# Reference for isomorphism between parabolic and cuspidal cohomology of the Siegel variety

I'm asking for a reference where I can find proof of isomorphism

$$H^{3}_{\text{cusp}}(Y(U),F_{\lambda})\simeq H^{3}_{\text{par}}(Y(U),F_{\lambda}),$$

where $$Y(U)$$ is the level $$U$$ shimura variety of $$\mathbf{GSp(4)}$$ and $$F_{\lambda}$$ the local system associated to the weight $$\lambda$$.

• Do you want such a result in all degrees of the cohomology? Commented Apr 23, 2023 at 10:50
• oohh no only for degree 3, I'm going to edit thank you. Commented Apr 24, 2023 at 16:42
• What are your definitions of parabolic and cuspidal cohomology? Commented Apr 24, 2023 at 17:30
• I am trying to answer correctly: Parabolic is the image of $H_{c}(Y(U))\rightarrow H(Y(U))$ and Cuspidal is the $\frak{g},K$- cohomology of the space of cuspidal forms (which can be defined using the cohomology of the completion of $Y(U)$). Commented Apr 25, 2023 at 14:07
• But actually, since yesterday (after @random123's comment) I tried to find it on my own and found it in a paper I've known for a long time now... Which makes me ashamed, but to my credit it is hidden inside two pages of calculation of cohomology spaces. You can find it on page 5-6 (293-294) of Taylor's "On the l-adic cohomology of Siegel threefold". Commented Apr 25, 2023 at 14:13

First of all, it is a known fact that it remains to show that $$H^{k}_{\text{cusp}}=H^{k}_{(2)}$$ where the right hand side denotes the $$L^{ 2}$$-cohomology. Now, according to Matsushima's formula, it remains to show that any representation appearing in $$H_{(2)}$$ is cupsidal and appears with the same multiplicity as in $$H_{\text{cusp}}$$. The Vogan-Zuckerman classification tells you that if $$\pi_{\infty}$$ appears in $$H_{(2)}^{3}$$ then it is tempered (this is why being with $$\mathbf{GSp}_{4}$$ is important but it is unclear for me, see bellow). Finally, a theorem of Wallach claims that for such $$\pi_{\infty}$$ the representations $$\pi_{\infty}\pi_{f}$$ are cuspidal and $$\pi_{\infty}$$ occur with the multiplicity desired.
I will simply conclude with a few words on the Vogan-Zuckerman classification. The paper quoted is D. Vogan, G. Zuckermann "Unit representations with non-zero cohomology". I'm having trouble finding where in this article is a result that allows us to prove that any $$L^{2}$$-cohomological representation of $$\mathbf{GSp}_{4}$$ is tempered. Maybe someone can comment or add an answer.