Group actions and invariants in algebras of generic matrices

We show that the fixed elements for the natural GL_m-action on the universal division algebra UD(m,n) of m generic n x n matrices form a division subalgebra of degree n, assuming n >= 3 and 2 <= m <= n^2 - 2. This allows us to describe the asymptotic behavior of the dimension of the space of SL_m-invariant homogeneous central polynomials p(X_1,…,X_m) for n x n matrices. Here the base field is assumed to be of characteristic zero.