A model is a fundamental concept in representation theory and integral representations. Typically, a model for a representation arises from an equivariant functional, which allows one to realize the representation in a space of complex-valued functions with some natural geometric properties. The useful cases are when the functional is unique up to scaling, and the class of representations affording the model is broad. One important example is the Whittaker model, which has had a profound impact on the study of representations with a vast number of applications, perhaps most notably Shahidi's theory of local coefficients.
Let F be a local field of characteristic 0. In this short note we discuss a new class of models, (k, c) models, for representations of GL kc = GL kc (F ), which first appeared in the construction of the generalized doubling integral ( [CFGK19]) in the context of generalized Speh representations. Our main result: Theorem 4, is that the local generalized Speh representation (defined in [Jac84]) of GL kc corresponding to a unitary generic representation τ of GL k admits a unique (k, c) model. Our result is in fact stronger: We construct a map ρ c from irreducible generic representations τ of GL k to (k, c) representations ρ c (τ ). This general context is essential for the analysis of the local generalized doubling integrals ( [CFGK19,GK]) when data are ramified or archimedean. We also discuss two realizations of the (k, c) functional, which are also important for the study of such integrals.
The main application of Theorem 4 concerns the generalized doubling integrals. This theorem completes the proof that the global integral of [CFGK19] is Eulerian. See Corollary 5. In loc. cit. uniqueness was only proved when data are unramified, thus producing an "almost Eulerian" integral, i.e., only separating out the unramified places (cf. [CFGK19, (3.1)]). The existence of an Euler product is important for the development of the local theory, and Theorem 4 also plays a key role there, in the functional equation (see [GK]).
Let F be a local field of characteristic 0. Identify linear algebraic F -groups with their F points, i.e., GL l = GL l (F ). Fix the Borel subgroup B GL l = T GL l ⋉ N GL l of upper triangular invertible matrices in GL l , where T GL l is the diagonal torus. For a d parts composition β = (β 1 , . . . , β d ) of l, P β = M β ⋉ V β denotes the corresponding standard parabolic subgroup, where V β < N GL l . The unipotent subgroup opposite to V β is denoted V - β and δ P β is the modulus character of P β . For an integer c ≥ 0, βc = (β 1 c, . . . , β d c) is a composition of lc. Let w β be the permutation matrix consisting of blocks of identity matrices I β 1 , . . . , I β d , with I β i ∈ GL β i on its anti-diagonal, beginning with I β 1 on the top right, then
). We use τ β to denote a representation of M β , where τ β = ⊗ d i=1 τ i (τ i is then a representation of GL β i ). Let Mat a×b and Mat a denote the spaces of a×b or a×a matrices. For g ∈ Mat a×b , t g is the transpose of g. The trace map is denoted tr. For x, y
All representations in this work are by definition complex and smooth. A generic representation of GL l will be admissible, by definition. Over archimedean fields, by an admissible representation we mean admissible Fréchet of moderate growth. We use the smooth and normalized induction functor.
Let U < R < GL l be closed subgroups such that U is a unipotent subgroup, and fix a character ψ of U. For a representation σ of R on a space V, the Jacquet module J U,ψ (π) is the quotient of V by the subspace spanned by {π(u)ξ -ψ(u)ξ ∶ ξ ∈ V, u ∈ U} over p-adic fields, and by the closure of this subspace for archimedean fields. Then J U,ψ (π) is a representation of R and we normalize the action as in [BZ77,1.8].
When the field is p-adic, an entire (resp., meromorphic) function f (ζ 1 , . . . , ζ m ) ∶ C m → C will always be an element of C[q ∓ζ 1 , . . . , q ∓ζm ], (resp., C(q -ζ 1 , . . . , q -ζm )).
2.1. Definition. Let k, c ≥ 1 be integers. A partition σ = (a 1 , . . . , a l ) of kc such that a i > 0 for all i identifies a subgroup V (σ) < N GL kc as follows. Consider the multi-set of integers
and let p σ be the kc-tuple obtained by arranging Λ σ in decreasing order. For any x ∈ F * , put x pσ = diag(x pσ(1) , . . . , x pσ(kc) ) ∈ T GL kc . The one-parameter subgroup {x pσ ∶ x ∈ F * } acts on the Lie algebra of N GL kc by conjugation, and V (σ) is the subgroup generated by the weight subspaces of weight at least 2. (This is not the subgroup V β defined for compositions). Let G σ < GL kc denote the centralizer of {x pσ ∶ x ∈ F * }, it acts on the set of characters of V (σ). Under this action there is a unique
in G σ is of the same type as G σ,ψo over C. For further details see [Gin06], [CM93,5] and [Car93]. Let V (σ) gen denote the set of generic characters of V (σ). If σ ′ is another partition of kc, write σ ′ ≿ σ if σ ′ is greater than or non-comparable with σ under the natural partial ordering.
For σ = (k c ), V (σ) = V (c k ) and M (c k ) acts transitively on V (σ) gen . We fix ψ ∈ V (σ) gen by taking a nontrivial additive character ψ of F and extending it to a character of V (c k ) by
We say that an admissible representation ρ of GL kc is m-weakly (k, c), if it satisfies the vanishing condition of Definition 1 and m = dim Hom V (c k ) (ρ, ψ) satisfies 1 ≤ m < ∞. Also note that ρ is (1, c) if and only if it is a character of GL c .
A (k, c) functional on ρ (with respect to ψ) is a nonzero element of Hom V (c k ) (ρ, ψ). If ρ is of type (k, c), the space of such functionals is one dimensional. The resulting model (which is unique by definition) is called a (k, c) model, and denoted W ψ (ρ). If λ is a fixed (k, c) functional, W ψ (ρ) is the space of functions g ↦ λ(ρ(g)ξ) where g ∈ GL kc and ξ is a vector in the space of ρ. Then W ψ (ρ) is a quotient of ρ, and when ρ is irreducible W ψ (ρ) ≅ ρ.
The following is an heredity-type result for (k, c) representations.
Proposition 2. For 1 ≤ i ≤ d, let ρ i be a (k i , c) representation. If F is archimedean we further assume k 1 = . . . = k d = 1 and for each i, ρ i = τ i ○ det for a quasi-character τ i of F * and det defined on GL c . Then ρ = Ind
Proof. We need to prove Hom V (β ′ ) (ρ, ψ ′ ) = 0 for any partition β ′ ≿ (k c ) and character ψ ′ ∈ V (β ′ ) gen , and dim Hom V (c k ) (ρ, ψ) = 1. We consider m i -weakly representations at the end of the proof.
We will use the theory of detivatives of Bernstein and Zelevinsky [BZ76, BZ77] over p-adic fields, its partial extension to archimedean fields by Aizenbud et. al. [AGS15a,AGS15b], and the relation between derivatives and degenerate Whittaker models developed (over both fields) by Gomez et. al. [GGS17]. Let P l be the subgroup of matrices g ∈ GL l with the last row (0, . . . , 0, 1) (P l < P (l-1,1) ) and let ψ l be the character of V (l-1,1) given by ψ l (( I l-1 v 1 )) = ψ(v l-1 ). Then we have the functor Φ -from (smooth) representations of P l to representations of P l-1 given by Φ -(ϱ) = J V (l-1,1) ,ψ l (ϱ). For 0 < r ≤ l, the r-th derivative of a representation ϱ of GL l is defined over p-adic fields by ϱ (r) = J V (l-1,1) ((Φ -) r-1 (ϱ| P l )), and over archimedean fields by ϱ (r) = ((Φ -) r-1 (ϱ| P l ))| GL n-r (more precisely this is the pre-derivative, we use the term derivative for uniformity). Also ϱ (0) = ϱ. The highest derivative of ϱ is the representation ϱ (r 0 ) such that ϱ (r 0 ) ≠ 0 and ϱ (r) = 0 for all r > r 0 .
According to the definition of (k i , c) representations and [GGS17, Theorems E, F] (which also apply over p-adic fields), the highest derivative of ρ i is ρ
), where over archimedean fields ρ The only difference regarding m i -weakly representations, is that after taking the highest derivative of ρ for c times, we obtain a ∏ i m i dimensional space. Then again by [GGS17, Theorems E, F], dim Hom
Remark 3. The result for p-adic fields is stronger, because we have a general "Leibniz rule" for derivatives ([BZ77, Lemma 4.5]).
Let τ be an irreducible generic representation of GL k (for k = 1, this means τ is a quasi-character of F * ). It is of type (k, 1), by the uniqueness of Whittaker models and because (k) is the maximal unipotent orbit for GL k . For any c, we construct a (k, c) representation ρ c (τ ) as follows. First assume τ is unitary. For ζ ∈ C c , consider the intertwining operator
Given a section ξ of Ind
) )ξ is defined for Re(ζ) in a suitable cone by the absolutely convergent integral
then by meromorphic continuation to C c . Let ρ c (τ ) be the image of this operator at
Since τ is unitary, this image is well defined and irreducible by Jacquet [Jac84, Proposition 2.2] (see also [MW89,I.11]) and ρ c (τ ) is the unique irreducible quotient of
and the unique irreducible subrepresentation of
Note that when τ is unitary, the definition as the image of an intertwining operator agrees with the definition (2.4). Indeed in this case by Tadić [Tad86] and Vogan [Vog86] we have
In particular, e.g., ρ c (Ind
where the l.h.s. (left-hand side) is defined as the image of the intertwining operator. Hence if a 1 > . . . > a d are the d ≤ d ′ distinct numbers among r 1 , . . . , r d ′ , τ i is the tempered representation parabolically induced from ⊗ 1≤j≤d ′ ∶ r j =a i τ ′ j and β = (a 1 , . . . , a d ),
which agrees with (2.4). Moreover, (2.6) implies that (2.4) also holds when a 1 ≥ . . . ≥ a d . For example, if τ is irreducible unramified tempered, τ = Ind GL k B GL k (⊗ k i=1 τ i ) for unramified unitary characters τ i of F * . By definition ρ c (τ ) is the unique irreducible unramified quotient of (2.2), but by [MW89, I.11], ρ c (τ ) = Ind GL kc
We extend the definition to certain unramified principal series, which are not necessarily irreducible. Assume τ = Ind GL k
where τ i are as above (e.g., unitary) but a 1 ≥ . . . ≥ a k (τ is not a general unramified principal series because of the order). In this case τ still admits a unique Whittaker functional. We define ρ c (τ ) = Ind GL kc
, which is a (k, c) representation by Proposition 2. Transitivity of induction and the example in the last paragraph imply that this definition coincides with (2.4), when τ is irreducible (in which case the order of a i does not matter). Proof. First assume F is non-archimedean. We start by proving the result for squareintegrable representations τ . By Zelevinsky [Zel80], τ can be described as the unique irreducible subrepresentation of
To this end, by [BSS90, SS90] (see also [GS13, Corollary 4.2.5]), each ρ c (τ i ) with
itself is also a quotient of a degenerate principal series Ind GL kc
Corollary 5. The global generalized doubling integrals of [CFGK19,GK] are Eulerian, for decomposable data.
Proof. The proof follows from Theorem 4 and the results of [CFGK19,GK]. In order to provide some details, we switch, in this proof alone, to a global setting. Let F be a global number field with a ring of adeles A and fix a nontrivial additive character ψ of F /A.
Let τ be an irreducible cuspidal automorphic representation of GL k (A), and denote the generalized Speh representation of GL kc (A) corresponding to τ by ρ c (τ )
where ξ is an automorphic form in the space of ρ c (τ ) and ψ is defined by (2.1). By Theorem 4 this functional is Eulerian: One can choose for each place ν of F a local (k, c) functional λ ν on ρ c (τ ν ), such that for any decomposable vector ξ = ⊗ ν ξ ν , Λ(ξ) = ∏ ν λ ν (ξ ν ).
The generalized doubling integral was defined in [CFGK19,GK] for several reductive groups G. Since the details of the construction are similar, we take G = SO c . Define H = SO 2kc . Fix a Siegel parabolic subgroup P < H and a maximal compact subgroup K < H. Let E(h; s, f ) be the Eisenstein series attached to a standard K-finite section f of Ind
One can choose a unipotent subgroup U < H and a generic character ψ U of U(F )/U(A), such that the Fourier coefficient E U,ψ U of the series along (U, ψ U ) is an automorphic form on G(A) × G(A).
Let π be an irreducible cuspidal automorphic representation of G(A), and let ϕ 1 and ϕ 2 be two cusp forms in the space of π. The global integral is defined by
where g ↦ ι g is an involution of G(A) and (g 1 , g 2 ) is the embedding of G × G in H. By [GK, 3.2 and (3.8)] (see also [CFGK19, Theorem 1]), for Re(s) ≫ 0 we have
Here U 0 < U, ⟨, ⟩ is the standard inner product and δ ∈ G(F ). Consider decomposable vectors f = ⊗ ν f ′ ν , ϕ 1 and ϕ 2 . Then (by Theorem 4) Λ○f = ∏ ν f ν where for each ν, f ν = λ ν ○f ′ ν belongs to the space of Ind H(Fν ) P (Fν ) (| det | s-1/2 W ψν (ρ c (τ ν ))), and ⟨ϕ 1 , π(g)ϕ 2 ⟩ = ∏ ν ω ν (g ν ) where ω ν is a matrix coefficient of π ∨ ν . The local integral Z(s, ω ν , f ν ) is of the form (2.8) but with local data and we integrate over G(F ν ) and U 0 (F ν ). We obtain Z(s, ϕ 1 , ϕ 2 , f) = ∏ ν Z(s, ω ν , f ν ), as claimed.
For an admissible representation ϱ of GL l , let ϱ * (g) = ϱ(J l t g -1 J l ) where
Proof. The first assertion follows because ρ c (τ ) is a quotient of (2.2), hence both ρ c (τ ) ∨ and ρ c (τ ∨ ) are irreducible subrepresentations of Ind
, but there is a unique such. The general case follows from the definition, the tempered case and the fact that for any composition β of l, (Ind
While (2.4) may be reducible, it is still of finite length and admits a central character. We mention that since the Jacquet functor is exact over non-archimedean fields, and the generalized Whittaker functor of [GGS17] is exact over archimedean fields ([GGS17, Corollary G]), ρ c (τ ) admits a unique irreducible subquotient which is a (k, c) representation.
3.1. Explicit (k, c) functionals from compositions of k. Let τ be an irreducible generic representation of GL k , where k > 1. If τ is not supercuspidal, it is a quotient of Ind GL k P β (τ β ) for a nontrivial composition β of k and an irreducible generic representation τ β (e.g., one can take τ β to be supercuspidal). A standard technique for realizing the Whittaker model of τ is to write down the Jacquet integral on the induced representation (this integral stabilizes in the p-adic case). Since Ind GL k P β (τ β ) also affords a unique Whittaker model, the functional on the induced representation factors through τ . One may also twist the inducing data τ β using auxiliary complex parameters, to obtain an absolutely convergent integral which admits an analytic continuation in these parameters. See e.g., [Sha78,JPSS83,Sou93,Sou00].
We generalize this idea to some extent, for (k, c) functionals.
. . ≥ a d and each τ 0,i is square-integrable, or τ is the essentially square-integrable quotient of Ind GL k P β (τ β ) and τ β is irreducible supercuspidal (this includes the case β = (1 k ), i.e., τ β is a character of T GL k , over any local field), then ρ c (τ ) is a quotient of Ind GL kc P βc (⊗ i ρ c (τ i )).
Proof. In the first case, this is true by (2.4) and (2.6). For the essentially square-integrable case, over an archimedean field the result follows from [GS13, Corollary 4.2.5] (see the proof of Theorem 4), and if F is p-adic from [Tad86, Theorem 7.1].
Take β as in the lemma. If F is archimedean assume β = (1 k ), i.e., ρ c (τ ) is a quotient of a degenerate principal series, which is always possible (see the proof of Theorem 4). Denote β = (β 1 , . . . , β d ), β ′ = (β d , . . . , β 1 ) and consider the following Jacquet integral
where ξ lies in the space of Ind GL kc P βc (⊗ d i=1 W ψ (ρ c (τ i ))) and regarded as a complex-valued function, and ψ is the restriction of (2.1) to V β ′ c . The integral (3.1) is formally a (k, c) functional on the full induced space. Twist the inducing data and induce from (k, c) functionals from compositions of c. Let τ be an irreducible generic representation of GL k , and assume an unramified twist of τ is unitary. In this section we construct (k, c) functionals on ρ c (τ ) using compositions of c. Fix 0 < l < c. Since now both ρ l (τ ) and ρ c-l (τ ) embed in the corresponding spaces (2.3), ρ c (τ ) is a subrepresentation of
Both (k, l) and (k, cl) models exist by Theorem 4. We may regard vectors in the space of (3.2) as complex-valued functions. We construct a (k, c) functional on (3.2), and prove it does not vanish on any of its subrepresentations, in particular on ρ c (τ ).
), where v 1 i,j ∈ Mat l and v 4 i,j ∈ Mat c-l . For t ∈ {1, . . . , 4}, let V t < V (c k ) be the subgroup obtained by deleting the blocks v t ′ i,j for all i < j and t ′ ≠ t, and
Example 8. For c = 2 (then l = 1) and k = 3,
Consider the functional on the space of (3.2),
This is formally a (k, c) functional. Indeed, the conjugation v ↦ κ v of v ∈ V (c k ) takes the blocks v 1 i,j onto the subgroup V (l k ) embedded in the top left kl × kl block of M (kl,k(c-l)) . Then these blocks transform by the (k, l) functional realizing W ψ (ρ l (τ )); v 2 i,j is taken to V (kl,k(c-l)) and ξ is left-invariant on this group; and after the conjugation, the blocks v 4 i,j form the subgroup V ((c-l) k ) embedded in the bottom right k(cl) × k(cl) block, and transform by the (k, cl) functional realizing W ψ (ρ c-l (τ )). Thus V (c k ) transforms under (2.1). Also note that this conjugation takes
be obtained from y be zeroing out all the blocks in (3.4) except v 3 i,j . Also let X < V (kl,k(c-l)) be the subgroup of matrices x whose top right kl × k(cl) block is ⎛ ⎝
Define x i,j similarly to y i,j . Let X i,j and Y i,j be the respective subgroups of elements. Then ξ(y i,j x i,j ) = ψ(tr(v 3 i,j x 3 i,j ))ξ(y i,j ). (3.5)
We show that (3.3) can be used to realize W ψ (ρ c (τ )).
Lemma 9. For 0 < l < c, realize ρ c (τ ) as a subrepresentation of (3.2). The integral (3.3) is absolutely convergent, and is a (nonzero) (k, c) functional on ρ c (τ ).
Proof. The proof technique is called "root elimination", see e.g., [Sou93, Proposition 6.1], [Sou93, 7.2] and [Jac09, 6.1] (also the proof of [LR05, Lemma 8]). We argue by eliminating each y i,j separately, handling the diagonals left to right, bottom to top: starting with y k-1,k , next y k-2,k-1 , ..., up to y 1,2 , then y k-2,k , y k-3,k-1 , etc., with y 1,k handled last. Let W be a subrepresentation of (3.2). For ξ in the space of W and a Schwartz function φ on Mat l×(c-l) (over p-adic fields, Schwartz functions are in particular compactly supported), define
ξ(gy i,j ) φ(v 3 i,j ) dy i,j .
Here φ is the Fourier transform of φ with respect to ψ ○ tr. By smoothness over p-adic fields, or by the Dixmier-Malliavin Theorem [DM78] over archimedean fields, any ξ is a linear combination of functions ξ i,j , and also of functions ξ ′ i,j . Using (3.5), the definition of the Fourier transform and the fact that V is abelian we obtain, for (i, j) = (k -1, k),
where v ○ ∈ V does not contain the block of v 3 i,j . We re-denote ξ = ξ ′ i,j , then proceed similarly with (i, j) = (k -2, k -1). This shows that the integrand is a Schwartz function on κ V , thus the integral (3.3) is absolutely convergent. At the same time, the integral does not vanish on W because in this process we can obtain ξ(I kc ). Over archimedean fields the same argument also implies that (3.3) is continuous (see [Sou95, 5, Lemma 2, p. 199]).
Over p-adic fields we provide a second argument for the nonvanishing part. Choose ξ 0 in the space of W with ξ 0 (I kc ) ≠ 0. Define for a (large) compact subgroup X < X, the function
which clearly also belongs to the space of W. We show that for a sufficiently large X , ∫ V ξ 1 ( κ v) dv = ξ(I kc ). Put y = κ v. We prove ξ 1 (y) = 0, unless y belongs to a small compact neighborhood of the identity, and then ξ 1 (y) = ξ(I kc ). We argue by eliminating each y i,j separately, in the order stated above. Assume we have zeroed out all blocks on the diagonals to the left of y i,j , and below y i,j on its diagonal. Let B denote the set of indices (i ′ , j ′ ) of the remaining y i ′ ,j ′ and B ○ = B -(i, j). Denote X i,j = X ∩ X i,j and assume that if
and the (k, l) and (k, cl) characters (2.1) are trivial on u. Therefore by (3.5),
The second integral vanishes unless v 3 i,j is sufficiently small, then this integral becomes a nonzero measure constant. Moreover, if X i,j is sufficiently large with respect to ξ and X i ′ ,j ′ for all (i ′ , j ′ ) ∈ B ○ , then for any x ∈ X ○ , y i,j x = xz where z belongs to a small neighborhood of the identity in GL kc , on which ξ is invariant on the right. Therefore we may remove y i,j This representation contains (3.6) as an unramified subrepresentation. We show m(ζ, κ)ξ 0 (ζ, ⋅) satisfies the required properties for the prescribed ζ. We may decompose m(ζ, κ) into rank-1 intertwining operators on spaces of the form
According to the Gindikin-Karpelevich formula ([Cas80, Theorem 3.1]), each intertwining operator takes the normalized unramified vector in this space to a constant multiple of the normalized unramified vector in its image, and this constant is given by 1
.
Since Re(-ζ i + ζ j ) ≤ 0 and -l + l ′ ≤ -1, if the quotient has a zero or pole, then 1q -s τ i (ϖ)τ -1 j (ϖ) = 0 for Re(s) ≥ 1, contradicting our assumption. Therefore m(ζ, κ)ξ 0 (ζ, ⋅) is well defined and nonzero, and because it is unramified, it also belongs to (3.6).
Integral (3.1) is also absolutely convergent for Re(ζ) in a cone Re(ζ 1 ) ≫ . . . ≫ Re(ζ k ), which depends only on the inducing characters. The proof is that of the known result for similar intertwining integrals.
Lemma 11. In the domain of absolute convergence of (3.1) and in general by meromorphic continuation, for any meromorphic section ξ on V (τ, c),
Here m(ζ, κ)ξ belongs to the space obtained from (3.7) by applying the (k, l) and (k, cl) functionals (3.1) on the respective factors of P (kl,k(c-l)) .
Proof. Using matrix multiplication we see that w (c k ) = κ -1 diag(w (l k ) , w ((c-l) k ) )κ. The character ψ is trivial on V 2 . Thus in its domain of absolute convergence integral (3.1) equals
The integrals dv 1 dv 4 constitute the applications of (k, l) and (k, cl) functionals, e.g., κ V 1 = diag(V (l k ) , I k(c-l) ) (see after (3.3)). Now combine this with the proof of Lemma 10.
Combining this result for ξ 0 with Lemma 10, we obtain a result analogous to Lemma 9.
3.4. Equivariance property under GL △ c . Let g ↦ g △ be the diagonal embedding of GL c in GL kc . Since dim Hom V (c k ) (ρ c (τ ), ψ) = 1, a (k, c) functional on ρ c (τ ) translates under the action of GL △ c by a character. The following proposition explicates this character. Lemma 12. Let λ be a (k, c) functional on ρ c (τ ) and ξ be a vector in the space of ρ c (τ ). For any g ∈ GL c , λ(ρ c (τ )(g △ )ξ) = τ (det(g)I k )λ(ξ).
Proof. The claim clearly holds for c = 1, since then g △ belongs to the center of GL k . Let c > 1. We prove separately that λ(ρ c (τ )(t △ )ξ) = τ (det(t))λ(ξ) for all t ∈ T GLc and λ(ρ c (τ )(g △ )ξ) = λ(ξ) for all g ∈ SL c . Since all (k, c) functionals are proportional, we may prove each equivariance property using a particular choice of functional.
First take t = diag(t 1 , . . . , t c ). Assume τ is irreducible essentially tempered. Consider the (k, c) functional (3.3) with l = 1 < c. Conjugate V by t △ , then κ (t △ ) = diag(t 1 I k , t ′ △ ) with t ′ = diag(t 2 , . . . , t c ) and t ′ △ ∈ GL k(c-1) . The change to the measure of V is δ -1/2+1/(2k) P (k,k(c-1)) ( κ t) and the result now follows using induction. The case of irreducible generic τ is reduced to the essentially tempered case using (3.1) and note that δ P (βc) (t △ ) = 1 for any composition β of k. If τ is a reducible unramified principal series we again compute using (3.1).
It remains to consider g ∈ SL c . By definition the Jacquet module of ρ c (τ ) with respect to V (c k ) and (2.1) is one dimensional (whether τ is irreducible or unramified principal series), hence SL △ c acts trivially on the Jacquet module, and the result follows. The above property is useful for the study of integrals involving (k, c) models. For example, let π 1 and π 2 be irreducible admissible representations of GL c , τ 1 and τ 2 be irreducible generic representations of GL k , and s ∈ C. Let V (s, τ 1 × τ 2 ) be the space of the representation Ind GL 2kc P (kc,kc) (| det | s W ψ (ρ c (τ 1 )) ⊗ | det | -s W ψ (ρ c (τ 2 ))). Denote U = V (c k-1 ,2c,c k-1 ) and fix a character ψ U of U whose stabilizer in M (c k-1 ,2c,c k-1 ) is isomorphic to GL c × GL c . Let D = U ⋊ (GL c × GL c ). The study of the generalized doubling integral in this setup involves the space Hom D (V (s, τ 1 × τ 2 ), ψ U ⊗ π 1 ⊗ π 2 ) (see [CFGK19,GK]). Lemma 12 can be used to determine the requirements on the central chatecters of π i and τ j , in order to ensure this space is nontrivial. This lemma is also important for the determination of the equivariance properties of the doubling integral for representations of SO 2c+1 × GL k with respect to varying the character ψ of F . In that case, if ψ is replaced by ψ b where ψ b (x) = ψ(bx), b ∈ F * , Lemma 12 is applied with g = diag(I c , b -1 , I c ).