In the paper by L.Clozel [in this book](https://www.jmilne.org/math/Books/AA1988a.pdf) (a French text), there is this conjecture (conjecture 4.5 p139)


**Conjecture**: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote by $E$ it's definition field. Then there exists a degree $n$ motive $M$ over a finite extension of $E$ such that the euler factors of the two L-function coïncide up to a translation of $\frac{1-n}{2}$.


**First question**: How can we generalize this conjecture for the group $GSp\_{4}$?


On the over hand, in [this paper](https://webusers.imj-prg.fr/%7Emichael.harris/occult.pdf), M.Harris associate to $\pi\_{f}$ the finite part of a cuspidal representation $\pi$ of $GSp\_{4}$ a motive $M(\pi\_{f})$ as follows
$$M(\pi\_{f})=H^{3}\_{!}(Sh,\tilde{V}\_{\rho})[\pi\_{f}]$$
It is done p.5, and note by the way that he does not tell us that it "is a motive associated to $\pi$", I just believe that it is what he mean because of the name of the section 1 and notation..


**Second question**: Assume $\pi$ is an algebraic representation of $GSp\_{4}$, does the motive $M(\pi\_{f})$ satisfy the conjecture stated in Clozel's paper and why??


**Last question**: Does the compatible system of Galois representation of $M(\pi\_{f})$ coïncide with the representation in the Theorem 1 p.3 of [this paper](http://www.numdam.org/item/AST_2005__302__67_0.pdf) by Weissauer?


**Bonus question**: What about the more general case of $GSp\_{2n}$ (including $Gl\_{2}$!) ????



Motive associated to a cuspidal representation of $GSp_{4}$
The formula you quote from Harris defines a Galois representation, not a motive. We expect that there is a motive whose etale realisation is Harris' space, but that is not immediate.


The problems are: firstly, the Siegel threefold is not proper; secondly, the projector cutting out M(pi\_f) might not be an idempotent in the Chow ring.


The first problem was [solved by Wildeshaus](https://arxiv.org/abs/1706.02743) using his theory of "interior motives". This gives a Grothendieck motive attached to pi, but sadly not a Chow motive, because of the second problem above. It is not known if Wildeshaus' Grothendieck motive can be upgraded to a Chow motive.


(Sorry for lack of nice formatting, typing on my phone.)


