function construction - Can SetDelayed (:=) be compiled?

Is it possible to put an If statement as below within a compile (see below)? I received a warning about SetDelayed.

  eef = Compile[{{μ, _Real}, {NNN, _Integer}}, 
   Module[{NN = NNN}, 
     kd[m_, n_] := If[m == n, 1, 0];
     cosfun[m_, n_] := 
       If[m == n, 
          0, 
          (1 - Cos[(m + n) π])/( (m + n) π) 
          + ((Cos[(m - n) π] - 1)/( (m - n) π))

       ];

      mm = Table[kd[m, n] (MM - B2 k k - B1 (m Pi/L)^2),  
              {m, 1, NN}, {n, 1, NN}];
      kz = Table[cosfun[m, n] (-I A2 m Pi/L), {m, 1, NN}, {n, 1, NN}];
      kxM = Table[kd[m, n] A1 k, {m, 1, NN}, {n, 1, NN}];
      μM = Table[kd[m, n] μ, {m, 1, NN}, {n, 1, NN}];

      HH = ArrayFlatten[{
             {μM + mm, 0 mm, kz, kxM}, 

             {0 mm, μM + mm, kxM, -kz}, 
             {kz, kxM, μM - mm, 0 mm}, 
             {kxM, -kz, 0 mm, μM - mm}
           }];

      ees = Table[Eigenvalues[HH], {k, -.1, .1, .01}]

     ] (* End Module  *)
    ]  (* End Compile *)

where A1, A2, B1, B2, and MM are global variables.

Answer

Compile can't handle SetDelayed.

In your specific case, you might be able to avoid the need for SetDelayed altogether through the use of Boole.

kd[m_, n_] :=If[m == n, 1, 0] can be replaced by Boole[n==m]. (or as FJRA pointed out in comments, by KroneckerDelta.)

cosfun[m_, n_] := 
 If[m == n, 0, (1 - Cos[(m + n) π])/((m + n) π) + 
  ((Cos[(m - n) π] - 1)/((m - n) π))]

can be replaced by

Boole[m == n] (1 - Cos[(m + n) π])/((m +  n) π) + 
 ((Cos[(m - n) π] - 1)/((m - n) π)) (-I A2 m Pi/L)

Giving

eenew = Compile[{{μ, _Real}, {NNN, _Integer}}, 
  Module[{NN = NNN}, (* do you even need this placeholder for the input? *)
   mm = Table[Boole[m == n] (MM - B2 k k - B1 (m Pi/L)^2), {m, 1, NN}, {n, 1, NN}]; 
   kz =  Table[Boole[m != n] (1 -  Cos[(m + n) π])/((m + n) π) + 
     ((Cos[(m - n) π] - 1)/((m - n) π)) (-I A2 m Pi/L), 
     {m, 1, NN}, {n, 1, NN}]; 

   kxM =   Table[Boole[m == n] A1 k, {m, 1, NN}, {n, 1, NN}]; 
  μM = Table[Boole[m == n] μ, {m, 1, NN}, {n, 1, NN}]; 
   HH =    ArrayFlatten[{{μM + mm, 0 mm, kz, kxM}, 
      {0 mm, μM + mm, kxM, -kz}, {kz, kxM, μM - mm,0 mm}, 
      {kxM, -kz, 0 mm, μM - mm}}]; 
  ees = Table[Eigenvalues[HH], {k, -.1, .1, .01}]]] 

What is k for in this last line? There is no way for the iterator to matter for HH.

This doesn't give the error involving SetDelayed anymore.