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An etale cohomology group $W$ of some irreducible components, which is thesmooth compactification of an affine curve$(X^{q^2}-X)^{q-1}=(Y^{q(q+1)}-Y^{q+1})^{q-1},$ in the stable reduction theLubin-Tate curve of level two is related to the Lusztig theory over finiterings. In this paper, we investigate a relationship between the cohomologygroup $W$ and the Lusztig theory.

On the cohomology of the Lubin-Tate curve of level 2 and the Lusztig theory over finite rings