Representation Theory for Division Algebras over Local Fields ( Tamely Ramified Case

Aside from intrinsic interest there are three (related) reasons for studying the unitary representations of £-adic division algebras. (1) They provide heuristics for more difficult (e.g., noncompact) £-adic groups. (2) They would be basic building blocks in a theory of representations of reductive algebraic £-adic groups based on the philosophy of cusp forms. (3) There seem to be deep relations of representations of division algebras with representations of Gl„, having implications in the theory of automorphic forms. This was pointed up in JacquetLanglands [2]. We announce here, for the tamely ramified case, a classification of the representations (Theorem 1), and a result related to (3) (Theorem 2). I would like to thank R. P. Langlands for some stimulating conversations, and in particular for telling me of the likelihood of Theorem 2. Let F be a locally compact non-archimedean field of residual characteristic p. Let R be its maximal order, TT a prime element, F its multiplicative group, and U=1+TTRQF. Let D be a division algebra over F. Let 5, II, D, V be its maximal order, and so forth. We will say D is tamely ramified if its degree, n, is prime to p. This is the same as to say all its commutative subfields are tamely ramified over F. Let F' be a finite extension, with maximal order R, prime 7r', t/ ' = l + 7 r ' # ' and multiplicative group F'. Let N(F'/F):F'->F be the norm map. Let ^ be a character of F' and A C F' a subgroup. We will say x// is nondegenerate on A if there is no proper subextension F", FQF"CF', such that ker N(F'/F")C\AQkev if/HA. Suppose now F' is tamely ramified over F. We will say a character \[/ of F' is admissible if (a) \f/ is nondegenerate on F ' x , and (b) if on U', \f/=\l/" o N(F'/F"), where ^ " is nondegenerate on [7"ÇF"x, then F' is unramified over F".