This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way. 


However, what bother me is that Hecke operators are also mentioned in the automorphic *representations* language. Those remain very mysterious to me: what are they exactly? Is there a way to define them more explicitly than using the coefficients appearing in the Dirichlet series of the L-function associated to the representation? (What is, even if it appears explicit, unpractical).


My main aim is to grasp the Hecke eigenvalues of a representation, for instance using trace formulas. In the "function" language, since Hecke operators are defined as the action of double classes, we can take the Hecke operators themselves as test-functions and obtain some "Eichler-Selberg"-type trace formulas. 



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> But on the "representation" language, is there something doing the similar work?
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More precisely my question could be: is there a function $f\_p$ in the Hecke algebra of a local/global group $G(F)$ such that $\mathrm{tr} \ \pi(f\_p) = \lambda\_\pi(p)$ where $\pi$ is an admissible/automorphic representation of $G(F)$ and $\lambda\_p(\pi)$ the Hecke eigenvalue of $\pi$?


Any clue or reference is welcome!



What is the matter with Hecke operators?
If $K$ is a local field and $\mathcal O$ is its ring of integers, we say an irreducible representation of $G(K)$ is unramified if it contains a vector invariant under $G(\mathcal O)$. It is known that such a representation naturally has a unique vector invariant under $G(\mathcal O)$. The Hecke eigenvalue of a double coset in $G(\mathcal O)\backslash G(K)/G(\mathcal O)$ on this representation is simply the eigenvalue of the operator defined by averaging over that double coset on this vector. (Because the vector is the only $G(\mathcal O)$-invariant vector in the representation, it is necessarily sent to a multiple of itself by this operator).


In fact, this operator vanishes on the orthogonal complement of this $G(\mathcal O)$-invariant vector, so its trace is equal to this eigenvalue. On any ramified representation, the trace of this operator is zero. I think this answers your last question.


Of course when we say "the" Hecke operator we mean a particular double coset. I think this is only ambiguous in $GL\_n$, where we take the coset generated by a diagonal matrix where one entry is a uniformizer and the rest are units - as I'm guessing you know, based on your familiarity with the double coset theory in the functions world.


