Central characters for smooth irreducible modular representations of GL_2(Q_p)
aa r X i v : . [ m a t h . R T ] J u l CENTRAL CHARACTERS FOR SMOOTH IRREDUCIBLEMODULAR REPRESENTATIONS OF GL ( Q p ) by Laurent Berger
Abstract . —
We prove that every smooth irreducible F p -linear representation of GL ( Q p )admits a central character. Introduction
Let Π be a representation of GL ( Q p ). We say that Π is smooth, if the stabilizer of any v ∈ Π is an open subgroup of GL ( Q p ). We say that Π admits a central character, if every z ∈ Z(GL ( Q p )) acts on Π by a scalar. The smooth irreducible representations of GL ( Q p )over an algebraically closed field of characteristic p , admitting a central character, havebeen studied by Barthel-Livn´e in [ BL94, BL95 ] and by Breuil in [
Bre03 ]. The purposeof this note is to prove the following theorem.
Theorem A . — If Π is a smooth irreducible F p -linear representation of GL ( Q p ) , then Π admits a central character. The idea of the proof of theorem A is as follows. If Π does not admit a central character,and if f = (cid:0) p p (cid:1) , then for any nonzero polynomial Q ( X ) ∈ F p [ X ], the map Q ( f ) : Π → Πis bijective, so that Π has the structure of a F p ( X )-vector space. The representation Π istherefore a smooth irreducible F p ( X )-linear representation of GL ( Q p ), which now admitsa central character, since f acts by multiplication by X . It remains to apply Barthel-Livn´e and Breuil’s classification, which gives the structure of the components of Π afterextending scalars to a finite extension K of F p ( X ). A corollary of this classification is thatthese components are all “defined” over a subring R of K , where R is a finitely generated F p -algebra. This can be used to show that Π is not of finite length, a contradiction. Mathematics Subject Classification . —
Key words and phrases . —
Smooth representation; admissible representation; parabolic induction;supersingular representation; central character; Schur’s lemma.
LAURENT BERGER
Acknowledgments . I am grateful to C. Breuil, G. Chenevier, P. Colmez, G. Henniart,M. Schein and M.-F. Vign´eras for helpful comments.
1. Barthel-Livn´e and Breuil’s classification
Let E be a field of characteristic p . In this section, we recall the explicit classificationof smooth irreducible E -linear representations of GL ( Q p ), admitting a central character.We denote the center of GL ( Q p ) by Z. If r >
0, then Sym r E is a representation ofGL ( F p ) which gives rise, by inflation, to a representation of GL ( Z p ). We extend it toGL ( Z p ) Z by letting (cid:0) p p (cid:1) act trivially. Consider the representationind GL ( Q p )GL ( Z p ) Z Sym r E . The Hecke algebra End E [GL ( Q p )] (cid:16) ind GL ( Q p )GL ( Z p ) Z Sym r E (cid:17) is isomorphic to E [ T ] where T is a Hecke operator, which corresponds to the double classGL ( Z p ) Z · (cid:0) p
00 1 (cid:1) · GL ( Z p ). If χ : Q × p → E × is a smooth character, and if λ ∈ E , thenlet π ( r, λ, χ ) = ind GL ( Q p )GL ( Z p ) Z Sym r E T − λ ⊗ ( χ ◦ det) . This is a smooth representation of GL ( Q p ), with central character ω r χ (where ω : Z × p → F × p is the “reduction mod p ” map). Let µ λ : Q × p → E × be given by µ λ | Z × p = 1,and µ λ ( p ) = λ . If λ = ±
1, then we have two exact sequences:0 → Sp ⊗ ( χµ λ ◦ det) → π (0 , λ, χ ) → χµ λ ◦ det → , → χµ λ ◦ det → π ( p − , λ, χ ) → Sp ⊗ ( χµ λ ◦ det) → , where the representation Sp is the “special” representation. Theorem 1.1 . — If E is algebraically closed, then the smooth irreducible E -linear rep-resentations of GL ( Q p ) , admitting a central character, are as follows: χ ◦ det ;
2. Sp ⊗ ( χ ◦ det) ; π ( r, λ, χ ) , where r ∈ { , . . . , p − } and ( r, λ ) / ∈ { (0 , ± , ( p − , ± } . This theorem is proved in [
BL95 ] and [
BL94 ], which treat the case λ = 0, and in[ Bre03 ], which treats the case λ = 0.We now explain what happens if E is not algebraically closed. ENTRAL CHARACTERS FOR REPRESENTATIONS OF GL ( Q p ) Proposition 1.2 . — If Π is a smooth irreducible E -linear representation of GL ( Q p ) ,admitting a central character, then there exists a finite extension K/E such that (Π ⊗ E K ) ß is a direct sum of K -linear representations of the type described in theorem 1.1.Proof . — Barthel and Livn´e’s methods show (as is observed in § Paˇs10 ]) that Πis a quotient of Σ = ind GL ( Q p )GL ( Z p ) Z Sym r E P ( T ) ⊗ ( χ ◦ det) , for some integer r ∈ { , . . . , p − } , character χ : Q × p → E × , and polynomial P ( Y ) ∈ E [ Y ]. Let K be a splitting field of P ( Y ), write P ( Y ) = ( Y − λ ) · · · ( Y − λ d ), and let P i ( Y ) = ( Y − λ ) · · · ( Y − λ i ) for i = 0 , . . . , d . The representations P i − ( T )Σ /P i ( T )Σ arethen isomorphic to π ( r, λ i , χ ), for i = 1 , . . . , d .We finish this section by recalling that if λ = 0, then the representations π ( r, λ, χ ) areparabolic inductions (when λ = 0, they are called supersingular). Let χ and χ : Q × p → E × be two smooth characters, and consider the parabolic induction ind GL ( Q p )B ( Q p ) ( χ ⊗ χ ).The following result is proved in [ BL94 ] and [
BL95 ]. Theorem 1.3 . — If λ ∈ K \ { ± } , and if r ∈ { , . . . , p − } , then π ( r, λ, χ ) isisomorphic to ind GL ( Q p )B ( Q p ) ( χµ /λ , χω r µ λ ) .
2. Proof of the theorem
We now give the proof of theorem A. Let Π be a smooth irreducible F p -linear repre-sentation of GL ( Q p ). We have Π (1+ p Z p ) · Id = 0 (since a p -group acting on a F p -vectorspace always has nontrivial fixed points), so that if Π is irreducible, then (1 + p Z p ) · Idacts trivially on Π. If g ∈ Z × p · Id, then g p − = Id on Π, so that Π = ⊕ ω ∈ F p Π g = ω · Id . SinceΠ is irreducible, this implies that the elements of Z × p · Id act by scalars.If f = (cid:0) p p (cid:1) , then for any nonzero polynomial Q ( X ) ∈ F p [ X ], the kernel and imageof the map Q ( f ) : Π → Π are subrepresentations of Π. If Q ( f ) = 0 on a nontrivialsubspace of Π, then f admits an eigenvector for an eigenvalue λ ∈ F p . This implies thatΠ = Π f = λ · Id , so that Π does admit a central character. If this is not the case, then Q ( f )is bijective for every nonzero polynomial Q ( X ) ∈ F p [ X ], so that Π has the structure of a F p ( X )-vector space, and is a F p ( X )-linear smooth irreducible representation of GL ( Q p ),admitting a central character.Let E = F p ( X ). Proposition 1.2 gives us a finite extension K of E , such that (Π ⊗ E K ) ß is a direct sum of K -linear representations of the type described in theorem 1.1. The F p -linear representation underlying (Π ⊗ E K ) ß is isomorphic to Π [ K : E ] , and hence of length LAURENT BERGER [ K : E ]. We now prove that none of the K -linear representations of the type describedin theorem 1.1 are of finite length, when viewed as F p -linear representations.Let Σ be one such representation, and let λ ∈ K be the corresponding Hecke eigenvalue. Proposition 2.1 . —
There exists a subring R of K , which is a finitely generated F p -algebra, such that Σ = Σ R ⊗ R K , where Σ R is an R -linear representation of GL ( Q p ) .Proof . — If λ ∈ F p , then theorem 1.1 shows thatΣ = ind GL ( Q p )GL ( Z p ) Z Sym r F p T − λ ⊗ F p K ( χ ◦ det) , or Sp ⊗ F p K ( χ ◦ det) , or K ( χ ◦ det) . We can then take R = F p [ χ ( p ) ± ], and Σ R = (ind GL ( Q p )GL ( Z p ) Z Sym r F p / ( T − λ )) ⊗ F p R ( χ ◦ det),or Sp ⊗ F p R ( χ ◦ det), or R ( χ ◦ det), respectively.If λ / ∈ F p , then by theorem 1.3, we haveΣ = ind GL ( Q p )B ( Q p ) ( χµ /λ , χω r µ λ ) . We can take R = F p [ λ ± , χ ( p ) ± ], and let Σ R be the set of functions f ∈ Σ with valuesin R .Let β ∈ F p be such that ( X − β ) / ∈ R × , so that ( X − β ) j Σ R = ( X − β ) j +1 Σ R for all j ∈ Z . The representation Σ contains ∪ j ∈ Z ( X − β ) j Σ R , so that the underlying F p -linearrepresentation is not of finite length, which is a contradiction. This finishes the proof oftheorem A. References [BL94]
L. Barthel & R. Livn´e – “Irreducible modular representations of GL of a localfield”, Duke Math. J. (1994), no. 2, p. 261–292.[BL95] , “Modular representations of GL of a local field: the ordinary, unramified case”, J. Number Theory (1995), no. 1, p. 1–27.[Bre03] C. Breuil – “Sur quelques repr´esentations modulaires et p -adiques de GL ( Q p ): I”, Compositio Math. (2003), no. 2, p. 165–188.[Paˇs10]
V. Paˇsk¯unas – “The image of Colmez’s Montr´eal functor”, preprint, 2010.
July 2011
Laurent Berger , UMPA, ENS de Lyon, UMR 5669 du CNRS, Universit´e de Lyon