§ 9. The formula of Hermann Weyl

In this paragraph, we retain the general notations of §6 and §7.

1. Characters of finite dimensional $\mathfrak{g}$ -modules

Let $(e^{\lambda})_{\lambda\in\mathfrak{h}^*}$ be the canonical basis of the ring $\mathbf{Z}[\mathfrak{h}^*]$ . Give the space $\mathbf{Z}^{\mathfrak{h}^*}$ of all maps from $\mathfrak{h}^*$ to $\mathbf{Z}$ the product topology of the discrete topologies on the factors. If $\varphi \in \mathbf{Z}^{\mathfrak{h}^*}$ , the family $(\varphi (\nu )e^{\nu})_{\nu\in\mathfrak{h}^*}$ is summable, and

\varphi =\sum_{\nu\in\mathfrak{h}^*}\varphi (\nu )e^{\nu}

Let $\mathbf{Z}\langle P\rangle$ be the set of $\varphi \in \mathbf{Z}^{\mathfrak{h}^*}$ whose support is contained in a finite union of sets of the form $\nu -P_+$ , where $\nu \in \mathfrak{h}^*$ . Then $\mathbf{Z}[P]\subset \mathbf{Z}\langle P\rangle \subset \mathbf{Z}^{\mathfrak{h}^*}$ . Define on $\mathbf{Z}\langle P\rangle$ a ring structure extending that of $\mathbf{Z}[P]$ by putting, for $\varphi , \psi \in \mathbf{Z}\langle P\rangle$ and $\nu \in \mathfrak{h}^*$ ,

(\varphi \psi )(\nu ) =\sum_{\mu\in\mathfrak{h}^*}\varphi (\mu)\psi (\nu -\mu)

(the family $(\varphi (\mu)\psi (\nu -\mu))_{\mu\in\mathfrak{h}^*}$ has finite support, in view of the condition satisfied by the supports of $\varphi$ and $\psi$ ). If $\varphi =\sum_{\nu}x_{\nu}e^{\nu}$ and $\psi =\sum_{\nu}y_{\nu}e^{\nu}$ , then $\varphi \psi =\sum_{\nu ,\mu}x_{\nu}y_\mu e^{\nu+\mu}$ .

Let $\nu \in \mathfrak{h}^*$ . A partition of $\nu$ into positive roots is a family $(n_{\alpha})_{\alpha\in R_+}$ , where the $n_{\alpha}$ are integers $\geq 0$ such that $\nu =\sum_{\alpha\in R_+}n_{\alpha}\alpha$ . We denote by $\mathfrak{P}(\nu )$ the number of partitions of $\nu$ into positive roots. We have

\mathfrak{P}(\nu )>0\Leftarrow \Rightarrow \nu \in Q_+

In this paragraph, we denote by K the following element of $\mathbf{Z}\langle P\rangle :$

K =\sum_{\gamma\in Q_+}\mathfrak{P}(\gamma )e^{-\gamma}

Now recall (Chap. VI, §3, no. 3, Prop. 2) that

d=\prod_{\alpha\in R_+}(e^{\alpha /2}-e^{-\alpha /2}) =\sum_{w\in W}\varepsilon (w)e^{w\rho}

is an anti-invariant element of $\mathbf{Z}[P]$ .

Lemma 1 015T

In the ring $\mathbf{Z}\langle P\rangle$ , we have $K.\prod_{\alpha\in R_+}(1-e^{-\alpha}) = Ke^{-\rho}d= 1$ .

Indeed,

K =\prod_{\alpha\in R_+}(e^0+e^{-\alpha}+e^{-2\alpha}+\cdots )

Ke^{-\rho}d=\prod_{\alpha\in R_+}(1 +e^{-\alpha}+e^{-2\alpha}+\cdots )\prod_{\alpha\in R_+}(1-e^{-\alpha}) = 1

Lemma 2 015U

Let $\lambda \in \mathfrak{h}^*$ . The module $Z(\lambda ) ($ §6, no. 3) admits a character that is an element of $\mathbf{Z}\langle P\rangle$ , and we have $d$ . ch $Z(\lambda ) =e^{\lambda+\rho}$ .

Let $\alpha_1, . . . , \alpha_q$ be distinct elements of $R_+$ . The $X_{-\alpha_1}^{n_1}X_{-\alpha_2}^{n_2}. . . X_{-\alpha_q}^{n_q}\otimes 1$ form a basis of $Z(\lambda ) ($ §6, Prop. 6 (iii)). For $h\in \mathfrak{h}$ , we have

h.(X_{-\alpha_1}^{n_1}X_{-\alpha_2}^{n_2}. . .X_{-\alpha_q}^{n_q}\otimes 1)

= [h, X_{-\alpha_1}^{n_1}. . . X_{-\alpha_q}^{n_q}]\otimes 1 + (X_{-\alpha_1}^{n_1}. . . X_{-\alpha_q}^{n_q})\otimes h.1

= (\lambda -n_1\alpha_1- \cdots -n_q\alpha_q)(h)(X_{-\alpha_1}^{n_1}. . . X_{-\alpha_q}^{n_q}\otimes 1)

Thus, the dimension of $Z(\lambda )^{\lambda-\mu}$ is $\mathfrak{P}(\mu)$ . This proves that ch $Z(\lambda )$ is defined, is an element of $\mathbf{Z}\langle P\rangle$ , and that

ch $Z(\lambda ) =\sum_\mu\mathfrak{P}(\mu)e^{\lambda-\mu}= Ke^{\lambda}$ .

It now suffices to apply Lemma 1.

Lemma 3 015V

Let M be a $\mathfrak{g}$ -module which admits a character ch(M) whose support is contained in a finite union of the sets $\mu-P_+$ . Let U be the enveloping algebra of $\mathfrak{g}$ , Z the centre of U $,\lambda_0\in \mathfrak{h}^*$ , and $\chi_{\lambda_0}$ the corresponding homomorphism from Z to $k($ §8, Cor. 1 of Th. 2). Assume that, for all $z\in Z,z_M$ is the homothety with ratio $\chi_{\lambda_0}(z)$ . Let $D_M$ be the set of $\lambda \in W(\lambda_0+\rho )-\rho$ such that $\lambda + Q_+$ meets Supp(ch M). Then ch(M) is a $\mathbf{Z}$ -linear combination of the ch $Z(\lambda )$ for $\lambda \in D_M$ .

If Supp(ch M) is empty, the lemma is clear. Assume that Supp(ch $M$ ) $\not=\emptyset$ . Let $\lambda$ be a maximal element of this support, and put dim $M^{\lambda}=m$ . There exists a $\mathfrak{g}$ -homomorphism $\varphi$ from $(Z(\lambda ))^m$ to M which maps $(Z(\lambda )^{\lambda})^m$ bijectively onto $M^{\lambda}($ §6, no. 3, Prop. 6 (i)). Thus, the central character of $Z(\lambda )$ is $\chi_{\lambda_0}$ , so $\lambda \in W(\lambda_0+\rho )-\rho ($ §8, no. 5, Cor. 1 of Th. 2). This proves that $D_M\not=\emptyset$ , and allows us to argue by induction on Card $D_M$ . Let L and N be the kernel and cokernel of $\varphi$ . Then we have an exact sequence of $\mathfrak{g}$ -homomorphisms:

0\rightarrow L\rightarrow (Z(\lambda ))^m\rightarrow M\rightarrow N\rightarrow 0

ch(M) $=-$ ch(L) $+m$ ch $Z(\lambda ) +$ ch(N)

(§7, no. 7, formula (6)). The sets Supp(ch L) and Supp(ch N) are contained in a finite union of sets $\mu-P_+$ . For $z\in Z,z_L$ and $z_N$ are homotheties with ratio $\chi_{\lambda_0}(z)$ . Clearly, $D_N\subset D_M$ . On the other hand, $(\lambda + Q_+)\cap$ Supp(ch $M$ ) $=\{\lambda \}$ , and $\lambda \notin$ Supp(ch N), so $\lambda \notin D_N$ and

Card $D_N<$ Card $D_M$ .

On the other hand, L is a submodule of $(Z(\lambda ))^m$ ; if $\lambda '\in D_L$ , then $\lambda '+ Q_+$ meets Supp(ch $L$ ) $\subset$ Supp ch $Z(\lambda )$ , so $\lambda \in \lambda '+ Q_+($ §6, no. 1, Prop. 1 (ii)); it follows that $D_L\subset D_M$ . Since $L\cap (Z(\lambda )^{\lambda})^m= 0$ , we have $\lambda \notin D_L$ , so

Card $D_L<$ Card $D_M$ .

It now suffices to apply the induction hypothesis.

Theorem 1 (Character Formula of H. Weyl) 015W

Let M be a finite dimensional simple $\mathfrak{g}$ -module, and $\lambda$ its highest weight. Then

$(\sum_{w\in W}\varepsilon (w)e^{w\rho})$ .ch $M =\sum_{w\in W}\varepsilon (w)e^{w(\lambda+\rho)}$ .

With the notations of Lemma 3, the central character of M is $\chi_{\lambda}($ §6, no. 4, Prop. 7). Hence, by Lemmas 2 and $3,d$ .ch M is a $\mathbf{Z}$ -linear combination of the $e^{\mu+\rho}$ such that

\mu+\rho \in W(\lambda +\rho )

On the other hand, by §7, no. 7, Lemma $7,d$ .ch M is anti-invariant, and its unique maximal term is $e^{\lambda+\rho}$ , hence the theorem.

Example 015X

Take $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k),\mathfrak{h}=kH$ . Let $\alpha$ be the root of $(\mathfrak{g},\mathfrak{h})$ such that $\alpha (H) = 2$ . The $\mathfrak{g}$ -module $V(m)$ has highest weight $(m/2)\alpha$ . Hence

ch(V( $m$ )) $= (e^{(m/2)\alpha+^1_2\alpha}-e^{-(m/2)\alpha-^1_2\alpha})/(e^{^1_2\alpha}-e^{-^1_2\alpha})$

=e^{-(m/2)\alpha}.(e^{(m+1)\alpha}-1)/(e^{\alpha}-1)

=e^{-(m/2)\alpha}(e^{m\alpha}+e^{(m-1)\alpha}+\cdots + 1)

=e^{(m/2)\alpha}+e^{(m-2)\alpha /2}+\cdots +e^{-(m/2)\alpha}

which also follows easily from §1, no. 2, Prop. 2.

2. Dimensions of simple $\mathfrak{g}$ -modules

If $\mu\in \mathfrak{h}^*$ , put $J(e^\mu) =\sum_{w\in W}\varepsilon (w)e^{w\mu}$ , cf. Chap. VI, §3, no. 3.

Theorem 2 015Y

Let E be a finite dimensional simple $\mathfrak{g}$ -module, $\lambda$ its highest weight and $(\cdot | \cdot )$ a W-invariant non-degenerate positive symmetric bilinear form on $\mathfrak{h}^*$ . Then:

\prod\langle \lambda +\rho , H_{\alpha}\rangle \prod((\lambda |\alpha ))

dim E = = 1 +.

_{\alpha\in R_+}\langle \rho , H_{\alpha}\rangle_{\alpha\in R_+}\overline{(\rho|\alpha)}

Let T be an indeterminate. For all $\nu \in P$ , denote by $f_{\nu}$ the homomorphism from $\mathbf{Z}[P]$ to $\mathbf{R}[[T]]$ that takes $e^\mu$ to $e^{(\nu|\mu)T}$ for all $\mu\in P$ . Then dim E is the constant term of the series $f_{\nu}$ (ch E).

For all $\mu, \nu \in P$ , we have

f_{\nu}(J(e^\mu)) =\sum_{w\in W}\varepsilon (w)e^{(\nu|w\mu)T}

=\sum_{w\in W}\varepsilon (w)e^{(w^{-1}\nu|\mu)T}=f_\mu(J(e^{\nu}))

In particular, in view of Chap. VI, §3, no. 3, formula (3),

f_{\rho}(J(e^\mu)) =f_\mu(J(e^{\rho})) =e^{(\mu|\rho)T}\prod_{\alpha\in R_+}(1-e^{-(\mu|\alpha)T})

Hence, setting Card(R $_+$ ) $= N$ ,

$f_{\rho}(J(e^\mu))\equiv T^N\prod_{\alpha\in R}(\mu|\alpha )$ (mod $T^{N+1}\mathbf{R}[[T]]$ ).

The equality $J(e^{\lambda+\rho}) =$ ch(E) $.J(e^{\rho})$ (Th. 1) thus implies that

$T^N\prod_{\alpha\in R_+}(\lambda +\rho |\alpha )\equiv f_{\rho}$ (ch $E$ ) $.T^N\prod_{\alpha\in R_+}(\rho |\alpha )$ (mod $T^{N+1}\mathbf{R}[[T]]$ )

dim $E =((\prod_{\alpha\in R_+}(\lambda +\rho |\alpha ))/((\prod_{\alpha\in R_+}(\rho |\alpha ))=\prod_{\alpha\in R_+}(1 +\frac{(\lambda|\alpha)}{(\rho|\alpha)})$ .

Now, if $\alpha \in R_+,\alpha$ can be identified with an element of $\mathfrak{h}_{\mathbf{R}}$ proportional to $H_{\alpha}$ , so

(\lambda +\rho |\alpha )/(\rho |\alpha ) =\langle \lambda +\rho , H_{\alpha}\rangle /\langle \rho , H_{\alpha}\rangle

Example 1 015Z

In the Example of no. 1, we find that

dim $V(m) =(\frac{m}{2}\alpha +\frac{\alpha}{2})(H_{\alpha})/\frac{\alpha}{2}(H_{\alpha}) =m+ 1$ ,

which we knew in §1.

Example 2 0160

Take $\mathfrak{g}$ to be the splittable simple Lie algebra of type $G_2$ and adopt the notations of Chap. VI, Plate IX. Give $\mathfrak{h}^*_{\mathbf{R}}$ the W-invariant positive symmetric form $(\cdot | \cdot )$ such that $(\alpha_1|\alpha_1) = 1$ . Then $\rho =\varpi_1+\varpi_2$ and

(\varpi_1|\alpha_1) =\frac{1}{2},(\varpi_1|\alpha_2) = 0,(\varpi_1|\alpha_2+\alpha_1) =\frac{1}{2}

(\varpi_1|\alpha_2+ 2\alpha_1) = 1,(\varpi_1|\alpha_2+ 3\alpha_1) =\frac{3}{2},(\varpi_1|2\alpha_2+ 3\alpha_1) =\frac{3}{2}

(\varpi_2|\alpha_1) = 0,(\varpi_2|\alpha_2) =\frac{3}{2},(\varpi_2|\alpha_2+\alpha_1) =\frac{3}{2}

(\varpi_2|\alpha_2+ 2\alpha_1) =\frac{3}{2},(\varpi_2|\alpha_2+ 3\alpha_1) =\frac{3}{2},(\varpi_2|2\alpha_2+ 3\alpha_1) = 3

Thus, if $n_1, n_2$ are integers $\geq 0$ , the dimension of the simple representation of highest weight $n_1\varpi_1+n_2\varpi_2$ is

(1 +\frac{n_1/2}{1\overline{2}})((1 +\frac{3n_2/2}{3\overline{2}})((1 +\frac{n_1/2 + 3n_2/2}{_{1\overline{2}} + \frac{3}{2}})((1 +\frac{n_1 + 3n_2/2}{1 +^{3\overline{2}}})

\times (1 +\frac{3n_1/2 + 3n_2/2}{_{3\overline{2}} + \frac{3}{2}})((1 +3n_1\frac{/_3}{2}2 + 3+ 3n_2)

= (1 +n_1)(1 +n_2)(1 +\frac{n_1 + 3n_2}{4})((1 +\frac{2n_1 + 3n_2}{5})((1 +\frac{n_1 + n_2}{2})

\times (1 +\frac{n_1 + 2n_2}{3})

(1+n_1)(1+n_2)(2+n_1+n_2)(3+n_1+2n_2)(4+n_1+3n_2)(5+2n_1+3n_2)

In particular, the fundamental representation of highest weight $\varpi_1$ (resp. $\varpi_2$ ) is of dimension 7 (resp. 14).

3. Multiplicities of weights of simple $\mathfrak{g}$ -modules

Proposition 1 0161

Let $\omega \in P_{++}$ . For all $\lambda \in P$ , the multiplicity of $\lambda$ in $E(\omega )$ is

m_{\lambda}=\sum_{w\in W}\varepsilon (w)\mathfrak{P}(w(\omega +\rho )-(\lambda +\rho ))

By Th. 1 and Lemma 1,

ch $E(\omega ) = Ke^{-\rho}d$ ch $E(\omega ) = Ke^{-\rho}\sum_{w\in W}\varepsilon (w)e^{w(\omega+\rho)}$

ch $E(\omega ) =\sum_{w\in W,\gamma\in Q_+}\varepsilon (w)\mathfrak{P}(\gamma )e^{-\rho+w(\omega+\rho)-\gamma}$

and

m_{\lambda}=\sum\varepsilon (w)\mathfrak{P}(\gamma )

w\in W,\gamma \in Q_+,\gamma =-\lambda -\rho +w(\omega +\rho )

Corollary 0162

If $\lambda$ is a weight of $E(\omega )$ distinct from $\omega$ ,

m_{\lambda}=-\sum_{w\in W,w\not=1}\varepsilon (w)m_{\lambda+\rho-w\rho}

Apply Prop. 1 with $\omega = 0$ . If $\mu\in P$ -- $\{0\}$ , we find that

0 =\sum_{w\in W}\varepsilon (w)\mathfrak{P}(w\rho +\mu-\rho )

hence

\mathfrak{P}(\mu) =-\sum_{w\in W,w\not=1}\varepsilon (w)\mathfrak{P}(\mu+w\rho -\rho ) \tag{1}

Prop. 1 also gives

m_{\lambda}=-\sum_{w\in W}\varepsilon (w)\sum_{w'\in W,w'\not=1}\varepsilon (w')\mathfrak{P}(w(\omega +\rho )-(\lambda +\rho ) +w'\rho -\rho )

since $w(\omega +\rho )\not=\lambda +\rho$ for all $w\in W ($ §7, Prop. 5 (iii)). Hence,

m_{\lambda}=-\sum_{w'\in W,w'\not=1}\varepsilon (w')\sum_{w\in W}\varepsilon (w)\mathfrak{P}(w(\omega +\rho )-(\lambda +\rho -w'\rho +\rho ))

$=-\sum_{w'\in W,w'\not=1}\varepsilon (w')m_{\lambda+\rho-w'\rho}$ (Prop $.1$ ).

4. Decomposition of tensor products of simple $\mathfrak{g}$ -modules

Proposition 2 0163

Let $\lambda , \mu\in P_{++}$ . In $\mathscr{R}(\mathfrak{g})$ , we have

[\lambda ].[\mu] =\sum_{\nu\in P_{++}}m(\lambda , \mu, \nu )[\nu ]

with

m(\lambda , \mu, \nu ) =\sum_{w,w'\in W}\varepsilon (ww')\mathfrak{P}(w(\lambda +\rho ) +w'(\mu+\rho )-(\nu + 2\rho ))

Let $E,F$ be finite dimensional simple $\mathfrak{g}$ -modules of highest weights $\lambda , \mu$ . Let $l_{\nu}$ be the length of the isotypical component of $E\otimes F$ of highest weight $\nu$ . It suffices to show that

l_{\nu}=\sum_{w,w'\in W}\varepsilon (ww')\mathfrak{P}(w(\lambda +\rho ) +w'(\mu+\rho )-(\nu + 2\rho )) \tag{2}

Put $c_1=$ ch(E) $=\sum_{\sigma\in P}m_{\sigma}e^{\sigma},c_2=$ ch(F), and $d= J(e^{\rho})$ , where J is defined as in no. 2. We have $\sum_{\xi\in P_{++}}l_{\xi}$ ch[ $\xi ] =$ ch(E $\otimes F$ ) $=c_1c_2$

so, after multiplying by $d$ and using Th. 1,

\sum_{\xi\in P_{++}}l_{\xi}J(e^{\xi+\rho}) =c_1J(e^{\mu+\rho}) =(\sum_{\sigma\in P}m_{\sigma}e^{\sigma})((\sum_{w\in W}\varepsilon (w)e^{w(\mu+\rho)}) \tag{3}

=\sum_{\tau\in P}(\sum_{w\in W}\varepsilon (w)m_{\tau+\rho-w(\mu+\rho)})e^{\tau+\rho}

Now, if $\xi \in P_{++},\xi +\rho$ belongs to the chamber defined by B (Chap. VI, §1, no. 10); thus, for all $w\in W$ distinct from 1, we have $w(\xi +\rho )\notin P_{++}$ . Consequently, the coefficient of $e^{\nu+\rho}$ in $\sum_{\xi\in P_{++}}l_{\xi}J(e^{\xi+\rho})$ is equal to $l_{\nu}$ . In view of (3), we obtain

l_{\nu}=\sum_{w\in W}\varepsilon (w)m_{\nu+\rho-w(\mu+\rho)}

that is, by Prop. 1,

l_{\nu}=\sum_{w,w'\in W}\varepsilon (w)\varepsilon (w')\mathfrak{P}(w'(\lambda +\rho )-(\nu +\rho -w(\mu+\rho ) +\rho ))

which proves (2).

Example 0164

We return to the Example of no. 1. Let $\lambda = (n/2)\alpha , \mu= (p/2)\alpha$ , $\nu = (q/2)\alpha$ with $n\geq p$ . We have

m(\lambda , \mu, \nu ) =\mathfrak{P}(\frac{n}{2}\alpha +\frac{\alpha}{2}+\frac{p}{2}\alpha +\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

-\mathfrak{P}(\frac{n}{2}\alpha +\frac{\alpha}{2}-\frac{p}{2}\alpha -\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

-\mathfrak{P}(-\frac{n}{2}\alpha -\frac{\alpha}{2}+\frac{p}{2}\alpha +\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

+\mathfrak{P}(-\frac{n}{2}\alpha -\frac{\alpha}{2}-\frac{p}{2}\alpha -\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

=\mathfrak{P}(\frac{n + p-q}{2}\alpha )-\mathfrak{P}(\frac{n-p-q- 2}{2}\alpha )

This is zero if $n+p+q$ is not divisible by 2, or if $q\geq n+p$ . If

q=n+p-2r

with $r$ an integer $\geq 0$ , we have

m(\lambda , \mu, \nu ) =\mathfrak{P}(r\alpha )-\mathfrak{P}((r-p-1)\alpha )

hence $m(\lambda , \mu, \nu ) = 1$ if $r\leq p$ and $m(\lambda , \mu, \nu ) = 0$ if $r > p$ . Finally, the $\mathfrak{g}$ -module $V(n)\otimes V(p)$ is isomorphic to

V(n+p)\oplus V(n+p-2)\oplus V(n+p-4)\oplus \cdots \oplus V(n-p)

(Clebsch-Gordan formula).

Exercises

All the $\mathfrak{g}$ -modules considered are assumed to be finite dimensional.

See the exercises for § 9.