Local universal lifting spaces of mod l Galois representations

Let p and l be distinct primes, K a finite extension of Qp, and ρ‾:Gal(K‾/K)→GLn(F‾l) a mod l Galois representation. In this paper, we show that the generic fiber of universal lifting space of ρ‾ is equidimensional of dimension n2. We also characterize the irreducible components of the generic fiber of the universal lifting space which represent the liftings ρ's of ρ‾ with unipotent ρ|IK's, when ρ‾|IK is trivial or n≤4 and the square of the order of the residue field of K is not equal to 1 mod l.