The Weyl formula for matrix functions

Let C[m x m\ be the space of square [m x m] matrices, and let C"[m x m] be the direct product of n copies of the space C[m x m]. In this work, a new integral representation for the local residue in the space C"[m x m] is given, based on the Bochner-Hua-Loken integral formula for the matrix polydisk. Moreover, a matrix polyhedron in this space is defined. It is worth noting that the classical Weyl integral representation in the space C” is related to transformation formula for Grothendieck’s local residue and can be derived using this formula from the multiple Cauchy integral representation for the polydisk. We apply the same approach to the local residue in the space C"[m x m] and obtain a generalization of the Bochner-Hua-Loken integral representation for the matrix polydisk, which shares the same nature as the well-known Weyl integral representations in polyhedra. In the obtained Weyl integral representation, the integral is taken over the skeleton of the polyhedron, and it is reduced to the classical Weyl formula for the polyhedron in C [от x от] when n = 1. Furthermore, a modification of the Weyl integral formula is derived, in which the integral is taken over the face of the polyhedron in the space C"[m x m].