X(T) such that $\langle \lambda , K_{\alpha}\rangle \geq 0$ for every positive root $\alpha$. The elements of $X_{++}$ are said to be dominant. They form a fundamental domain for the operation of W on X(T) (Chap. VI, §1, no. 10). We have $X_{++}\subset X_+$.

If G is simply-connected, for each simple root $\alpha$ there exists an element $\varpi_{\alpha}$ of X(T) such that $\langle \varpi_{\beta}, K_{\alpha}\rangle =\delta_{\alpha \beta}$ for every simple root $\beta$, that is, $s_{\alpha}(\varpi_{\alpha}) =$ $\varpi_{\alpha}-\alpha ,s_{\beta}(\varpi_{\alpha}) =\varpi_{\alpha}$ for every simple root $\beta \not=\alpha$; the $\varpi_{\alpha}$ are called the fundamental dominant weights; they form a basis of the commutative group X(T) and of the commutative monoid $X_{++}$; more precisely, every element $\lambda$ of X(T) can be written in the form $\lambda =\sum_{\alpha}\langle \lambda , K_{\alpha}\rangle \varpi_{\alpha}$.

Denote by $\rho$ the element of $X(T)\otimes \mathbf{Q}$ such that

We have $\langle \rho , K_{\alpha}\rangle = 1$ for every simple root $\alpha$ (Chap. VI, §1, no. 10, Prop. 29). If G is simply-connected, $\rho$ is the sum of the fundamental dominant weights.

If there exists a representation $\tau$ of G such that $L(\tau )_{(\mathbf{C})}=\varphi$, then $\varphi$ is semi-simple because G is connected and $\tau$ is semi-simple (Chap. III, §6, no. 5, Cor. 2 of Prop. 13), and the weights of $\mathfrak{t}_{\mathbf{C}}$ on V belong to the image of $\delta$. Thus, the condition is necessary; we shall prove that it is sufficient. If $\varphi$ is semi-simple, V is the direct sum of the $V_\mu(\mathfrak{t}_{\mathbf{C}})$, where $\mu$ belongs to the set of weights of $\mathfrak{t}_{\mathbf{C}}$ on V (Chap. VII, §2, no. 4, Cor. 3 of Th. 2); if all the weights belong to the image of $\delta$, there exists a representation $\tau_T$ of T on V such that $L(\tau_T)_{(\mathbf{C})}=\varphi |\mathfrak{t}_{\mathbf{C}}:$ indeed, it suffices to put $\tau_T(t)v=t^{\lambda}v$ for $t\in T$ and $v\in V_{\delta(\lambda)}(\mathfrak{t}_{\mathbf{C}})$. The lemma now follows from Prop. 8 of §2, no. 6.

b) Two irreducible representations of G are equivalent if and only if their highest weights are equal.

c) For every dominant element $\lambda$ of X(T), there exists an irreducible representation of G of highest weight $\lambda$.

By Lemma 2, the equivalence classes of irreducible representations of G correspond bijectively to the classes of finite dimensional irreducible representations of $\mathfrak{g}$ whose weights belong to $\delta (X(T))$.

Denote by $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$ the centre and by $\mathscr{D}\mathfrak{g}_{\mathbf{C}}$ the derived Lie algebra of $\mathfrak{g}_{\mathbf{C}}$, so that $\mathfrak{g}_{\mathbf{C}}=\mathscr{C}\mathfrak{g}_{\mathbf{C}}\oplus \mathscr{D}\mathfrak{g}_{\mathbf{C}}$. For every bilinear form $\mu$ on $\mathfrak{t}_{\mathbf{C}}\cap \mathscr{D}\mathfrak{g}_{\mathbf{C}}$, denote by $E(\mu)$ the simple $\mathscr{D}\mathfrak{g}_{\mathbf{C}}$-module introduced in Chap. VIII, §6, no. 3; for every linear form $\nu$ on $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$, denote by $\mathbf{C}(\nu )$ the 1-dimensional $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$-module over $\mathbf{C}$ associated to it. Then the $\mathfrak{g}_{\mathbf{C}}$-modules $\mathbf{C}(\nu )\otimes E(\mu)$ are simple, and by Chap. VIII, §7, no. 2, Cor. 2 of Th. 1 and Algebra, Chap. VIII, §11, no. 1, Th. 1, every finite dimensional simple $\mathfrak{g}_{\mathbf{C}}$-module is isomorphic to one of the modules $\mathbf{C}(\nu )\otimes E(\mu)$; moreover (loc. cit.)$\mathbf{C}(\nu )\otimes E(\mu)$ is finite dimensional if and only if $\mu(H_{\alpha})$ is a positive integer for every simple root $\alpha$. If we denote by $\nu +\mu$ the linear form on $\mathfrak{t}_{\mathbf{C}}$ that induces $\nu$ on $\mathscr{C}\mathfrak{g}_{\mathbf{C}}$ and $\mu$ on $\mathfrak{t}_{\mathbf{C}}\cap \mathscr{D}\mathfrak{g}_{\mathbf{C}}$, then $(\nu +\mu)(H_{\alpha}) =\mu(H_{\alpha})$; moreover, the weights of $\mathbf{C}(\nu )\otimes E(\mu)$ are the $\nu +\lambda$, where $\lambda$ is any weight of $E(\mu)$, and are thus of the form $\nu +\mu-\theta$, with $\theta \in \delta (X_+)$ (Chap. VIII, §6, no. 2, Lemma 2).

We conclude that the $\mathfrak{g}$-module $\mathbf{C}(\nu )\otimes E(\mu)$ is finite dimensional if and only if $(\nu +\mu)(H_{\alpha})$ is a positive integer for every simple root $\alpha$, and that its weights belong to $\delta (X(T))$ if and only if $\nu +\mu$ belongs to $\delta (X(T))$. The conjunction of these two conditions means that $\nu +\mu$ belongs to $\delta (X_{++})$; in that case, $\nu +\mu$ is the highest weight of $\mathbf{C}(\nu )\otimes E(\mu)$. Thus, we have constructed for every dominant weight $\lambda$ of X(T) an irreducible representation of G of highest weight $\lambda$, and have thus obtained, up to equivalence, all the irreducible representations of G. Since the vectors of weight $\nu +\mu$ in $\mathbf{C}(\nu )\otimes E(\mu)$ form a subspace of dimension 1, the proof is complete.

Observe first that every element of X(T) is equal to the difference of two dominant weights: more precisely, let $\varpi$ be an element of $X_{++}$ such that $\langle \varpi , K_{\alpha}\rangle >0$ for every simple root $\alpha$; for all $\lambda \in X(T)$ there exists a positive integer $n$ such that $\langle \lambda +n\varpi , K_{\alpha}\rangle \geq 0$ for every simple root $\alpha$, that is (no. 1, Lemma $1$)$\lambda +n\varpi \in X_{++}$.

Consequently, there exists a finite family $(\lambda_i)_{i\in I}$ of elements of $X_{++}$ generating the $\mathbf{Z}$-module X(T). For $i\in I$, let $\tau_i$ be an irreducible representation of G of highest weight $\lambda_i$ (Th. 1); let the representation $\tau$ be the direct sum of the $\tau_i$. By construction the set $P(\tau ,T)$ of weights of $\tau$ (relative to T) generates the $\mathbf{Z}$-module X(T). It now follows from Prop. 6 of §4, no. 3 that the homomorphism $\tau$ is injective, hence the corollary.

be the element of the Weyl group such that $w_0(R_+) = R_-$ (Chap. VI, § 6, Cor. 3 of Prop. 17). There are three possible cases:

a$)w_0(\lambda ) =-\lambda$ and $m$ is even. Then there exists a G-invariant separating symmetric bilinear form on V; the representation $\tau$ is of real type (Appendix II).

b$)w_0(\lambda )\not=-\lambda$. Every G-invariant bilinear form on V is zero; the representation $\tau$ is of complex type (loc. cit.).

c$)w_0(\lambda ) =-\lambda$ and $m$ is odd. Then there exists a G-invariant separating alternating bilinear form on V; the representation $\tau$ is of quaternionic type (loc. cit.).

If the restriction of $\tau$ to $C(G)_0$ is not trivial, we are in case b).

A bilinear form B on V is invariant under G if and only if it is invariant under $\mathfrak{g}_{\mathbf{C}}$ (Chap. III, §6, no. 5, Cor. 3). Thus, if G is semi-simple, the proposition follows from Chap. VIII, §7, no. 5, Prop. 12 and Prop. 3 of Appendix II.

In the general case, put $C(G)_0= S$, and identify $X(T/S)$ with a subgroup of X(T) (stable under W). If $\tau (S) =\{1_V\},\tau$ induces by passage to the quotient a representation $\tau ': G/S\rightarrow \mathbf{G}\mathbf{L}(V)$ of highest weight $\lambda$; in this case the proposition follows from the preceding, applied to $G/S$.

Assume that $\tau (S)\not=\{1_V\}$. There exists a non-zero element $\nu$ of X(S) such that $\tau (s) =\nu (s)_V$ for all $s\in S$ (Remark 3). Then $\nu$ is the image of $\lambda$ under the restriction homomorphism $X(T)\rightarrow X(S)$; since W operates trivially on X(S), the equality $w_0(\lambda ) =-\lambda$ implies that $\nu =-\nu$, which is impossible: hence $w_0(\lambda )\not=-\lambda$. On the other hand, if B is a G-invariant bilinear form on V, we have, for all $x, y$ in V and $s$ in S,

which implies that B = 0, hence the proposition.

Let $\mathfrak{S}_{\mathbf{R}}(G)$ be the set of classes of irreducible continuous representations of G on finite dimensional real vector spaces. Prop. 1 and the results of Appendix II give a bijection $\Phi : X_{++}/\Sigma \rightarrow \mathfrak{S}_{\mathbf{R}}$(G), where $\Sigma$ denotes the subgroup $\{1,-w_0\}$ of Aut(X(T)). More precisely, let $\lambda \in X_{++}$, and let $E_{\lambda}$ be a representation of G of highest weight $\lambda$; then

$\Phi (\{\lambda ,-w_0(\lambda )\}) = E_{\lambda[\mathbf{R}]}$ if $\lambda \not=-w_0(\lambda )$ or if $\sum_{\alpha\in R_+}\langle \lambda , K_{\alpha}\rangle \notin2\mathbf{Z}$

$\Phi (\{\lambda ,-w_0(\lambda )\}) = E'_{\lambda}$ if $\lambda =-w_0(\lambda )$ and $\sum_{\alpha\in R_+}\langle \lambda , K_{\alpha}\rangle \in 2\mathbf{Z}$

where $E'_{\lambda}$ is an $\mathbf{R}$-structure on $E_{\lambda}$ invariant under G.

For $\lambda \in X_{++}$, denote by $[\lambda ]$ the class in R(G) of the irreducible representation of highest weight $\lambda$. Since the family $([\lambda ])_{\lambda\in X_{++}}$ is a basis of the $\mathbf{Z}$-module R(G), it suffices to prove the following assertion:

The family (Ch[$\lambda ]$)$_{\lambda\in X_{++}}$ is a basis of the $\mathbf{Z}$-module $\mathbf{Z}[X(T)]^W$.

For every element $u=\sum_{\lambda}a_{\lambda}e^{\lambda}$ of $\mathbf{Z}[X(T)]$, a term $a_{\lambda}e^{\lambda}$ in $u$ is called maximal if $\lambda$ is a maximal element of the set of $\mu\in X(T)$ such that $a_\mu\not= 0$. Th. 1 implies that Ch[$\lambda ]$ has a unique maximal term, namely $e^{\lambda}$. Thus, the proposition follows from the following lemma.

The proof is identical to that of Prop. 3 of Chap. VI, §3, no. 4, replacing A by $\mathbf{Z}, P$ by X(T) and $P\cap \overline{C}$ by $X_{++}$.

Let $\Theta (G)$ (resp. $\Theta (T)$) be the $\mathbf{C}$-algebra of continuous representative functions on G (resp. T), and let $Z\Theta (G)$ (resp. $\Theta (T)^W$) be the subalgebra consisting of the central (resp. W-invariant) functions. The restriction map $\Theta (G)\rightarrow \Theta (T)$ induces a homomorphism of rings $r: Z\Theta (G)\rightarrow \Theta (T)^W$. On the other hand, the map that associates to a representation $\tau$ its character (that is, the function $g \rightarrow$ Tr $\tau (g)$) extends to a homomorphism of $\mathbf{C}$-algebras Tr$:\mathbf{C}\otimes_{\mathbf{Z}}R(G)\rightarrow Z\Theta (G)$ which, by Spectral Theories, is an isomorphism. Similarly, the canonical injection $X(T)\rightarrow \Theta (T)$ induces an isomorphism of $\mathbf{C}$-algebras $\iota :\mathbf{C}[X(T)]\rightarrow \Theta$(T), which induces an isomorphism $\iota :\mathbf{C}[X(T)]^W\rightarrow \Theta (T)^W$. The diagram

is commutative: indeed, for every representation $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ and all $t\in T$,

Tr $\tau (t) =\sum_{\lambda\in X(T)}$(dim $V_{\lambda}(T)$)$\lambda (t) =\iota$(Ch$\tau$ )$(t)$,

that is, $(r\circ$ Tr)[$\tau ] = (\iota \circ$ Ch)[$\tau ]$.

We can now deduce from the proposition the following result.

The proof is identical to that of Chap. VI, §3, no. 3, Prop. 1.

By Chap. VI, loc. cit., Prop. 2,

so

By Cor. 2 of Th. 1 (§6, no. 2), we deduce:

For all $\lambda \in X_{++}$, denote by $\chi_{\lambda}$ the character of an irreducible representation of G of highest weight $\lambda$.

The function $J(\rho ).\chi_{\lambda}|T$ is anti-invariant under W, and is a linear combination with integer coefficients of elements of X(T). Thus, by Chap. VI, §3, no. 3, Prop. 1, it can be written as $\sum_\mu a_\mu J(\mu\rho )$, where $\mu$ belongs to $X_{++}$, and the $a_\mu$ are integers all but finitely-many of which are zero; since $\int_G|\chi_{\lambda}(g)|^2dg= 1$

(Spectral Theories), it follows from Prop. 3 and Lemma 4 that $\sum_\mu(a_\mu)^2= 1$; thus, the $a_\mu$ are all zero, except one which must be equal to 1 or $-1$. But the coefficient of $\lambda$ in $\chi_{\alpha}|T$ is equal to 1 (Th. 1), hence the coefficient of $\lambda \rho$ in $J(\rho ).\chi_{\lambda}|T$ is equal to 1 (Chap. VI, §3, no. 3, Remark 2), which implies that $a_{\lambda \rho}= 1$, hence the theorem.

$(\sum_{w\in W}\varepsilon (w)e^{w\rho})$ Ch[$\lambda ] =\sum_{w\in W}\varepsilon (w)e^{w\lambda}e^{w\rho}$ for all $\lambda \in X_{++}$.

This follows from the theorem and commutative diagram (2) (no. 3).

Indeed, $J(\rho )(t)$ is non-zero, since $t$ is regular (formula (4)).

If $\varphi$ is a central function on G, the restriction of $\varphi$ to T is invariant under W, so $J(\rho ).\varphi |T$ is anti-invariant under W. Further, by Spectral Theories and Th. 1, the family $(\chi_{\lambda})_{\lambda\in X_{++}}$ is an algebraic basis of the space of central representative functions on G and an orthonormal basis of the space ZL$^2(G)$ of classes of square-integrable central functions on G.

Thus, from Prop. 3 and Th. 2 we deduce:

Indeed, by Lemma 4 and Th. 2,

But the function $t \rightarrow \varphi (t)J(\rho )(t)$ is anti-invariant and $\frac{1}{w(G)}J(\lambda \rho )$ is the orthogonal projection of $\lambda \rho$ onto the subspace of anti-invariant elements of $L^2$(T), so

finally, by formula (3), we have $\rho (t)J(\rho )(t) =\prod_{\alpha >0}(1-\alpha (t)^{-1})$, hence the corollary.

Put $\gamma =\frac{1}{2}\sum_{\alpha >0}K_{\alpha}$, so $\delta (\alpha )(\gamma ) = 2\pi i$ for every simple root $\alpha$ (Chap. VI, §1, no. 10, Prop. 29). The line $\mathbf{R}\gamma$ is not contained in any of the hyperplanes Ker $\delta (\alpha )$, so exp($z\gamma$ ) is a regular element of G for all sufficiently small $z\in \mathbf{R}^*$; for all $\mu\in X(T)$ and all $z\in \mathbf{R}$, we have

$J(\mu$)(exp($z\gamma$ )) $=\sum_{w\in W}\varepsilon (w)e^{z\delta(\mu)(w^{-1}\gamma)}$.

Let us accept provisionally the following lemma:

$J(\mu$)(exp($z\gamma$ )) $=e^{z\delta(\mu)(\gamma)}\prod_{\alpha >0}(1-e^{-z\delta(\mu)(K_{\alpha})})$.

Thus, the function $J(\mu$)(exp($z\gamma$ )) is the product of a function that tends to 1 when $z$ tends to 0 and of

where N = Card $R_+$.

Assume first of all that $\rho \in X(T)$; applying Cor. 2 of Th. 2 we see that, as $z$ tends to $0,\chi_{\lambda}(z\gamma )$ tends to

hence the theorem in this case.

In the general case, it suffices to remark that, in proving Th. 3, we can always replace G by a suitable connected covering and thus reduce to the preceding case.

We now prove Lemma 5. Let $z\in \mathbf{C}$; denote by $\varphi_z$ the map from $\mathfrak{t}$ to the $\mathbf{C}$-algebra Map(X(T)$,\mathbf{C}$) of maps from X(T) to $\mathbf{C}$ that associates to $H\in \mathfrak{t}$ the map

$\varphi_z(H) :\mu \rightarrow \mu$(exp $zH$) $=e^{z\delta(\mu)(H)}$.

We have $\varphi_z(H+H') =\varphi_z(H)\varphi_z(H')$, so there exists a homomorphism of rings

$\psi_z:\mathbf{Z}[\mathfrak{t}]\rightarrow$ Map(X(T)$,\mathbf{C}$)

such that $\psi_z(e^H)(\mu) =e^{z\delta(\mu)(H)}$. On the other hand, by Chap. VI, §3, no. 3, Prop. 2, we have the following relation in $\mathbf{Z}[\mathfrak{t}]:$

Applying the homomorphism $\psi_z$, and using the equality

we deduce the stated formula.

a) sup$_{\lambda\in X_{++}}d(\lambda )/\|\lambda +\rho \|^N<\infty$, where $N = 1/$2(dim $G-$ dim T).

b) If G is semi-simple, inf$_{\lambda\in X_{++}}d(\lambda )/\|\lambda +\rho \|>0$.

a) For all $\alpha \in R_+$, there exists $A_{\alpha}>0$ with $|\langle \lambda +\rho , K_{\alpha}\rangle | \leq A_{\alpha}\|\lambda +\rho \|$, hence $d(\lambda )/\|\lambda +\rho \|^N\leq \prod_{\alpha >0}A_{\alpha}/\langle \rho , K_{\alpha}\rangle$.

b) Assume that G is semi-simple, denote by $\beta_1, . . . , \beta_r$ the simple roots and put $N_i=K_{\beta_i}$. Then

since $\langle \lambda +\rho , N_i\rangle \geq \langle \rho , N_i\rangle = 1$, this implies that

$d(\lambda )\geq$ sup$_i|\langle \lambda +\rho , N_i\rangle |$.

If G is semi-simple, $x \rightarrow$ sup$|\langle x, N_i\rangle |$ is a norm on $X(T)\otimes \mathbf{R}$, necessarily equivalent to the given norm, hence b).

Cor. $1b)$ implies that the set $X_d$ of elements $\lambda$ of $X_{++}$ such that $d(\lambda )\leq d$ is finite. For all $\lambda$ in $X_d$, let $V_{\lambda}$ be an irreducible representation of highest weight $\lambda$; every representation of dimension $\leq d$ is isomorphic to a direct sum $\bigoplus_{\lambda\in X_d}V^{n_{\lambda}}_{\lambda}$, with $n_{\lambda}\leq d$, hence the corollary.