Characters of $F_4(\mathbb{C})$ at involutions

Abstract

The exceptional group $F_4(\mathbb{C})$ has two conjugacy classes of involutions: an involution $t$ with centralizer $\mathrm{Spin}(9,\mathbb{C})$, and the principal involution, with centralizer $(\mathrm{SL}(2,\mathbb{C})\times \mathrm{Sp}(6,\mathbb{C}))/\{\pm 1\}$. We determine the values of all irreducible characters of $F_4(\mathbb{C})$ at $t$. For the irreducible representation with highest weight $a{\omega }_1+b{\omega }_2+c{\omega }_3+d{\omega }_4$ (in Bourbaki's labelling), we show that the character vanishes at $t$ if and only if $c$ and $d$ are both odd. In all other cases its value is $\pm 2^{-7}$ times the dimension of an explicit irreducible representation of $\mathrm{Spin}(9,\mathbb{C})$. The proof is a limiting argument in the Weyl character formula, combined with a decomposition over the three cosets of $W(B_4)$ in $W(F_4)$ and the triality action of $W(F_4)=W(D_4)⋊S_3$ on the weight lattice modulo the $D_4$-lattice. At the principal involution, Nadimpalli, Pattanayak and Prasad showed that the character value is $\pm 1/2$ times the dimension of an irreducible representation of $\mathrm{SL}(2,\mathbb{C})\times \mathrm{Spin}(7,\mathbb{C})$. Comparing the two results, we find that the characters of $F_4(\mathbb{C})$ vanish at the two involutions for exactly the same highest weights. Together these results determine all irreducible characters of $F_4(\mathbb{C})$ at all involutions. We also explain how our formula at $t$ relates to the generalized Weyl character formula of Gross, Kostant, Ramond and Sternberg.