I am a new learner of Iwasawa theory and currently reading the famous [paper](https://link.springer.com/article/10.1007/s00222-013-0448-1) by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some people tend to work over the *general* symplectic group $\mathrm{GSp}\_{2n}$ and *general* unitary group $\mathrm{GU}\_{r,s}$ (i.e. the ones with similitude factor) rather than the groups $\mathrm{Sp}\_{2n}$ and $\mathrm{U}\_{r,s}$ without similitudes. Yet meanwhile, some people often work with the ones without similitude, especially when doing quite explicit calculations (e.g. the pullback formula in Skinner-Urban's paper). Still, at least from the perspective of readers as naive and stupid as me, some authors just "randomly" switch the groups they are working with, without any hint.


So **I wonder** that what are the essential differences between the ones with similitude and those without similitudes? Why do people prefer one over the other?


I am so sorry if this post does not fit this advanced site and thank you all for commenting and answering!



Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?
*Symplectic case*: Here are two reasons (not necessarily the only ones) why $\operatorname{GSp}\_{2n}$ is more convenient to work with than $\operatorname{Sp}\_{2n}$.


* Firstly: there is **no Shimura datum with underlying group $\operatorname{Sp}\_{2n}$**; you need the "extra room" provided by $\operatorname{GSp}\_{2n}$. In practice, this means that although you can define Siegel Shimura varieties as varieties over $\overline{\mathbb{Q}}$ working purely with $\operatorname{Sp}$, there is no sensible way of describing canonical models over number fields until you bring in the larger group
* Secondly: The general symplectic group has a **richer theory of Hecke operators**, because you can consider the action of matrices like $\operatorname{diag}(1, ..., 1, p, ..., p)$, which normalise $\operatorname{Sp}\_{2n}$ but aren't contained in it. Having this larger Hecke algebra means that, although strong multiplicity one doesn't hold for either group (except for $n = 1$), it is "closer to being true" for $\operatorname{GSp}$ than $\operatorname{Sp}$ (i.e. the $L$-packets are smaller).


*Unitary case*: Here the issue is a bit more nuanced because there are various ways of choosing the Shimura datum: roughly, you can choose a Shimura datum for $U(a, b)$ which looks like $z \mapsto \operatorname{diag}(z / \bar{z}, \dots, z / \bar{z}, 1, \dots, 1)$, or you can choose one for $\operatorname{GU}(a, b)$ which looks like $\operatorname{diag}(z, \dots, z, \bar{z}, \dots, \bar{z})$. Each of these has its merits for different things: working with **$\operatorname{GU}(a, b)$ gives you a nicer moduli-space interpretation** for your Shimura variety (it is of PEL type), so its canonical models are easier to describe; but it is the **$U(a, b)$ one which has the nicer Hecke theory**.


I have the impression that there is a shift under way in the recent literature. 10 years ago, most of the arithmetic theory of automorphic forms on unitary groups (Shimura varities, Galois representations etc) was developed using $GU(a, b)$, as in the Skinner-Urban paper, or the Harris–Taylor book on local Langlands. However, more recently, people seem to have got better at working with non-PEL Shimura data; thus the recent work on unitary special cycles (arithmetic Gan–Gross–Prasad conjectures, etc) is mostly being done in the context of actual unitary groups. [This paper of Rapoport–Smithling–Zhang](https://arxiv.org/pdf/1906.12346.pdf) and [this paper of Jetchev](https://arxiv.org/pdf/1410.6692.pdf) have some useful discussion of the relations between the $U$ and $GU$ Shimura varieties.


