An isomorphism theorem for Lusztig algebras

In [Lu6] Lusztig defined a certain algebra $H,$ which is a direct sum of various algebras $H_{mathfrak{o}}.$ We establish an explicit algebra isomorphism between each algebra $H_{mathfrak{o}}$ and some matrix algebra with coefficients in the tensor product of an (affine or finite) Hecke algebra and a group algebra. We give an application.