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

In this article, we show in the ADEADE 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=MX=M conjecture for type An(1) and Dn(1).