Abstract: Over a field of characteristic p≥3,firstly,according to the Z-degree structure of P^~(2)and the standard Cartan decomposition of the 0-degree P^~(2)₀,for a given weight λ and the maximal vector v_λ, finite-dimensional irreducible modules M(λ)of P^~(2)₀ are constructed, and then its Kac structure is given. Secondly, using the definition of the first cohomology group, the first cohomology group from P^~(2)to its Kac module is transformed into the weight-derivation of the non-inner derivation of P^~(2)to Kac module. Finally, the first cohomology group of P^~(2)with coefficients to a class of Kac modules is determined.

Key words: Kac module, derivation, cohomology, weight