  In this paper, for the Stokes eigenvalue problem in $d$-dimensional case
$(d=2,3)$, we present an a posteriori error estimate of residual type of the
mixed discontinuous Galerkin finite element method using $P_{k}-P_{k-1}$
element $(k\geq 1)$. We give the a posteriori error estimators for approximate
eigenpairs, prove their reliability and efficiency for eigenfunctions, and also
analyze their reliability for eigenvalues. We implement adaptive calculation,
and the numerical results confirm our theoretical predictions and show that our
method can achieve the optimal convergence order $O(dof^{-2k/d})$.
