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The irreducible orthogonal and symplectic Galois representations of a p- adic field. (The tame case). (English) Zbl 0546.12009
Let F be a finite extension of $${\mathbb{Q}}_ p$$ and n a positive number prime to p. Then all irrreducible n-dimensional Galois representations of F are parameterized by characters of the multiplicative groups of degree n extensions of F. An $$\epsilon$$ -factor preserving correspondence with admissible irreducible cuspidal representations of GL(n,F) was given by the author in his thesis. This short note determines which irreducible n- dimensional Galois representations of F have real-valued character and which are orthogonal or symplectic representations. Given the above parametrization, the proof is a pleasant exercise with Frobenius reciprocity and local class field theory.
Reviewer: G.Henniart

##### MSC:
 11S20 Galois theory 22E50 Representations of Lie and linear algebraic groups over local fields 11S37 Langlands-Weil conjectures, nonabelian class field theory 11S31 Class field theory; $$p$$-adic formal groups
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##### References:
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