Fusion products of Kirillov-Reshetikhin modules and the X = M conjecture

In this article, we show in the ADE case that the fusion product of Kirillov-Reshetikhin modules for a current algebra, whose character is expressed in terms of fermionic forms, can be constructed from one-dimensional modules by using Joseph functors. As a consequence, we obtain some identity between fermionic forms and Demazure operators. Since the same identity is also known to hold for one-dimensional sums of nonexceptional type, we can show from these results the X = M conjecture for type $A_n^{(1)}$ and $D_n^{(1)}$.