This paper develops a detailed theory of Gauss sums for central simple algebras A (of finite degree) of a finite extension F of \({\mathbb{Q}}_ p\). The case where \(A=F\) is the well known theory of Abelian congruence Gauss sums, and the authors’ previous paper [Gauss sums and p-adic division algebras (Lect. Notes Math. 987) (1983; Zbl 0507.12008)] dealt with the case of division algebras over F. Here the general case is defined and thoroughly studied. For this the authors first introduce after Benz, the so-called principal orders of A; the normalizers G(\({\mathfrak A})\), in \(A^{\times}\), of principal orders \({\mathfrak A}\) of A are precisely the maximal compact-mod-center subgroups of \(A^{\times}.\)
Let us fix a non-trivial additive character of F. Then to any irreducible admissible representation \(\rho\) of G(\({\mathfrak A})\) (over an algebraically closed field B of characteristic not p) is attached a Gauss sum \(\tau\) (\(\rho)\). The absolute value of this Gauss sum (when \(B={\mathbb{C}})\) is computed and a functional equation between \(\tau\) (\(\rho)\) and \(\tau (\rho^ v)\) is proved; reduction modulo a prime number \(\ell \neq p\) is also investigated, as well as the behaviour of \(\tau\) (\(\rho)\) under Galois actions. Finally the relationship of those Gauss sums with local constants (\(\epsilon\)-factors) for supercuspidal admissible irreducible representations of \(A^{\times}\) is studied, leading to the conjecture that any such representation of \(A^{\times}\) contains some ”non- degenerate” representation of some G(\({\mathfrak A})\) as above. Throughout the paper the parallelism with Galois Gauss sums is stressed.