Abstract
Abstract. Let F be a p-adic field of characteristic 0, and let M be an F -Levi subgroup of a connected reductive
GLn i for positive integers r and n i . We prove that the Plancherel measure for any unitary supercuspidal representation of M (F ) is identically transferred under the local Jacquet-Langlands type correspondence between M and its F -inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups SO 4n and Sp 4n under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple F -group of type E 6 or E 7 , and a connected reductive F -group of type An, Bn, Cn or Dn.
Introduction
The theory of Plancherel measures is well known for real groups [1, 22, 29] . Arthur gave an explicit formula of the Plancherel measure in terms of Artin factors [1] . It follows from the local Langlands correspondence for real groups [36] that Plancherel measures are invariant on L-packets, and they are preserved by inner forms. Plancherel measures thus turn out to be identical if they are associated to the same L-parameter.
For p-adic groups, however, the behavior of the Plancherel measure is not completely understood. Although the Lefschetz principle [20] conjectures that what is true for real groups is also true for p-adic groups, the L-packet invariance of the Plancherel measure is currently known only for some cases [27, 42, 15, 16, 17, 2, 14, 10] . Also, it was proved that the Plancherel measures are preserved by p-adic inner forms for the following cases in characteristic 0. Arthur and Clozel proved the argument for discrete series representations under the local Jacquet-Langlands correspondence between GL n and its inner forms [3] . The author also verified it for depth-zero supercuspidal representations associated to tempered and tame regular semi-simple elliptic L-parameters with an unramified central character between an unramified group and its inner forms [10] . The approaches in both of two results are based on p-adic harmonic analysis. For the groups SO 4n and Sp 4n , using a local to global argument, Muić and Savin proved that Plancherel measures for unitary supercuspidal representations are preserved under the local Jacquet-Langlands correspondence between the Siegel Levi subgroup and its inner forms [37] . In a similar way, Gan and Tantono identically transferred Plancherel measures attached to supercuspidal representations having the same L-parameter from the Levi subgroup GL r × GSp 4 of GSp 2r+4 to its inner forms [17] .
The purpose of this paper is to prove that Plancherel measures attached to unitary supercuspidal representations are preserved under the local Jacquet-Langlands type correspondence (defined below), assuming a working hypothesis that Plancherel measures are invariant on a certain set. To be precise, let F denote a p-adic field of characteristic 0, and let M be an F -Levi subgroup of a connected reductive F -split group G such that
GL ni for positive integers r and n i . Let G ′ be an F -inner form of G, and let M ′ be an F -Levi subgroup of G ′ that is an F -inner form of M . Then, M ′ satisfies the following property
1991 Mathematics Subject Classification. Primary 22E50, Secondary 11F70, 22E55, 22E35. Key words and phrases. Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence.
Here D di denotes a central division algebra of dimension d Given τ ∈ E
• u (M (F )), we have σ ∈ E
• u ( M (F )) such that τ is isomorphic to an irreducible constituent of the restriction σ| M(F ) . Denote by Π σ (M (F )) the set of equivalence classes of all irreducible constituents of σ| M(F ) . Note that the set Π σ (M (F )) does not depend on the choice of σ and is contained in E • u (M (F )). On the other hand, we have a unique σ ′ ∈ E
• u ( M ′ (F )) corresponding to σ under the local Jacquet-Langlands correspondence. Let Π σ ′ (M ′ (F )) have the corresponding meaning for the F -inner form M ′ . We say that any two representations σ ∈ Π σ (M (F )) and σ ′ ∈ Π σ ′ (M ′ (F )) are under the local Jacquet-Langlands type (JL-type) correspondence. This will be defined more generally in Definition 4.5.
Fix a representative w ∈ G(F ) of a Weyl elementw such thatw(θ) ⊆ ∆. Here ∆ denotes the set of simple roots of A 0 in G, A 0 denotes the split component of a minimal F -Levi subgroup M 0 of G, and θ denotes the subset of ∆ such that M = M θ . We denote by a
for any ν ∈ a * M,C ≃ a * M ′ ,C . As a corollary to Theorem 1.2, we obtain the invariance of Plancherel measures on the set Π σ (M (F )).
Our approach to prove Theorem 1.2 is a general version of the local to global argument by Muić and Savin in [37] . First, given local data F , G, G ′ , M , M ′ , M and M ′ above, the following theorem allows us to construct a number field F, F-groups G, G ′ , M, M ′ , M and M ′ with prescribed local behavior.
Theorem 1.3 (Theorem 5.4). Let F be a p-adic field of characteristic 0, let G be a connected reductive quasi-split F -group and let G ′ be its F -inner form. Then, there exist a number field F, a non-empty finite set S of finite places of F, a connected reductive quasi-split F-group G and its F-inner form G
′ such that
∈ S including all the archimedean places,
This theorem is proved by local and global cohomological results in [30, 38] . Second, we find a finite set V ⊇ S and two cuspidal automorphic representations as described in the following Proposition. We prove Proposition 1.4 by using the result of Henniart [24] , the global Jacquet-Langlands correspondence [6] and the result of Hiraga and Saito [25] .
Next, we obtain equalities (6.2) and (6.3) in Section 6, which consist of a product of Plancherel measures at places in V and a quotient of the local Langlands L-functions for unramified representations. The property (c) of Proposition 1.4 allows us to cancel all local factors appearing outside V . For the places in V − S, we use the results of Keys and Shahidi in [27] and of Arthur [1] . We thus retain only Plancherel measures appearing inside S as in equation (6.4) . The hypothesis in Theorem 1.2 is needed here to identify all the Plancherel measures attached to π
Finally, from the fact that Plancherel measures are holomorphic and non-negative on the unitary axis, we deduce Theorem 1.2.
We remark that, our construction in Theorem 1.3 can be applied to any F -Levi subgroup M of any connected reductive group over a p-adic field of characteristic 0, including the case when M ≃ GL n in [37] (cf. Remark 5.6). Further, Proposition 1.4 extends the global Jacquet-Langlands correspondence for GL n to a connected reductive F -split group M satisfying condition (1.1).
As applications of Theorem 1.2, we transfer the reducibility of the induced representations and the edges of complementary series from unitary supercuspidal representations of maximal F -Levi subgroups under the local JL-type correspondence. We also prove that the reducibility and the edges are invariant on the set Π σ (M (F )). Further, our work can be applied to a simply connected simple F -group of type E 6 or E 7 , and a connected reductive F -group of type A n , B n , C n or D n . This paper is organized as follows. In Section 2, we recall basic notions, terminologies and known results. After reviewing the local and global Jacquet-Langlands correspondences in Section 3, we define the local JL-type correspondence in Section 4. In Section 5, we set up a generalized local to global argument. We prove in Section 6 that Plancherel measures are preserved under the local JL-type correspondence, then we present some applications in Section 7. In Section 8, we generalize the result in Section 6 to any Levi subgroup under some assumptions. Appendix A gives a few examples of an F -Levi subgroup M and its F -inner form satisfying condition (1.1).
Acknowledgements. We are deeply indebted to Professor Freydoon Shahidi for his constant support and profound insights. We are also grateful to Professor Gordan Savin for his valuable advice and corrections to the manuscript. Thanks are due to Professor Dipendra Prasad for his helpful correspondences and comments on a local to global argument. This work is a part of the author's Ph.D. dissertation at Purdue University.
Preliminaries
We recall basic notions, terminologies and known results. We mainly refer to [19, 21, 30, 42, 44, 49] .
2.1. Notation and Conventions. Throughout this paper, F denotes a p-adic field of characteristic 0 and F denotes a number field unless otherwise stated. Fix algebraic closuresF andF. We shall use the ordinary capital letters G, M , etc. for groups defined over a local field and the boldfaced capital letters G, M, etc. for groups defined over a global field.
By abuse of terminology, we identify the set of isomorphism classes with the set of representatives. Given a connected reductive group G over F , we use the following notation: Irr(G(F )) denotes the set of isomorphism classes of admissible representations of G(F ); E 2 (G(F )) denotes the set of essentially squareintegrable representations in Irr(G(F )); Irr u (G(F )) denotes the set of unitary representations in Irr(G(F )); E Denote by X * (M ) F the group of F -rational characters of M . We denote by
We define the homomorphism
Here | · | F denotes the normalized absolute value on F . Note that one can extend H M to G(F ) using the Iwasawa decomposition.
For σ ∈ Irr(M (F )) and ν ∈ a * M,C , we denote by I(ν, σ) the normalized induced representation
Here ½ is the trivial representation of N (F ). The space V (ν, σ) of I(ν, σ) consists of locally constant functions
for m ∈ M (F ), n ∈ N (F ) and g ∈ G(F ). Here ρ P denotes the half sum of all positive roots in N . We often write i G,M (ν, σ) for I(ν, σ) in order to specify groups. We fix a representative w ∈ G(F ) ofw ∈ W G such thatw(θ) ⊆ ∆. Set Nw := N 0 ∩ wN − w −1 . We fix a Haar measure dn on Nw. Given f ∈ V (ν, σ), for g ∈ G(F ), the standard intertwining operator is defined as
We set
where dn is a normalized Haar measure onNw(F ) andNw := w
we define the Plancherel measure attached to ν, σ andw as a non-zero complex number µ M (ν, σ, w) such that
Remark 2.2. The Plancherel µ M (ν, σ, w) depends only on ν, σ andw. It is independent of the choices of any Haar measure and of any representative w ofw [42, p.280] . Further, as a function ν → µ M (ν, σ, w) on a * M,C , it extends to a meromorphic function on all of a * M,C . Moreover, it is non-negative and holomorphic on √ −1a * M [21, Theorem 20] . We review two useful properties of the Plancherel measure. Let Φ(P, A M ) denote the set of reduced roots of P with respect to A M . For α ∈ Φ(P, A M ), A α := (ker α ∩ A M )
• denotes the identity component of (ker α ∩ A M ) regarding α as an element in a * M . Set M α := Z G (A α ) and P α := M α ∩ P . Note that M α contains M and P α is a maximal F -parabolic subgroup of M α with its Levi decomposition M N α and the split component A M . The Plancherel measure µ α (ν, σ, w) and the function γ α (P α |M ) can be defined by replacing G with P α in Definition 2.1 and equation (2.1), respectively. The following is the product formula of the rank-one Plancherel measures [21, Theorem 24] .
Next, suppose that G is a connected reductive group over F such that
where the subscript der means the derived group. Let M denote an
The following is the compatibility of the Plancherel measure with the restriction. 
Proof. Since G and G have the same derived group, we have
Hence, the proposition follows from the following property (cf. [43, p.293] ):
Remark 2.5. Remark 2.2, Propositions 2.3 and 2.4 reduce the study of Plancherel measures to the case of irreducible unitary admissible representations of maximal F -Levi subgroups of a semi-simple group G.
2.3.
Inner Forms. Let G and G ′ be connected reductive groups over a p-adic field F of characteristic 0. We say that G and G ′ are F -inner forms with respect to anF -isomorphism ϕ :
. We note [31, p.280 ] that there is a bijection between H 1 (F, G ad ) and the set of isomorphism classes of F -inner forms of G, obtained by sending the isomorphism class of a pair (G ′ , ϕ) to the class of the 1-cocycle τ → ϕ • τ (ϕ) −1 . We note that a Gal(F /F )-stable G ad (F )-orbit of ϕ gives the same isomorphism class of a pair (G ′ , ϕ). We often omit the reference to ϕ when there is no danger of confusion. We note that this notion holds for any field F with an algebraic separable closureF .
Suppose that G is quasi-split over F . Let G ′ be an F -inner form of G with respect to anF -isomorphism
In this case, we often say that M and M ′ are corresponding. In fact, the split components
HereĜ denotes the connected Langlands dual group (L-group) of G, Z(Ĝ) denotes the center ofĜ, Γ F = Gal(F /F ), π 0 ( · ) denotes the group of connected components, and ( · ) D denotes the Pontryagin dual, that is, Hom( · , R/Z). We note that A(G) is a finite abelian group when G = G ad . To see this, since the center of a simply-connected semisimple complex
Lie group is finite, Z( G ad ) = Z(Ĝ sc ) turns out to be a finite abelian group. HereĜ sc denotes the simply connected cover of the derived group ofĜ. 
We note that if the adjoint group G ad of G is simply connected (e.g., G 2 , F 4 , or E 8 ), there is no non-quasi-split F -inner form due to Proposition 2.6. Example 2.7. Let G be either GL n or SL n over a p-adic field F of characteristic 0. Then the set of isomorphism classes of F -inner forms of G is in bijection with the subgroup Br(F ) n of n-torsion elements in the Brauer group Br(F ). Indeed, we have
Here µ n is the algebraic group of n th root of unity.
Example 2.8. Let G be either GL n or SL n over a number field F. Then the set of isomorphism classes of F-inner forms of G is also in bijection with Br(F) n . Indeed, since 
The injection turns out to be surjective due to [38, Theorem 6.20 ].
Local and Global Jacquet-Langlands Correspondences for GL n
In this section, we recall the Jacquet-Langlands correspondence for GL n over a p-adic field F of characteristic 0 and a number field F. We mainly refer to [6] .
The local and global Jacquet-Langlands correspondences were initially found by Jacquet and Langlands [26] for the case GL 2 in any characteristic. The local generalization to GL n in zero characteristic was established by Rogawski [39] and independently by Deligne, Kazhdan and Vigneras [12] . Badulescu completed the proof of the local correspondence for GL n in positive characteristic [5] . On the other hand, the global generalization of the correspondence to GL n was proved only for a number field by Badulescu [6] . For some particular cases in zero characteristic, Flath treated the local and global correspondences for GL 3 [13] , and Snowden presented a new purely local proof of the local Jacquet-Langlands correspondence for GL 2 [47] .
3.1. Local Jacquet-Langlands Correspondence. Let G be GL n over a p-adic field F of characteristic 0 and let
, where D denotes a central division algebra of dimension d 2 over F and n = md. For semisimple elements g ∈ G(F ) and g ′ ∈ G ′ (F ), we write g ↔ g ′ if both are regular (i.e., all roots inF of the characteristic polynomial are distinct) and have the same characteristic polynomial. We write G(F ) reg for the set of regular semisimple elements in G(F ). We denote by C ∞ c (G(F )) the Hecke algebra of locally constant functions on G(F ) with compact support. Fix a Haar measure dg on G(F ). For any ρ ∈ Irr(G(F )), there is a unique locally constant function Θ ρ on G(F ) reg which is invariant under conjugation by G(F ) such that 
We denote by R(G) the Grothendieck group of admissible representations of finite length of G(F ). So, R(G) is a free abelian group with basis Irr(G(F )). Let R(G ′ ) be the Grothendieck group for the F -inner form G
′ . In what follows, we extend the correspondence C to a Z-morphism from R(G) to R(G ′ ). We refer to [6, Section 2.7] .
Let B be the collection of all normalized (twisted by δ
where L is a standard Levi subgroup of G and σ ∈ E 2 (L(F )). Let B ′ have the corresponding meaning for the F -inner form G ′ . We notice that any element Σ ∈ B (resp. Σ ′ ∈ B ′ ) has a unique irreducible quotient by the Langlands classification (see [32, Theorem 1.2.5]). We denote it by Lg(Σ) (resp. Lg(Σ ′ )). We note that the set B is a basis of R(G) and the map Σ → Lg(Σ) is a bijection from B onto Irr(G(F )). The same is true for the We have the following correspondence between the Grothendieck groups.
)) = 0. Thus, Example 3.3 yields
Remark 3.9. It follows from Proposition 3.
We have the following correspondence for d-compatible irreducible unitary representations. 
Definition 3.11. By sending u → u ′ , we define a map |LJ| from the set of irreducible unitary d-compatible representations of G(F ) to the set of irreducible unitary representations of G ′ (F ).
Remark 3.12. The restriction of |LJ| to the set E 2 u (G(F )) equals C.
Example 3.13. For the case that G = GL 2 , the representation i G,T (δ
) is not d-compatible due to Example 3.7. We also note that |LJ|(
Remark 3.14. All statements in this section admit an obvious generalization to the case that G is a product of a general linear groups.
3.2. Global Jacquet-Langlands Correspondence. Let G be GL n over a number field F and let For all v / ∈ S, we abuse the notation |LJ| v for the identity map from Irr u (G(F v )) to Irr u (G ′ (F v )). The global Jacquet-Langlands correspondence is as follows. 
Moreover, the image of Φ is exactly the set of D-compatible discrete series of G(A).
We note that there was an assumption on the set S in [6] and it has been removed in [4] . Let the space
The following proposition describes the behavior of cuspidal automorphic representations under the map Φ. Remark 3.19. All statements in this section admit an obvious generalization to the case that G is a product of a general linear groups.
Local Jacquet-Langlands Type Correspondence
In this section, we define the local Jacquet-Langlands type (JL-type) correspondence which is a general version of the local Jacquet-Langlands correspondence.
Restriction of Representations.
We recall the results of Tadić in [48] . Throughout Section 4.1, G and G denote connected reductive groups over a p-adic field F of characteristic 0, such that
where the subscript der means the derived group. Given σ ∈ Irr(G(F )), we denote by Π σ (G(F )) the set of equivalence classes of all irreducible constituents of σ| G(F ) . Remark 4.4. Let G = SL n over F or its F -inner form, and let G = GL n over F or its F -inner form. It then turns out [18, 25] that any L-packet is of the form Π σ (G(F )) for some σ ∈ Irr( G(F )).
Definition of the Local Jacquet-Langlands Type
Definition 4.5. Given σ ∈ Irr(G(F )) and σ ′ ∈ Irr(G ′ (F )), we say that σ and σ ′ are under the local Jacquet-Langlands type (JL-type) correspondence if there exist σ ∈ Irr( G(F )) and σ ′ ∈ Irr( G ′ (F )) such that (a) σ and σ ′ are isomorphic to irreducible constituents of the restrictions σ| G(F ) and σ F ) ) are under the local JL-type correspondence. 
A Local to Global Argument
In this section, we set up a local to global argument which will be used in Section 6. This generalizes the method of Muić and Savin in [37] .
Construction of Global Data from Local Data.
Given local data, we first construct global data with prescribed local behavior. Proof. This is an easy consequence of Dirichlet's density theorem.
Proposition 5.3. Let F be a p-adic field of characteristic 0. For any positive integer l, there exist a number field F and a finite set T of finite places of F, with the cardinality l, such that
Proof. Given F , we fix a number field F 0 and a place v 0 as defined in Lemma 5.1, so that F 0 v0 ≃ F . From Lemma 5.2, we have infinitely many odd primes q such that v 0 splits completely over F 0 ( √ q * ). Denote by T 0 the set of all such primes q. For any positive integer l, we choose a positive integer r such that 2 r ≥ l.
. We note that [F :
By taking T to be any subset of {v|F v ≃ F } with cardinality l, we complete the proof.
The following theorem allows us to construct a number field and connected reductive groups with prescribed local behavior.
Theorem 5.4. Let G be a connected reductive quasi-split group over a p-adic field F of characteristic 0, and let G ′ be an F -inner form of G with respect to anF -isomorphism ϕ : G ′ → G. Then, there exist a number field F, a non-empty finite set S of finite places of F, a connected reductive group G defined over F and its
Proof. Let l denote a sufficiently large multiple of the cardinality |A(G ad )| of A(G ad ) (see Section 2.3 for the definition of A(G ad )). From Proposition 5.3, we obtain a number field F and a finite set T of places with the cardinality l such that F v ≃ F for all v ∈ T . We note that the set A(G ad ) is a finite abelian group. Next we choose a connected reductive quasi-split group G defined over F. It follows that
We note that A(G ad ) is a finite abelian group, and there is a surjective homomorphism v A(G ad v ) → A(G ad ) for any place v of F (see [30, 2.3] ). Here v denotes the subset of the direct product consisting of (x v ) such that x v = 1 for all but a finite number of v. Since the integer l = |T | is taken to be sufficiently large, we can assume that the cardinality |A(G ad )| is smaller than l due to the surjective homomorphism. Choose a subset S of T such that the cardinality |S| equals a multiple of |A(G ad )|. We recall the following lemma from [30 
The bottom map is given by local maps defined in Proposition 2.6, which are isomorphisms for all finite places of F. Thus, the commutative diagram implies that the morphism β G is equal to sum of local 1-cocycles.
Since G ′ is an F -inner form of G with respect to anF -isomorphism ϕ : G ′ → G, we have a 1-cocycle
be a nontrivial 1-cocycle such that a τv = ϕ τv for all v ∈ S and a τv = 1 for all v / ∈ S. Since A(G ad ) is a finite abelian group and |S| is a multiple of |A(G ad )|, we get
From the exactness in Lemma 5.5, we have a nontrivial 1-cocycle
This completes the proof of Theorem 5.4.
Remark 5.6. Theorem 5.4 is an analogue of the well-known result:
To be precise, this exact sequence explains how to obtain a central division algebra over F from a given central division algebra over F . When an F -group G satisfies condition (4.1), this is a manner to construct an F-group G and its F-inner form G ′ with prescribed local behavior. Theorem 5.4 fully extends this notion to any connected reductive group over F .
Given an irreducible unitary supercuspidal representation, the following proposition tells us how to construct a cuspidal automorphic representation with specified local behavior at a finite set of places. 
Local and Global Compatibility in Restriction.
In this section, we establish the local and global compatibility in the restriction of representations from a group to its subgroup sharing the same derived group. Let F denote a p-adic field of characteristic 0 with the ring of integers O F . Proposition 5.9. Let G and G be unramified groups over F such that
Given an unramified representation τ of G(F ), its restriction τ | G(F ) to G(F ) has a unique unramified constituent with respect to G(O F ).

Proof. Fix a Borel subgroup B = T U of G. Then we have a Borel subgroup
we have a G(F )-embedding of τ into an unramified principal series i G, Bχ , whereχ is an unramified character of T (F ). Consider the restriction (i G, Bχ )| G(F ) . We note that G(F ) = T (F )G(F ), and f (tg) =χ(t)f (g) fort ∈ T (F ) and g ∈ G(F ). It follows that if f | G(F ) = 0 for f ∈ i G, Bχ , then f = 0. By sending f → f | G(F ) , we thus have a G(F )-equivariant embedding
Here χ is the restriction ofχ to T (F ). Since χ is an unramified character of T (F ), we note that i G,B χ has a unique non-trivial spherical vector with respect to G(O F ) up to scalar. Hence, there is exactly one irreducible constituent of i G,B χ which contains the unique non-trivial spherical vector. This completes the proof.
Let F denote a number field, and let A denote the adele ring of F. Let G and G be connected reductive groups over F such that 
To construct the tensor product, we choose a finite set T 0 of places and a non-zero spherical vector x 0 with respect to G(O v ) for each v / ∈ T 0 . We can find a finite set T of places and a vector
There is a surjective map
is a finite direct sum of irreducible constituents of G(F v ). So, the representation of G T on V T /U T is irreducible and is of the form ⊗ v∈T π v , where π v is an irreducible constituent of π v | G(Fv) . Thus, the constituent π is of the form Here π runs through all irreducible constituents of the restriction π| G(A) which are cuspidal automorphic representations of G(A), and m π is the multiplicity of π.
Remark 5.12. Let π = ⊗π v be a cuspidal automorphic representation of G(A). Theorem 4.13 in [25] verifies that there exists a cuspidal automorphic representation π = ⊗ π v of G(A) such that π is an irreducible constituent of π| G(A) . From Proposition 5.10, we note that π v is an irreducible constituent of π v | G(Fv) for all v.
Transfer for Unitary Supercuspidal Representations under the Local JL-Type Correspondence
Using a local to global argument in Section 5, we prove that Plancherel measures attached to unitary supercuspidal representations are preserved under the local JL-type correspondence, assuming a working hypothesis that Plancherel measures are invariant on a certain finite set. Throughout Section 6, F denotes a p-adic field of characteristic 0, and M denotes an F -Levi subgroup of a connected reductive F -split group G such that
GL ni for positive integers r and n i . Let G ′ be an F -inner form of G, and let M ′ be an F -Levi subgroup of G ′ that is an F -inner form of M . Then, M ′ satisfies the following property
. We denote by θ and θ ′ the subsets of ∆ and ∆ ′ such that M = M θ and M = M θ ′ , respectively. We fix representatives w ∈ G(F ) and
6.1. Statement of Theorem. In this section, we state the main result and its contributions. Let σ ∈ E
• u (M (F )) and σ ′ ∈ E
• u (M ′ (F )) be under the local JL-type correspondence. Since both σ and σ ′ are supercuspidal, by Remark 4.2 and Definition 4.5, we have σ ∈ E
(a) σ and σ ′ are isomorphic to irreducible constituents of the restrictions σ| M(F ) and σ
We recall the sets Π σ (M (F )) and Π σ ′ (M ′ (F )) of equivalence classes of all irreducible constituents of σ| M(F ) and σ ′ | M ′ (F ) , respectively.
Working Hypothesis 6.1. Let σ The following states our main result. 
for any ν ∈ a * M,C ≃ a * M ′ ,C . Remark 6.4. If the central character of σ ′ is unramified and the set Π σ ′ (M ′ (F )) is associated to a tempered and tame regular semi-simple elliptic L-parameter, the assumption is no longer needed due to [10] .
We note that any two members τ ∈ Π σ (M (F )) and 
for all ν ∈ a * M,C . Remark 6.6. Since Π σ (M (F )) can be considered as an L-packet on M (cf. Remark 4.4), Proposition 6.5 supports the conjecture that Plancherel measures are invariant on L-packets. We also refer the reader to [2, 10, 15, 16, 17] for other cases. Step 2 ) Find two cuspidal automorphic representations as described in Proposition 6.9.
(Step 3 ) Use Langlands' functional equation on Eisenstein series. This is a more general version of the method of Muić and Savin in [37] . Along with their Siegel Levi subgroups (≃ GL n ) of the groups SO 2n and Sp 2n , our construction in (Step 1 ) treats any F -Levi subgroup of connected reductive groups over a p-adic field of characteristic 0. Further, ( Step 2 ) can be applied to any connected reductive group M satisfying condition (6.1).
The rest of this section is devoted to the proof of Theorem 6.3. Due to Remarks 2.2 and 2.5, it suffices to consider the case when G is semisimple (so is G ′ ) and M is maximal (so is M ′ ). It then turns out that
We start with the following Lemma which is immediately a consequence of Theorem 5.4. 
Note that M(F) and M ′ (F) are of the forms From now on, we fix a number field F and a finite set S of finite places of F as defined in Lemma 6.8, so that F v ≃ F for all v ∈ S and S consists of all non-split places. Next, we find two following cuspidal We claim that π is in the image of the map Φ defined in Theorem 3.16. To see this, we need to show that π is D-compatible. For all v ∈ S, since π v is in E
It follows from remark 3.9 that π v is d v -compatible for all v ∈ S. Hence, from Definition 3.15, π is D-compatible. We note from Proposition 3.17 that π ′ is cuspidal since π is cuspidal. Therefore, we have a unique cuspidal automorphic representation π
Since both π v and π ′ v are supercuspidal for v ∈ S and S consists of non-split places, we note that |LJ| v ( π v ) = C( π v ) for all v ∈ S by Remark 3.12 and π
Now we consider the restriction
The assertions ( 
. This completes the proof of Proposition 6.9
We consider the standard global intertwining operator M (s, π) := ⊗ v A(s, π v , w) from the global induced representation I(s, π) to I(−s, w(π)). We refer to [11, Sections 4 and 7] . For f ∈ I(s, π), we note that
v have the corresponding meaning for the cuspidal automorphic representation π ′ of M ′ (A). Due to Proposition 6.9, we identify f 
It then follows that
Here c v (·, ·) is a quotient of the local Langlands L-functions for unramified representations (see [41, (2 
For all v ∈ S, we note that:
by Working Hypothesis 6.1. Hence, we deduce from equations (6.2) and (6.3) that
Here m denotes the cardinality of S. Since Plancherel measures are holomorphic and non-negative along the unitary axis Re(s) = 0, we therefore have that
for all s ∈ C. This completes the proof of Theorem 6.3.
Applications
In this section, we present some applications of Theorem 6.3. We continue with the notation in Section 6. Throughout Section 7, we assume that M and M ′ are maximal. For σ ∈ Irr(M (F )) and σ ′ ∈ Irr(M ′ (F )), we define W (σ) := {w ∈ W M : w σ ≃ σ} and W (σ In what follows, we prove that both the reducibility of i G,M (ν, σ) and the edges of complementary series are invariant on the set Π σ (M (F )) when M is maximal. In a similar way to Corollary 7.3, we have the following. 
A Generalization
Let F be a p-adic field of characteristic 0, and let M be an F -Levi subgroup of a connected reductive F -group G. Let G ′ be an F -inner form of G, and let M ′ be an F -Levi subgroup of G ′ that is an F -inner form of M . In this section, we extend Theorem 6.3 to the case that unitary supercuspidal representations of M (F ) and M ′ (F ) have the same L-parameter under the following assumption. 
for ν ∈ a (a) (E 6 − 1) Let G be a simply connected group of type E 6 . Set θ = ∆ − {α 3 }, where α 3 = e 3 − e 4 . From [28, 41] we have [34] ) Let G be a simply connected group of type E 6 . Set θ = ∆ − {α 6 }, where α 6 = e 4 + e 5 + e 6 + ǫ. From [28, 41] we have (a) (E 7 − 1) Let G be a simply connected group of type E 7 . Set θ = ∆ − {α 4 }, where α 4 = e 4 − e 5 . From [28, 41] we have
So, M (F ) ֒→ GL 1 (F ) × GL 1 (D 2 ) × GL 3 (F ) × GL 2 (D 2 ) = M ′ (F ). (b) (E 7 − 4) Let G be a simply connected group of type E 7 . Set θ = ∆ − {α 5 }, where α 5 = e 5 − e 6 .
From [28, 41] we have
So, M (F ) ֒→ GL 1 (F ) × GL 3 (D 2 ) × GL 2 (F ) = M ′ (F ).
(7) E 8 , F 4 and G 2 cases Any connected reductive algebraic F -group G of type E 8 , F 4 , or G 2 does not have non quasi-split F -inner forms of G (see Proposition 2.6).

{'paper_id': '119627655', 'title': 'Transfer of Plancherel Measures for Unitary Supercuspidal Representations between p-adic Inner Forms', 's2_url': 'https://api.semanticscholar.org/CorpusID:119627655', 'source': 'S2ORC'}