Consider a finite Chevalley group G p defined over a field of p elements, p a prime. The projective indecomposable characters in characteristic p are indexed by the p n -restricted weights. If a weight is p-restricted, it is also p n -restricted. If β½ is the corresponding projective , n Ε½ n . indecomposable character of G p , we can ask how the decomposition of β½ into ordinary irreducible characters varies with n.
, n
The ordinary irreducible characters are obtained from the DeligneαLusztig characters. In this paper we will show how to write the DeligneαLusztig characters in terms of what we call a p-adic coding, so that we can answer the above question. We will show how to associate with each component of the p-adic coding a matrix, so that the trace of the product of the corresponding matrices gives the multiplicity of the corresponding character in β½ .
, n
To avoid some technical complications we shall assume here that we are in the generic case, so that all the DeligneαLusztig characters occurring in the decomposition of β½ are irreducible. We shall also assume that β½ , n , n Ε½ . can be expressed as a tensor product of G p -projective characters raised to a power of the Frobenius. However, the methods can be extended to non-generic cases; we shall deal with this elsewhere.
In fact, our result is a general one, and we apply it to decompose any Ε½ . projective character which can be expressed as a tensor product of G pprojective characters raised to a power of the Frobenius. The results apply to twisted groups as well, and illustrate the relationship between the decomposition numbers in the twisted and untwisted case.
We apply our method to determine the decomposition of β½ for
, n
Ε½ n . Ε½ n . Ε½ . SL p and SU p , a generic weight for SL p . We also show how 3 3 3
Ε½ n . generic Cartan invariants of G p can be calculated using our methods, Ε½ n . Ε½ n .