Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?

Question

I am a new learner of Iwasawa theory and currently reading the famous paper 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!

Answer 1

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 and this paper of Jetchev have some useful discussion of the relations between the $U$ and $GU$ Shimura varieties.