Draft: March 23, 2011
ON THE VANISHING OF CERTAIN EXTENSIONS OFADMISSIBLE SMOOTH REPRESENTATIONS IN
CHARACTERISTIC p
MATTHEW EMERTON

Let k be a finite field of characteristic p and G be a p-adic analytic group. We
augwork in the category Modfg(k) [2], which is antiequivalent to the category ofGadmissible smooth G-representations over k.
For any i ≥ 0, we write E i (–) := Extik[[H]] (–, k[[H]]), for some fixed compact
open subgroup H of G. Up to natural isomorphism, these functors are independentaugof the choice of H, and taken together form a δ functor from Modfg(k) to itself.
Gfg aug
We recall that an object M of ModG (k) is said to be of codimension ≥ c (resp.
codimension c) if E i (M ) = 0 for i < c (and in addition E c (M ) 6= 0).
aug
For any object M of Modfg(k), there is a double duality fourth quadrant
Gspectral sequenceE2i,j := E −i E −j (M ) =⇒ M,
which is natural in M . In particular, if M is non-zero of codimension c then there
is a natural non-zero double duality map M → E c E c (M ). (In particular, E c (M )is again of codimension c.)aug
Proposition 1. If M is an object of Modfg(k) of codimension c for which
GcE (M ) is irreducible, and for which the double duality map M → E c E c (M ) is anaug
isomorphism, then Ext1 (N, M ) = 0 for all objects N of Modfg(k) of codimenG
sion > c.

Proof. Consider an extension 0 → M → E → N → 0, where N has codimension > c. Applying the delta functor E • , we obtain the exact sequenceδ0 −→ E c (E) → E c (M ) −→ E c+1 (N ).
Since E c (M ) is irreducible (by assumption) of codimension c, while E c+1 (N ) is ofcodimension > c, we see the connecting homomorphism δ necessarily vanishes, and∼so E c (E) −→ E c (M ). Applying E c to this isomorphism, we obtain an isomorphism∼∼
c cM −→ E E (M ) −→ E c E c (E), the first isomorphism holding by assumption. The
double duality map E → E c E c (E) thus yields a splitting of the given extension. Example 2. Let G be a p-adic reductive group and B be a Borel subgroup of G, andwrite αu : B → F×p to denote the mod p reduction of the unit part of the character
α describing the action of B on det n, where n is Lie algebra of the unipotent∨radical of B. If χ is a character of B, then (IndGB χ) has codimension equal to∨∨−1 uc := dim B, and E c (IndG= IndG
α, which is irreducible if χ is chosenB χ)Bχ∨generically. Thus for a generic character χ, we have Ext1 N, (IndG= 0 whenB χ)
N has codimension > dim B.
The author was supported in part by NSF grants DMS-0701315 and DMS-1003339.
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2MATTHEW EMERTON

Example 3. We consider the previous example in the particular case when G =GL2 (Qp ), so that dim B = 3. (To fix ideas, we take B to the Borel subgroup ofaug
upper triangular matrices.) An object of Modfg(k) has codimension 4 if and onlyGif it is finite-dimensional over k, and so the previous result shows that for genericcharacters χ, there are no non-trivial extensions of IndG
B χ by a finite dimensionalrepresentation of G (a result which can also be proved by a computation withordinary parts [3, Prop. 4.3.13]).
Of course, this is not true for every character χ. For example, there is a non−1
trivial extension of IndG
by the trivial representation.
This does not conBε⊗ε
G−1 ∨∨
3
tradict the preceding result, because E (IndB ε ⊗ ε ) = (IndGB 1 ⊗ 1) (the−1ucharacter ε ⊗ ε is precisely the character α considered above) and the representation IndG
B 1 ⊗ 1 is not irreducible.
Example 4. Suppose that f ≥ 2, and let ρ : GQpf → GL2 (k) be reducible but
indecomposable, and suitably generic. Breuil and Paškūnas [1] have constructed afamily of admissible smooth representations π(ρ) of GL2 (Qpf ) which are associatedto ρ in a certain sense, as well as a family of admissible smooth representationsπ(ρss ) which are similarly associated to the semi-simplification ρss of ρ. Furthermore, π(ρss ) can be taken to be the direct sum of two principal series representations(each of which is the induction of a generic character, since ρ was assumed to begeneric) and f − 1 supersingular representations.
It does not seem to be known, but is generally hoped, that π(ρ) and π(ρss ) can
be furthermore chosen so that π(ρss ) = π(ρ)ss . If this is the case, then at least
one of the principal series constituents of π(ρss ) will admit an extension by at least
one of the supersingular constituents. Our proposition implies that the dual of anysuch supersingular constituent has codimension at most f .
Acknowledgments. It is a pleasure to thank V. Paškūnas for a stimulating conversation in which the germ of the computation underlying the proof of our proposition
was worked out, and to thank both him and C. Breuil for helpful correspondence
on the subject of this note.
References[1] Breuil C., Paškūnas V., Towards a modulo p Langlands correspondence for GL2 , to appear
in the Memoirs of the Amer. Math. Soc.
[2] Emerton M., Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Astérisque 331 (2010), 355–402.
[3] Emerton M., Ordinary parts of admissible representations of p-adic reductive groups II.
Derived functors , Astérisque 331 (2010), 403–459.
Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston,IL 60208
E-mail address: emerton@math.northwestern.edu