Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general.  We then give a block decomposition of the supercuspidal subcategory Rep_k(SL_n(F))_{SC}, by introducing a partition on each GL_n(F)-equivalent supercuspidal class by type theory, which gives a prediction of the block decomposition of Rep_k(SL_n(F)). We give an example where a block of Rep_k(SL_2(F)) is defined with several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.