In the case of PEL-type Shimura data Ancona has recently described a lift ofµHG defined on all of RepF(G) [Anc15]. But, as defined, this construction depends on the choice of PEL-datum and it is not immediately clear that it is well behaved with respect to pullbacks.
In this section we briefly recall Ancona’s construction, but in the language of mixed Shimura varieties.
Notation 4.9.1. Given an algebraB/Q, we write BF for B⊗QF. Similarly if W
is aB-module, thenWF denotes W ⊗QF.
Definition 4.9.2.A PEL-datum is a tuple (B,∗, V,h , i, h) consisting of: a semi- simpleQ-algebraBwith a positive (anti-)involution∗onB, that is an anti-commutative
involution such that TrBR/R(bb
∗) > 0 for all 0 6= b ∈ B
R. Together with a finite
dimensional B-module V equipped with an alternating non-degenerate Q-valued
pairingh , ion V such that, forb∈B, u, v∈V
hbu, vi=hu, b∗vi,
and finally a choice ofR-algebra homomorphismh:C→EndBR(VR) such that hh(z)u, vi=hu, h(¯z)vi ∀z∈C, u, v ∈V
hu, h(i)uiis positive definite,
Let G be the algebraic group whose R-points, for any Q-algebra R, are defined by G(R) = g∈AutBR(VR)
∃µ(g) ∈ R× such that hgu, gvi = µ(g)hu, vi
for all u, v∈V ⊗R
.
For z ∈ C×, we automatically have that h(z) ∈ G(R). We also denote by h the
induced mapS→GR. Then (G, h) satifies (1.5.1), (1.5.2) and (1.5.3) of [Del71] and
so defines a Shimura datum (see [Kot92, Lem. 4.1]). A Shimura datum (G, h) which arises in this way is said to be ofPEL-type and the corresponding (B,∗, V,h , i, h) is said to be aPEL-datum for (G, h).
Lemma 4.9.3.For any (G, h) of PEL-type, the centre of G is an almost–direct product of aQ-split torus and anR-anisotropic torus.
Proof. We want to show that the largest anisotropic subtorus of Z(G) remains anisotropic over R. The largest anisotropic subtorus of Z(G) is contained in the
kernel of the multiplier character µ:G→ Gm and so must be contained in Z(G1).
We claim that (Z(G1)◦)R is always anisotropic.
Any semisimple R-algebra with positive involution is a product of simple
factors of one of the following three types: (Mn(R), A7→At), (Mn(C), A7→A¯t) or
(Mn(H), A7→A¯t) where Hdenotes the quaternions and where ¯Adenotes coefficien-
twise complex conjugation in theMn(C)-case and the involutiona+bi+cj+dij7→ a−bi−cj−dij in theMn(H)-case (for example [Kot92, p. 386]).
In particular, all symplecticBR-modules split as an orthogonal direct sum of submodules only acted on non-trivially by a single simple factor of one of the above types and G1 splits accordingly. As such, it suffices assume that BR is simple of
each of the three above types. Moreover, we are able to reduce to the case of BR
isomorphic toR,C orHby an easy Morita equivalence argument.
We shall make repeated use of the following result of Kottwitz. Given any semisimple R-algebra (B,∗) and two triples (V,h , i, h) and (V0,h , i0, h0), that
together with (B,∗) satisfy the conditions of Definition 4.9.2 with Rin place of Q,
then if V and V0 are isomorphic as B ⊗RC-modules, with C acting via h and h0
respectively, then (V,h, i) and (V0h, i0) are isomorphic as symplectic (B,∗)-modules
First assume that (BR,∗) = (R,∗= id). Then W =R⊕2,h , i= 0 1 −1 0 ! , h(i) = 0 −1 1 0 !!
is a triple as above with correspondingBR⊗RC-moduleC. As a result, as a symplec-
tic (BR,∗)-module VR must split as an orthogonal direct sum of terms isomorphic toW. By definition, G1(R) for W⊕n is Sp2n. In particular, it has finite centre so
that (Z(G)◦)R is anisotropic.
Now assume that (BR,∗) = (C,∗=z7→z¯). In this case, BR⊗RC∼=C×C
has two irreducible modules. The two triples given by (C,trC/R(xiy¯), h(i) =i) with
z ∈ C acting by multiplication by z and ¯z respectively correspond to these two irreducibles. Both have the same underlying symplectic (BR,∗)-module soVR must be isomorphic to Cn. For this symplectic module, G1(R) consists of elements of
GLn(C) which also lie in Sp2n(R). This is precisely the unitary group Un(R). Now Z(Un(R)) =U1(R), which is indeed anisotropic.
Finally, for the quaternion case we shall assume that (BR,∗) = (Hop,∗) (with Hop an expositional choice). In this case,BR⊗RC∼=M2(C) has a unique irreducible
module which is ofR-dimension 4. This is realised by the triple (H,trH/R(xjy˜), h(i) =
j) where Hop acts by right multiplication and y 7→ y˜ is the involution given by y = a+bi+cj +dij 7→ a+bi−cj +dij. In this case, EndHop(H) ∼= H with
H acting by left multiplication. On EndHop(H), taking adjoints with respect to
tr(xjy˜) coincides with the map y 7→ y˜. The embedding H ,→ M2(C) which sends
i7→ i 0 0 −i ! and j7→ 0 1 −1 0 !
defines a splitting of H⊗RC∼=M2(C). The
involution of M2(C) induced by y 7→ y˜ is then matrix transposition. As a result, G1(C) = {c ∈ AutHop(H) | cc
∗ = id} = O
2(C) is the orthogonal group. More
generally, for H⊕n we then have G1(C) = O2n(C), which does indeed have finite
centre.
If we fix a PEL-datum for (G, h), then we say that V ∈ Rep(G) is the
standard representation ofG. Shimura data with a fixed choice of PEL-datum have an explicit moduli interpretation (see [Mil17, Sec. 8]).
Example 4.9.4.We give an example of two distinct PEL-data for the same Shimura datum. First consider the PEL-datum (Q,∗,Q⊕2,h , i, h), where ∗ = id, h , i is
the alternating pairing represented by J := 0 1
−1 0 !
a −b
b a
!
. The corresponding Shimura datum is then the usual datum (GL2,H)
defined in Example 4.2.2.
There is also a PEL-datum (M2(Q),∗ = (−)t,Q⊕4,h , i), where the involu-
tion is transposition, M2(Q) acts diagonally on Q⊕4 = Q⊕2⊕Q⊕2 acting on each
factor in the standard way, the pairing is represented by 0 I2
−I2 0 ! , and h is given bya+ bi7→ aI2 −bI2 bI2 aI2 !
. ThenGis isomorphic to GL2, which is embed-
ded within GL4(Q) via a b
c d
!
7→ aI2 bI2
cI2 dI2
!
, so that the associated Shimura datum is again (GL2,H). This is an example of the Morita equivalence used in the
proof of Lemma 4.9.3.
Remark 4.9.5.Suppose (B,∗, V,h , i, h) is a PEL-datum. Then h is uniquely determined, up to G conjugacy, by (B,∗, V,h , i) [Kot92, Lem. 4.3]. If we assume that B is simple and linear or symplectic in the sense of [Mil17, Sec. 8], then any 4-tuple (B,∗, V,h , i) satisfying the relevant parts of Definition 4.9.2 admits an h
(which is necessarily unique up to conjugacy) [Mil17, Prop. 8.12].
Proposition 4.9.6.Given a Shimura datum (G, h) with a choice of PEL-datum
(B,∗, V,h , i, h), then for all fieldsF/Q, all objects of RepF(G) are, up to isomor-
phism, direct summands of some space of the form Lk
i=1V ⊗ak
F ⊗V
⊗bk
F .
Proof. As V is a faithful G-representation, this follows from the proof of [DM82, Prop. 2.20].
Theorem 4.9.7 ([Anc15, Thm. 6.1]). Given a Shimura datum (G, h) with a PEL- datum (B,∗, V,h , i, h), let K be a neat open compact subgroup of G(Af) and L a
ˆ
Z-lattice ofVF (considered as a representation overQ). Then for anyn∈N, there is
a canonical inclusion of rings a: EndRepF(G)(V
⊗n
F ),→EndHomMF/S(h
1(S VF,K)
∨⊗n)
such that the diagram
EndRepF(G)(V ⊗n F ) EndHomM/S(h1(SVF,K) ∨⊗n) EndVHS/S(C)(µHG(VF)⊗n) µHG a Hi B
commutes. Here, we have used the isomorphismϕVF:H
1
B((SVF,K)(C))
∨→µH
G(VF)
of Lemma 4.8.1 to identify End(µHG(VF)⊗n) and End(H1((SVF,K)(C))
Remark 4.9.8.Ancona’s strategy is to lift endomorphisms ofVF itself (in our pre-
sentation, this is via functoriality of mixed Shimura varieties) and permutations of
VF⊗n in the obvious way and then additionally lift cycles arising from the polarisa- tion via the Poincar´e-Lefschetz isomorphisms (which have been described motivi- cally). Ancona then shows that endomorphisms of the above kinds generate all of EndRep
F(G)(V
⊗n
F ) in the case of PEL-type Shimura varieties. This is not true for ar-
bitrary Shimura varieties, and to obtain such a result more generally would require identifying more algebraic cycles.
Construction 4.9.9.There is a ⊗-functor AncG: RepF(G)→HomMF/S defined
as follows: set AncG(VF⊗n) =h1(SVF,K)
∨⊗n and let Anc
G(α) forα ∈End(VF⊗n) be
defined via the map of Theorem 4.9.7. By Hom-tensor adjunction, Theorem 4.9.7 also defines a motivic lift of the map 1 → V ⊗V∨. More generally, to define the image of elements of Hom(VF⊗a⊗VF∨⊗b, VF⊗c⊗VF∨⊗d) it suffices to fix the image of Hom(VF⊗(a+d), VF⊗(b+c)), but for weight reasons this is zero unlessa−b =c−d, in which case it is covered by Theorem 4.9.7.
This also allows us to define, for any choice of idempotent e the image of a direct summand e·(L
V⊗an
F ⊗V
∨⊗bn
F ). Since every element of W ∈ RepF(G)
is of this form by Proposition 4.9.6, if we pick a fixed isomorphism θW: W
∼ → eW·(LV ⊗aW,n F ⊗V ∨⊗bW,n
F ) for eachW, then we can compatibly extend AncG to all
of RepF(G). Finally, by composition with the section of Theorem 4.4.6, we obtain a functor RepF(G)→CHMF/S, which we also denote AncG.
Lemma 4.9.10. The construction ofAncGis, up to natural isomorphism, indepen-
dent of all choices made.
Proof. Fix W ∈RepF(G) and two summands isomorphic to W of a tensor space,
e·L Vak F ⊗V ∨⊗bk F ,e 0·L V⊗a 0 k F ⊗V ∨⊗b0 k
F . We must provide an isomorphism
e·Mh1(SVF,K) ∨ak ⊗h1(S VF,K) ⊗bk →e0·Mh1(S VF,K) ∨a0k⊗h1(S VF,K) ⊗b0k.
Given the compatibility of the K¨unneth formula with mixed Shimura varieties, we may assume that W is irreducible and there is a corresponding isomorphism e·
(VF⊗a⊗VF∨⊗b)→e·(VF⊗a0⊗VF∨⊗b0).
As before, it suffices to assume that b = b0 = 0. For weight reasons, we must then have that a = a0. Finally, since Lemma 4.9.7 lifts all elements of EndRepF(G)(V
⊗a
F ), we obtain a motivic lift of the isomorphism between the two
desired natural isomorphism.
Remark 4.9.11.Let (G,X) be a Shimura datum with a chosen PEL-datum for which all objects of Rep(G)AV are direct summands of V⊕n for varying n. Then the argument given above can be adapted to show that AncG extends µmotG up to
natural isomorphism. If the PEL-datum is of “symplectic type”, then this always holds (see Section 4.12). This can also be checked to hold much more generally.