Degrees of maps between locally symmetric spaces

Let X   be a locally symmetric space Γ\G/KΓ\G/K where G is a connected non-compact semisimple real Lie group with trivial centre, K is a maximal compact subgroup of G  , and Γ⊂GΓ⊂G is a torsion-free irreducible lattice in G  . Let Y=Λ\H/LY=Λ\H/L be another such space having the same dimension as X. Suppose that real rank of G   is at least 2. We show that any f:X→Yf:X→Y is either null-homotopic or it is homotopic to a covering projection of degree an integer that depends only on Γ and Λ. As a corollary we obtain that the set [X,Y][X,Y] of homotopy classes of maps from X to Y is finite.We obtain results on the (non-)existence of orientation reversing diffeomorphisms on X as well as the fixed point property for X.