  We introduce an alternate set of generators for the Hecka algebra, and give
an explicit formula for the action of these operators on Fourier coefficients.
With this, we compute the eigenvalues of Hecke operators acting on average
Siegel theta series with half-integral weight (provided the prime associated to
the operators does not divide the level of the theta series). Next, we bound
the eigenvalues of these operators in terms of bounds on Fourier coefficients.
Then we show that the half-integral weight Kitaoka subspace is stable under all
Hecke operators. Finally, we observe that an obvious isomorphism between Siegel
modular forms of weight $k+1/2$ and "even" Jacobi modular forms of weight $k+1$
is Hecke-invariant (here the level and character are arbitrary).
