Extensions between irreducible representations of a p-adic GL(n)

Marie-France Vignéras

Let H be the group of points of a connected reductive group over a local non archimedean field F. Let ω be a character of the center of H. Let C := Mod_{ω} H be the category of complex representations of H which are smooth (the stabilizer of a vector is an open subgroup of H), with central character ω. It is known that C has enough injectives and projectives, and we can define Ext_{C}^{i}(V, V′) for two representations V, V′ ∈ C, using a projective resolution (P^{i})_{i≥0} of V, or an injective resolution (I^{i})_{i≥0} of V′. The cohomology of the complex Hom_{C}(P^{i}, V′) and of the complex Hom_{C}(V, I^{i}) are the same, and are equal to Ext_{C}^{i}(V, V′) by definition.

Question Let V, V′ ∈ C irreducible, with V essentially square integrable (essentially because of the center), and V′ essentially tempered. Is is true that

Ext_{C}^{i}(V, V′) = Ext_{C}^{i}(V′, V) = 0

for all integers i > 0 ?

This question is motivated by the orthogonal decomposition of the Schwartz algebra of H given by the Plancherel formula ([Sil th.3 page 4679] for example). I tried to prove without success that the answer was yes, some years ago while writing [Vig1]. The answer (yes) is an exercise for GL(n, F) for any integer n > 1.

It can be worth to publish it.

Let H = G := GL(n, F). Let V ∈ C irreducible essentially square integrable. We can describe all the irreducible V′ ∈ C such that Ext_{C}^{i}(V′, V) ≠ 0 for at least one integer i ≥ 0. For such a V′, there is a unique i such that Ext_{C}^{i}(V′, V) ≃ C, and is zero otherwise. If V′ ≄ V, then V′ does not have a Whittaker model. An irreducible essentially tempered representation has a Whittaker model. For all irreducible tempered representation V′ not isomorphic to V, we get Ext_{C}^{∗}(V′, V) = 0. Using duality, we get Ext_{C}^{∗}(V, V′) = 0.

The computation of Ext_{C}^{∗}(V′, V) for V irreducible essentially square integrable and V′ irreducible, is a corollary of the classification of square integrable representations by Zelevinski, the theory of simple types by Bushnell and Kutzko, the Zelevinski involution by Aubert, Schneider and Stuhler, the computation of Ext_{C}^{∗}(1, V′) by Casselman.

We give a very short proof of Ext_{C}^{∗}(V, V′) = 0 for V, V′ ∈ C, irreducible tempered and not isomorphic, suggested by Waldspurger. The group G has the particularity to have at most one irreducible tempered representation with a given infinitesimal character (i.e. cuspidal support), and Ext_{C}^{∗}(V, V′) = 0 for two irreducible representations V, V′ of G having different infinitesimal characters. This second fact is very general, and uses the interpretation by Yoneda of Ext_{C}^{n}(V, V′) by n-extensions, as in the real case.

The author was supported by a grant of the von Humboldt foundation, and this work was done in the wonderful atmosphere of the Max Planck Institute in Bonn in the fall of 1996.

1 We set G := GL(n, F) and C = Mod G (we do not fix the central character). From Bernstein [Z 9.3], any V ∈ C irreducible essentially square integrable is a Steinberg representation St_{k}(ρ) where ρ is an irreducible cuspidal representation of GL(r, F) for some integer r > 0, and rk = n. The Steinberg representation St_{k}(ρ) is the unique irreducible subquotient with a Whittaker model in the natural representation of G in the space of locally constant functions f : G → ⊗^{k}ρ such that f(mug) = ⊗^{k}ρ(m)f(g) for any g ∈ G and any element mu (m ∈ M, u ∈ U), in a parabolic subgroup of G with Levi component M isomorphic to GL(r, F)^{k}, and unipotent radical U. When r = 1 and ρ = 1 is the trivial character 1 of F^{∗}, St_{n}(1) = St is the usual Steinberg representation.

A block in the abelian category C is an indecomposable abelian subcategory which is a direct factor. There are no non trivial homomorphisms between two different blocks. The blocks are classified by the semi-simple types of Bushnell-Kutzko [BK2, BK3], and also by the irreducible cuspidal representations of Levi subgroups modulo G-conjugation, and twist by unramified characters [BD].

The semi-simple type of a block is a distinguished irreducible representation σ of a distinguished open compact subgroup K of G, such that the functor

F_{σ} : V → Hom_{G}(ind_{G,K} σ, V)

is an equivalence of categories between the block and the category of right End_{G} ind_{G,K} σ-modules.

Let I be an Iwahori subgroup (unique modulo G-conjugation). A representation V ∈ C generated by the I-invariant vectors V^{I}, is called unipotent. The unipotent representations form a block, of semisimple type the trivial representation of I. Set F_{I} = Hom_{G}(ind_{G,I} 1, −).

Let (e, f, d), efd = r, be the invariants of ρ [Vig1 III.5]. Let q be the order of the residual field of F. Let F′ be any local non archimedean field, with residual field of order q′ = q^{fd}. We set G′ = GL(k, F′) and C′ = Mod G′. Denote I′ an Iwahori subgroup of G′.

Bushnell and Kutzko [BK1 7.6.18] have shown that there is a natural algebra isomorphism [BK1 7.6.18, 7.6.21]

i : End_{G′} ind_{G′,I′} 1 → End_{G} ind_{G,K} σ.

We get a functor Φ which is an equivalence of categories, from the unipotent block in C′ to the block in C containing St_{k}(ρ) such that

i^{∗} ∘ F_{σ} ∘ Φ′ = Hom_{G′}(ind_{G′,I′} 1, −).

For any Levi subgroup M′ of G′, there is a similar functor Φ′ which is an equivalence from the unipotent block of M′ to a block in a Levi subgroup M of G. This is compatible with the normalized parabolic induction i_{G′,M′} and i_{G,M}, or restriction r_{M′,G′} and r_{M,G} , along Q′ = M′Q′_{o} and Q = MQ_{o}, where Q′_{o} and Q_{o} are suitable Borel subgroups of G′ and G:

Φ ∘ i_{G′,M′} = i_{G,M} ∘ Φ′, Φ′ ∘ r_{M′,G′} = r_{M,G} ∘ Φ

This is a consequence of [BK1 7.6.21].

Proposition The functor Φ sends an essentially square integrable (resp. unitary, having a Whittaker model, essentially tempered) irreducible unipotent representation of G′ to an essentially square integrable (resp. unitary, having a Whittaker model, essentially tempered) irreducible representation of G.

For essentially square integrable see [BK1 7.7]. For unitary see [BK1 7.6.25]. The irreducible representations of G with a Whittaker model are induced from essentially square integrable representations of Levi subgroups [Z 9.11]. The assertion for the Whittaker model follows from this and the compatibility of Φ′, Φ with the induction. The tempered irreducible representations of G are induced from square integrable representations [Sil 4.5.11]. Hence the assertion for essentially tempered representations.

2 We want to prove a vanishing result for Ext^{1}, between characters of affine Hecke algebras, directly and in an elementary way. In fact, the best method to compute Ext^{∗} between modules for affine Hecke algebras, is to use the dictionnary with representations. This paragraph could be skipped.

The Hecke algebra End_{G} ind_{G,I} 1 is naturally isomorphic to the affine Hecke algebra H_{C}(n, q) of type A_{n−1} and parameter q [BK1 5.6.6].

The Hecke C-algebra H_{C}^{o}(n, x) of type A_{n−1} with parameter x ∈ C^{∗}, x ≠ 0, 1, is the C-algebra generated by (s_{1}, …, s_{n−1}) with the relations

(s_{i} + 1)(s_{i} − x) = 0 (1 ≤ i ≤ n − 1),

s_{i}s_{j} = s_{j}s_{i} (1 ≤ i, j ≤ n − 1, |j − i| ≠ 1)
s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} (1 ≤ i ≤ n − 2).

The affine Hecke C-algebra H_{C}(n, x) of type A_{n−1} with parameter x is generated by H_{C}^{o}(n, x) and t with tt^{−1} = t^{−1}t = 1, ts_{i} = s_{i−1}t (1 < i < n), t^{2}s_{1} = s_{n−1}t^{2}. Note that this description [BK 5.4 page 177] is not the Bernstein description.

The finite algebra H_{C}^{o}(n, x) is isomorphic to the group algebra C[S_{n}] of the symmetric group S_{n} and has two characters. For the character sign, the image of all the s_{i} is −1. For the trivial character, the image of all the s_{i} is x. The two characters extend to characters of H_{C}(n, x), the image of t beeing an arbitrary not zero complex element. The H_{C}(k, q)-module F (St) is a sign character of H_{C}(n, q).

The center of G is naturally identified with F^{∗} diagonally enbedded in G. The center of H_{C}(n, x) contains t^{n}. The central character of St is trivial. The category of unipotent representations of G with trivial central character is isomorphic by the functor F_{I} defined in (1) to the category Mod H_{C}(n, q)_{1} of right modules of the quotient H_{C}(n, q)_{1} of H_{C}(n, q) by the two-sided ideal generated by t^{n} − 1 [Vig2 I.3.14].

2.1 Lemma Let χ, χ′ ∈ C := Mod H_{C}(n, q)_{1} two characters. Then Ext_{C}^{1}(χ, χ′) = 0.

Indeed the algebras H_{C}^{o}(n, q) and C[t], t^{n} = 1, are semisimple (but the quotient H_{C}(n, q)_{1} is not semisimple). If V ∈ C is an extension of χ by χ, then hv = χ(h)v for all h ∈ H_{C}^{o}(n, q), v ∈ V, and t acting semisimply, V ≃ χ ⊕ χ.

There is another proof when n = 2 in [DPrasad p.175 proof of the lemma 7]. Note that if we were not fixing the center, we could have extensions. I do not know how to compute directly Ext^{i} when i > 1.

When V is an extension of two different characters χ′ ≠ χ in Mod H_{C}(n, q)_{1} or in Mod H_{C}(n, q), one sees that V ≃ χ ⊕ χ′ by restriction to the commuting algebras H_{C}^{o}(n, q) and C[t].

2.2 From (2.1), Ext^{1}(St, St) = 0 in Mod_{1} G. Any irreducible square integrable unipotent representation V of G is the twist St ⊗χ_{x} of St by an unramified character of G

χ_{x}(g) = x^{val det g}, g ∈ G,

for some x ∈ C^{∗}, where val : (F)^{∗} → Z is the valuation of F, sending an uniformizing parameter to 1. The central character of St ⊗χ_{x} is the character χ_{kx} of F^{∗}. It is trivial if and only if St ⊗χ_{x} ≃ St .

The twist by a character χ of G does not change the value of Ext^{∗}. If V, V′ ∈ C := Mod_{ω} G, then V ⊗ χ, V′ ⊗ χ ∈ C_{χ} := Mod_{ωω(χ)} G where ω(χ) is the restriction of χ to the center of G. We have :

Ext_{C}^{∗}(V, V′) ≃ Ext_{C_{χ}}^{∗}(V ⊗ χ, V′ ⊗ χ).

Using the functor F_{I} of (1), we get :

Proposition Let V, V′ irreducible and essentially square integrable in the category C := Mod_{ω} G. Then

Ext_{C}^{1}(V, V′) = 0.

There is another proof due to Silberger of this result, valid for a general reductive group [Sil2]. To compute some Ext_{C}^{i}(V, V′) when i > 1, we use the results of Casselman [Cas].

3 Let H as in the introduction. Let C := Mod_{1} H be the category of representations of H with trivial character. Denote by St_{Q} ∈ C the Steinberg representation defined by a parabolic subgroup Q of H [BW 4.6 page 308]. If τ_{Q} is the natural representation of H on the complex space of locally constant left Q-invariant functions H → C, then St_{Q} is the quotient of τ_{Q} by the subrepresentation generated by the natural images of τ_{Q′} in τ_{Q}, for all parabolic subgroups Q′ of H which contain Q. We have St_{H} = 1. When Q = Q_{o} is minimal, then St_{Q_{o}} = St is the usual Steinberg representation. The representations St_{Q} are irreducible and not isomorphic.

The parabolic rank of Q is the rank of a maximal split torus in the center of a Levi component of Q. We denote

m_{Q} = parabolic rank of Q − parabolic rank of H .

This an integer ≥ 0.

Theorem [BW 5.1 th.4.12 page 313] Let V ∈ C := Mod_{1} H irreducible such that Ext^{∗}(1, V) ≠ 0. Then there exists a parabolic subgroup Q of H such that V ≃ St_{Q}. Moreover Ext_{C}^{m}(1, St_{Q}) ≃ C if m = m_{Q}, and is zero otherwise.

Remarque Suppose H = G := GL(n, F), and C := Mod_{1} G. We have Ext_{C}^{o}(1, 1) ≃ C and Ext_{C}^{m}(1, 1) = 0 for any integer m ≥ 1. The representation τ_{Q} ∈ C has a unique irreducible subquotient with a Whittaker model, this unique

subquotient is isomorphic to St [Z 9.7]. In particular, when Q ≠ Q_{o} the representation St_{Q} does not have a Whittaker model. Hence Ext_{C}^{∗}(1, V) = 0 for any irreducible representation V ≠ St with a Whittaker model.

4 Zelevinski involution Let G as in (1). The Zelevinski involution τ in Mod G has the following properties :

a) τ respects the property of beeing irreducible [A 2.3, 2.9].

b) τ exchanges the trivial and the usual Steinberg representation [Z 9.2].

c) τ(− ⊗ χ) = τ(−) ⊗ χ commutes with the twist by a character χ of G [Z 9.1].

d) τ respects the cuspidal support [Z 9.1].

e) τ is an exact contravariant functor and respects the cuspidal support [SS 3.1], hence respects the representations with a given central character.

Set C := Mod G or C := Mod_{ω} G, where ω is a character of the center of G. By e) we have for any V, V′ ∈ C

Ext_{C}^{∗}(V, V′) ≃ Ext_{C}^{∗}(τ(V′), τ(V)).

With the notations of (3), the representation τ(St_{Q}) is not isomorphic to St when Q ≠ G by b), and is a subquotient of τ_{Q_{o}} by d). Hence τ(St_{Q}) does not have a Whittaker model when Q ≠ G, in particular is not essentially tempered. We deduce from (3):

Theorem Let V, V′ ∈ C := Mod_{w} G, irreducible, such that V ≃ St ⊗ χ is unipotent and essentially square integrable as in 2), and Ext_{C}^{∗}(V′, V) ≠ 0. Then there exists a parabolic subgroup Q of G′ such that V′ ≃ τ(St′_{Q}) ⊗ χ. For V′ = τ(St_{Q}) ⊗ χ, we have Ext_{C}^{m}(V′, V) ≃ C^{∗} if m = m_{Q} as in 3), and zero otherwise.

In particular, if V is a unipotent Steinberg representation, and if V′ ≄ V is essentially tempered, then

Ext_{C}^{o}(V, V) ≃ C, Ext_{C}^{i}(V, V) = Ext_{C}^{i}(V′, V) = 0

for all integers i > 0. We will prove also

(4.1)  Ext_{C}^{i}(V, V′) = 0

using duality as follows.

5 Duality Let (H, ω) as in the introduction. The contragredient V → V^{∗} is a contravariant exact functor in Mod H, which sends a projective representation to an injective representation [Vig2 I.4.18]. A representation V is called admissible when V^{∗∗} ≃ V. When V is admissible, and (P_{i}) → V is a projective resolution of V, then V^{∗} → (P_{i}^{∗}) is an injective resolution of V^{∗}, and Hom(P_{i}, W) ≃ Hom(W^{∗}, P_{i}^{∗}) canonically [Vig2 I4.13]. If V ∈ Mod_{ω} H, then V^{∗} ∈ Mod_{ω^{−1}} H. Set C := C^{∗} := Mod H or C := Mod_{ω} H, C^{∗} := Mod_{ω^{−1}} H.

Proposition Let V, W ∈ C admissible of contragredient V^{∗}, W^{∗} ∈ C^{∗}, one has Ext_{C}^{∗}(V, W) ≃ Ext_{C^{∗}}^{∗}(W^{∗}, V^{∗}).

The contragredient respects the property of beeing essentially square integrable and of beeing essentially tempered. We deduce (4.1). Hence the answer to the question in the introduction is yes, for G = GL(n, F).

There is another proof, suggested by Waldspurger, using that the essentially tempered irreducible representations of G have different cuspidal support. This comes from the classification of Zelevinki [Z], which shows that tempered irreducible representations are not degenerate (1), and that not degenerate irreducible representations have different cuspidal support.

6 Let (H, w), C as in (5). There is a natural equivalence between the two bifunctors on C,

Ext_{C}^{n}(A, B) and Yext_{C}^{n}(A, B)

given by the Yoneda n-extensions of A by B modulo an equivalence relation ≡. The proofs are the same than in the category of (left) modules for a ring [M III.6.4, III.8.2].

An n-extension X of A by B is an exact sequence starting at B and ending at A,

X : 0 → B → X_{n} → … → X_{1} → A → 0.

A morphism γ : X → Y between two n-extensions starting with β and ending with α is a commutative diagram

X : 0 → B → X_{n} → … → X_{1} → A → 0
↓ γ ↓ β ↓ ↓ ↓ α
Y : 0 → D → Y_{n} → … → Y_{1} → C → 0

The equivalence relation ≡ in the set of n-extensions of A by B, is generated by the relation : there exists a morphism γ : X → Y starting and ending with the identity.

An n-extension X ending at A can be spliced with an m-extension Y starting at A, to give an n + mextension X ∘ Y starting like X, ending like Y. If α : A′ → A, one defines by pull-back an extension Xα starting like X, ending at A′. If Z is an m-extension starting by A′, one defines by push-out an m-extension αZ starting at A, ending like Z. By definition of the equivalence relation, one has

Xα ∘ Z ≡ X ∘ αZ.

A morphism γ : X → Y starting with β and ending with α gives an equivalence [M III.5.1]

βX ≡ Y α.

An element z of the center of C defines an endomorphism of X. If z acts on A and on B by multiplication by two different scalars z_{a} ≠ z_{B} ∈ R, we deduce that the image of X in Yext^{n}(A, B) ≃ Ext^{n}(A, B) is 0.

For A, B ∈ C irreducible of different cuspidal support, there is an element z in the center of C which acts by the identity on A and is zero on B′. This comes from the description of the center by Bernstein [BD]. We get the following theorem.

6.1 Theorem Let V, V′ ∈ C irreducible of different cuspidal support. Then Ext_{C}^{∗}(V, V′) = 0.

6.2 Corollary Suppose that H = GL(n, F). Let V, V′ ∈ C irreducible not degenerate, and V ≠ V′. Then Ext_{C}^{∗}(V, V′) = 0.

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