4
$\begingroup$

Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for which there exists a homomorphism $h:\mathbb{S}\to G$ satisfying Deligne's axioms:

1) The possible weights of $\mathbb{S}$ acting on $Lie(G)$ via $Ad\circ h$ are $z/\bar{z}, 1, \bar{z}/z$.

2) $inn(h(i))$ is a Cartan involution of $G^{ad}$.

3) $G^{ad}$ has no compact factor.

I'm aware of Deligne's classification where the same question is answered for $G$ a real simple adjoint algebraic group, but I'm not sure on how to go from there to a general reductive group (or even a simple group not of adjoint type).

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
2
$\begingroup$

Deligne effectively describes the connected Shimura data. Every connected Shimura datum arises from a Shimura datum, and there are a number of results saying how to choose the Shimura datum to have good properties. For example: for any connected Shimura datum $(G,X)$ there is a Shimura datum $(G_1,X_1)$ such that (a) $(G_1,X_1)^+=(G,X)$; (b) the weight of $(G_1,X_1)$ is defined over $\mathbb{Q}$; (c) the center of $G_1$ is a product of tori of the form $Res_{L/K}(\mathbb{G}_m)$ with $L$ a CM-field Galois over $\mathbb{Q}$. Invent math, 92 (1988), p.127.

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ Invent...? Could you say who the author is? $\endgroup$
    – Donu Arapura
    Commented Jun 19, 2018 at 12:50
  • $\begingroup$ A quick check on SpringerLink reveals that the intended citation was to Milne, Automorphic vector bundles on connected Shimura varieties, link.springer.com/article/10.1007/BF01393994 (more specifically, Prop A.2 in the appendix of this paper). $\endgroup$
    – David Loeffler
    Commented Jun 19, 2018 at 19:16
  • 2
    $\begingroup$ This doesn't quite answer the question though: it shows that every conn. Sh. datum $(G, X)$ arises from a Shimura datum, and that can be chosen to be of a rather nice form; but it doesn't tell us how to describe all Shimura data $(G_1, X_1)$ such that $(G_1, X_1)^+ = (G, X)$, which is the content of the question. $\endgroup$
    – David Loeffler
    Commented Jun 19, 2018 at 19:25

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .