Weyl Character and Multiplicity Formulas

Fix the finite Weyl root-system data of Finite Weyl root system, lattice and chamber conventions: the real span $E$ of the roots with its positive definite form, a positive system ${\mathrm{\Phi }}^{+}$ with base ${\alpha }_1,\dots ,{\alpha }_r$, the root group $Q={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}\alpha \subseteq P$, the weight lattice $P\subseteq E$, and the positive cone $Q_{+}={\sum }_{i=1}^r{\mathbb{Z}}_{\ge 0}{\alpha }_i$ (the set written $Q^{+}$ in The Grothendieck group and character of O). A downward cone is a set $\lambda -Q_{+}=\{\lambda -\beta :\beta \in Q_{+}\}$ with $\lambda \in {\mathfrak{h}}^{\ast }$.

The completed formal character ring $\mathcal{R}$ is the set of formal sums $f={\sum }_{\mu \in {\mathfrak{h}}^{\ast }}{c}_{\mu }{e}^{\mu },c_{\mu }\in \mathbb{Z},$ whose support $\mathrm{supp}f=\{\mu :c_{\mu }\mathrm{\ne }0\}$ is contained in a finite union of downward cones. Addition is coefficientwise, the product is the convolution $(fg{)}_{\eta }={\sum }_{\mu +\nu =\eta }{c}_{\mu }{d}_{\nu },$ and the unit is $e^0$, so that $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. The product is well defined: if $\mathrm{supp}f$ lies in ${\bigcup }_{i=1}^m({\lambda }_i-Q_{+})$ and $\mathrm{supp}g$ in ${\bigcup }_{j=1}^n({\mu }_j-Q_{+})$, then a pair of exponents contributing to $\eta$ lies in some ${\lambda }_i-Q_{+}$ and some ${\mu }_j-Q_{+}$, and the solutions $\beta ,\gamma \in Q_{+}$ of $\beta +\gamma ={\lambda }_i+{\mu }_j-\eta$ are the elements of the box $0\le {\beta }_k\le ({\lambda }_i+{\mu }_j-\eta {)}_k$ in the simple-root coordinates, a finite set (empty unless ${\lambda }_i+{\mu }_j-\eta \in Q_{+}$). The coefficients are integers, and a finite union of downward cones is again such a union, so addition and multiplication make $\mathcal{R}$ a commutative $\mathbb{Z}$-algebra. In the notation of The Grothendieck group and character of O this is the ring denoted $\mathcal{R}$ there, where the character homomorphism of category $\mathcal{O}$ takes its values; the elements of finite support form the group ring $\mathbb{Z}[{\mathfrak{h}}^{\ast }]$ of the additive group ${\mathfrak{h}}^{\ast }$ and contain the subring $\mathbb{Z}[P]$ generated by the $e^{\mu }$ with $\mu \in P$.

By The formal character of a Verma module, $\mathrm{ch}M(\lambda )=e^{\lambda }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$ is an element of $\mathcal{R}$: the geometric series ${\sum }_{k\ge 0}{e}^{-k\alpha }$ is supported in the downward cone $-Q_{+}$, and a finite product of elements of $\mathcal{R}$ lies in $\mathcal{R}$ by the convolution formula, while $e^{\lambda }$ is a single monomial. No convergence of any formal sum is asserted: all sums are formal, coefficients are compared coefficientwise, and every finite sum, product or finite product of geometric series below is interpreted in $\mathcal{R}$ by the rules just recorded.

Assume the Axiom of Choice (The Axiom of Choice). Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\mathrm{\Phi }$ with a chosen positive system ${\mathrm{\Phi }}^{+}$, and let $C\in U(\mathfrak{g})$ be the quadratic Casimir element of The quadratic Casimir element. Let $h_1,\dots ,h_r$ and $h^1,\dots ,h^r$ be bases of $\mathfrak{h}$ dual with respect to the Killing form $B$ of The Killing form of a semisimple Lie algebra, so that $B(h_j,h^k)={\delta }_{jk}$, and for every $\alpha \in {\mathrm{\Phi }}^{+}$ choose $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $f_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ with $B(e_{\alpha },f_{\alpha })=1$; then $[e_{\alpha },f_{\alpha }]=H_{\alpha }$ is the Killing-dual vector of $\alpha$ (Opposite root spaces bracket to the Killing-dual line). Then $C={\sum }_{j=1}^r{h}_j{h}^j+{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}(e_{\alpha }{f}_{\alpha }+f_{\alpha }{e}_{\alpha })$ in $U(\mathfrak{g})$, and for every dominant integral $\lambda \in {\mathrm{\Lambda }}^{+}$ and every $\mu \in {\mathfrak{h}}^{\ast }$, writing $m_{\lambda }(\mu )=\dim L(\lambda {)}_{\mu }$ for the multiplicity of $\mu$ as a weight of the finite-dimensional simple module $L(\lambda )$ (Highest-weight classification), ${\mathrm{tr}}_{L(\lambda {)}_{\mu }}(C)=m_{\lambda }(\mu )(\mu ,\mu )+{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{\mathrm{tr}}_{L(\lambda {)}_{\mu }}(e_{\alpha }{f}_{\alpha }+f_{\alpha }{e}_{\alpha })\mathrm{.}$ Since $C$ acts on $L(\lambda )$ by the scalar $(\lambda ,\lambda +2\rho )$ (The quadratic Casimir eigenvalue on a highest-weight module is $(\lambda ,\lambda +2\rho )$), where $\rho$ is the Weyl vector (The Weyl vector rho for a chosen positive system) and the pairing on ${\mathfrak{h}}^{\ast }$ is the one induced by $B$, also ${\mathrm{tr}}_{L(\lambda {)}_{\mu }}(C)=m_{\lambda }(\mu )(\lambda ,\lambda +2\rho )$, hence $((\lambda ,\lambda +2\rho )-(\mu ,\mu ))m_{\lambda }(\mu )={\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{\mathrm{tr}}_{L(\lambda {)}_{\mu }}(e_{\alpha }{f}_{\alpha }+f_{\alpha }{e}_{\alpha })\mathrm{.}$

Let $\mathrm{\Phi }$ be the root system with positive system ${\mathrm{\Phi }}^{+}$, simple roots ${\alpha }_1,\dots ,{\alpha }_r$, Weyl group $W$ and Weyl vector $\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha$ (Finite Weyl root system, lattice and chamber conventions, The Weyl vector rho for a chosen positive system, Root reflections and the Weyl group action). For every $w\in W$, $\rho -w\rho ={\sum }_{\begin{array}{c}\alpha \in {\mathrm{\Phi }}^{+}\\ w^{-1}\alpha \in {\mathrm{\Phi }}^{-}\end{array}}\alpha ,$ the sum running over the positive roots whose image under $w^{-1}$ is a negative root. In particular $\rho -w\rho \in Q_{+}$: it is a nonnegative integral combination of the simple roots.

Let $W$ be the Weyl group of the root system $\mathrm{\Phi }$ with its real span $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$ and length function $\mathrm{\ell }$ (Finite Weyl root system, lattice and chamber conventions, Root reflections and the Weyl group action). Then $(-1{)}^{\mathrm{\ell }(uv)}=(-1{)}^{\mathrm{\ell }(u)}(-1{)}^{\mathrm{\ell }(v)}(u,v\in W),$ so that $w\mapsto (-1{)}^{\mathrm{\ell }(w)}$ is a group homomorphism $W\to \{\pm 1\}$, and this homomorphism is the determinant of the action of $W$ on $E$: $(-1{)}^{\mathrm{\ell }(w)}=\det (w)$ for every $w\in W$. In particular $(-1{)}^{\mathrm{\ell }(w^{-1})}=(-1{)}^{\mathrm{\ell }(w)}$.

Assume the Axiom of Choice (The Axiom of Choice). Let $V$ be a finite-dimensional $\mathfrak{g}$-module, where $\mathfrak{g}$ is a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, and let $V={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{V}_{\mu }$ be its weight-space decomposition, with $V_{\mu }$ the weight space of Weight and weight space (Finite-dimensional modules decompose into weight spaces). The formal character of $V$ is $\mathrm{ch}V:={\sum }_{\mu \in {\mathfrak{h}}^{\ast }}(\dim V_{\mu })e^{\mu }\in \mathcal{R},$ the finite sum with integer coefficients taken in the completed formal character ring $\mathcal{R}$ of The completed formal character ring; equivalently, $\mathrm{ch}V$ is the coefficient family $\mu \mapsto \dim V_{\mu }$ on ${\mathfrak{h}}^{\ast }$, which has finite support by the cited decomposition.

We write $m_{\lambda }(\mu ):=\dim L(\lambda {)}_{\mu }$ for the multiplicity of $\mu$ as a weight of the finite-dimensional simple module $L(\lambda )$ of highest weight $\lambda \in {\mathrm{\Lambda }}^{+}$ (Highest-weight classification), so that $\mathrm{ch}L(\lambda )={\sum }_{\mu }{m}_{\lambda }(\mu )e^{\mu }$; the coefficients $m_{\lambda }(\mu )$ are nonnegative integers and $m_{\lambda }(\mu )=0$ for all but finitely many $\mu$.

Let $W$ be the Weyl group of the root system of $\mathfrak{g}$, acting on ${\mathfrak{h}}^{\ast }$ by the root reflections of Root reflections and the Weyl group action; by The Weyl group is finite and faithful and Finite Weyl root system, lattice and chamber conventions, $W$ is a finite group and it permutes the roots, hence preserves the root lattice $Q$ and the weight lattice $P$ (Finite Weyl positive roots and simple reflections).

Action on finite support. Every $w\in W$ maps the finite subsets of ${\mathfrak{h}}^{\ast }$ to finite subsets, so the assignment $w\cdot e^{\mu }:=e^{w\mu }$ extends uniquely to a $\mathbb{Z}$-algebra automorphism $f\mapsto w\cdot f$ of the group ring of finite-support elements of the completed character ring $\mathcal{R}$ (The completed formal character ring), with inverse $w^{-1}\cdot (-)$; it restricts to a $\mathbb{Z}$-algebra automorphism of $\mathbb{Z}[P]$, because $W$ preserves the weight lattice $P$. Caveat. If $\mathrm{\Phi }\mathrm{\ne }\mathrm{\varnothing }$, this formula does not define an action on all of $\mathcal{R}$. For a simple root ${\alpha }_i$, the series ${\sum }_{k\ge 0}{e}^{-k{\alpha }_i}$ lies in $\mathcal{R}$, whereas its image under $s_i$ has support $\{k{\alpha }_i:k\ge 0\}$, outside every finite union of downward cones: in each cone the $i$th simple-root coordinate is bounded above. If $\mathrm{\Phi }=\mathrm{\varnothing }$, then $Q_{+}=\{0\}$, every cone is a point, $\mathcal{R}$ has only finite-support elements and $W=\{1\}$ acts trivially. Only finite-support elements are acted on below.

The alternation operator. For $\nu \in {\mathfrak{h}}^{\ast }$ define the Weyl alternation operator by $A(\nu ):={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w\nu },$ the sum being finite because $W$ is finite; here $\mathrm{\ell }$ is the length function of Finite Weyl root system, lattice and chamber conventions turned into the sign homomorphism of The sign of the Weyl length is multiplicative. By The sign of the Weyl length is multiplicative the coefficients $(-1{)}^{\mathrm{\ell }(w)}$ define the determinant sign of $w$ acting on the real span $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$ of the roots. Since the sum is finite, $A(\nu )$ is a finite-support element of $\mathcal{R}$ for every $\nu$, and when $\nu \in P$ it lies in $\mathbb{Z}[P]\subseteq \mathcal{R}$.

Assume the Axiom of Choice. Keep the notation of The Casimir comparison on a weight space: $\mathfrak{g}$ is a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and positive system ${\mathrm{\Phi }}^{+}$, the vectors $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $f_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ satisfy $B(e_{\alpha },f_{\alpha })=1$, so that $[e_{\alpha },f_{\alpha }]=H_{\alpha }$ is the Killing-dual vector of $\alpha$, and $m_{\lambda }(\mu )=\dim L(\lambda {)}_{\mu }$ for $\lambda \in {\mathrm{\Lambda }}^{+}$. Then for every $\lambda \in {\mathrm{\Lambda }}^{+}$, $\mu \in {\mathfrak{h}}^{\ast }$ and $\alpha \in {\mathrm{\Phi }}^{+}$, ${\mathrm{tr}}_{L(\lambda {)}_{\mu }}(e_{\alpha }{f}_{\alpha }+f_{\alpha }{e}_{\alpha })=m_{\lambda }(\mu )(\mu ,\alpha )+2{\sum }_{j\ge 1}{m}_{\lambda }(\mu +j\alpha )(\mu +j\alpha ,\alpha ),$ the sum being finite because $L(\lambda )$ has only finitely many weights.

Assume the Axiom of Choice (The Axiom of Choice). Let $\lambda \in {\mathrm{\Lambda }}^{+}$ be a dominant integral weight and let $\mu$ be a weight of the finite-dimensional simple module $L(\lambda )$ (Highest-weight classification, Weight and weight space). With $\rho$ the Weyl vector (The Weyl vector rho for a chosen positive system), $(\mu +\rho ,\mu +\rho )\le (\lambda +\rho ,\lambda +\rho ),$ with equality if and only if $\mu =\lambda$. Consequently $(\lambda +\rho ,\lambda +\rho )-(\mu +\rho ,\mu +\rho )$ is strictly positive for every weight $\mu \mathrm{\ne }\lambda$ of $L(\lambda )$.

Let $u\in \mathcal{R}$ be supported in $-Q_{+}\setminus \{0\}$, so the coefficient of $e^0$ in $u$ is $0$ and every exponent in the support of $u$ has strictly negative height, where heights are taken in the simple-root coordinates of Height and highest root and Simple roots form a signed integral basis. Then $1+u$ is invertible in $\mathcal{R}$, with inverse $(1+u{)}^{-1}={\sum }_{k\ge 0}(-u{)}^k,$ the sum being coefficientwise finite because $u^k$ is supported in weights of height at most $-k$. In particular:

(i) $e^{\mu }$ is invertible with inverse $e^{-\mu }$ for every $\mu \in {\mathfrak{h}}^{\ast }$;

(ii) the finite product ${\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha })$ equals $1+u_0$ for an element $u_0$ supported in $-Q_{+}\setminus \{0\}$, and is therefore invertible, with ${\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}{\sum }_{k\ge 0}{e}^{-k\alpha }$;

(iii) the product $e^{\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha })$ is invertible, with inverse $e^{-\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$;

(iv) for the Weyl vector $\rho$ the alternant $A(\rho )$ of The Weyl alternation operator factors as $A(\rho )=e^{\rho }(1+u^{\mathrm{\prime }})$ with $u^{\mathrm{\prime }}$ supported in $-Q_{+}\setminus \{0\}$, and so is invertible with inverse $e^{-\rho }(1+u^{\mathrm{\prime }}{)}^{-1}$.

The identification of the inverse in (iv) with the inverse in (iii) is the content of the Weyl denominator identity proved later and is not asserted here.