Base change for ramified unitary groups: the strongly ramified case

We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.


Introduction
The local Langlands correspondence for a general linear group over a non-Archimedean local field F is, roughly speaking, a parametrization of its irreducible admissible representations in terms of representations of the Weil-Deligne group of F .An extensive study of this correspondence involves certain important invariants, for example L-and epsilon-factors (see [HT01,Hen00] for the characteristic 0 case, [LRS93] for the positive characteristic case, and [Sch13] for a recent proof).Using moreover the processes of automorphic induction and base change [HH95,AC89], one obtains an explicit description of the correspondence in the essentially tame case [BH05a,BH05b,BH10].
After [Rog90,Moeg07,Art13,Mok15,KMSW], we know that the local Langlands correspondence for classical groups is closely related to that for general linear groups, at least for discrete series representations: their L-packets are parametrized by multiplicity-free (conjugate-)self-dual representations of the Weil-Deligne group.Therefore, in principle, we can understand the correspondence for classical groups via general linear groups.For unitary groups, this theory is explained in the context of base change in [Moeg07] (for special orthogonal groups and symplectic groups, the theory is sometimes called local transfer [ACS16]).More precisely, the Langlands parameter of an L-packet of a discrete series of a unitary group is the same as that of its base change, a representation of a general linear group.Hence describing Langlands parameters for L-packets of discrete series is equivalent to describing their base changes.This approach was adopted in [AL05, AL10,Bla10,Bla08] for supercuspidal representations of small unitary groups.
In our present paper, we describe a special case of base change for ramified unitary groups, which complements the previous method in [Tam18] for describing the local Langlands correspondence for packets of supercuspidal representations of unramified quasi-split unitary groups.In the previous case, we determined first the inertial class of the base change using the method developed in [BHS18] (for symplectic groups), and then the exact base change by using Asai L-functions [Sha90,Gol94].In the following paragraphs, we will first describe our new results and the methodology for ramified unitary groups, and then explain why the previous method fails.
Let F • be a non-Archimedean local field of residual characteristic p where p is odd, F/F • be a quadratic extension, V be a vector space over F , and G = GL F (V ).Suppose that V is equipped with a Hermitian form defined with respect to F/F • , and let G be the corresponding unitary group, the fixed-point subgroup of the Hermitian involution σ of G. Take a supercuspidal representation π of G, compactly induced from a cuspidal type containing a simple character, or in other words, such that its underlying self-dual semi-simple stratum s is indeed simple (see [BK93,Ste05,MS14] for the related definitions).Suppose that E is the field datum of the stratum, which is invariant under the Galois involution defined by σ, with fixed-point subfield E • .In our paper, we require that [E : F ] = dim F V and E/E • is quadratic ramified.
We call this the strongly ramified case.The latter condition actually forces F/F • to be also quadratic ramified.For the final computation, we additionally assume that E/F is tamely ramified.
We refer the full detail of constructing supercuspidal representations by cuspidal types to [BK93,Ste08] and only specify that, under the above conditions, a cuspidal type is indeed a character (a representation of degree 1).Suppose that the cuspidal type of a supercuspidal representation π of G is of the form ρκ 0 , where κ 0 is the p-primary beta-extension of a simple character θ, in the sense of [BHS18], and ρ is a level zero character, which is a character of {±1} in our situation.We now take θ the self-dual simple character for G whose restriction to the σ-fixed-point subgroup is the square of θ (see (2.2)), and κ0 its unique self-dual p-primary beta-extension (see Proposition 3.2).We also take ρ a self-dual level zero character of E × to form a cuspidal type ρκ 0 and compactly induce it to a supercuspidal π of G.
The following theorem is the main result of our present paper, giving the conditions for π to be the base change of π, which is the only member in its L-packet in our case.Recall that we have assumed the 'strongly ramified' condition and also that E/F is tamely ramified.
Theorem 1.1.Under the above assumptions, suppose that the simple characters θ and θ are related as above (or see (2.2)), and the level zero characters ρ and ρ are related by and ρ(̟ E ) = ρ(−1)ǫ P z (̟ E , s), where • µ E is the subgroup of E × of roots of unity with order coprime to p, and is the quadratic character of µ E ; • ̟ E is a chosen uniformizer of E, and ǫ P z (̟ E , s) is a sign attached to a quadratic Gauss sum (see (4.21)), defined by ̟ E and the simple stratum s associated to θ.
Then π is the base change of π.
For example, when dim V = 1, then E = F .The sign ǫ P z (̟ E , s) is 1, and the relation in Theorem 1.1 becomes ρ(x) = ρ(x σ x) for all x ∈ F × , which is exactly the base change for characters of U 1 .The main idea here is that: when dim V > 1, we have to modify our base change formula for the level zero component by taking the internal structure of the simple character into account.This idea is also exhibited in the description of local Langlands correspondences for general linear groups in the essentially tame case [BH05a, BH05b, BH10], for quasi-split unramified unitary groups [Tam18], and for symplectic groups [BHS18].In all these cases, the modifications are incarnated by certain characters (called rectifiers in the first case and amending characters in the second case) of elliptic tori defined by the field data of simple characters.
We now describe the methodology for the theorem.The first step, following a series of papers of the first author [BB02,Blo07,Blo12], is to study the reducibility points of the parabolic induction where W is the Hermitian space W = V − ⊕ V ⊕ V + , with each V − and V + just V as a vector space and V − ⊕ V + hyperbolic, and so the unitary group G W contains a parabolic subgroup P with Levi component M = G × G.By [Sil80], when this parabolic induction is reducible at a point s ∈ R ≥0 , then π is conjugate-self-dual, in which case we can use [Moeg07] to show that indeed in our case, s = 1, and π is the base change of π.
We can apply the theory of covering types [BK98] to get a preliminary information about the reducibility: using the type λ P in G W covering λ ⊠ λ the product of a pair of types λ and λ for π and π respectively, together with the categorial equivalence between the Bernstein component of the inertial class of π ⊠ π in G W and the module category of the Hecke algebra H(G W , λ P ), we can single out two candidates for π of self-dual representations in the inertial class of the base change of π, the two differing from each other by an unramified character of a finite order, such that π| det The second step is to further study the structure of the Hecke algebra H(G W , λ P ) as well as its modules.By [BK98,MS14], when λ is conjugate-self-dual, this Hecke algebra has two generators, denoted by T y and T z in this paper, respectively satisfying a quadratic relation of the same form where each ǫ w is a sign.By computing the eigenvalues of these two generators, or equivalently the coefficients b w , on the module corresponding to π| det | s ⋊ π, we show that, when π| det | s ⋊ π is reducible at s = 1 and ρ| µ E is given as in Theorem 1.1, the signs ǫ y , ǫ z , and ǫ y ǫ z are respectively ρ(−1), ǫ P z (̟ E , s), and ρ(̟ E ) (modulo some irrelevant factors which cancel with each other in the comparison).The detailed version of this result can be derived from Theorem 3.4 and Corollary 3.5.
It turns out that the coefficient b y is easy to compute, while b z involves some calculations similar to [Kim01,Kimnt] for large p and [BB02] for Sp 4 , related to the structure of the simple stratum s.For simplicity of our paper, we further assume that E/F is tamely ramified.This is the condition assumed in [How77,MR99] and the series [BH05a, BH05b, BH10] (see also the warning in [BK93, (2.2.6)]), as well as in [Yu01] more generally.It also facilitates comparisons, by the second author in [Tam16a,Tam16b], between the essentially tame local Langlands correspondence for inner forms of general linear groups [BH05a,BH05b,BH10,BH11a] with the twisted endoscopy theory [LS87,KS99].
Finally, we show that π is indeed the base change of π using a finiteness result from Moeglin for the possibilities of π such that π| det | s ⋊ π is reducible for some s ∈ R ≥0 ([Moeg07, 4.Prop], [Moeg14, Th.3.2.1] for quasi-split groups, and Moeglin-Renard [MR18, 8.3.5] for non-quasi-split groups).The result is obtained from Arthur's endoscopic character relations [Art96] and their generalizations in twisted endoscopy [MW16, XI.], which require that char(F ) is 0. It is possible that we could apply an approach analogous to [BHS18, Th. 2.5] to compute the reducibilities of π| det | s ⋊ π for all π and obtain a finiteness result similar to Moeglin's without the characteristic requirement, but in our strongly ramified case we rather take the above shortcut using Moeglin to simplify the discussion.(See [GV17] for a result on the local Langlands correspondence in positive characteristic for split classical groups without using reducibilities of induced representations.) We now briefly explain why the previous method in [Tam18] fails in the strongly ramified case.According to [Moeg07], there is a notion of parity of a conjugate-self-dual supercuspidal representation of GL n , either conjugate-orthogonal (+) or conjugate-symplectic (−) but not both.In the nonstrongly ramified case, the two conjugate-self-dual candidates have opposite parities because they differ by an unramified character χ such that χ • N E/F is conjugate-symplectic.In this case we can determine the correct base change between the two by computing their parities using Asai L-functions for example [Hen10].However, in the strongly ramified case, χ • N E/F is then conjugate-orthogonal, and so the two conjugate-self-dual candidates have the same parity (see the appendix in Section 3.4 for a detailed discussion).This explains why the previous method no longer works, and we have to rely on the complete structure of the modules over the Hecke algebra.

Acknowledgements
To be added.

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
The Galois group Gal(F/F • ) is generated by an involutive automorphism c.
If Λ : Z → S is a sequence into a set S, 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.

Unitary groups
Let V be an F -vector space.We denote Ã = ÃV = End F (V ) and G = GV = Aut F (V ).Suppose that V is equipped with a non-degenerate (F/F • , ǫ)-Hermitian form h = h V , where ǫ = ǫ V = ±1.If X → X is the conjugate-adjoint on Ã defined by h, we define the adjoint anti-involution α X = − X.Note that α (XY ) = − α Y α X, X, Y ∈ Ã.
We also have the corresponding involution σ : X → X−1 on G.The subgroup G = Gσ is a (connected) unitary group that we consider throughout the paper and whose Lie algebra is A = Ãα .
Given an o F -lattice L in V , we denote by we always normalize a self-dual lattice sequence such that d = 1 and its o F -period e(Λ/o F ) is even. (2.1) Hence ÃΛ := P0 Λ is a hereditary order in Ã, with Jacobson radical PΛ := P1 Λ .We denote by v Λ the valuation on Ã associated to Λ.If Λ is self-dual, then each Pk Λ is α-invariant, in which case we put We also define ŨΛ = Ũ 0 Suppose that Λ is a self-dual lattice sequence of the form then U Λ is a maximal compact subgroup, and U 0 Λ is the underlying maximal parahoric subgroup.When F/F • is unramified, U 0 Λ 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 a symplectic group and a special orthogonal group, both defined over k F = k F• and can be possibly trivial.

Cuspidal types
We recall from [BK93,Ste08] 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 k F ).We denote by ÃE and GE respectively the centralizer of β in Ã and G, and for k ∈ Z, denote Pk As in [BK93, Chapter 3] associated to s we construct • subrings H = HΛ,β ⊆ J = JΛ,β of Ã and the two-sided fractional ideals Hk = H ∩ Pk Λ and Jk = J ∩ Pk Λ , for all k ∈ Z; • C(s) := C(Λ, 0, β) the set of simple characters of H1 ; • associated to each simple character θ ∈ C(s) the Heisenberg representation η = ηθ of J1 , an irreducible representation that restricts to a multiple of θ on H1 ; • a beta-extension κ of η to the subgroup J = JΛ,β = ŨΛ,E J1 (note that among these extensions there is a unique one κ0 whose determinant has a p-power order, called the p-primary betaextension); 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 J/ J1 ∼ = GL n/[E:F ] (k E ).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 G J 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 H1 = J1 = Ũ 1 Λ .We take κ of ŨΛ to be trivial, and choose ρ to be inflated from a cuspidal representation of ŨΛ / Ũ 1 Λ ∼ = 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 G J 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 H1 , J1 , 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 κ0 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 ÃE and Pk Λ,E , resp.GE , ŨΛ,E , and Ũ k Λ,E , are invariant under α, resp.σ, we define When s is skew and semisimple, following [Ste08], we construct • 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) which turns out to be bijective since the restriction operation satisfies the properties of Glaubermann correspondence (see [Ste08,Sec 5.3] for details); • 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 ; • 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 a p-power order, 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 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 G J λ.If s is null, then by definition C(s) contains only the trivial character of U 1 Λ .We again take κ to be trivial, and so π = cInd G J ρ is a level zero supercuspidal representation.Remark 2.1.G E/E• has a compact center except precisely when it has a factor isomorphic to the split SO 2 , (which is just a GL 1 ), which does not happen in our case.Also, for the parahoric subgroup U 0 Λ,E/E• to be maximal, it is not enough to just assume that the order P 0 Λ,E is maximal, since there exists such a maximal order whose corresponding parahoric subgroup is not maximal.See [MS14,Appendix] for details.

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 ǫ-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 The ǫ-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 ÃW = 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 GZ− × 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, σ g) for g ∈ G, h ∈ G, and abbreviate a matrix of the form We will proceed to construct covering types for G W .To simplify the discussions in this paper, as well as to focus on the case we are interested in, let's assume the following from now on.
• V = Z + , so that we can identify Z − with Z + by the Hermitian form h = h V . • 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 Given a skew simple or null stratum s = [Λ, r, 0, β], we hence have corresponding skew semi-simple 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 ÃW as the block-diagonal matrix with diagonal blocks (β, β, β).We now follow [BK93,Ste08] to define, for L = m or M w , with w ∈ {y, z}, L , such that they satisfy a transfer relation with each other, as in [BK93, (3.6)], [Ste05, Sec 3.5]; • beta-extensions κ M w of θ M w on J M w , and κ w m of θ m on J m , such that κ M w and κ w m are compatible in the sense of [Ste08, Lemma 4.3]; • unique p-primary beta extension (J M w , κ M w ,0 ), and the one (J m , κ w m,0 ) compatible with κ M w ,0 (note that κ w m,0 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 1 m , J 1 m , 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 [Ste08, Cor 5.11 and Prop 5.5] that H 1 M := H1 × H 1 and where θΛ+ and θ Λ are simple characters related under the correspondence (2.2), i.e., θ Λ = ( θΛ+ To construct a covering type, we define • the following subgroups and also denote J ± P = J P ∩ U ± ; • a simple character θ P on H 1 P by extending θ m trivially to J 1 m ∩ U , and the unique representation η P on J 1 P containing θ P (by [Ste08, Lemma 5.12]); • for w ∈ {y, z}, an extension κ w P of η P to J 1 P that is the restriction of κ w m to the subspace of (J 1 m ∩ U )-fixed vectors of η m (which can be also characterized by the relation κ w m ∼ = Ind Jm JP κ w P ).
With our construction of m, the decomposition W = V ⊥ (Z − ⊕ Z + ) is moreover exactly subordinate to s m .By [Ste08, Prop 6.3], if we write κ w P |J M = κw ⊠ κ w , then κw and κ w are beta-extensions of θ and θ respectively, and κw is self-dual (with respect to σ) [Ste08, Cor 6.10].
We then choose a (product of) cuspidal representation ρ M = ρ ⊠ ρ of J P , inflated from J P /J 1 P ∼ = J M /J 1 M .(At this moment, there is no relation between ρ and ρ.) Define which is a covering type over λ w M := λ w P | JM by [MS14].Finally, we denote by π w M = πw ⊠ π w a supercuspidal representation of M = GZ × G V containing the maximal type λ w M = λw ⊠ λ w , where λw = κw ⊗ ρ and λ w = κ w ⊗ ρ.

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 [BK98].
• 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 G P τ := Ind G P (τ ∆ 1/2 P ), where τ ranges over representations in s and ∆ P is the modulus character of P .
• 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 with an associative C-algebra structure under the convolution The support of every element in H(G, λ) lies in the intertwining set We call (K, λ) an s-type if, for an irreducible representation τ of G, τ | K contains λ ⇔ the inertial class of the cuspidal support of τ is s, 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 w P , for w ∈ {y, z}).In particular, we have

we have an equivalence of categories
τ → Hom JP (λ P , τ ); • an injective morphism of algebras [BK98, (8.3, 8.4)] giving rise to the natural functor such that the following diagram commutes.
We will be interested in when the parabolic induction is reducible.By [Sil80], when this happens, π must be self-dual, and there is a unique real s π,π ≥ 0 such that (2.5) is reducible at s = ±s π,π .A simple argument shows that there are exactly two selfdual representations in the inertial class of π giving this reducibility.Say one of those is π and the other π′ , a twist of π by an unramified character, and denote by f π = f π′ the order of the stabilizer of π in the group of unramified characters of G.The points of reducibility for π are of the form and those for π′ takes 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.

Structures of Hecke algebras
We continue from the constructions in Subsection 2.4, and briefly recall the structure of the above Hecke algebras, referring the detail to [BK98,Ste08,MS14].
where Z, chosen up to a scalar, is supported on a single coset ̟ E JΛ , where ̟ E is a uniformizer of E (note that this coset is independent of the choice of We now fix w to be either one of {y, z} and abbreviate κ P = κ w P s and λ P = κ P ⊗ ρ M , such that λ P | JM = 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 that [Ste08,Cor 6.16 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 in [Ste08] or [Blo12]), each of which is a generator for the normalizer group N GW ([M, π ⊠ π]) mod M such that s y J − P s y ⊂ J + P and s z J + P s −1 z ⊂ J − P , and moreover ζ := s y s z = i M (̟ E I, I) which is a P -positive element in the sense that (If we have chosen P − instead of P to define our covering type, then we have to switch y and z.)Each of the two generators T w , for w ∈ {y, z}, has support on a single double coset J P s w J P and is defined up to a scalar.They satisfy and certain quadratic relations for certain real numbers b w and c w .Here ½ is the unit in H(G W , λ P ), which is the function supported on J P with ½(1) = I λP , the identity operator on the representation space of λ P .
For direct formulas for the coefficients, we follow [BB02, Sec 1] and obtain and similarly (2.10) Later in (2.16) we will choose s w such that s 2 w ∈ J M , so that we may normalize each T w , up to a sign, such that which is equivalent to requiring that and so that c y = [J + P : s y J − P s y ] and c z = [s z J − P s −1 z : J + P ] are both positive.
Indeed, by [Ste08, Section 7.1], each T w comes from the generator of H(U M w ,E , ρ ⊠ ρ) which is then reduced to a Hecke algebra for the finite reductive quotient (not necessarily connected) By Lusztig, under suitable normalizations, the quadratic relations can be written as (2.12) for certain integers r w ≥ 0. The values of r w can be determined by the method in [LS16, BHS18] combined with Lusztig's classification [Lus77].In our present situation, we can also compute these values by comparing the coefficients, or more precisely the eigenvalues, between (2.12) and (2.8), which will be done in Section 4.

Modules over Hecke algebras
It is worthwhile to mention a result, in (2.18) below, from [Blo12] about the real parts of the points of reducibility, i.e., the values of s 1 and s 2 in (2.6), which are enough to determine the inertial Jordan blocks (c.f.[BHS18]) of a supercuspidal π of G with type λ.The result will be recalled after some prerequisites on modules over Hecke algebras.
Suppose that π = cInd G J λ, where λ extends λ of J.We abbreviate H G = H( G, λ) and recall the equivalence of categories where the H G-action on τ is given by (Here V λ is the representation space of λ, and recall that we fixed the Haar measure on G such that the measure of JΛ is 1.)In particular, we have (2.13) The group of unramified characters of G acts on R [ G,π] ( G) by and on which induces an action on Mod-H G naturally.The equivalence map M G is hence equivariant under this action.
We now turn to our constructed type λ P in G W and abbreviate H M = H(M, λ P | JM ) and H GW = H(G W , λ P ).For s ∈ C, we denote by D s the simple right H M -module necessarily 1-dimensional.We then denote by X s be the right H G -module where the last isomorphism is given by the commutative diagram (2.4).
Suppose that we choose t P normalized by the relation where ∆ P is the modular character of P and ζ = s y s z = i M (̟ E I, I) as in the previous subsection, then by combining (2.13), (2.14), and (2.7) (evaluating at ̟ E ), the action of t P (Z) on X s is scalar, given by (2.15) We now impose an assumption that • the operators λ(̟ E ), T y (s y ), and T z (s z ) on V λ have finite orders.
This assumption can be easily satisfied when λ is self-dual and s y and s z are chosen to be simple enough; for example, we can and do pick which by (2.11) means that T y (s y ) 2 = 1 and T z (s z ) 2 = ρ(−1).We also use the fact We now compare the two quadratic relations (2.8) and (2.12), and obtain the value of b w , for w ∈ {y, z}, b w = ±c 1/2 w (q Hence the eigenvalues of T w are ±{c 1/2 w q rw/2 E , −c 1/2 w q −rw/2 E } and the possible products of eigenvalues of T y and those of T z are (2.17) When π| det | s ⋊ π, and hence X s , is reducible, the eigenvalue of Z is a product of one of those of T y and one of those of T z .By comparing the absolute values of (2.15) and (2.17), we obtain (2.18) In Section 4, we will provide more detail about the products of the eigenvalues, and obtain the conditions on the cuspidal types for which reducibilities happen.

Moeglin's results
In this subsection, we provide two results due to Moeglin, for which we require the characteristic of F to be 0. Given a supercuspidal representation π of a unitary group G = G V , then where the sum π ranges over all supercuspidal representations π, each of which is a supercuspidal representation of a general linear group GVπ for some space V π , such that π| det | s ⋊ π is reducible at s π ∈ 1 2 Z with s π ≥ 1.In particular, in the extreme case (which is studied in this paper), if there exists such a π with dim V π = dim V , then it is the unique such representation that gives rise to the reducibility above with s π = 1, in which case π is the base change of π.
We now switch to the Galois side and look at Langlands parameters.We recall that c is the generator of Gal(F/F • ).We call an (semi-simple, smooth) irreducible representation φ of for all w ∈ W F , and φ∨ is the contragredient of φ.Hence an irreducible representation φ is conjugate-self-dual if there exists a non-degenerate bilinear form B on V := C deg φ such that, We also define a parity on φ: call φ conjugate-orthogonal (resp.conjugate-symplectic) if, furthermore, where sgn( φ) = 1 (resp.−1).The parity of φ, as an irreducible representation of W F , can also be described using Asai L-functions: if r A is the Asai representation of GL(V) [Moeg02, A.2.1], then φ is conjugate-orthogonal if and only if has a pole at s = 0.

Ramified unitary groups
We now specify F/F • to be a quadratic ramified extension, generated by a uniformizer ̟ F such that

The strongly ramified case
Given a simple or null stratum s = [Λ, r, 0, β], which is assumed to be skew, we assume the following condition on the field extension E/E • , where E • is the subfield of E fixed by c = −α.
E/E • is a quadratic ramified extension. (3.1) The reason of imposing this condition has been explained at the end of the introduction section.This condition implies that e = e(E/F ) is an odd integer, and if K is an intermediate subfield between E and F , and K • is the fixed-field of K by the involution −α, then K/K • is quadratic ramified.
The numbers {r j } d j=0 are the critical exponents of s.For the moment we do not require the definition, but the simplicity of [ Bj+1 , r j , r j − 1, δ j ] implies that v Bj+1 (δ j ) = −r j .Note that if s is skew, we can apply [Ste01, (1.10)] and do choose all s j to be also skew.If we also choose the tame corestriction s j to be α-equivariant, then we can also assume [ Bj+1 , r j , r j − 1, δ j ] to be skew.Proposition 3.1.When E/E • is ramified, all {r j } d j=1 are odd.
Proof.Let [ Bj+1 , r j , r j −1, δ j ] be the simple stratum equivalent to the derived stratum [ Bj+1 , r j , r j − 1, s j+1 (γ j −γ j+1 )], which is assumed to be skew; in particular, we have δ j = α δ j .Hence −α defines an involution on the field F [δ j ] whose restriction to F is c, the Galois conjugation of F/F • .We denote its fixed field by F [δ j ] • .By [BK93, (2.2.8)] and since e(E/F ) is odd, e(F [δ j ]/F ]) is also odd.This implies that ) is odd, and so is This proposition facilitates our calculations remarkably.First of all, we immediately see that J1 = H1 from their constructions [BK93, Sec.3.1].We also have J1 M y = H1 M y since M y is a dilation and shift of a lattice sequence equivalent to Λ, and as for J1 m = H1 m we just note that the critical exponents of (β, β, β) are {3r j } d j=1 .The Heisenberg representation containing a simple character θ is just θ itself, and so any beta-extension of θ is a character.In particular, the unique p-primary beta extension κ0 is trivial on µ E .Finally, we choose our covering type λ P = κ P ⊗ ρ M such that κ P = κ y P , since in this case κ P | JM = κ0 ⊠ κ 0 , i.e., it is p-primary.
If moreover θ is self-dual, which is assumed from now on, then the p-primary beta extension κ0 is a self-dual character.We fix a uniformizer ̟ E• of E • , and choose another one ̟ E for E satisfying Proposition 3.2.Suppose that θ is a self-dual simple character, and let κ0 be its p-primary betaextension.There is a unique representation (a character) κ0 of J characterized by the following conditions: Moreover, such an extension κ0 is self-dual, i.e., σ κ0 = κ0 .
Proof.Clearly there is a unique representation of F × J satisfying (i) and (ii).Since ̟ e E ∈ F × U E , the value of κ0 (̟ E ) e is known, and so is κ0 (̟ E ) because of (iii) and that e is odd.The last statement is also clear since We also take a self-dual level-zero type ρ of J = E × J, i.e., it is an extension of a representation ρ of J inflated from an irreducible cuspidal representation J/ J1 ∼ = GL f0 (k E ), where such that ρ is self-dual if and only if ξρ is, in which case it means that either f 0 = 1 or else, by [MR99, Lemmas 2.1 and 5.1], the involution −α on E extends to E f0 which is unramified over the the fixed field E f0 • = (E f0 ) −α , and in particular f 0 is even.
We postpone the discussion of the case f 0 > 1 to the appendix 3.4 of this section and proceed to our main results under the condition f 0 = 1.The reason of imposing this condition is accounted for in Proposition 3.12: if f 0 > 1, the two candidates for the base change do not have the same parity, therefore the decision between them can be made by computing Asai L-functions, as was briefly explained in the last paragraph of the introduction.
So we look at the case when so that both κ0 and ρ = ξρ are characters.In this case, κ0 | µ E is trivial and κ0 (̟ E ) 2 = 1.A self-dual extended maximal simple type λ is a twist of κ0 by a self-dual character ρ of E × trivial on U 1 E , which implies that ρ, defined on µ E , is at most quadratic and ρ(̟ E ) 2 = ρ(−1), so that ρ(̟ E ) is a 4-th root of unity.We summarize the above conditions in the following terminology.Definition 3.3.We call the extended type λ for G, as well as its induced supercuspidal representation π, strongly ramified when both (3.1) and (3.2) are satisfied, and similarly for λ and π for G.
We now recall from [Lus77, Sec.5] (or see [LS16,Sec. 7.6]) the values of the integers r w , with w ∈ {y, z}, that appear in the quadratic relations in (2.12) at the end of subsection 2.5.From our construction we have and also From (2.18) the positive real parts of the points of reducibility are {0, 1} if ρ is trivial, and {1/2, 1/2} otherwise.
Hence if we view each generator T w as an endomorphism of the module X s corresponding to the parabolic induction π| det | s ⋊ π, then the above results, combining with the quadratic relation (2.8), imply that where ǫ w is a sign.We will provide the precise value for the sign ǫ w in the next subsection, with calculation given in Section 4.

Reducibility results
To state the main result on the eigenvalues for the quadratic relations, we impose the following assumptions which are used in Section 4.
• We only consider a strongly ramified supercuspidal representation π of G, i.e., if (θ, ρ) is a pair consisting of a simple character and a level zero cuspidal representation defining π, then that E/E • is ramified and dim V /[E : F ] = 1.
• The extension E/F is tamely ramified, which allows us to assume that ] for all j, forming a tower of intermediate extensions between E and F .
We also take a supercuspidal representation π of G constructed by ( θ, ρ) satisfying the same conditions as π above, with θ = ( θ| H 1 ) 1/2 .The following theorem will be proven in Section 4: Theorem 3.4.Suppose that π and π satisfy the above conditions.
(ii) The coefficient b z of the quadratic relation of T z is given as follows.
(a) When where ǫ P z (̟ E , s) is a sign, associated to a quadratic Gauss sum and determined by the choice of ̟ E and the simple stratum s.
For the purpose of base change, the case when ρ| , the two eigenvalues of T z are We now choose π = π( θ, ρ) with Corollary 3.6.The points of reducibility of π| det | s ⋊ π are ±1 and πi log q E .
Proof.Since the eigenvalue of t P (Z) is the product of eigenvalues of T y and T z , the comparison in The corollary follows by solving s.
The following remark explains Corollaries 3.5 and 3.6 are independent of the various choices made throughout the progress.
Remark 3.8.Note that both T w (s w ), for w ∈ {y, z}, have been cancelled out in (3.4).Indeed the reducibility points, as well as the base change result in Theorem 3.9 below, are independent of the normalizations of T w (and also that of Z) and the choices of s w .Moreover, as explained in Subsection 2.5, if we have chosen P − instead of P , then we need to switch the roles of y and z.In this case, ǫ P z (̟ E , s) should then be denoted by ǫ P − y (̟ E , s).Our corollaries are clearly independent of choosing P or P − .

The main result for base change
Let's first look at a simple case when the representations are characters, in which case Note that the restriction ρ| µ F has no relation with the level-zero part of ρ, i.e., the character of the finite reductive quotient of U 1 , which is O 1 = {±1}.However, if we change ρ by twisting a quadratic unramified character, then ρ(−1) is changed to another sign.
Theorem 3.9.Suppose that char(F ) = 0, and the simple characters θ and θ and the tamely ramified characters ρ and ρ are related as follows.
Then π is the base change of π.
This can be easily deduced from the reducibility result in the previous subsection, together with the finiteness result of Moeglin (2.19).One may notice that (3.5) is a special case of the theorem.
Remark 3.10.[Moeg07, 7.1] implies that if in general a discrete series parameter, when viewed as a representation of the Weil-Deligne group, has k irreducible components, then its corresponding L-packet contains 2 k−1 members.In our case when the base change representation is supercuspidal, then its parameter is an irreducible representation, and the L-packet is a singleton.

Appendix: the strongly ramified case and parity
This appendix is a sequel to subsection 3.1, and does not intervene with our main results.
We provide a result on the parity of a conjugate self-dual supercuspidal representation.To this end, we have to switch to the Galois side via the local Langlands correspondence for GL n , and assume that char(F ) = 0. We call a conjugate self-dual supercuspidal representation π of a general linear group conjugate-orthogonal (resp.conjugate-symplectic) if the Asai L-function [Sha90] for π, has a pole at s = 0.By [Hen10], we know that π is conjugate-orthogonal (resp.conjugate-symplectic) if its Langlands parameter is so (see 2.20) and (2.21)).
Proposition 3.12.When F/F • is ramified, a conjugate self-dual supercuspidal representation π and its twist π′ := π| det | πi/f0 log qE are of the same parity if and only if π is strongly ramified.
Proof.Let T /F be the maximal tamely ramified subextension of E/F , and T f0 be the unramified extension of T of degree f 0 with fixed field • is ramified if and only if π is strongly ramified.From [BH11b, Ch.1], suppose the Langlands parameter of π takes the form Ind T /F (α where α is an irreducible representation of W T , and ξ, as a character of T f0× , is viewed as a character of W T f 0 .The Langlands parameter of π′ is Since the parity of a representation is preserved under induction, the result then holds by the remark before the proposition, and the fact that |det T f 0 | πi/ log q T f 0 is conjugate-symplectic when f 0 is even.

The coefficients
The whole section is devoted to prove Theorem 3.4.It suffices to compute the values of b w for w ∈ {y, z}.We do not require that char(F ) = 0.
We begin by recalling the explicit form of J 1 P .First of all, we can express the rings as follows H = ÃΛ,E + P(r0/2)+ Λ,E1 By [Blo06, Prop.1], In the strongly ramified case, since all exponents r i = 2s i + 1, for i = 0, . . ., d, are odd, we have JΛ = HΛ .We can show that Note that we have chosen certain convenient normalizations of T w , for w ∈ {y, z}, and Z in the relation t P (Z) = T y * T z to simplify our calculations.Following [BB02], we may choose T w such that which can be easily shown to be equivalent to requiring that T w (s w ) 2 = λ P (s 2 w ).

Computation of b y
We use (2.9) to compute b y , so we compute T y ((X, Y ) − ) for (X, Y ) − ∈ s y J + P s y ∩ J P s y J P /J − P .We also recall (2.3) which we use repeatedly: We take ∈ s y J + P s y /J − P , then from (4.1) we have, Also, we write supp(T y ) = J P s y J P = J + P J M s y J + P , so that (X, Y ) − ∈ s y J + P s y ∩ J P s y J P can be written as Lemma 4.1.Each coset in (s y J + P s y ∩ J P s y J P )/J − P has a representative of the form (X, Y 0 (I + Y ′ )) − with Y 0 ∈ µ E and X ∈ (o E \ p E ) mod p E such that 2Y 0 ≡ −X 2 mod p E , and I + Y ′ ∈ J1 .
Proof.The coset space J0 / H1 containing X, when viewed as a k F -vector space, takes the form (o E /p E ) ⊕ ( J1 / H1 ).If r j = 2s j + 1 are odd, then J1 = H1 , and so we can choose X ∈ o E mod p E .If furthermore (X, Y ) − ∈ J P s y J P , then Y is forced to belong to J by (4.3), which allows us to choose Y 0 ∈ µ E .The relation (4.2) implies that 2Y 0 ≡ −X 2 mod p E , so that X / ∈ p E , and Y 0 is uniquely determined by X. Hence the lemma follows.
The following lemma shows that it is indeed a constant.
Proof.Since now X ∈ J is invertible, we have and so On the other hand, that 2Y 0 ≡ −X 2 mod p E implies that Since θ and θ are related by (2.2), we have and the lemma follows.

Computation of b z
We use (2.10) to compute b z , so we compute T z ((X, Y ) + ) for (X, Y ) + ∈ s −1 z J − P s z ∩ J P s z J P /J + P .We take We also write supp(T z ) = J P s z J P = J − P J M s z J − P , so that (X, Y ) + ∈ J P s z J P ∩ s z J − P s z can be written as If we write Y = y̟ −1 E (I + Y ′ ) with y ∈ o E mod p E and I + Y ′ ∈ J1 = H1 , then from (4.4) we can assume that y ∈ µ E .We hence obtain (4.5) In the following subsections, we will expand and simplify θ(I + Y ′ ) and θ(I − α XY −1 X).
Since θ and θ are related by (2.2), we can and do assume similar relations between θj and θ j , and also between ξj and ξ j , i.e., In the following subsections, we call the factors of θ involving ξj the multiplicative parts of θ, and those involving ψ cj the additive parts, and similarly for θ.

Cancellation of multiplicative parts
We recall that we have written Y = y̟ −1 E (I + Y ′ ), with y ∈ µ E and I + Y ′ ∈ H1 .We first compare the multiplicative parts of θ(I + Y ′ ) and θ(I − α XY −1 X) by rewriting and so we are actually comparing θ(I − α XY −1 X) and θ(I − X α XY −1 ).
Starting from the proposition below, we have to assume that E/F is tamely ramified.
This condition allows us to assume that each field E j = F [γ j ], is contained in E j−1 for all j, forming a tower of intermediate extensions between E and F .
Proof.Let's temporarily write W = α XY −1 .We have to compare θ(I − W X) and θ(I − XW ).Let's first expand X = X 0 + • • • + X d+1 , and similarly for W .For each i = 0, . . ., d + 1, we write X ≤i = X 0 + • • • + X i , and similarly for W ≤i .The part of the character θ(I − W X) involving ξ i is and similarly for the part of θ(I − X α XY −1 ) involving ξ i , with W and X exchanged.Due to the identity det (I − W X) = det (I − XW ) these two parts are the same.

Choosing representatives
In this subsection, we will expand X and Y = y̟ −1 E (I + Y ′ ) such that the additive parts of θ(I + Y ′ ) and of θ(I − α XY −1 X) admit many simplifications.

The coset spaces
H1 respectively containing X and Y , when viewed as k F -vector spaces, take the form ).Since each r j = 2s j + 1 is odd, the summand W ′ z,j is trivial.We expand X = d+1 j=0 X j and Y ′ = d+1 j=0 Y ′ j accordingly, first requiring that With these fixed, we choose auxiliary data such that X α X = Y − α Y still holds.Eventually our main results are independent of these auxiliary choices, see (4.19) for example.
We require some notations.Let E i = F [γ i ] and Bi the centralizer of γ i in Ã.For i ≥ 1, we denote by B⊥ i−1 the orthogonal complement of Bi−1 in Bi relative to the non-degenerate symmetric form (X, Y ) → tr Bi/Ei (XY ).We also write 1 − α : Ã → Ã for the map x → x − α x, whose image is denoted by Ã−α .
Lemma 4.4.Each coset in (s −1 z J − P s z ∩ J P s z J P )/J + P , has a representative (X, y̟ −1 for j ≥ 1, with a decomposition (Note that P 0 = 0.) Proof.We can choose X j , for j ≥ 1, as stated using the commutative diagram We see that the top row is an α-equivariant isomorphism of E j -spaces, which induces at the bottom row an α-equivariant isomorphism Pk Λ,Ej ∩ B⊥ j−1 ∼ = Pk Λ,Ej / Pk Λ,Ej−1 of o Ej -lattices for all k ∈ Z.For choosing P j , we notice that the left hand side of (4.12) lies in B⊥ j−1 by (4.13) as well as in the (−α)-fixed point subspaces B−α j := Bj ∩ Ã−α , and the restriction and summing it up for all j yields X α X = Y − α Y .
Here is a very simple consequence which will be frequently used later on: suppose that T is either X j or P j , for j ≥ 1, as in Lemma 4.4, and U is a product of elements in B i for i < j, then tr Bj /Ej (T U ) = 0. (4.13)

Simplifying the additive parts
We will simplify the additive parts of θ(I + Y ′ ) and θ(I − α XY −1 X).Since each factor of the additive parts is a value of the character ψ • tr A/F , we will show that some of the inputs either lie in PΛ or have trace 0 by (4.13), so that their character values are 1.
Lemma 4.5.The additive part of Proof.In our calculation below, we have to switch between the additive expansion of Y ′ ∈ H1 Λ given by Y ′ = Note that each factor above can be rewritten as This expression admits a lot of simplification by writing Y ′ i = P i + Q i as in Lemma 4.4.First of all, we have ψ • tr A/F ( c j 2 (I + Y * ) 0 i−1 P i ) = 1 since each summand has trace 0 by (4.13), and similarly since cj 2 Q i ∈ ÃΛ and all other summands (involving Y * ) lie in PΛ .By the same reason, this term is non-trivial only when i = j + 1.We have similarly Therefore, (4.15) is equal to and the lemma follows using (4.11).
Lemma 4.6.The additive part of θ(I − α XY −1 X) is Proof.If we expand I − α XY −1 X = (I + W * ) d+1 0 using (4.7) for some W k ∈ Ps k−1 +1 Λ,E k , then the additive part of θ(I − α XY −1 X) can be written as We can expand I − α XY −1 X as so that we can write I − α XY −1 X = I + d+1 k=0 W ′ k according to the additive expansion of H 1 Λ , where We hence have We now fix k and simplify a sub-product from (4.17The indices for this summand are (i, j, S) such that max{i, j, S} = l > k.In the following cases, this (i, j, S)-summand has zero trace.
(i) If max S < l and i = j, then since one of i and j is l, the summand has trace 0 by (4.13).
(ii) If max S = l and exactly one of i and j is also l, then the summand lies in PΛ .
(iii) If both i, j < l, then max S = l.We write the summand as We then change Y l into (I + Y * ) 0 l−1 Y ′ l using (4.14) and decompose Y ′ l = P l + Q l .We then see that any summand involving P l has trace 0, and any of those involving Q l lies in PΛ .
(iv) The remaining case is i = j = l.The summand lies in PΛ except when l = k + 1 and S = ∅, which is − c k 2 α X k+1 y −1 ̟ E X k+1 .
Therefore, we obtain l>k and the lemma follows by multiplying the above equalities for all k together.By Lemmas 4.5 and 4.6, the additive part of θ(I + Y ′ )θ(I − α XY −1 X) is equal to Note that it is independent of the auxiliary X 0 , as expected from (4.9).
): k<l ψ c k /2 (I + W l ) by first expressing each of its factors asψ c k /2 (I + W l ) = ψ • tr A/F ( c k 2 (I + W * ) 0 l−1 W ′ l )and using (4.18) to further expand the input for ψ • tr A/F into summands of the form− c k 2 (I + W * ) 0 l−1 α X i ( Y * ) S y −1 ̟ E X j .

α
X i Y l ( Y * ) S−{l} y −1 ̟ E X j and further expand Y l = ∞ m=1 (−Y l ) m .Any summand involving (−Y l ) m with m ≥ 2 lies in PΛ , i.e., we remain to consider summands of the formc k 2 (I + W * ) 0 l−1 α X i Y l ( Y * ) S−{l} y −1 ̟ E X j