Another look at J. A. Todd’s covariants for the binary (3,1) form.

The talk will consist of a report on joint work
with A. Van Groningen related to a 1945 paper of J. A. Todd
concerning a problem in classical invariant theory. In this
paper, the author obtains a complete system of generators
for the covariants in the polynomial functions on a certain
eight dimensional irreducible representation, $V$, of
$K=SL(2)\times SL(2)$. The space $V$ consists of the double
binary form of degree (3,1). Expanding on Todd’s result, we
obtain an asymptotic estimate for the graded multiplicities
in the $K$-harmonic polynomial functions on $V$. This point
of view uses the fact that $V$ is the complexified Cartan
complement corresponding to the maximal compact subgroup of
simply connected split $G_2$. The analysis involves the
branching rule from the rank 2 complex symplectic Lie
algebra to the principally embedded $\mathfrak{sl}_2$.