Base change for ramified unitary groups: The strongly ramified case

1.1 Notations

Let F ∙ be a non-Archimedean local field, with ring of integers 𝔬 F ∙ , its maximal ideal 𝔭 F ∙ , and residue field 𝐤 F ∙ = 𝔬 F ∙ / 𝔭 F ∙ of cardinality q ∙ and odd characteristic p. Let F / F ∙ be a quadratic extension, whose residue field 𝐤 F = 𝔬 F / 𝔭 F has q elements, such that q = q ∙ 2 in the unramified case, and q = q ∙ in the ramified case. Let 𝝁 F be the subgroup of roots of unity of F whose orders are coprime to p.

We denote U F := 𝔬 F × and U F k := 1 + 𝔭 F × for k ∈ ℤ ≥ 1 . If N F / F ∙ : F → F ∙ is the norm map, we denote U F / F ∙ := 𝔬 F × ∩ ker ⁡ N F / F ∙ and U F / F ∙ k := U F k ∩ ker ⁡ N F / F ∙ for k ∈ ℤ ≥ 1 . The Galois group Gal ⁢ ( F / F ∙ ) is generated by an involutive automorphism c .

If Λ : ℤ → S is a sequence into a set S, then we extend Λ from ℤ to ℝ by putting Λ ⁢ ( r ) = Λ ⁢ ( ⌈ r ⌉ ) and Λ ⁢ ( r + ) = Λ ⁢ ( ⌈ r + ⌉ ) , where ⌈ r ⌉ and ⌈ r + ⌉ are the smallest integers ≥ r and > r , respectively.

2.1 Unitary groups

Let V be an F-vector space. We denote A ~ = A ~ V = End F ⁢ ( V ) and G ~ = G ~ V = Aut F ⁢ ( V ) . Suppose that V is equipped with a non-degenerate ( F / F ∙ , ϵ ) -Hermitian form h = h V , where ϵ = ϵ V = ± 1 . If X ↦ X ¯ is the conjugate-adjoint on A ~ defined by h, we define the adjoint anti-involution X α = - X ¯ . Note that

We also have the corresponding involution σ : X ↦ X ¯ - 1 on G ~ . The subgroup G = G V = G ~ σ is a (connected) unitary group that we consider throughout the paper and whose Lie algebra is A = A ~ α .

Given an 𝔬 F -lattice L in V, we denote by L * its dual { v ∈ V : h ⁢ ( v , L ) ⊂ 𝔭 F } . We call an 𝔬 F -lattice sequence Λ in V self-dual if there exists d ∈ ℤ such that Λ ⁢ ( k ) * = Λ ⁢ ( d - k ) for all k ∈ ℤ . As in [50], we always normalize a self-dual lattice sequence such that

Given an 𝔬 F -lattice sequence Λ in V, we define an 𝔬 F -lattice sequence 𝔓 ~ Λ k , for k ∈ ℤ , in A ~ by

Hence 𝔄 ~ Λ := 𝔓 ~ Λ 0 is a hereditary order in A ~ , with Jacobson radical 𝔓 ~ Λ := 𝔓 ~ Λ 1 . We denote by v Λ the valuation on A ~ associated to Λ. If Λ is self-dual, then each 𝔓 ~ Λ k is α-invariant, in which case we put 𝔓 Λ = 𝔓 ~ Λ α .

We also define U ~ Λ = U ~ Λ 0 = 𝔄 ~ Λ × and U ~ Λ k = 1 + 𝔓 ~ Λ k for k ∈ ℤ ≥ 1 . If Λ is self-dual, then each U ~ Λ k is σ-invariant, in which case we put U Λ = U ~ Λ σ and U Λ k = ( U ~ Λ k ) σ . The quotient 𝖦 Λ = U Λ / U Λ 1 is the group of 𝐤 F ∙ -points of a reductive group 𝗚 Λ defined over 𝐤 F ∙ , which is disconnected in general. We denote by U Λ 0 the inverse image of 𝖦 Λ 0 in U Λ , where 𝖦 Λ 0 is the subgroup of 𝐤 F ∙ -points of the identity component 𝗚 Λ 0 of 𝗚 Λ .

Suppose that Λ is a self-dual lattice sequence of the form

Then U Λ is a maximal compact subgroup, and U Λ 0 is the underlying maximal parahoric subgroup. When F / F ∙ is unramified, 𝖦 Λ 0 is a product of at most two unitary groups relative to 𝐤 F / 𝐤 F ∙ ; while when F / F ∙ is ramified it is a product of at most one symplectic group and at most one special orthogonal group, both defined over 𝐤 F = 𝐤 F ∙ (see [50, Section 3.3]).

2.2 Cuspidal types

We recall from [19, 50] the constructions of cuspidal types for general linear groups and unitary groups. The compact inductions of these types are irreducible supercuspidal representations.

Let 𝐬 = [ Λ , r , 0 , β ] be either a simple stratum or the null stratum [ Λ , 0 , 0 , 0 ] where, in the former case, we denote E = F ⁢ [ β ] , which is a field contained in A ~ , such that Λ is an 𝔬 E -lattice chain, while in the latter case, we put E = F and Λ ⁢ ( k ) = Mat n ⁢ ( 𝔭 F k ) . We denote by A ~ E and G ~ E , respectively, the centralizer of β in A ~ and G ~ , and for k ∈ ℤ , denote 𝔓 ~ Λ , E k = 𝔓 ~ Λ k ∩ A ~ E and U ~ Λ , E k = U ~ Λ k ∩ G ~ E . As in [19, Chapter 3], associated to 𝐬 we construct

1. subrings ℌ ~ = ℌ ~ Λ , β ⊆ 𝔍 ~ = 𝔍 ~ Λ , β of A ~ and the two-sided fractional ideals ℌ ~ k = ℌ ~ ∩ 𝔓 ~ Λ k and 𝔍 ~ k = 𝔍 ~ ∩ 𝔓 ~ Λ k , for all k ∈ ℤ ,

2. subgroups H ~ k = H ~ Λ , β k = ℌ ~ ∩ U ~ Λ k ⊂ J ~ k = J ~ Λ , β k = 𝔍 ~ ∩ U ~ Λ k of G ~ , for all k ∈ ℤ > 0 ,

3. 𝒞 ~ ⁢ ( 𝐬 ) := 𝒞 ~ ⁢ ( Λ , 0 , β ) the set of simple characters of H ~ 1 ,

4. associated to each simple character θ ~ ∈ 𝒞 ~ ⁢ ( 𝐬 ) the Heisenberg representation η ~ = η ~ θ ~ of J ~ 1 , an irreducible representation that restricts to a multiple of θ ~ on H ~ 1 ,

5. a beta-extension κ ~ of η ~ to the subgroup J ~ = J ~ Λ , β = U ~ Λ , E ⁢ J ~ 1 (note that among these extensions there is a unique one κ ~ 0 whose determinant has order a power of p, called the p-primary beta-extension).

To construct a supercuspidal representation of G ~ , we require that Λ is principal, which is assumed from now on, and also e ⁢ ( Λ / 𝔬 E ) = 2 (note the convention in (2.1)). We take an irreducible representation ρ ~ of J ~ inflated from a cuspidal representation of

The maximal simple type λ ~ = κ ~ ⊗ ρ ~ extends to an irreducible representation 𝝀 ~ of 𝐉 ~ = E × ⁢ J ~ , which is then compactly induced to an irreducible supercuspidal representation π ~ = cInd 𝐉 ~ G ~ ⁡ 𝝀 ~ .

Note that if 𝐬 is null, then by convention we assume that e ⁢ ( Λ / 𝔬 F ) = 2 , and 𝒞 ~ ⁢ ( 𝐬 ) contains only the trivial character of H 1 ~ = J ~ 1 = U ~ Λ 1 . We take κ ~ of U ~ Λ to be trivial, and choose ρ ~ to be inflated from a cuspidal representation of U ~ Λ / U ~ Λ 1 ≅ GL n ⁢ ( 𝐤 F ) . The extension 𝝀 ~ on 𝐉 ~ = F × ⁢ J ~ of λ ~ = ρ ~ can be chosen by fixing a central character, and so π ~ = cInd 𝐉 ~ G ~ ⁡ 𝝀 ~ is a level zero supercuspidal representation.

Every supercuspidal representation of G ~ is obtained by the way above. Moreover, the maximal simple type λ ~ is determined by π ~ up to conjugacy, and so is the extended type 𝝀 ~ containing λ ~ .

If V is a Hermitian space, as in Section 2.1, we call a simple stratum 𝐬 skew if Λ is self-dual and β ∈ A . In this case, all subgroups H ~ 1 , J ~ 1 , J ~ , and 𝐉 ~ are σ-invariant. If we choose θ ~ , κ ~ , ρ ~ , and the extension of 𝝀 ~ to be self-dual, i.e., σ-invariant, then so is π ~ . Note that the p-primary beta-extension κ ~ 0 of θ ~ is self-dual because of its uniqueness. If β ≠ 0 , then - α restricts to a Galois involution on E. We denote the fixed field by E ∙ . Since A ~ E and 𝔓 ~ Λ , E k , resp. G ~ E , U ~ Λ , E , and U ~ Λ , E k , are invariant under α, resp. σ, we define A E / E ∙ = A ~ E α , G E / E ∙ = G ~ E σ , and 𝔓 Λ , E / E ∙ k , U Λ , E / E ∙ , U Λ , E / E ∙ k similarly.

When 𝐬 is skew and semisimple, following [50], we construct

1. subgroups H 1 := ( H ~ 1 ) σ ⊆ J 1 := ( J ~ 1 ) σ ⊂ J := J ~ σ ,

2. a set 𝒞 ⁢ ( 𝐬 ) := 𝒞 ⁢ ( Λ , 0 , β ) of semisimple characters of H 1 , defined as follows: denote the subset of 𝒞 ~ ⁢ ( 𝐬 ) of self-dual semisimple characters by 𝒞 ~ ⁢ ( 𝐬 ) σ , and define 𝒞 ⁢ ( 𝐬 ) to be the image set of the following map (well-defined since the group H 1 is a pro-p subgroup where p is odd):

   (2.2) 𝒞 ~ ⁢ ( 𝐬 ) σ → 𝒞 ⁢ ( 𝐬 ) , θ ~ ↦ ( θ ~ | H 1 ) 1 / 2 ,

   which turns out to be bijective since the restriction operation satisfies the properties of Glauberman correspondence (see [50, Section 5.3] for details),

3. associated to each semisimple character θ ∈ 𝒞 ⁢ ( 𝐬 ) the Heisenberg representation η = η θ of J 1 , an irreducible representation that restricts to a multiple of θ on H 1 ,

4. a beta-extension κ of η to the subgroup J = U Λ , E / E ∙ ⁢ J 1 (note that among these extensions there is a unique one κ 0 whose determinant has order a power of p, called the p-primary beta-extension).

To construct a supercuspidal representation of G , from now on we suppose that G E / E ∙ has a compact center and the parahoric subgroup U Λ , E / E ∙ 0 is maximal in G E / E ∙ . We take an irreducible representation ρ of J inflated from a cuspidal representation ρ ¯ of 𝖦 = J / J 1 , a (possibly non-connected) classical group over the finite field 𝐤 F ∙ . (A cuspidal representation of a disconnected 𝖦 means that its restriction to 𝖦 0 is a sum of conjugates of a cuspidal representation.) The product λ = κ ⊗ ρ is then a cuspidal type, and is compactly induced to an irreducible supercuspidal representation π = cInd J G ⁢ λ . If 𝐬 is null, then by definition 𝒞 ⁢ ( 𝐬 ) contains only the trivial character of U Λ 1 . We again take κ to be trivial, and so π = cInd J G ⁢ ρ is a level zero supercuspidal representation.

2.3 Covering types

To proceed, we require a larger unitary group G W defined on the space W = V ⊥ Z . Here Z = Z - ⊕ Z + is an ϵ-Hermitian space, with Z + being a finite-dimensional vector space over F, and Z - is the dual space of Z + with respect to the form

for z - , w - ∈ Z - and z + , w + ∈ Z + . The ϵ-Hermitian form on W is h W = h ⊥ h Z . We denote the Lie algebra of G W by A W , the subspace of fixed points of A ~ W = End F ⁢ ( W ) by the adjoint anti-involution defined by h W .

We denote by M the Levi subgroup of block diagonal matrices in G W , isomorphic to G ~ Z - × G V . We denote by P the subgroup of block upper triangular matrices leaving invariant the flag Z - ⊂ V ⊂ W , with unipotent radical U and the opposite U - . We denote by i M : M → G W the embedding ( g , h ) ↦ diag ⁢ ( g , h , g σ ) for g ∈ G ~ , h ∈ G , and abbreviate

We can check that these matrices satisfy the relation

If Λ + is an 𝔬 F -lattice chain in Z + , we define

If β + ∈ A ~ Z + , we define β - ∈ A ~ Z - by

In this general setting, let λ V be a cuspidal type in G V and let λ ~ Z - be a cuspidal type in G ~ Z - as in Section 2.2. A covering type in G W for the cuspidal type λ ~ Z - ⊠ λ V in M is constructed in [36], providing a type for the Bernstein component of the smooth dual of G W that contains a representation whose cuspidal support comes from a supercuspidal representation of M containing λ ~ Z - ⊠ λ V . The use of covering types in the study of parabolic induction is briefly recalled in the next subsection. In the context of base change or of the description of L-packets, most covering types are irrelevant; the relevant ones are those in which the cuspidal type λ ~ Z - bears a strong relation with λ V . In the present work we focus on the case when the stratum 𝐬 underlying λ V is skew simple. In this case, and when studying base change, the relevant covering types are those for which the space Z - has the same dimension as V and the cuspidal type λ ~ Z - is built from a simple character that transfers in V to the square of the simple character underlying λ V (or see (2.4) for the precise relation).

So we proceed to construct covering types for G W precisely in that case.

We define a self-dual 𝔬 E -lattice sequence 𝔪 in W = V ⊥ Z of 𝔬 E -period 6 (hence of 𝔬 F -period 6 ⁢ e ⁢ ( E / F ) ) by

and two maximal self-dual 𝔬 E -lattice sequences 𝔐 w in W, where w ∈ { y , z } and both of 𝔬 E -period 2, such that the set of lattices in the sequence 𝔐 y is

while those in 𝔐 z is

Given a skew simple or null stratum 𝐬 = [ Λ , r , 0 , β ] , we hence have corresponding skew semisimple strata 𝐬 𝔪 = [ 𝔪 , r 𝔪 , 0 , β ] and 𝐬 w = [ 𝔐 w , r w , 0 , β ] , with w ∈ { y , z } , where β is embedded into A ~ W as the block-diagonal matrix with diagonal blocks ( β , β , β ) . We now follow [19, 50] to define, for ℒ = 𝔪 or 𝔐 w , with w ∈ { y , z } ,

1. subgroups H 𝔏 1 ⊆ J 𝔏 1 ⊂ J 𝔏 in G W ,

2. semisimple characters θ 𝔏 on H 𝔏 1 , such that they satisfy a transfer relation with each other, as in [19, (3.6)], [49, Section 3.5],

3. beta-extensions κ 𝔐 w of θ 𝔐 w on J 𝔐 w , and κ 𝔪 w of θ 𝔪 on J 𝔪 , such that κ 𝔐 w and κ 𝔪 w are compatible in the sense of [50, Lemma 4.3],

4. unique p-primary beta extension ( J 𝔐 w , κ 𝔐 w , 0 ) , and the one ( J 𝔪 , κ 𝔪 , 0 w ) compatible with κ 𝔐 w , 0 (note that κ 𝔪 , 0 w is in general not the p-primary beta extension κ 𝔪 , 0 of θ 𝔪 ).

Since the decomposition W = V ⊥ ( Z - ⊕ Z + ) is properly subordinate to the stratum 𝐬 𝔪 , the subgroups H 𝔪 1 , J 𝔪 1 , and J 𝔪 admit an Iwahori decomposition of the form

for K being one of the compact subgroups just mentioned, and the product can be taken in any order. We can then show from [50, Corollary 5.11 and Proposition 5.5] that H M 1 := H ~ 1 × H 1 and

where θ ~ Λ - and θ Λ are simple characters related under the correspondence (2.2), i.e.,

To construct a covering type, we define

1. the following subgroups:

   H P 1 = H ~ 𝔪 1 ⁢ ( J ~ 𝔪 1 ∩ U ) ∩ G W = H 𝔪 1 ⁢ ( J 𝔪 1 ∩ U ) , J P 1 = H ~ 𝔪 1 ⁢ ( J ~ 𝔪 1 ∩ P ) ∩ G W = H 𝔪 1 ⁢ ( J 𝔪 1 ∩ P ) , J P = H ~ 𝔪 1 ⁢ ( J ~ 𝔪 ∩ P ) ∩ G W = H 𝔪 1 ⁢ ( J 𝔪 ∩ P ) ,

   and also denote J P ± = J P ∩ U ± ,

2. a simple character θ P on H P 1 by extending θ 𝔪 trivially to J 𝔪 1 ∩ U , and the unique representation η P on J P 1 containing θ P (by [50, Lemma 5.12]),

3. for w ∈ { y , z } , an extension κ P w of η P to J P 1 that is the restriction of κ 𝔪 w to the subspace of ( J 𝔪 1 ∩ U ) -fixed vectors of η 𝔪 (which can be also characterized by the relation κ 𝔪 w ≅ Ind J P J 𝔪 ⁡ κ P w ).

With our construction of 𝔪 , the decomposition W = V ⊥ ( Z - ⊕ Z + ) is moreover exactly subordinate to 𝐬 𝔪 . By [50, Proposition 6.3], if we write κ P w | J M = κ ~ w ⊠ κ w , then κ ~ w and κ w are beta-extensions of θ ~ and θ, respectively, and κ ~ w is self-dual (with respect to σ), see [50, Corollary 6.10].

We then choose a (product of) cuspidal representation ρ M = ρ ~ ⊠ ρ of J P , inflated from J P / J P 1 ≅ J M / J M 1 . (At this moment, there is no relation between ρ ~ and ρ.) Define

which is a covering type over λ M w := λ P w | J M by [36]. Finally, we denote by

a supercuspidal representation of M = G ~ Z × G V containing the maximal type

where λ ~ w = κ ~ w ⊗ ρ ~ and λ w = κ w ⊗ ρ .

2.4 Covers and Hecke algebras

At the beginning of this subsection, we use G to denote the F-points of a connected reductive group over F, and will later switch back to denote a unitary group as in previous subsections. We recall the following notions from [20].

1. Suppose that G contains a parabolic subgroup P with Levi component M. If π M is a supercuspidal representation of M, we denote by 𝔰 = [ M , π M ] G the inertial class of π M , and by ℛ 𝔰 ⁢ ( G ) the full subcategory of representations of G whose irreducible subquotients are those of the normalized parabolic induction ι P G ⁢ τ := Ind P G ⁡ ( τ ⁢ Δ P 1 / 2 ) , where τ ranges over representations in 𝔰 and Δ P is the modulus character of P.

2. Suppose that K is a compact open subgroup of G. We fix a Haar measure on G such that K has volume 1. Given a representation λ of K on a finite-dimensional ℂ -vector space W, we denote by ℋ ⁢ ( G , λ ) the associated Hecke algebra, which is the space of compactly supported functions f : G → End ℂ ⁢ ( W ) satisfying

   f ⁢ ( k 1 ⁢ g ⁢ k 2 ) = λ ⁢ ( k 1 ) ∘ f ⁢ ( g ) ∘ λ ⁢ ( k 2 )   for all  ⁢ k 1 , k 2 ∈ K ⁢  and  ⁢ g ∈ G ,

   with an associative ℂ -algebra structure under the convolution

   f 1 * f 2 ⁢ ( g ) = ∫ G f 1 ⁢ ( x ) ⁢ f 2 ⁢ ( x - 1 ⁢ g ) ⁢ 𝑑 x   for all  ⁢ g ∈ G .

   The support of every element in ℋ ⁢ ( G , λ ) lies in the intertwining set

   I G ⁢ ( λ ) := { g ∈ G : Hom K ∩ g ⁢ K ⁢ g - 1 ⁢ ( λ , λ g ) ≠ 0 } .

We call ( K , λ ) an 𝔰 -type if, for an irreducible representation τ of G,

in which case we have an equivalence of categories

We now return to our notations in the previous subsection, suppose that π M contains a type ( J M , λ M ) , and ( J P , λ P ) is a G W -cover of ( J M , λ M ) (e.g., λ P = λ P w , for w ∈ { y , z } ). In particular, we have

Since ( J M , λ M ) is an [ M , π M ] -type, we obtain:

1. that ( J P , λ P ) is a [ M , π M ] G W -type, i.e., we have an equivalence of categories

   ℳ : ℛ [ M , π M ] ⁢ ( G W ) → Mod ⁢ - ⁢ ℋ ⁢ ( G W , λ P ) , τ ↦ Hom J P ⁢ ( λ P , τ ) ,

2. an injective morphism of algebras [20, (8.3), (8.4)]

   t P : ℋ ⁢ ( M , λ M ) → ℋ ⁢ ( G W , λ P )

   giving rise to the natural functor

   ( t P ) * : Mod ⁢ - ⁢ ℋ ⁢ ( M , λ M ) → Mod ⁢ - ⁢ ℋ ⁢ ( G W , λ P ) , D ↦ Hom ℋ ⁢ ( M , λ M ) ⁢ ( ℋ ⁢ ( G W , λ P ) , D ) ,

   such that the following diagram commutes:

   (2.5)

   

We will be interested in when the parabolic induction

is reducible, where π ~ is a supercuspidal representation of G ~ Z and π is a supercuspidal representation of G V . We stress that in the theory of covers one thinks in terms of inertial classes, hence the parameter s is complex, whereas in classical references about such reducibilities, the parameter s is real. In particular, by [47], if (2.6) is reducible for some s, which can be assumed to be real by twisting π ~ by a unitary character, then there is a real a such that π ~ ⁢ | det | a is self-dual and, assuming π ~ itself is self-dual, there is a unique real s π ~ , π ≥ 0 , indeed a half integer, such that (2.6) is reducible at s = ± s π ~ , π . We keep assuming π ~ self-dual. Let f π ~ be the order of the stabilizer of π ~ in the group of unramified characters of G ~ Z . By [19, Lemma 6.2.5] there are exactly two self-dual representations in the inertial class of π ~ , namely π ~ and π ~ ′ , the twist of π ~ by the unramified character | det | π ⁢ i f π ~ ⁢ log ⁡ q . The complex points of reducibility for π ~ are of the form

and those for π ~ ′ take the same form, with s 1 and s 2 exchanged. To obtain more information about these values, we will study the structures of Hecke algebras and their modules in the next section. We will also recall results of Mœglin about them in Section 2.7.

2.5 Structure of the Hecke algebras

We continue from the constructions in Section 2.4, and briefly recall the structure of the above Hecke algebras, referring the detail to [20, 50, 36].

For G ~ , if E = F ⁢ [ β ] is the field datum of 𝐬 , then I G ~ ⁢ ( λ ~ ) = 𝐉 ~ with 𝐉 ~ / J ~ ≅ E × / U E ≅ 〈 ϖ E 〉 , where ϖ E is a uniformizer of E. The Hecke algebra ℋ ⁢ ( G ~ , λ ~ ) is isomorphic to ℂ ⁢ [ Z , Z - 1 ] , where Z, chosen up to a scalar, is supported on the single coset ϖ E ⁢ J ~ Λ (note that this coset is independent of the choice of ϖ E ). For G , since I G ⁢ ( λ ) = J Λ , we have ℋ ⁢ ( G , λ ) ≅ ℂ . Therefore, ℋ ⁢ ( M , λ ~ ⊠ λ ) ≅ ℂ ⁢ [ Z , Z - 1 ] .

We now fix w to be either one of { y , z } and abbreviate κ P = κ P w and λ P = κ P ⊗ ρ M , such that λ P | J M = λ ~ ⊠ λ . Consider the Hecke algebra ℋ ⁢ ( G W , λ P ) . If λ ~ is not self-dual, then

The interesting case happens when λ ~ is self-dual, in which case it is known (see [50, Corollary 6.16] and [12, Proposition 3.3]) that

The Hecke algebra ℋ ⁢ ( G W , λ P ) has two invertible generators T w , for w ∈ { y , z } . To describe them precisely, we choose two elements s y and s z (for example, we may choose s 1 and s 1 ϖ in [50] or [12]), each of which is a generator for the normalizer group N G W ⁢ ( [ M , π ~ ⊠ π ] ) mod M such that