Past exam of the representation theory of semisimple Lie algebras course of the University of Oxford / 2010 / 3

For $f,g\in \mathcal{E}$, define their convolution byThe support restrictions make each sum finite: for fixed upper-cone representatives, the simple root coefficients in the two downward displacements are bounded nonnegative integers. This also proves closure in $\mathcal{E}$ and associativity by finite reordering at each coefficient. The unit is ${\delta }_0$, and ${\delta }_{\mu }\ast {\delta }_{\nu }={\delta }_{\mu +\nu }$.

With the source's negative root partition function, interpret $Q_{+}={\sum }_{\alpha \in B}{\mathbb{Z}}_{\ge 0}\alpha$ and allow zero coefficients in the partitions. Write $e^{\lambda }$ for ${\delta }_{\lambda }$, with multiplication meaning convolution. ThenThe denominator needed for the asserted character identities isThe repeated identical subscripts in the printed denominator would instead make it zero, so those literal factors cannot satisfy an inverse identity. Also the required inverse uses the negative $\rho$ shift. Geometric-series cancellation givesWith a positive shift the left side would be ${\delta }_{2\rho }$, which displays the second printed sign problem even after the denominator is repaired.

Poincaré-Birkhoff-Witt theorem gives ${\pi }_{M^{\xi }}={\delta }_{\xi -\rho }\ast K$, since each ordered negative root monomial lowers the highest weight by its corresponding partition. ThereforeThe remaining parts use $D$ for the intended Weyl denominator, while the private TeX preserves the printed signs.

For a finite-support formal character, use the same pullback rule $(wu)(\xi )=u(w^{-1}\xi )$; equivalently $w(e^{\lambda })=e^{w\lambda }$. This specifies the action on the denominator, which is a finite delta sum rather than a polynomial function. The whole infinite-support completion need not be Weyl-stable, so no such claim is required.

A simple reflection $s_{\alpha }$ permutes the positive roots other than $\alpha$ and sends $\alpha$ to $-\alpha$. In the product defining $D$, all factors are therefore permuted except the $\alpha$ factor, whose sign reverses. Thus $s_{\alpha }D=-D$. Simple reflections generate $W$, and each has determinant $-1$, so

Use the intended finite-dimensional irreducible highest weight module $A^{\varphi }$, with dominant integral highest weight $\varphi$. Its highest weight multiplicity is one. An arbitrary direct sum having that same largest representation weight would not in general satisfy this formula. Set $f={\pi }_{A^{\varphi }}$ and use the corrected denominator $D$ of part (a).

Restriction to each simple root ${\mathfrak{sl}}_2$ decomposes into symmetric finite representation weight strings, so $f$ is Weyl-invariant. Consequently $Df$ is Weyl-antisymmetric. We prove that no lower Weyl alternants occur, rather than assuming the desired character formula.

First establish the denominator identity itself. The finite sum $D$ has support in $\rho -Q_{+}$ and coefficient one at $\rho$. Antisymmetry kills all singular representation weight coefficients, since a reflection fixing the representation weight reverses their sign. Every remaining Weyl orbit has a unique regular dominant representative $\eta$. If it occurs in $D$, then $\eta \le \rho$. Since $\eta$ is integral and regular dominant, $\eta -\rho$ is dominant integral; it also belongs to $-Q_{+}$. A dominant vector $v=-\sum n_i{\alpha }_i$ satisfies $(v,v)=-\sum n_i(v,{\alpha }_i)\le 0$, so it must be zero. Thus the only possible orbit is $W\rho$. The leading coefficient givesFor the formal Laplacian $\mathcal{D}(e^{\lambda })=\Vert \lambda {\Vert }^2{e}^{\lambda }$, this implies $\mathcal{D}D=\Vert \rho {\Vert }^{2}D$.

Now choose root vectors dual under an invariant form, so $[E_{\alpha },E_{-\alpha }]=H_{\alpha }$ with $H_{\alpha }$ corresponding to the vector $\alpha$, and take the quadratic Casimirfor an orthonormal Cartan basis. On a highest vector its scalar is $c_{\varphi }=\Vert \varphi +\rho {\Vert }^2-\Vert \rho {\Vert }^2$, hence it has this scalar throughout the cyclic highest weight module.

Here is the computation of the weighted trace of its radial identity. For a positive root, let $X$ be the formal weighted trace of $E_{-\alpha }{E}_{\alpha }$ and $Y$ that of $E_{\alpha }{E}_{-\alpha }$. Cyclicity of trace and representation weight shifting give $Y=e^{\alpha }X$, while their difference is the weighted trace of $H_{\alpha }$, namely $(\alpha ,\nabla f)$. ThusTaking the trace of $C$ yieldsThe product expression for $D$ givesUsing $2\rho ={\sum }_{\alpha >0}\alpha$ and the product rule for the formal Laplacian therefore givesHence every representation weight in the support of $Df$ has norm $\Vert \varphi +\rho \Vert$. Its support is also contained in $\varphi +\rho -Q_{+}$, and it is antisymmetric. Any occurring regular dominant weight $\eta$ thus satisfies $\eta \le \varphi +\rho$. If the inequality were strict, thenbecause the first factor is a nonzero nonnegative sum of simple roots and the second is strictly dominant. This contradicts the equal-norm condition. Only the orbit $W(\varphi +\rho )$ remains. Its leading coefficient is one, proving

Convolve the identity of part (c) with the inverse denominator $K\ast {\delta }_{-\rho }$ found in part (a). Associativity givesTranslation by a Kronecker delta obeys $(K\ast {\delta }_{\lambda })(\psi )=K(\psi -\lambda )$. Taking the coefficient at $\psi$ therefore yieldsThis is the Kostant multiplicity formula in the paper's convention that $K$ counts negative root partitions. With a partition function counting positive root sums, each argument would have the opposite sign.