Galois self-dual cuspidal types and Asai local factors

Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $\sigma$ be its non-trivial automorphism. We show that any $\sigma$-self-dual cuspidal representation of ${\rm GL}_n(F)$ contains a $\sigma$-self-dual Bushnell--Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai $L$-function of a ${\rm GL}_n(F_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands--Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.


1.1
Let F/F o be a quadratic extension of locally compact non-archimedean fields of odd residual characteristic p and let σ denote the non-trivial element of Gal(F/F o ). Let G denote the general linear group GL n (F), make σ act on G componentwise and let G σ be the σ-fixed points subgroup GL n (F o ). In [17], using the Rankin-Selberg method, Flicker has associated Asai local factors to any generic irreducible (smooth, complex) representation of G. Let N denote the subgroup of upper triangular unipotent matrices in G, and ψ be a non-degenerate character of N trivial on N σ = N∩ G σ . Given a generic irreducible representation of G, let W(π, ψ) denote its Whittaker model with respect to the Whittaker datum (N, ψ), that is the unique subrepresentation of the smooth induced representation Ind G N (ψ) which is isomorphic to π. Given a function W ∈ W(π, ψ), a smooth compactly supported complex function Φ on F n o and a complex number s ∈ C, the associated local Asai integral is where |·| o is the normalized absolute value of F o and dg is a right invariant measure on N σ \G σ . This integral is convergent when the real part of s is large enough, and it is a rational function in q −s o , where q o is the cardinality of the residual field of F o . When one varies the functions W and Φ, these integrals generate a fractional ideal of C[q s o , q −s o ]. The Asai L-function L As (s, π) of π is defined as a generator, suitably normalized, of this fractional ideal. It does not depend on the choice of the non-degenerate character ψ.

1.2
Now consider a cuspidal (irreducible, smooth, complex) representation π of G, and suppose that its Asai L-function L As (s, π) is non-trivial. By [34], this happens if and only if π has a distinguished unramified twist, that is, an unramified twist carrying a non-zero G σ -invariant linear form. In this case, the Asai L-function can be described explicitly (see Proposition 7.5). We prove that it can be realized as a single Asai integral: Theorem 1.1 (Theorem 7.14 and Corollary 7.15). Let π be a cuspidal representation of G having a distinguished unramified twist. Then there is an explicit function W 0 ∈ W(π, ψ) such that This thus provides an integral formula for the Asai L-function of π. As an application of this theorem, we compute the associated root number: the Asai L-functions of π and its contragredient π ∨ are related by a functional equation (8.3), in which appears a local Asai epsilon factor ǫ As (s, π, ψ o , δ) depending on a non-trivial character ψ o of F o and a non-zero scalar δ ∈ F × such that tr F/Fo (δ) = 0. We prove the following theorem.
Our proof of this theorem is purely local and relies on Theorem 1.1.
Theorem 1.8 (Theorem 7.14). Let π be a distinguished cuspidal representation of G, and (J, λ) be a generic σ-self-dual type contained in π. Then there is a unique right invariant measure on N σ \G σ such that I As (s, Φ 0 , W λ ) = L As (s, π) where Φ 0 is the characteristic function of the lattice O n o in F n o .
Let us briefly explain how we prove this theorem. Following the method of [29], we compute the local Asai integral and get I As (s, Φ 0 , W λ ) = 1 where e o a positive integer attached to the generic σ-self-dual type (J, λ) in Paragraph 5.4. On the other hand, starting from Proposition 7.5 giving a formula for L As (s, π), and using the dichotomy theorem (Paragraph 7.1) together with Proposition 7.2, we find the same expression for L As (s, π).

1.6
We now explain how we prove Theorem 1.2. First, thanks to the functional equation (8.3) together with the fact that π is distinguished, the Asai root number ǫ As (1/2, π, ψ o , δ) must be equal to either 1 or −1. Then, by using the explicit integral expression for L As (s, π) provided by our test vectors, we prove that ǫ As (1/2, π, ψ o , δ) is positive.
where the product is taken over all places v of k o , where ǫ RS As (s, Π v , ψ o,v ) is defined by Definition 1.9 when v is inert, and is the Jacquet-Piatetski-Shapiro-Shalika epsilon factor ǫ RS As (s, π 1 , π 2 , ψ o,v ) when v is split and Π v identifies with π 1 ⊗ π 2 as representations of GL n (k v ) ≃ GL n (k o,v ) × GL n (k o,v ).
We define the global factor ǫ LS As (s, Π) similarly. Using the equality (that we prove when F has characteristic p in Appendix A) of the Flicker and Langlands-Shahidi Asai L-functions of Π v for all v, the comparison of the global functional equations gives: Theorem 1.10 (Theorem 9.26). Let Π be a cuspidal automorphic representation of GL n (A). Then ǫ RS As (s, Π) = ǫ LS As (s, Π).
Realizing any cuspidal representation of G as a local component of some cuspidal automorphic representation of GL n (A) with prescribed ramification at other places, and combining with Theorems 1.10 and 9.13, we get Theorem 1.3.

1.8
Finally, we must explain the interconnection between [45] and the present paper. The starting point of both papers is the σ-self-dual type theorem for cuspidal R-representations, namely Theorem 1.5, which is proved in Section 4 below.
Starting from this theorem, and independently from the rest of this paper, one gives in [45] a necessary and sufficient condition of distinction for σ-self-dual supercuspidal R-representations. In particular, for complex representations, in which case all cuspidal representations are supercuspidal, this implies the two results stated in Paragraph 7.1 (i.e. Theorem 7.1 and Proposition 7.2) which we use in the proof of Theorem 1.1. We also use Proposition 5.8, which is also proved in [45], for any σ-self-dual cuspidal R-representation of G.

Acknowledgements
We thank Raphaël Beuzart-Plessis for noticing and providing a correction to a gap in an argument of a previous version of Section 9, and for useful conversations.
We thank Jiandi Zou for noticing a small mistake in a previous version of the proof of Lemma 4.17.

Notation
Let F/F o be a quadratic extension of locally compact non-archimedean fields of residual characteristic p = 2. Write σ for the non-trivial F o -automorphism of F. Let R be an algebraically closed field of characteristic ℓ different from p; note that ℓ can be 0. We will say we are in the "modular case" when we consider the case where ℓ > 0.
We also denote by ω F/Fo the character of F × o whose kernel contains the subgroup of F/F o -norms and is non-trivial if ℓ = 2.
Let G denote the locally profinite group GL n (F), with n 1, equipped with the involution σ acting componentwise. Its σ-fixed points is the closed subgroup G σ = GL n (F o ). We will identify the centre of G with F × , and that of G σ with F × o .
By a representation of a locally profinite group, we mean a smooth representation on a R-module. Given a representation π of a closed subgroup H of G, we write π ∨ for the smooth contragredient of π and π σ for the representation π • σ of σ(H). We also write Ind G H (π) for the smooth induction of π to G, and ind G H (π) for the compact induction of π to G. If χ is a character of H, we write πχ for the representation g → χ(g)π(g).
A pair (K, π), consisting of an open subgroup K of G and a smooth irreducible representation π of K, is called σ-self-dual if K is σ-stable and π σ is isomorphic to π ∨ . When K = G, we will just talk about π being σ-self-dual.
Let χ be a character of F × o . A pair (K, π), consisting of an open subgroup K of G and an irreducible representation π of K, is called χ-distinguished if where det denotes the determinant on G and K σ = K ∩ G σ . We say that (K, π) is distinguished if it is 1-distinguished, that is, distinguished by the trivial character of F × o . When K = G, we will just talk about π being χ-distinguished.
Given g ∈ G and a subset X ⊆ G, we set X g = {g −1 xg | x ∈ X}. If f is a function on X, we write f g for the function x → f (gxg −1 ) on X g .
For any finite extension E of F o and any integer n 1, we write N n (E) for the subgroup of GL n (E) made of all upper triangular unipotent matrices and P n (E) for the standard mirabolic subgroup of all matrices in GL n (E) with final row 0 · · · 0 1 . Throughout the paper, by a cuspidal representation of G, we mean a cuspidal irreducible (smooth) representation of G.

Preliminaries on simple types
We recall the main results on simple strata, characters and types [12,8,11,37] that we will need.

Simple strata
Let [a, β] be a simple stratum in the F-algebra M n (F) of n × n matrices with entries in F for some n 1. Recall that a is a hereditary O-order in M n (F) and β is a matrix in M n (F) such that: (i) the F-algebra E = F[β] is a field, whose degree over F is denoted d; (ii) the multiplicative group E × normalizes a; plus an additional technical condition (see [12, (1.5.5)]). The centralizer of E in M n (F), denoted B, is an E-algebra isomorphic to M m (E), where n = md. The intersection a ∩ B, denoted b, is a hereditary O E -order in B. We write p a for the Jacobson radical of a and U 1 (a) for the compact open pro-p-subgroup 1 + p a of G = GL n (F). of a × and a finite set C(a, β) of characters of H 1 (a, β) called simple characters. This set depends on the choice of a character of F, trivial on p but not on O, which we assume to be σ-stable and is fixed from now on. Such a choice is possible since p = 2. Write J(a, β) for the compact mod centre subgroup of G generated by J(a, β) and the normalizer of b in B × . Proposition 3.1 ([11, 2.1]). We have the following properties: (i) The group J(a, β) is the unique maximal compact subgroup of J(a, β).
(ii) The group J 1 (a, β) is the unique maximal normal pro-p-subgroup of J(a, β).
(iii) The group J(a, β) is generated by J 1 (a, β) and b × , and we have: (iv) The normalizer of any simple character θ ∈ C(a, β) in G is equal to J(a, β).
Given a simple character θ ∈ C(a, β), the degree of E/F, its ramification order and its residue class degree only depend on the endo-class of θ. These integers are called the degree, ramification order and residue class degree of this endo-class. The field extension E/F is not uniquely determined, but its maximal tamely ramified sub-extension is uniquely determined, up to an F-isomorphism, by the endo-class of θ. This tamely ramified sub-extension is called the tame parameter field of the endo-class [11, 2.2, 2.4].
Let E(F) denote the set of all endo-classes of simple characters in all general linear groups over F. Given a finite tamely ramified extension T of F, there is a surjective map: with finite fibers, called the restriction map [11, 2.3]. Given Θ ∈ E(F), the endo-classes Ψ ∈ E(T) which restrict to Θ are called the T/F-lifts of Θ. If Θ has tame parameter field T, then Aut F (T) acts transitively and faithfully on the set of T/F-lifts of Θ [11, 2.3, 2.4].

Simple types and cuspidal representations
Let us write G = GL n (F) for some n 1. A family of pairs (J, λ) called simple types, made of a compact open subgroup J of G and an irreducible representation λ of J, has been constructed in [12] (see also [37] for the modular case).
By construction, given a simple type (J, λ) in G, there are a simple stratum [a, β] and a simple character θ ∈ C(a, β) such that J(a, β) = J and θ is contained in the restriction of λ to H 1 (a, β). Such a simple character is said to be attached to λ. Definition 3.2. When the hereditary order b = a ∩ B is a maximal order in B, we say that the simple stratum [a, β] and the simple characters in C(a, β) are maximal. A simple type with a maximal attached simple character is called a maximal simple type.
When the simple stratum [a, β] is maximal, and given a homomorphism B ≃ M m (E) of E-algebras identifying b with the standard maximal order, one has group isomorphisms: The following proposition gives a description of cuspidal (irreducible) representations of G in terms of maximal simple types.
Proposition 3.4. Let π be a cuspidal representation of G.
(i) There is a maximal simple type (J, λ) such that λ occurs as a subrepresentation of the restriction of π to J. This simple type is uniquely determined up to G-conjugacy.
(ii) The simple character θ attached to λ is uniquely determined up to G-conjugacy. Its endo-class Θ is called the endo-class of π.
(iii) If θ ′ ∈ C(a ′ , β ′ ) is a simple character in G, then the restriction of π to H 1 (a ′ , β ′ ) contains θ ′ if and only if θ ′ is maximal and has endo-class Θ, that is, if and only if θ, θ ′ are G-conjugate.
(iv) Let [a, β] be a maximal simple stratum such that J = J(a, β) and θ ∈ C(a, β). The simple type λ extends uniquely to a representation λ of the normalizer J = J(a, β) of θ in G such that the compact induction of λ to G is isomorphic to π.
A pair (J, λ) constructed in this way is called an extended maximal simple type in G. Compact induction induces a bijection between G-conjugacy classes of extended maximal simple types and isomorphism classes of cuspidal representations of G ( [12, 6.2] and [37,Theorem 3.11]).
4 The σ-self-dual type theorem We state our first main theorem. We fix an integer n 1 and write G = GL n (F).
Theorem 4.1. Let π be a cuspidal representation of G. Then π σ ≃ π ∨ if and only if π contains an extended maximal simple type (J, λ) such that J is σ-stable and λ σ ≃ λ ∨ .
In other words, a cuspidal representation of G is σ-self-dual if and only if it contains a σ-self-dual extended maximal simple type.
If (J, λ) is an extended maximal simple type for the cuspidal representation π, then (σ(J), λ σ ) is an extended maximal simple type for π σ and (J, λ ∨ ) is an extended maximal simple type for π ∨ . Thus, if π contains an extended maximal simple type (J, λ) such that J is σ-stable and λ σ , λ ∨ are isomorphic, then π σ , π ∨ are isomorphic. The rest of Section 4 is devoted to the proof of the converse statement.

The endo-class
Start with a cuspidal representation π of G, and suppose that π σ ≃ π ∨ . Let Θ be its endo-class over F. Associated with it, there are its degree d = deg(Θ) and its tame parameter field T: this is a tamely ramified finite extension of F, unique up to F-isomorphism (see §3.2).
Theorem 4.2. Let Θ ∈ E(F) be an endo-class of degree dividing n such that Θ σ is equal to Θ ∨ , and let θ ∈ C(a, β) be a simple character in G of endo-class Θ. There are a simple stratum [a ′ , β ′ ] and a simple character θ ′ ∈ C(a ′ , β ′ ) such that: Before proving Theorem 4.2, we show how it implies Theorem 4.1. By applying Theorem 4.2 to any simple character ϑ contained in π, which is maximal by Proposition 3.4(iii), we get a maximal simple character θ ∈ C(a, β), conjugate to ϑ, such that a is σ-stable and σ(β) = −β and: Thus θ is contained in π and its normalizer J in G is σ-stable. Let (J, λ) be an extended maximal simple type for π with attached simple character θ. Since π is σ-self-dual, it contains both (J, λ) and (J, λ ∨σ ). By Proposition 3.4, this implies that they are conjugate by an element g ∈ G, that is, g normalizes J and λ ∨σ is isomorphic to λ g . Now consider the simple characters θ −1 • σ = θ and θ g . Both of them are contained in λ g . Restricting λ g to the intersection: we get a direct sum of copies of θ containing the restriction of θ g to (4.3). It follows that g intertwines θ. By [12,Theorem 3.3.2], which describes the intertwining set of a simple character, we have g ∈ JB × J. We thus may assume that g ∈ B × . By uniqueness of the maximal compact subgroup in J, the identity J g = J gives us J g = J. Intersecting with B × gives b ×g = b × . It follows that g normalizes the order b. We thus have g ∈ J, thus λ σ ≃ λ ∨ . Theorem 4.1 is proved.
Remark 4.4. Assuming that Theorem 4.2 holds, and using Intertwining Implies Conjugacy [12,Theorem 5.7.1], the same argument shows that, if π is a σ-self-dual irreducible representation of G that contains a simple type, then π contains a σ-self-dual simple type.
Remark 4.5. However, an arbitrary σ-self-dual irreducible representation of G may not contain a σ-self-dual semisimple type. See [14,37] for the notion of semisimple type and Paragraph 4.9 for a counterexample.
It thus remains to prove Theorem 4.2. For this, one can forget about the representation π.

A prelude
We first show how to deal with the (second part of the) third condition of Theorem 4.2. Recall (see [12]) that a stratum [a, v, It is here that we use the fact that the character of F fixed in Paragraph 3.1 is σ-stable.
Lemma 4.8. Let Θ be a σ-self-dual endo-class and T/F be its tame parameter field.
Proof. The tame parameter field of Θ ∨ is T, and that of Θ σ is the field T endowed with the map x → σ(x) from F to T. The assumption on Θ implies that these tame parameter fields are F-isomorphic. Thus there exists an F o -automorphism of T whose restriction to F is σ.
Then Ψ ∨ is a T/F-lift of Θ ∨ , and the bijection α → Ψ α between automorphisms of T/F o and T/F o -lifts of Θ induces a bijection between F o -automorphisms of T extending σ and T/F-lifts of Θ σ . Thus there is a unique F o -automorphism α of T extending σ such that Ψ ∨ = Ψ α . Since Ψ ∨∨ = Ψ, we deduce that Ψ α 2 = Ψ. That α 2 is trivial follows from the fact that α 2 is in Aut F (T), which acts faithfully on the set of T/F-lifts of Θ.  Proof. The canonical homomorphism is an isomorphism if and only if F does not embed in T o as an F o -algebra. Assume that there is such an embedding. Since F is Galois over F o , its image is F.
Write t for the degree of T over F.
There is an embedding of F-algebras ι : T ֒→ M t (F) such that: for all x ∈ T. In particular, the image of ι in M t (F) is σ-stable.
. Then ι = ι o ⊗ F has the required property, thanks to Lemma 4.10.
Remark 4.12. The natural group homomorphism: (where the semi-direct product is defined with respect to α) is an isomorphism.

The maximal and totally wild case
In this paragraph, we will assume that d = n and T = F. Proposition 4.13. Let θ be a simple character in G with endo-class Θ. There is a simple character θ ′ ∈ C(a ′ , β ′ ) which is G-conjugate to θ, such that a ′ is σ-stable and θ ′ • σ = θ ′−1 .
Let [a, β] be a simple stratum such that θ ∈ C(a, β). We may and will assume that the principal order a is standard (that is, a is made of matrices with coefficients in O and its reduction mod p is made of upper block triangular matrices), thus σ-stable. The extension F[β] is totally wildly ramified over F. In particular, a is a minimal order in M n (F).
Write U = a × , which is the standard Iwahori subgroup of G. For all i 1, write U i = 1 + p i a , which is a normal subgroup of U. Then U/U 1 ≃ k ×n is abelian, of order prime to p, and U i /U i+1 is an abelian p-group for all i 1.
Note that J = O × J 1 since F[β] is totally ramified over F. Thus the image of J in U/U 1 ≃ k ×n is the image of the diagonal embedding of k × in k ×n . Let M be the torus made of all diagonal matrices of G. Proof. There are u 1 , . . . , u n ∈ k × such that u mod U 1 is equal to (u 1 , . . . , u n ) in U/U 1 ≃ k ×n . Changing u in the equivalence class O × u, we may assume that u 1 = 1.
The condition (4.14) says that uσ(u) mod U 1 is in the image of the diagonal embedding of k × in k ×n . Since u 1 = 1, this gives us u i σ(u i ) = 1 for all i ∈ {1, . . . , n}.
Assume first that F is unramified over F o . Then k is quadratic over k o and σ induces the non-trivial k o -automorphism of k. We search for y = (y 1 , . . . , y n ) ∈ k ×n such that uσ(y)y −1 = 1 in k ×n . This is possible by Hilbert's Theorem 90, since u i σ(u i ) = 1 for all i.
Assume now F is ramified over F o . Then σ is trivial on k = k o . We thus have u 2 i = 1 which implies u i ∈ {−1, 1}. Let ̟ be a uniformizer of F such that σ(̟) = −̟. Such a choice is possible since p = 2. We are searching for a y = (y 1 , . . . , y n ) ∈ F ×n such that σ(y)y −1 ∈ U and uσ(y)y −1 = 1 in k ×n . Let y i = 1 if u i = 1, and let y i = ̟ otherwise. This gives us a y ∈ M satisfying the required condition.
Let us write zuσ(y)y −1 ∈ U 1 for some y ∈ M and z ∈ O × given by Lemma 4.15. By replacing the stratum [a, β] by [a y , β y ], the simple character θ by θ y ∈ C(a y , β y ) and u by y −1 zuσ(y), which does not affect the fact that the order is σ-stable, we may and will assume that u ∈ U 1 . We write J 0 = J and J i = J ∩ U i for i 1. Let v ∈ U i for some i 1, and assume that vσ(v) ∈ J i . Then there are j ∈ J i and We thus have vσ(v) ≡ 1 mod V. The quotient W = U i /V is an abelian, finite and σ-stable p-group, and the first cohomology group of σ in W is trivial since p = 2. We thus have v ≡ xσ(x) −1 mod V for some element x ∈ U i . This gives us vσ(x)x −1 ∈ V ⊆ J i U i+1 as required.
, for i 0, satisfying the following conditions: Proof. Assume the triples (x k , j k , v k ) have been defined for all k < i, for some i 1. Applying thanks to Condition (ii), we obtain h i ∈ J i (a, β y i−1 ) and . Setting y i = y i−1 x i and θ i = θ y i , we get: Since θ x i i−1 is equal to θ i , we get the expected result.
Let x ∈ U 1 be the limit of y i = x 0 x 1 . . . x i and h ∈ J 1 that of j i . . . j 1 j 0 when i tends to infinity. We have: Passing to the limit, we get uσ(x)x −1 = h −1 ∈ J, as expected.

The maximal case
In this paragraph, we assume that d = n only. We generalize Proposition 4.13 to this situation.
Proof. Let E be the field extension F[β], and let T be the maximal tamely ramified extension of F in E. It is the tame parameter field for the endo-class Θ. The simple character θ determines a T/F-lift Ψ of Θ as in [8,Section 9]. Namely, let C denote the centralizer of T in M n (F). The intersection c = a ∩ C is a minimal order in C, giving rise to a simple stratum [c, β] in C. The restriction of θ to H 1 (c, β), denoted θ T , is a simple character associated to this simple stratum, called the interior T/F-lift of θ in [8]. Its endo-class, denoted Ψ, is a T/F-lift of Θ.
Lemma 4.8 gives us a unique F o -involution α of T such that α| F = σ and Ψ ∨ = Ψ α . Let us fix an F-embedding ι of T in M t (F) as in Corollary 4.11. Composing with the diagonal embedding of M t (F) in M n (F) gives us an F-embedding of T in M n (F) such that: By the Skolem-Noether theorem, this embedding is implemented by conjugating by some g ∈ G.
Thus, conjugating [a, β] and θ by g, we may assume that T is σ-stable and that the F o -involution σ of M n (F) induces α on T. Note that C is σ-stable and is canonically isomorphic to the T-algebra M n/t (T). The restriction of σ to C identifies with the involution α acting componentwise. From now on, we will abuse the notation and write σ instead of α.
We now apply Proposition 4.13 to the simple character θ T whose endo-class Ψ satisfies Ψ ∨ = Ψ σ . We thus get a y ∈ C × such that c y is σ-stable and the simple character ϑ = θ y T satisfies ϑ • σ = ϑ −1 . Since a → a × ∩ C × is injective on hereditary orders of M n (F) normalized by T × , we deduce that a y is σ-stable. Since interior T/F-lifting is injective, the simple character θ ′ = θ y satisfies the expected property θ ′ • σ = θ ′−1 .

The general case
In this paragraph, we prove Theorem 4.2 in the general case. Write n = md, with m 1.
Let θ ∈ C(a, β) be a simple character of endo-class Θ. By conjugating in G, we may assume that a is σ-stable.
Let us now embed M d (F) diagonally in the F-algebra M n (F). This gives us an F-algebra homomor- Since σ(β ′ ) = −β ′ , the field E ′ is stable by σ. The centralizer B ′ of E ′ in M n (F) naturally identifies with M m (E ′ ).
Let b ′ be a standard hereditary order in B ′ , and let a ′ be the unique hereditary order in M n (F) normalized by E ′× such that a ′ ∩ B ′ = b ′ . Then we have a simple stratum [a ′ , β ′ ] in M n (F). Let θ ′ ∈ C(a ′ , β ′ ) be the transfer of θ. Since a ′ is σ-stable and σ(β ′ ) = −β ′ , we have: Let M be the standard Levi subgroup of G isomorphic to GL d (F) × · · · × GL d (F). Write P for the standard parabolic subgroup of G generated by M and upper triangular matrices, and N for its unipotent radical. Let N − be the unipotent radical of the parabolic subgroup opposite to P with respect to M. By [12,Paragraph 7.1], we have: By [12,Proposition 7.1.19], the character θ ′ is trivial on H 1 (a ′ , β ′ ) ∩ N and H 1 (a ′ , β ′ ) ∩ N − , and the restriction of θ ′ to H 1 (a ′ , β ′ ) ∩ M is equal to θ ′ 0 ⊗ · · · ⊗ θ ′ 0 . As M, N, N − and H 1 (a ′ , β ′ ) are σ-stable, and by uniqueness of the Iwahori decomposition (4.19), we get θ ′ • σ = θ ′−1 . Finally, as F[β] and E ′ have the same ramification index over F (see §3.2) we may choose the order b ′ such that a and a ′ are conjugate. The transfer map from C(a, β) to C(a ′ , β ′ ) is thus implemented by conjugacy by an element of G. It follows that θ and θ ′ are G-conjugate. (i) the hereditary order a is σ-stable and σ(β) = −β; (ii) the element β has the block diagonal form: is thus equal to M m (E), equipped with the involution σ acting componentwise; (iii) the order b = a ∩ B is the standard maximal order of M m (E).
In conclusion, the following corollary refines Theorem 4.1.
Corollary 4.21. Let π be a σ-self-dual cuspidal representation of G. Then π contains a σ-self-dual type attached to a σ-standard stratum.
Remark 4.22. Let π be a σ-self-dual cuspidal representation of G, and θ ∈ C(a, β) be a simple character in π such that θ • σ = θ −1 and σ(β) = −β. Let E denote the field extension F[β] and write E o = E σ . Let T denote the maximal tamely ramified sub-extension of E/F, that is, the tame parameter field of the endo-class of π, and write T o = T σ .
(ii) The extensions E/E o and T/T o have the same ramification index.
For the first property, see Lemma 4.10 and its proof. The second one follows from the fact that E is totally wildly ramified over T and p is odd, thus [E : T] is odd.

Classification of σ-self-dual types
From now on, we will abbreviate σ-self-dual extended maximal simple type to σ-self-dual type. In this paragraph, we determine the G σ -orbits of σ-self-dual types in a σ-self-dual cuspidal representation of G.
Lemma 4.23. Let π be a cuspidal representation of G containing a σ-self-dual type (J, λ). The σ-self-dual types in π are the (J g , λ g ) for g ∈ G such that σ(g)g −1 ∈ J.
Proof. By Proposition 3.4, any (extended maximal simple) type contained in π is G-conjugate to (J, λ). Given g ∈ G, we have ( (ii) J = J(a, β) and the simple character θ associated to λ belongs to C(a, β).
We have σ(a g 0 ) = (a γ 0 ) g which is equal to a g 0 since J 0 is contained in the normalizer of a 0 . The result now follows from Proposition 4.7. and E o = E σ . Let g ∈ G and suppose that σ(g)g −1 ∈ J = J(a, β).
(ii) If E is ramified over E o , and ̟ E is a uniformizer of E, then: (a) there is a unique integer i such that 0 2i m and g ∈ Jt i G σ , where with ̟ E occurring i times; Proof. For any group Γ equipped with an action of σ, we will write H 1 (σ, Γ) for the first cohomology set of σ in Γ. Write γ = σ(g)g −1 . The identity σ(γ) = γ −1 implies that γ has valuation 0 in J. We thus have γ ∈ J = J(a, β). Write J 1 = J 1 (a, β) and identify J/J 1 with GL m (k E ), denoted G, as in (3.3). Let x denote the image of γ in G. It satisfies xσ(x) = 1.
If E is unramified over E o , then x = σ(y)y −1 for some y ∈ G, thus: for some a ∈ J. Since J 1 is a pro-p-group and p = 2, the first cohomology set H 1 (σ, J 1 ) is trivial. The left hand side of (4.27) can thus be written σ(j)j −1 for some j ∈ J 1 , thus we have g ∈ JG σ .
Suppose now that E is ramified over E o , so that σ acts trivially on k E . We may and will assume that ̟ E has been chosen such that σ(̟ E ) = −̟ E . Then x is conjugate in G to a class δJ 1 where: with −1 occurring i times for some i ∈ {0, . . . , m}. We thus have σ(a)γa −1 ∈ δJ 1 for some a ∈ J.
If we write a = ut r for some r ∈ Z and u ∈ J, then the images of δ k and (−1) r δ i in G are conjugate, thus either r is even and k = i, or r is odd and k = m − i.
where t i is defined by (4.26), form a set of representatives of the G σ -conjugacy classes of σ-self-dual types in π. The integer i is called the index of the G σ -conjugacy class. If one identifies the quotient then σ acts on GL m (k E ) by conjugacy by the diagonal element The inconvenience of the extension E/E o is that it is not canonically determined by π. We remedy this in the next paragraph.

The quadratic extension T/T o
Let Θ ∈ E(F) be an endo-class of degree d, such that Θ σ = Θ ∨ . By Theorem 4.2, given any multiple n of d, there are a maximal simple stratum [a, β] in M n (F) and a simple character θ ∈ C(a, β) of endo-class Θ such that θ • σ = θ −1 , the order a is σ-stable and σ(β) = −β. Thus E = F[β], its centralizer B and the maximal order b = a ∩ B are stable by σ.
Denote by E o the field of σ-fixed points in E, by T the maximal tamely ramified sub-extension of E over F, and set T o = T ∩ E o . Note that T is the tame parameter field of Θ, and that d is the degree [E : F]. We also write n = md.
Proof. Let [a ′ , β ′ ] be a maximal simple stratum in M n ′ (F) for some multiple n ′ of d, and let θ ′ be a simple character in C(a ′ , β ′ ) of endo-class Θ such that θ ′ • σ = θ ′−1 , the order a ′ is σ-stable and σ(β ′ ) = −β ′ . Associated with this, there are a tamely ramified extension T ′ of F and its σ-fixed points T ′ 0 .
Suppose first that θ ′ = θ. Write J 1 for the maximal normal compact open pro-p-subgroup of the Gnormalizer of θ. By [11, Proposition 2.6], one has T ′ = T x for some x ∈ J 1 . Since T ′ is stable by σ, the element y = σ(x)x −1 ∈ J 1 normalizes T, thus centralizes it by [11,Proposition 2.6]. Applying Hilbert's Theorem 90 to the element y in the centralizer G T of T in G implies that x ∈ G T G σ . It follows that T ′ is G σ -conjugate to T. The F o -isomorphism class of T/T o thus only depends on θ, not on the simple stratum [a, β] such that θ ∈ C(a, β).
Suppose now that n ′ = n. Since θ, θ ′ have the same endo-class, we have θ ′ = θ g for some g ∈ G.
Since they are both σ-self-dual, we have σ(g)g −1 ∈ J, where J is the G-normalizer of θ. By Lemma 4.25, we may even assume, up to G σ -conjugacy, that g ∈ B × , thus σ(g)g −1 ∈ B × centralizes T.
Thanks to the first case, we may also assume that a ′ = a g and β ′ = β g . We thus have T ′ = T g with σ(g)g −1 ∈ G T . By the same cohomological argument as above, we deduce that T ′ is G σ -conjugate to T.
We now consider the general case. Thanks to the first two cases and Corollary 4.21, we may assume, replacing θ, θ ′ by G-conjugate characters if necessary, that [a, β] and [a ′ , β ′ ] are σ-standard. We thus may transfer θ and θ ′ to GL d (F) without changing the F o -isomorphism classes of T/T o and T ′ /T ′ o . We are thus reduced to the previous case. Now let π be a σ-self-dual cuspidal representation of G. Its endo-class, denoted Θ, has degree dividing n and satisfies Θ σ = Θ ∨ . Associated with it, there is thus a quadratic extension T/T o , uniquely determined up to F o -isomorphism. Let us record this fact for future reference.  (i) If T is unramified over T o , the σ-self-dual types contained in π form a single G σ -conjugacy class.
(ii) If T is ramified over T o , the σ-self-dual types contained in π form exactly ⌊m/2⌋ + 1 different G σ -conjugacy classes, characterized by their index.

A counterexample in the semisimple case
We end this section by looking at a natural question which lies slightly outside the main thrust of this paper but which we find intriguing: namely, is there, for any σ-self-dual irreducible representation π, a σ-self-dual type contained in π. If one requires the type to be a semisimple type (in the sense of [14,37]) then the answer is no, as the following example shows.
Let χ be a tamely ramified character of F × such that the character χ(χ•σ) is ramified. We consider the representation π of GL 2 (F) obtained by applying the functor or normalized parabolic induction to the character χ ⊗ (χ −1 • σ) of the Levi subgroup F × × F × . This is an irreducible and σ-self-dual representation of level 0. By looking at its cuspidal support, one deduces that any semisimple type in π is conjugate to one of the following: (i) the pair (I, λ) where I is the standard Iwahori subgroup (the one whose reduction mod p F is made of upper triangular matrices) and λ is the character: Note that the latter one is conjugate to the first one by the element: where ̟ is a uniformizer of F. Thus any semisimple type in π is conjugate to (I, λ).
Remark 4.32. The example above shows that there is no σ-self-dual semisimple type for π. This also implies that there is no σ-self-dual type for π which is a cover of type for its cuspidal support (in the sense of [13]). However, writing K = GL 2 (O) for the standard maximal compact subgroup of GL 2 (F) and using the other notation above, the pair (K, ind K I λ) is a type for π, which is σ-selfdual. Thus the question of whether or not all irreducible σ-self-dual representations of G possess a σ-self-dual type remains as an interesting open question.

5.1
Let N be a σ-stable unipotent subgroup of G.
Lemma 5.1. The group N is a union of σ-stable pro-p subgroups.
Proof. We write N = i 0 N i as a nested union of compact subgroups N i which are open in N, so that N i ⊆ N j , for 0 i j. For any u ∈ N, there exist i, j 0 such that u ∈ N i and σ(u) ∈ N j . Then, taking k = max{i, j}, we have u ∈ N k ∩σ(N k ). Thus N = k 0 (N k ∩ σ(N k )), as required. (i) If the double coset NgK is σ-stable then it contains a σ-stable left K-coset.
Proof. (i) Suppose NgK is σ-stable, so that σ(g) = ugk, for some u ∈ N and k ∈ K. By Lemma 5.1, there is a σ-stable pro-p subgroup N 0 of N containing u, so that σ(g) ∈ N 0 gK. In particular, the double coset N 0 gK is σ-stable.
(iii) Suppose h ∈ (NK) σ . Then certainly hK is σ-stable. On the other hand, NhK = NK and K itself is also σ-stable so applying (ii) with g = 1 we get that every σ-stable left coset in NK lies in N σ K; thus h is in N σ K. Writing h = uk with u ∈ N σ and k ∈ K, the fact that h is σ-invariant implies k ∈ K σ , so h ∈ N σ K σ .

5.2
We suppose from now on that G is quasi-split. As before, a pair (K, τ ), consisting of an open subgroup K of G and an irreducible representation τ of K, is called σ-self-dual if σ(K) = K and τ σ ≃ τ ∨ .
A Whittaker datum for G is a pair (N, ψ) consisting of (the F o -points of) the unipotent radical N of an F o -Borel subgroup of G and a character ψ of N such that the stabilizer of ψ in G is ZN, where Z denotes the F o -points of the centre of G. If a Whittaker datum (N, ψ) is σ-self-dual then, since F o is not of characteristic two, ψ is trivial on N σ . Proposition 5.3 (cf. [9, Proposition 1.6]). Suppose that G is quasi-split. Let (N, ψ) be a σ-selfdual Whittaker datum in G and let π be an irreducible σ-self-dual cuspidal representation of G such that the space Hom N (π, ψ) is one-dimensional. Suppose that (J, ρ) is a σ-self-dual pair, with J a compact-mod-centre open subgroup of G, such that π ≃ ind G J ρ.
(i) There exists a σ-self-dual pair (J ′ , ρ ′ ) conjugate to (J, ρ) such that Proof. We follow the proof of [9, Proposition 1.6] which, although it is written only for G = GL n (F), is valid more generally. Let us write V ρ for the space of ρ, and H (G, ρ, ψ) for the space of functions , for all u ∈ N, g ∈ G and k ∈ J. By the main Theorem of [30] (which is valid also for R-representations), we have a natural G-isomorphism there is a unique double coset NgJ which supports a non-zero element of H (G, ρ, ψ) (that is, intertwines ψ with ρ), and moreover the space of ϕ ∈ H (G, ρ, ψ) supported on NgJ is one-dimensional -that is, Hom N g ∩J (ρ, ψ g ) is one-dimensional. Note that N g ∩ J is a compact subgroup of N g so is pro-p; in particular, the restriction of ρ to N g ∩ J is semisimple.
Applying σ and taking contragredients, we see that Hom N σ(g) ∩J (ψ σ(g) , ρ) is also non-zero; by semisimplicity, the same is true of Hom N σ(g) ∩J (ρ, ψ σ(g) ) so, by uniqueness, σ(g) lies in NgJ. Since the double coset NgJ is then σ-stable, Lemma 5.2 implies that it contains a σ-stable coset hJ, and that any σ-stable J-coset in NgJ lies in N σ hJ. Then the pair ( h J, h ρ) satisfies the hypotheses of (i), while the uniqueness statements in (ii) and (iii) also follow.

5.3
Finally in this subsection, we specialize to the case G = GL n (F), where F/F o is a quadratic extension and σ the Galois involution as in the rest of the paper. By the σ-self-dual type Theorem 4.1 together with [21, Corollary 1] (or [52,III.5.10] in the modular case), the hypotheses of Proposition 5.3 are satisfied for any irreducible σ-self-dual cuspidal representation π of GL n (F).
Remark 5.4. Note that [52, III.5.10] is for cuspidal representations with coefficients in an algebraic closure F ℓ of a finite field of characteristic ℓ = p only, but one can easily extend it to representations with coefficients in a general R of characteristic ℓ. Indeed, if π is a cuspidal R-representation, then, by twisting it by a character, we may assume that its central character has values in F ℓ ⊆ R. Then by [52,II.4] there is a cuspidal F ℓ -representation π 1 such that π is isomorphic to π 1 ⊗ F ℓ R. It now follows that the hypotheses of Proposition 5.3 are satisfied by π, since they are satisfied by π 1 .
Proposition 5.5. Let π be a σ-self-dual cuspidal representation of GL n (F), and let T/T o be the quadratic extension associated with it by Proposition 4.30. Let d be the degree of the endo-class of π, and write n = md.
(i) Let (N, ψ) be a σ-self-dual Whittaker datum in GL n (F). Then the representation π contains a σ-self-dual type (J, λ) such that The pair (J, λ) is uniquely determined up to conjugacy by N σ and Hom J∩N (λ, ψ) has dimension 1.
(ii) The set of all σ-self-dual types contained in π and satisfying (5.6) for some σ-self-dual Whittaker datum (N, ψ) is a single GL n (F o )-conjugacy class.
(iii) If T is unramified over T o , the conjugacy class in (ii) is the unique GL n (F o )-conjugacy class of σ-self-dual types in π.
(iv) If T is ramified over T o , the conjugacy class in (ii) is the unique GL n (F o )-conjugacy class of σ-self-dual types in π of index ⌊m/2⌋ (see Remark 4.28).
Proof. Assertion (i) follows from Proposition 5.3 and Assertion (ii) follows from (i) together with the fact that any two σ-self-dual Whittaker data in GL The result now follows from a simple cohomological argument.
We now prove (iv). By Proposition 4.31, there are ⌊m/2⌋+1 conjugacy class of σ-self-dual types in π and each conjugacy class has an index i as in Remark 4.28. If (J, λ) is a σ-self-dual type with index i then, identifying J/J 1 with GL m (k E ), the involution σ acts via conjugation by the diagonal element Now there is a g ∈ GL m (k E ) such that U g is equal to N, the standard maximal unipotent subgroup.
Since U and N are σ-stable, the element γ = σ(g)g −1 normalizes N. It thus can be written γ = n 0 t with n 0 ∈ N and t diagonal. Since γ −1 = σ(γ) = δγδ −1 , we have t −1 = t. Write δ ′ = tδ and let σ ′ be the involution of GL m (k E ) given by conjugacy by δ ′ . Then Thus, replacing g by h, we may assume that n 0 = 1. Moreover, if we identify U with N, then σ is replaced by σ ′ , that is, conjugacy by the diagonal matrix δ ′ .
Since the number of −1 and 1 differ by at most 1, and since i ⌊m/2⌋ by definition, it follows that i = ⌊m/2⌋. Definition 5.7. We call a type in the conjugacy class of Proposition 5.5(ii) a generic σ-self-dual type for π. Proposition 5.5 thus says that, when T is unramified over T o , any σ-self-dual type contained in π is generic, and, when T is ramified over T o , a σ-self-dual type contained in π is generic if and only if its index is ⌊m/2⌋.

5.4
We continue with the notation of Paragraph 5.3. The main result of this paragraph is Lemma 5.10, which will be useful in Sections 6 and 7.
Let π be a σ-self-dual cuspidal representation of G = GL n (F). By Proposition 5.5, this representation contains a generic σ-self-dual type (J, λ), uniquely determined up to G σ -conjugacy. Fix a maximal simple stratum [a, β] such that J = J(a, β) with a a σ-stable hereditary order and σ(β) = −β. Let E denote the F-extension F[β]. Let T be the maximal tamely ramified sub-extension of E over F, and let T 0 be its σ-fixed points. We also write m = n/ deg F (β). . Let π be a σ-self-dual cuspidal representation of GL n (F). If T/T o is ramified, then either m = 1 or m is even.
Remark 5.9. If T/T o is unramified, and with the additional hypothesis that π is supercuspidal, it is proved in [45,Proposition 8.14] that m is odd, but we will not need this result.
The parahoric subgroup a × of G is σ-stable; thus a × ∩G σ is a parahoric subgroup of G σ and has the form a  Suppose now that m = 2r for some r 1. Since (J, λ) has index r, we may assume that where each block has size r × r. Since E is ramified over E o , a simple calculation based on the fact that p −1 We have a similar description of a; if we let a E/F denote the unique O-order of End F (E) normalized by E × , then we have Remark 5.11. Let ̟ λ be an element of the principal order b o generating its Jacobson radical. We get that J ∩ G σ is generated by ̟ λ and J ∩ G σ .

Distinction and Whittaker functions
We return to the notation of the rest of the paper so that F/F o is a quadratic extension, G = GL n (F) for some n 1 and σ is the involution on G induced by the Galois involution.

Distinguished linear forms and Whittaker functions
In this subsection we begin to look at the question of distinction. Recalling that P = P n (F) denotes the standard mirabolic subgroup of G, we will prove the following analogue of a result of Ok [39].
Recall that saying that (J, λ) is distinguished means that the space on the right hand side is nonzero. The condition in the proposition that the σ-self-dual cuspidal representation π is distinguished is a priori weaker than this; however, see Remark 6.7.
In order to prove this proposition, we need a small lemma which again applies in a more general setting. Let G be a locally profinite group, let K be an open subgroup of G and let H ′ ⊆ H be closed subgroups of G. Let ρ be a smooth representation of K and let τ be a smooth representation of H. For g ∈ G, we write ind KgH K ρ for the subspace of ind G K ρ consisting of functions with support contained in KgH. Then the Mackey decomposition gives and, by Frobenius reciprocity applied to the natural projection in the opposite direction, we get natural maps We get similar maps with H replaced by H ′ and the following diagram commutes: where the vertical maps are given by natural inclusion.
Lemma 6.2. Suppose, in the situation above, that the inclusion ι 1 is an equality. Then the inclusion ι 0 is also an equality.
Proof of Proposition 6.1. We apply the lemma to our situation, where we recall that G = GL n (F), P = P n (F) is a σ-stable mirabolic subgroup, and (J, λ) is a σ-self-dual type -by which, we recall, we mean a σ-self-dual extended maximal simple type -with π = ind G J λ an irreducible distinguished σ-self-dual cuspidal representation of G. The result of Ok [39, Theorem 3.1.2] (see also [34, Proposition 2.1]), proved for any irreducible complex representation of G and which we generalize to any cuspidal representation of G with coefficients in R in Appendix B (see Proposition B.23), says that, in this situation, we have an equality Hom P σ (π, 1) = Hom G σ (π, 1).
We set G = G, H = G σ and H ′ = P σ , with τ = 1 the trivial representation of H, and (K, ρ) = (J, λ). Then the result follows at once from Lemma 6.2.
We turn now to Whittaker functions. Let N = N n (F) denote the standard maximal unipotent subgroup (consisting of upper triangular unipotent matrices) and let ψ be a σ-self-dual non-degenerate character of N. If π is any generic irreducible representation of G, recall also that its Whittaker model (with respect to ψ) is the subspace W(π, ψ) of Ind G N ψ which is the image of π under any non-zero map in the one-dimensional space Hom G (π, Ind G N ψ).
Now let π be an irreducible σ-self-dual cuspidal representation of G. By Theorem 4.1 and Proposition 5.5, it contains a σ-self-dual type (J, λ) such that Hom J∩N (λ, ψ) = 0. We use the usual notation for data associated to this type; in particular, we have the unique maximal simple character θ contained in λ and normalized by J, defined on the normal subgroup H 1 of J, as well as the normal subgroups J ⊇ J 1 of J.
Let U = (N ∩ J)H 1 and extend ψ to a character ψ λ of U as in [40,Definition 4.2]: We fix a normal compact open subgroup N of U contained in ker(ψ λ ) and define the Bessel function J λ : J → R of λ by where tr λ is the trace character of λ. This is independent of the choice of N . Note that this definition makes sense over R, since U is a pro-p-group.
We then define a function W λ : G → R supported in NJ by One checks that the function W λ is well defined, and that W λ (ng) = ψ(n)W λ (g) for all n ∈ N and g ∈ G.
We set M = (P ∩ J)J 1 and note that, by [40,Corollary 4.8], the subgroup P ∩ J = P ∩ J is contained in M. Let S λ denote the space of functions f : M → R such that f (um) = ψ λ (u)f (m) for all u ∈ U and m ∈ M. For each j ∈ J, we define an operator L(j) on S λ by for all f ∈ S λ and m ∈ M. This defines a representation L of J on S λ . We claim that this representation is isomorphic to λ. When R is the field of complex numbers, or more generally when R has characteristic 0, this is [40,Theorem 5.4]. Let us explain briefly how to deduce the modular case from the characteristic 0 case. Assume that R has characteristic ℓ > 0. First, by the same argument as in Remark 5.4, it is enough to prove the result when R is the field F ℓ . Then fix an extended maximal simple type λ with coefficients in Q ℓ whose reduction mod ℓ is isomorphic to λ (which is possible by [37, Proposition 2.39]). We thus have an isomorphism between λ and the representation on S λ defined as in (6.4). Reducing mod ℓ, we get the claimed result. In the sequel, we will identify the space of λ with S λ . It follows as in [40, Section 5.2] that the function W λ defined by (6.3) belongs to the Whittaker model W(π, ψ) of π. Note also (see [40,Proposition 5.3(iii)]) that the restriction of J λ to M lies in S λ .
Proposition 6.5. Let π be a σ-self-dual cuspidal representation of G, and let (J, λ) be a generic σ-self-dual type contained in π.
(i) Let dm be a right invariant measure on (J ∩ N σ )\(J ∩ P σ ). The linear form on λ defined by Proof. The form L λ is clearly J ∩ P σ -invariant by its definition. By [40,Proposition 5.3(iv)], the proof of which is written for complex representations but still works in the modular case, the function J λ is identically zero on the complement of U σ in M σ . On the other hand, for u ∈ U σ , we have J λ (u) = ψ λ (u) = 1, since ψ λ is a σ-self-dual character of a pro-p group U and p is odd. Hence the value L λ (J λ ) = dm((J ∩ N σ )\(J ∩ N σ )(H 1 ∩ P σ )) is non-zero, since H 1 is pro-p. The final statement follows immediately from the fact that J ∩ P = J ∩ P together with Proposition 6.1.
We deduce the following corollary from Proposition 6.5.
Corollary 6.6. Let π be a σ-self-dual cuspidal representation of G. Then π is distinguished if and only if any of its generic σ-self-dual types is distinguished.
Remark 6.7. Putting Corollary 6.6 and Proposition 5.5 together, we obtain a different proof of a result of [45] saying that a σ-self-dual cuspidal representation π of G is distinguished if and only if it contains a distinguished σ-self-dual type, and that, if the quadratic extension T/T o associated with π by Proposition 4.30 is ramified, any distinguished σ-self-dual type contained in π has index ⌊m/2⌋, where n = md and d is the degree of the endo-class of π.

Explicit Whittaker functions and restriction to GL n (F o )
We continue with the same notation, and write K = GL n (O) and K σ = GL n (O o ). In order to make computations, we need to be somewhat more careful with our choice of non-degenerate character ψ to ensure that corresponding generic σ-self-dual type is well-positioned with respect to the standard maximal compact subgroup K σ of G σ .
Let S n denote the group of permutation matrices in G σ . The Bruhat decomposition in the finite quotient of K σ by its pro-p unipotent radical, together with the Iwasawa decomposition of G σ , yields the Bruwasawa decomposition G σ = B σ S n I o , where B is the standard Borel subgroup of G and I o is the standard Iwahori subgroup of G σ . In particular, this decomposition implies that any parahoric subgroup of G σ is conjugate by N σ to a parahoric subgroup in the standard apartment, where N is the unipotent radical of B. Writing e 1 , . . . , e n for the standard basis of F n , and using the notation above, we get the following. Lemma 6.8. Let π be a σ-self-dual cuspidal representation of G. There are a σ-self-dual Whittaker datum (N, ψ) and a generic σ-self-dual type (J, λ) in π such that (i) the space Hom J∩N (λ, ψ) is non-zero; (ii) there is a numbering on the O o -lattice chain Λ o associated to (J, λ) such that where the a i : Z → Z are increasing functions satisfying (a) a i (k + e o ) = a i (k) + 1 for all k ∈ Z and a i (0) = 0, for i = 1, . . . , n, (b) a n (0) = · · · = a n (e o − 1) = 0.
Note that condition (ii) implies in particular that J σ ⊆ K σ (though it is not equivalent to this). It is also worth noting that it is not in general possible to find (J, λ) satisfying condition (i) and the stronger condition J ⊆ K (see Remark 6.9).
Proof. We pick a σ-self-dual Whittaker datum (N, ψ) where N = N n (F) is the standard maximal unipotent subgroup. By Proposition 5.5, we have a σ-self-dual type (J, λ) satisfying (i Since a o lies in the standard apartment, we can find t 1 , . . . , t n−1 ∈ F × o such that Conjugating both (J, λ) and the Whittaker datum (N, ψ) by t = diag(t 1 , .., t n−1 , 1) (which is in the diagonal torus of G σ ), we obtain the result.
Remark 6.9. Suppose F/F o is ramified, n = 2 and π is a σ-self-dual depth zero cuspidal representation of GL 2 (F). Then any generic σ-self-dual type (J, λ) in π has index 1 so J is GL 2 (F o )conjugate to t 1 Kt −1 1 where t 1 = diag(̟, 1) and ̟ is a uniformizer of F. In particular, J is not GL 2 (F o )-conjugate to (any subgroup of) K. Now let π be a σ-self-dual cuspidal representation of G and choose our non-degenerate character ψ and generic σ-self-dual type (J, λ) as in Lemma 6.8. We have an order a o as above. Note that, by Lemma 5.10, it is a principal order. We choose ̟ λ ∈ J σ as in Remark 5.11, so that J σ is generated by ̟ λ and J σ .
The following lemma shows a useful property of the Iwasawa decomposition of ̟ λ in G σ , which will be key to our computation to come. In that case, if we choose p i ∈ P σ and k i ∈ K σ such that Proof. Note first that P σ K σ consists precisely of those matrices whose last row lies in ( For i ∈ {0, . . . , e o −1}, we fix from now p i ∈ P σ and k i ∈ K σ such that ̟ i λ = p i k i , as in Lemma 6.10. A volume which appears in our computation follows from this, using that J σ ⊆ K σ thanks to our choice of basis, as in [29, Lemma 7.2]. Lemma 6.11. There is a unique right invariant measure dk on (P σ ∩ K σ )\K σ such that we have for all i ∈ {0, . . . , e o − 1}.
Proof. Let dk be any right invariant measure on (P σ ∩ K σ )\K σ . Following the first part of the proof of [29,Lemma 7.2], and thanks to Lemma 6.10, we have: for all i ∈ {0, . . . , e o − 1}. It thus remains to prove that dk can be chosen so that the measure of the quotient (P σ ∩ K σ )\(P σ ∩ K σ )J σ is 1, which follows from [52, I.2.8].
We recall from the previous section that we have an explicit Whittaker function W λ ∈ W(π, ψ) with support NJ. Proposition 6.12. For each l ∈ Z, let W l λ denote the function from G σ to R supported on the subset {g ∈ G σ ∩ NJ | | det(g)| o = q −l o } and coinciding with W λ on it.
Proof. Note that W λ | G σ is supported in G σ ∩ NJ, which is equal to N σ J σ by Lemma 5.2(iii). By definition of ̟ λ , the set N σ J σ is the disjoint union of the N σ ̟ i λ J σ for i ∈ Z, and then (i) follows from Lemma 6.10. The remaining parts follow exactly as in the proof of [29,Proposition 8.4].

Asai L-functions and test vectors
From now on, until the end of the paper, all representations are complex, that is, R is now the field C of complex numbers.

Distinction and dichotomy
We will need two further key results on distinction of σ-self-dual cuspidal complex representations, which we recall from [45]. Recall that ω F/Fo denotes the non-trivial character of F × o which is trivial on N F/Fo (F × ). The first result is dichotomy. It is proved for discrete series representations when F has characteristic 0 by Flicker [17], Kable [28] and Anandavardhanan, Kable and Tandon [2], and we prove in Appendix A (see Theorem A.2 below) that the global arguments of [28] and [2] remain valid when F has characteristic p. It is also proved by Sécherre [45] for cuspidal representations, in a purely local way, with no assumption on the characteristic of F (see also Remark 7.3). (i) π is σ-self-dual if and only if it is either distinguished or ω F/Fo -distinguished.
(ii) π cannot be both distinguished and ω F/Fo -distinguished.
Given a σ-self-dual cuspidal representation π of GL n (F), we denote by T/T o the quadratic extension associated to π by Proposition 4.30. Let d denote the degree of the endo-class of π. It is a divisor of n, and we write n = md. Proposition 7.2 ([45] Proposition 9.12). Let π be a distinguished cuspidal (complex) representation of GL n (F). Then π has an ω F/Fo -distinguished unramified twist if and only if either T/T o is unramified or m > 1.
Remark 7.3. These two results are proved in [45] in a more general setting: π is a supercuspidal representation of GL n (F) with coefficients in R, where R has characteristic different from p. Note that, when R has characteristic 0, any cuspidal representation is supercuspidal.

Definition of the integrals
As before, we suppose that ψ is a σ-self-dual non-degenerate character of N. Let π be a generic irreducible representation of G. For W a function in the Whittaker model W(π, ψ) of π and Φ in the space C ∞ c (F n o ) of locally constant functions on F n o with compact support, define the local Asai integral where τ is the row vector 0 . . . 0 1 and dg is a right invariant measure on N σ \G σ which will be fixed later in Paragraph 7.4. It turns out (see [28,Theorem 2]) that, for s ∈ C with sufficiently large real part, the integral (7.4) is a rational function in q −s o ; moreover, as W varies in W(π, ψ) and Φ varies in C ∞ c (F n o ), these functions generate a fractional ideal of C[q s o , q −s o ] which has a unique generator L As (s, π) which is an Euler factor (i.e. of the form 1/P(q −s o ) where P is a polynomial with constant term 1). Now let π be a cuspidal representation of G and let X(π) denote the set of unramified characters χ of G σ such that π is χ-distinguished. We recall the following description of the Asai L-function of a cuspidal representation, the proof of which is valid (as the rest of [34]) when F has positive characteristic as well: Let t(π) denote the torsion number of π, that is the number of unramified characters of F × such that π(χ • det) is isomorphic to π. Thanks to Theorem 7.1, we deduce the following formula.
Corollary 7.6. Let π be a distinguished cuspidal representation of G. Then if F/F o is ramified and no unramified twist of π is ω F/Fo -distinguished; Proof. Let R(π) denote the group of unramified characters of F × such that π(χ • det) is isomorphic to π. It is cyclic and has order t(π). Let us fix uniformizers ̟ and ̟ o of F and F o , respectively. Since π is distinguished, it is σ-self-dual. Let U(π) denote the subgroup of unramified characters χ of F × o such that π( χ • det) is σ-self-dual for any unramified character χ of F × extending χ. An unramified character χ belongs to U(π) if and only if Note that we have ω F/Fo ∈ U(π).
Let Y(π) denote the set of unramified characters χ of F × o such that π is ω F/Fo χ-distinguished. Then Theorem 7.1 says that U(π) decomposes as the disjoint union of X(π) and Y(π).

A decomposition of the integral
We continue with π a cuspidal (complex) representation of G. For computational convenience, we introduce a second integral: for W in the Whittaker model W(π, ψ) of π, we put where dp is a right invariant measure on N σ \P σ which will be fixed later in Proposition 7.13. Again, if s ∈ C has sufficiently large real part, I Now let dk be the measure on (P σ ∩ K σ )\K σ given by Lemma 6.11 and d × a be the Haar measure on F × o giving measure 1 to O × o . Then, as noticed in [17,Section 4], if s has a sufficiently large real part and if the function Φ ∈ C ∞ c (F n o ) is chosen to be K σ -invariant, there is a unique right invariant measure dg on N σ \G σ , depending only on the choice of dp, such that: As (s, k · W) dk (7.10) where ω π denotes the central character of π and g · W denotes the action of g ∈ G on W(π, ψ), that is (g · W)(x) = W(xg) for x ∈ G. From now on, we will assume that dg is chosen with respect to dp so that (7.10) holds.
Suppose that ω π is trivial when restricted to F × o , which is the case when π is distinguished. If Φ is the characteristic function by Tate's thesis [15]. Therefore, we have the following decomposition which we record as a lemma: Lemma 7.11. Let π be a distinguished cuspidal complex representation of G. Then, for all functions W ∈ W(π, ψ), we have For W ∈ W(π, ψ) and l ∈ Z, we write W l o for the function from G σ to C supported on the subset {g ∈ G σ | | det(g)| o = q −l o } and coinciding with W on it. Finally we decompose the integral given in Lemma 7.11 by the absolute value: Since π is cuspidal, the right hand term of the equality above is a finite sum [6]. We call the l th coefficient of the integral, and we record that: Lemma 7.12. Let π be a distinguished cuspidal complex representation of G. Then, for all functions W ∈ W(π, ψ), we have

Test vectors
Until the end of this section, π is a distinguished cuspidal representation of G and (J, λ) is a generic σ-self-dual type as in Lemma 6.8. Now we compute the Asai integral of the explicit Whittaker vector W λ showing it is a test vector, making use of the decomposition of Lemma 7.12. (ii) There is a unique right invariant measure dp on N σ \P σ such that Proof. By definition, we have where dk is the measure given by Lemma 6.11 and dp is a right invariant measure on N σ \P σ .
Part (i) of the proposition follows from Proposition 6.12(i).
Now assume that l = in/e o for some i ∈ {0, . . . , e o − 1}. We recall that we have our fixed decompo- Then it follows from Proposition 6.12(ii) that Let us compute the inner integral. By applying the change of variable p → pp −1 i and then Proposition 6.12(iii), we get where dm is the right invariant measure on (N σ ∩J σ )\(P σ ∩J σ ) corresponding to dp. Since ̟ i λ j ∈ J σ by Lemma 6.10, and thanks to Proposition 6.5(ii), this is equal to L λ (J λ ). Now let us fix dm so that L λ (J λ ) = 1, which is possible thanks to Proposition 6.5(i). This defines dp uniquely. Then our choice of dk gives us We now prove our main result on test vectors for Asai L-functions. Theorem 7.14. Suppose π is a distinguished cuspidal representation of G. Then where the right invariant measure dg defining the left hand side is chosen so that (7.10) holds and the measure dp defining (7.9) is the one given by Proposition 7.13.
Proof. By Lemma 7.11, we have By Proposition 7.13, we have The result then follows immediately from (7.8).
Corollary 7.15. Let π be a cuspidal representation of G such that L As (s, π) is not 1. Let χ be an unramified character of F × o such that π is χ-distinguished. Then I As (s, 1 O n o , ( χ • det)W λ ) = L As (s, π) for any unramified character χ of F × extending χ.

Flicker-Kable root numbers for cuspidal representations
In this section, using Theorem 7.14, we compute the local Asai root number, as defined by Flicker and Kable, of a cuspidal distinguished representation of G = GL n (F). Our methods here are purely local.
Given a generic irreducible complex representation π of G, its Asai integrals satisfy a local functional equation (see the appendix of [18] and [28,Theorem 3]): there is a unique element ǫ As (s, π, ψ o , δ) in the units of C[q s o , q −s o ], called the local Asai epsilon factor, such that, for all functions W ∈ W(π, ψ δ ) and Φ ∈ C ∞ c (F n o ), we have I As (1 − s, Φ, W) L As (1 − s, π ∨ ) = ǫ As (s, π, ψ o , δ) · I As (s, Φ, W) L As (s, π) (8.3) where: (i) Φ = Φ ψo denotes the Fourier transform of Φ with respect to the character ψ o ⊗ · · · ⊗ ψ o of F n o and its associated self-dual Haar measure, and (ii) W is the function in W(π ∨ , ψ −δ ) defined by where w 0 is the antidiagonal permutation matrix of maximal length and g * is the transpose of g −1 .
Notice that the epsilon factor defined above is the one used in [28]; it differs by a sign from the one defined in [18]. In the next section we will address the question of proper normalization.
Before stating the main result of this section, let us make one observation on Asai root numbers of distinguished generic representations of G. If π is such a representation, then applying the functional equation for I As (s, Ψ, W) and I As (1 − s, Φ, W) gives us as in [28,Theorem 3]. Since π is distinguished, its central character is trivial on F × o and π ∨ ≃ π σ . Since π and π σ have the same local Asai L-factor and ǫ As (s, π, ψ o , δ) = ǫ As (s, π σ , ψ o , −δ), we get It is expected that this number is 1 (cf. [1,Remark 4.4]). Here we prove it when π is a distinguished cuspidal representation.
Proof. Since we have already observed that the possible values for this epsilon factor are −1 and 1, we just need to show that ǫ As (1/2, π, ψ 0 , δ) is positive. To show this it is sufficient to show that ǫ As (0, π, ψ o , δ) is positive since Fix a Whittaker datum (N, ψ 1 ) and a σ-self-dual type (J, λ) as in Lemma 6.8. The symbol ∼ will stand for equality up to a positive constant. By Theorem 7.14, there is W λ ∈ W(π, ψ 1 ) such that where Φ 0 is the characteristic function of O n o in F n o . As ψ 1 (u) = ψ(tut −1 ) for some diagonal matrix t with coefficients in F × o , the function belongs to W(π, ψ). We may (and will) even assume that the bottom coefficient on the diagonal of t is 1. Applying the change of variable g → t −1 g, we check that Applying the functional equation, we get Let l and l ′ denote the linear forms on W(π, ψ) defined by Both these linear forms are defined by convergent integrals: by [6,Corollary 5.19] the supports in G σ of the integrands are compact mod N σ on G σ . They are G σ -invariant by [ On the other hand, we have the last equality by [2] again. In particular we get and the right hand side is positive thanks to our choice of Φ 0 . Hence we have ǫ As (0, π, ψ o , δ) > 0, which implies that ǫ As (1/2, π, ψ o , δ) = 1.
Remark 8.5. In the proof above, we used results written in characteristic 0 only. Let us explain why they are valid in characteristic p as well. First notice that, as Hom G σ (π, 1) and Hom P σ (π, 1) are equal by Ok [39, Theorem 3.1.2], the computation borrowed from the proof of [2, Theorem 1.4] holds for F of arbitrary characteristic. In [3,Theorem 7.2], and more generally in [3], the field F is assumed to have characteristic 0. In fact, appealing to [3, Theorem 6.3] is enough in the cuspidal case, since a distinguished cuspidal representation of G is always unitary (as its central character is). Now the only ingredient in the proof of [3, Theorem 6.3] which uses this rectriction on the characteristic of F is that the Godement-Jacquet epsilon factor ǫ(1/2, π, ψ) is equal to 1, for which [3] refers to [36], but the cuspidal case of this result is already in [39] and this reference does not assume the characteristic of F to be 0.

Comparing Asai epsilon factors
In this section, we compare, for π a generic unramified representation of G = GL n (F) (not necessarily distinguished), the Flicker-Kable Asai epsilon factor to the Asai epsilon factor of π defined via the Langlands-Shahidi method. This naturally leads to the normalization we give in Definition 9.10. Beuzart-Plessis came up to the same normalization in his forthcoming work on the refined global Gan-Gross-Prasad conjectures. Then, we show by a global argument that, for cuspidal representations, all these definitions of the Asai epsilon factor coincide. In particular we answer some questions posed in [1, Remark 4.4].

Changing the additive character
We denote by W F the Weil group of F with repect to a given separable closure F of F, and by W ′ F the corresponding Weil-Deligne group, that is, its direct product by SL(2, C). We use a similar notation for F o . We will write Ind ′ F/Fo and M ′ F/Fo for induction and multiplicative induction (defined for instance in [41,Section 7]) from W ′ F to W ′ Fo . We will also write Ind F/Fo and M F/Fo for induction and multiplicative induction from W F to W Fo .
Given an irreducible representation π of G, we denote by ρ(π) its Langlands parameter, which is a finite dimensional representation of W ′ F . Then, using local class field theory to identify characters of W ′ F and of F × , we have: When π = χ is a character of F × , which we identify with ρ(χ), this tells us that M ′ Given a generic irreducible representation π of G, we denote (i) by ǫ LS As (s, π, ψ o ) and L LS As (s, π) the Asai local factors attached to π via the Langlands-Shahidi method (see [48] when F has characteristic 0 and [31] when F has characteristic p), (ii) by ǫ Gal As (s, π, ψ o ) and L Gal As (s, π) the Langlands-Deligne local constants of the local Asai transfer M ′ F/Fo (ρ(π)) of the Langlands parameter of π (see [41,Section 7]).
When F has characteristic 0, the Asai local L-functions L As (s, π), L LS As (s, π) and L Gal As (s, π) are known to be all equal. We will see in the appendix (Theorem A.1) that this still holds in characteristic p.
By [48] when F has characteristic 0 and by [24] when F has characteristic p, when π is unramified and generic we have: ǫ LS As (s, π, ψ o ) = ǫ Gal As (s, π, ψ o ) whereas by [23] when F has characteristic 0 and [24] when F has characteristic p, when π is generic we have: ǫ LS As (s, π, ψ o ) = ζ · ǫ Gal As (s, π, ψ o ) where ζ is a root of unity independent from ψ o , which is expected to be 1, and known to be 1 when F has characteristic p.
Proof. We first give the proof of (i) for convenience of the reader; it follows verbatim the analogue for Rankin-Selberg L-factors in p. 7 of [27]. As before we set ψ = ψ δ . We introduce the matrix a = diag(t n−1 , . . . , t, 1).

It follows that
to get We thus get the relation: which gives us the expected result.
We will also need the following relation satisfied by ǫ As (s, π, ψ o , δ). Note that though ψ F,δ o,t = ψ F,tδ o , it is not true that ǫ As (s, π, ψ o,t , δ) = ǫ As (s, π, ψ o , tδ) since changing the character ψ o changes the Fourier transform in the functional equation. Here is what happens when one changes δ.
Proof. Going through the exact same computations as in the proof of Lemma 9.2, but taking the Fourier transform of Φ with respect to ψ o rather than ψ o,t in the computations, we arrive at whereas the relation I As (s, Φ, W a ) = µ(t) · |t| −n(n−1)s/2 o · I As (s, Φ, W) does not change. From this we obtain: which gives us the expected result.

Unramified representations
We are going to compute ǫ As (s, π, ψ o , δ) and ǫ LS As (s, π, ψ o , δ) when π is generic and unramified. From now on, Haar measures on any closed subgroup H of G will be normalized so that they give volume 1 to H∩K. This also normalizes all right invariant measures on quotients of the type U\H whenever U is an unimodular closed subgroup of H.
First, we perform a test vector computation similar to that done by Flicker when F/F o is unramified. We suppose that π is a generic unramified representation of G; we denote by W 0 the normalized spherical vector in W(π, ψ) and by Φ 0 the characteristic function of O n o .
Recall that the conductor of an additive character of a finite extension E of F o is the largest integer i such that it is trivial on p −i E . Proposition 9.5. Let π be a generic unramified representation of G and suppose that the character ψ F,δ o defined by (8.1) has conductor 0. Then I As (s, Φ 0 , W 0 ) = L As (s, π).
Proof. When F/F o is unramified, this is proved in [17,Section 3] where the unitarity assumption is unnecessary. In the ramified case, we have q = q o . We write π = µ 1 × · · · × µ n where the product notation stands for parabolic induction, and the characters µ 1 , . . . , µ n of F × are unramified. Let us fix a uniformizer ̟ of F such that ̟ o = ̟ 2 is a uniformizer of F o . For i = 1, . . . , n, set z i = µ i (̟). With notations as at p. 306 of [17], as ̟ λ o = ̟ 2λ , we find I As (s, Φ 0 , W 0 ) = λ q −s·tr(λ) s 2λ (z 1 , . . . , z n ) = λ s 2λ (z 1 q −s/2 , . . . , z n q −s/2 ) where the sum ranges over all partitions of length n and s 2λ is the Schur function (see [32, (3.1) p. 40]) associated to the partition 2λ obtained by multiplying the entries of λ by 2. By [32, Example 5a, p. 77], the sum above is equal to Now the Langlands parameter ρ(π) is the direct sum µ 1 ⊕ · · · ⊕ µ n . Since it is trivial on SL 2 (C) we consider it as a representation of W F only. Since µ i • σ = µ i for all i, we have by [41,Lemma 7.1]. Thus L Gal As (s, π) is equal to (9.6). The result follows from Theorem A.1.
Remark 9.7. At this point, we note that the authors of [3] appeal to Flicker's unramified computation even when F/F o is ramified, however Proposition 9.5 shows that there is no harm in doing that.
When F/F o is unramified, one can choose δ to be a unit, whereas when F/F o is ramified, one can choose δ to have valuation −1. In both cases, the character ψ F,δ o has conductor 0 if ψ o has conductor 0. In this case, the functional equation, together with Proposition 9.5, the fact that Φ 0 = Φ 0 and that W 0 is the normalized spherical vector in W(π ∨ , ψ −1 ), tells us that: Corollary 9.8. Suppose that π is a generic unramified representation of G, that ψ o has conductor 0 and δ has valuation 1 − e(F/F o ). Then ǫ As (s, π, ψ o , δ) = 1.
Let us compare this with the unramified situation for the Asai constant defined via the Langlands-Shahidi method. To do this we introduce the local Langlands constant λ(F/F o , ψ o ) (see for instance [10, (30.4.1)] for a definition). We note that λ(F/F o , ψ o ) is equal to ǫ(1/2, ω F/Fo , ψ o ), the Tate root number of the quadratic character ω F/Fo . We will freely use the relation [10, (30.4.2)]: where ρ a semi-simple representation of W F and Ind F/Fo denotes induction from W F to W Fo . We will also use the fact that if χ is an unramified character of E × for any finite extension E of F o and ψ E is a character of E of conductor 0, then ǫ(s, χ, ψ E ) = 1 (see the remark after (3.2.6.1) in [51]). More generally, we refer to [51] for the basic facts and relations concerning epsilon factors of characters that we will use in this section without necessarily recalling. Proposition 9.9. Suppose that π is a generic unramified representation of G, that ψ o has conductor 0 and δ has valuation 1 − e(F/F o ). Then ǫ LS As (s, π, ψ o ) = ǫ Gal As (s, π, ψ 0 ) = ω π (δ) 1−n · |δ| −n(n−1)(s−1/2)/2 · λ(F/F o , ψ o ) n(n−1)/2 where |δ| denotes the normalized absolute value of δ.
Proof. We use the notation of the proof of Proposition 9.5. We thus have ǫ Gal where we ignore the epsilon factors equal to 1. However ǫ Gal As (s, π, ψ o ) = ǫ Gal As 1 2 , | · | (s−1/2)/2 π, ψ o hence the previous equality gives the result.

Rankin-Selberg epsilon factors
Proposition 9.9, together with Corollary 9.8, suggests to introduce the following definition.
The following result, which is an immediate consequence of Lemma 9.4, was brought to our attention by Beuzart-Plessis.
Combining Proposition 9.9, Corollary 9.8 and Lemmas 9.11 and 9.12, we get the following result.
Theorem 9.13. For any generic unramified irreducible representation π of G, we have ǫ RS As (s, π, ψ o ) = ǫ LS As (s, π, ψ o ). (9.14) At the end of this section (see Theorem 9.29), we will show that (9.14) holds for cuspidal representations as well. Let π v be a generic irreducible representation of G v = GL n (k v ) and set N v = N n (k v ). We fix a nontrivial character ψ o,v of k o,v and an element δ v ∈ k × v such that tr kv/ko,v (δ v ) = 0, and set which is a non-trivial character of k v trivial on k o,v . We denote by | · | o,v the normalized absolute value on k o,v .
We first suppose that v is finite, and set q o,v for the cardinality of k o,v . Take W v ∈ W(π v , ψ v ) and Φ v ∈ C ∞ c (k n o,v ). By [26,Theorem 2.7], the integral is absolutely convergent when the real part of s is larger than a real number r v depending only on π v , it extends to an element of C[q s o,v , q −s o,v ], and these integrals span a fractional ideal of C[q s o,v , q −s o,v ] generated by a unique Euler factor denoted L As (s, π v ). Also, there is a unit in C[q s o,v , q −s o,v ], which we denote by ǫ As (s, π v , ψ o,v , δ v ) for the sake of coherent notations, such that where the Fourier transform of Φ v is defined with repect to the character ψ o,v .
Definition 9.15. We set Remark 9.16. Comparing with Definition 9.10 in the inert case, there is no Langlands constant appearing in Definition 9.15. However, note that the character ω kv/ko,v of k × o,v trivial on k v /k o,v -norms is trivial. In analogy with the inert case, we may set the Langlands constant λ(k v /k o,v , ψ o,v ) to be equal to ǫ(1/2, ω kv /ko,v , ψ o,v ), but this root number is equal to 1 by the classical properties of Tate epsilon factors.
A computation similar to the one carried out in the proof of Lemma 9.4 shows that this local factor ǫ RS As (s, π v , ψ o,v , δ v ) is independent of δ v hence we write When v is archimedean, the discussion above remains true up to the appropriate modifications (the L-factor is meromorphic rather than an Euler factor, and the epsilon factor is entire rather than a Laurent monomial) appealing to [27,Theorem 2.1] instead of [26,Theorem 2.7], and we define the local factor ǫ RS As (s, π v , ψ o,v , δ v ) as in Definition 9.15. Now we compare these epsilon factors to the epsilon factors of pairs defined by the authors of [26] and [27].
Lemma 9.17. Let φ be an isomorphism of k o,v -algebras between k v and k o,v ⊕ k o,v . It induces an isomorphism of groups between GL n (k v ) and GL n (k o,v ) × GL n (k o,v ), still denoted φ. Write π v • φ as a tensor product π 1,v ⊗ π 2,v of two generic irreducible representations of GL n (k o,v ). Then where ǫ RS (s, π 1,v , π 2,v , ψ o,v ) is the epsilon factor denoted ǫ(s, π 1,v , π 2,v , ψ o,v ) in [26,Theorem 2.7] if v is finite, and is the one canonically associated to the gamma factor of [27, Now replace φ by the other isomorphism φ ′ of k o,v -algebras such that φ ′ • φ −1 : (x, y) → (y, x) and replace δ v by −δ v = φ ′ (1, −1). We then get which proves the expected result.
We give another reason for the lemma above to be true for the possibly surprised reader.
Remark 9.18. It is in fact well known as a part of the local Langlands correspondence for as it is equal to the Langlands-Deligne constant Equality (9.19) can also be checked as follows. Using the notation of [26,Theorem 2.7], one has . The L-factors do not depend on the ordering of the representations, and a simple change of variable using the relation (9.3) gives [27]. Putting the different pieces together yields the equality we were looking for.
Remark 9.20. At p. 811 of [28], the author notices a sign ambiguity in the identification of the Asai epsilon factor with ǫ RS (s, π 1,v , π 2,v , ψ o,v ) due to the ordering of π 1,v and π 2,v . Lemma 9.17 or Remark 9.18 show that there is in fact no such ambiguity.
Remark 9.21. With the same assumptions as in Lemma 9.17, we also have the equalities L As (s, π v ) = L RS (s, π 1,v , π 2,v ) = L RS (s, π 2,v , π 1,v ) between local L-factors. Note that where the factors on the right hand side are the Langlands-Shahidi factors of [46]. It is known by [47] in the non-archimedean case, and by [46] in the archimedean case.

Global factors
As in the previous paragraph, k/k o is a quadratic extension of global fields of characteristic different from 2. We denote by A the ring of adeles of k and by A o that of k o . We suppose that all places of k o dividing 2, as well as all archimedean places in the number field case, are split in k.
We fix once and for all a non-trivial character ψ o of A o /k o and a non-zero element δ ∈ k such that tr k/k 0 (δ) = 0. Thus is a non-trivial character of A trivial on k + A o . Given a place v of k o , we denote by ψ o,v the local component of ψ o at v.
Let Π be a cuspidal automorphic representation of GL n (A) as in [7]. It decomposes as a restricted tensor product where v ranges over the set of all places of k o . When v is inert in k, then Π v is the local component of Π at the place of k above v. When v is split in k, and given an isomorphism Note that we have L As (s, Π v ) = L LS As (s, Π v ) = L Gal As (s, Π v ) for any place v of k o . See Theorem A.1 when v is inert, and Remark 9.21 when v is split.
The factors L As (s, Π v ) and ǫ RS As (s, Π v , ψ o,v ) have now been defined at all places of k o . We set where the products are taken over all places v of k o .
Note that by [48] and [31], the factor ǫ LS As (s, Π) is indeed independent of the character (9.22) and one has the functional equation L As (s, Π) = ǫ LS As (s, Π) · L As (1 − s, Π ∨ ). (9.23) In fact, we claim that with our normalization (see Definitions 9.10 and 9.15), we have L As (s, Π) = ǫ RS As (s, Π) · L As (1 − s, Π ∨ ). (9.24) Let us prove this claim. Whatever the place v of k o is, the local functional equation is of the form To be more precise, the left hand side term makes sense when the real part of −s is large enough, whereas the right hand side term makes sense when the real part of s is large enough, and both terms admit meromorphic continuations to C. It is these meromorphic continuation that are equal. Now let T be a finite set of places of k o , containing the set of archimedean places, such that for all v / ∈ T one has Taking the product of the equalities (9.25) for all v, we get On the other hand, we have (see Remark 9.16) However, the global root number ǫ(1/2, ω k/ko ) is equal to 1 by the dimension 1 case of the main result of [19]. By the assumption on T we get v∈T ǫ RS As (s, Π v , ψ o,v ) = ǫ RS As (s, Π) and (9.24) follows. In particular (9.23) and (9.24) imply: Theorem 9.26. Let Π be a cuspidal automorphic representation of GL n (A). Then ǫ RS As (s, Π) = ǫ LS As (s, Π).
Note that the functional equation of [28, Theorem 5] has a different epsilon factor and moreover is up to a sign. The presence of this sign is due to the fact that at an inert place v of k o , Kable takes the local factor ǫ As (s, Π v , ψ o,v , δ v ) whereas we take ǫ RS As (s, Π v , ψ o,v ). (i) k is split at all archimedean places (when k is a number field) and at all places dividing 2;

Cuspidal representations
(ii) one has k o,w ≃ F o and k w ≃ F.
We explain below how to realize a cuspidal representation π of G = GL n (F) as the local component at w of some suitable cuspidal automorphic representation of GL n (A). First, we realize its central character ω π as a local component of some character Ω of A × /k × .
Lemma 9.27. Let ω be a unitary character of F × , and u be a finite place of k o different from w. Then there exists a unitary automorphic character Ω : A × /k × → C × such that: (i) the local component of Ω at w is ω, where v ranges over all places of k o different from u, w and where k ×0 v is the maximal compact sub- w identifies with a locally compact subgroup of A × /k × U. By Pontryagin duality, ω extends to a unitary character Ω of A × /k × U, which satisfies the required conditions. Lemma 9.28. Let π be a unitary cuspidal representation of G, and u be a finite place of k o which is split in k. Then there is a cuspidal automorphic representation Π of GL n (A) such that: Proof. The proof follows that of [22, Appendice 1], adapted to our context, the reductive group of interest here being the restriction of GL n from k to k o . Let Ω be a unitary character as in Lemma 9.27 extending the central character ω π .
For each finite place v = u, w of k o , we let f v denote the complex function on GL n (k v ) supported on k × v GL n (O kv ) such that f v (zk) = Ω v (z) for all z ∈ k × v and k ∈ GL n (O kv ).
If k o is a number field and v is archimedean, we choose a smooth complex function f v on GL n (k v ), compactly supported mod the centre k × v , such that f v (1) = 1 and f v (zg) = Ω v (z)f v (g) for all elements z ∈ k × v and g ∈ GL n (k v ).
We let f w be a coefficient of π such that f w (1) = 1.
Finally we choose a smooth complex function f u on GL n (k u ), compactly supported mod the centre, such that f u (1) = 1 and f u (zg) = Ω u (z)f u (g) for all z ∈ k × u and g ∈ GL n (k u ), and of support small enough such that where f is the product of all the f v , as in [22, Appendice 1], top of p. 148.
We may also assume that f v (g −1 ) = f v (g) for all v and all g ∈ GL n (k v ).
Then there is a cuspidal automorphic representation Π of GL n (A) such that f v acts non-trivially on Π v for each place v of k o . In particular Π w ≃ π and Π v is unramified at every place different from w and u. Now let us consider a cuspidal representation π of G ≃ GL n (k w ). The character ω π | · | −s w is unitary for some s ∈ C, thus π 1 = π| det | −s w is unitary. Lemma 9.27 gives us a cuspidal automorphic representation Π 1 of GL n (A). Denoting by | · | the idelic norm on A × /k × , the cuspidal automorphic representation Π = Π 1 | det | s has a local component at w isomorphic to π, and all its local compotents at v = w, u are unramified.
We recalled in Remark 9.21 that when v is split (in particular when v = u), hence from Theorem 9.13 and Theorem 9.26, we get ǫ RS As (s, Π w , ψ o,w ) = ǫ LS As (s, Π w , ψ o,w ).
Thus we have proved: Theorem 9.29. Let π be a cuspidal representation of G = GL n (F) and ψ o be a non-trivial character of F o . Then ǫ RS As (s, π, ψ o ) = ǫ LS As (s, π, ψ o ).
Remark 9.31. When π is cuspidal and ω F/F 0 -distinguished, we may go in the opposite direction: applying [1, Theorem 1.1] together with (9.30) gives us the value of ǫ As (1/2, π, ψ o , δ) when π is cuspidal and ω F/F 0 -distinguished. Remark 9.32. It is shown in [1, Theorem 1.2] that the global Asai root number of a σ-self-dual cuspidal automorphic representation is 1. Hence, by Theorem 9.26, the same holds for the Asai factor defined via the Rankin-Selberg method. Globalizing local distinguished cuspidal representations as local components of distinguished cuspidal automorphic representations as in [43] or [20] and following the methods of [1], it is possible to prove that ǫ As (1/2, π, ψ o , δ) = 1 by global methods as well. However our proof in this paper has the advantage that it is purely local.
Remark 9.33. In his forthcoming work on the global refined Gan-Gross-Prasad conjectures for unitary groups, Beuzart-Plessis has already proved that ǫ LS As (s, π, ψ o ) and ǫ RS As (s, π, ψ o ) are equal up to a root of unity for any generic irreducible representation π. He explained to us that, by using a globalization argument similar to [25, Appendix A], one can deduce the equality of Asai epsilon factors for any tempered (hence generic) representation from the cuspidal case given by Theorem 9.29.

A Some remarks in positive characteristic
We use the notation of Section 2 and Paragraph 9.1. In particular, G denotes the group GL n (F) for some n 1, and we have defined Asai local L-factors L As (s, π), L LS As (s, π) and L Gal As (s, π) for all generic irreducible representations of G. We will first prove that these factors are all equal.
Theorem A.1. For any generic irreducible complex representation π of G, we have L As (s, π) = L Gal As (s, π) = L LS As (s, π).
Now we notice that the local results in [28,Section 3] hold in positive characteristic, and the global results of [28,Section 4] also hold in positive characteristic though written in characteristic 0 only. Indeed they refer to [17] which is for any global field. The main point is that [28,Theorem 5] is true for function fields, and its proof slightly simplifies because of the absence of archimedean places. This implies that, when F has characteristic p = 2, the equality L As (s, π) = L LS As (s, π) holds for any discrete series representation: the ingredients which make the proof of [5,Theorem 1.6] work are then all available. Once again, notice that its proof simplifies in the positive characteristic case as there are no archimedean places to worry about. Now notice that [34, Theorem 3.1] holds when F has characteristic p. Indeed its proof relies on [39, Theorem 3.1.2] which is for any non-archimedean local field of odd residual characteristic. Then the classification of generic distinguished representations in [35] relies only on the geometric lemma of Bernstein-Zelevinsky, the Bernstein-Zelevinsky explicit description of discrete series representations and their Jacquet modules, and the fact that a distinguished irreducible representation of G is σ-self-dual. All the aforementioned results are true in positive characteristic (different from 2 for the latter) hence the classification of [35] still holds when F has characteristic p.
Finally, the Cogdell-Piatetski-Shapiro method of derivatives to analyze the exceptional poles used in [33] still works in positive characteristic as well (for example the original paper [16] is written in arbitrary characteristic) hence the inductivity relation of L As (s, π) for any generic irreducible representation (see [33,Proposition 4.22]) follows. All in all, when F has characteristic p, we have L As (s, π) = L Gal As (s, π) for any generic irreducible representation.
We now prove that the dichotomy theorem of [28] and [2] holds when F has characteristic p.
Theorem A.2. Let π be a σ-self-dual discrete series representation of G. Then π is either distinguished or ω F/Fo -distinguished, but not both.
Similarly, using the formula p ′ (φ)(g) = Then, by uniqueness of the linear form L ′ corresponding to L , one has L ′ (y · φ) = χ(y)L ′ (φ), φ ∈ ind G H (τ ). (B.8) We arrive to the following result which we shall use many times hereunder.
Corollary B.9. The map L ′ → L ′ • p is an isomorphism of R-modules between:

B.2 A modular version of a result of Kable
In this subsection, we generalize a result of Kable ([28, Proposition 1]) to the case of smooth representations of GL n (F) with coefficients in a commutative ring with sufficiently many roots of unity of p-power order and in which p is invertible. In fact, we expand and simplify Kable's proof, appealing to Theorem B.4 when he appeals to Warner [53].
We go back to the main notation of the paper: G is the group GL n (F) where F/F o is a quadratic extension, σ is the Galois involution and P is the mirabolic subgroup of G. We also write G ′ for the group GL n−1 (F) considered as a subgroup of G in the usual way, and P ′ for the mirabolic subgroup of G ′ . Denoting by U the unipotent radical of P, one has the semi-direct product decomposition P = G ′ U.
We also assume that R is a commutative ring with unit, such that p is invertible in R and there is a non-trivial R-character ψ o of F o .
Let ψ U be the restriction to U of the standard σ-self-dual non-degenerate character ψ of N defined by (8.2) for some non-zero δ ∈ F × of trace 0.
Since p is invertible in R and G is locally pro-p, there is a non-zero right invariant measure dh on P ′ U with values in R, giving measure 1 to some compact open subgroup. Given any smooth representation τ of P ′ on an R-module V, we denote by τ ⊗ ψ U the representation of P ′ U defined by τ ⊗ ψ U : xu → ψ U (u)τ (x) for x ∈ P ′ and u ∈ U. Following [6], we set Φ + (τ ) = ind P P ′ U (τ ⊗ ψ U ).
This defines a functor from smooth R-representations of P ′ to smooth R-representations of P. Note that, since we use the unnormalized version of the functor Φ + as in [6], we do not have to worry about the existence of a square root of q in R.
We will write ν and ν o for the unramified characters g → | det(g)| and g → | det(g)| o , respectively.
Proposition B.10. For any smooth R-representation τ of P ′ and any character χ of P σ , one has an isomorphism: of R-modules.
Proof. First, we apply Corollary B.9 with G = P, H = P ′ U and ρ = τ ⊗ψ U . Since the character δ P ′ U associated with P ′ U is equal to ν 2 , we get an isomorphism of R-modules from Hom P σ (Φ + (τ ), χ) to the space of all linear forms T on C ∞ c (P, V) such that: for all g o ∈ G ′σ , u o ∈ U σ , g ∈ P ′ , u ∈ U and f ∈ C ∞ for all f ∈ C ∞ c (P, V), x ∈ P ′ G ′σ and p ′ ∈ P ′ , where the second equality follows from the fact that the character δ P ′ associated with P ′ is ν, and the third one from the fact that P ′ normalizes ψ U . It follows that A * induces an isomorphism of R-modules between: Now consider the map (x, y) → x −1 y from P ′ × G ′σ onto P ′ G ′σ . It identifies P ′ G ′σ with the homogeneous space P ′σ \(P ′ × G ′σ ) where P ′σ = P ′ ∩ G ′σ is diagonally embedded in P ′ × G ′σ . This thus identifies the space C ∞ c (P ′ G ′σ , V) with the compact induction ind P ′ ∩G ′σ P ′σ (1 ⊗ V) where 1 ⊗ V denotes the trivial representation of P ′σ on V. Namely, f ∈ C ∞ c (P ′ G ′σ , V) identifies with the function φ on P ′ ∩ G ′σ defined by φ(x, y) = f (x −1 y) for (x, y) ∈ P ′ ∩ G ′σ . This thus gives us an isomorphism of R-modules between: (i) the space of linear forms S on C ∞ c (P ′ G ′σ , V) satisfying (B.16) and (B.17), and (ii) the space of linear forms Q on ind P ′ ∩G ′σ P ′σ We now apply Corollary B.9 again, with G = P ′ × G ′σ , H = P ′σ and ρ = 1 ⊗ V. Since the character δ P ′σ associated with P ′σ is equal to ν o , we get an isomorphism of R-modules between: (i) the space of linear forms Q on ind P ′ ∩G ′σ for all φ ∈ C ∞ c (P ′ × G ′σ , V), p ′ o ∈ P ′σ and (p ′ , g ′ o ) ∈ P ′ × G ′σ .
Finally, one verifies that the map ϕ → ϕ(1, 1) from ind P ′ ×G ′σ P ′ ×G ′σ (τ ⊗χ −1 ) to V induces an isomorphism of R-modules between the space of linear forms t as above and Hom P ′σ (τ, χν o ), which ends the proof of the proposition.

B.3 A modular version of a result of Ok for cuspidal representations
In this subsection, we generalize a result of Ok ([39, Theorem 3.1.2]) on irreducible complex representations of G = GL n (F). More precisely, using Proposition B.10, we prove it for any cuspidal representation of G with coefficients in an algebraically closed field of characteristic different from p.
In this subsection, R is an algebraically closed field of characteristic different from p.
Proof. By [6] and [52, III.1], the restriction of π to P is isomorphic to ind P N (ψ), where ψ is the standard σ-self-dual non-degenerate character of N which has been fixed at the beginning of B.2. This induced representation can be written (Φ + ) n−1 Ψ + (1), where 1 denotes the trivial character of the trivial group, Ψ + (1) is the trivial character of the (trivial) mirabolic subgroup P 1 (F) and Φ + is the functor which has been defined in B.2. Applying n − 1 times Proposition B.10, we get the expected result.