aa r X i v : . [ m a t h . N T ] F e b DISTINCTION AND BASE CHANGE
U. K. ANANDAVARDHANANA bstract . An irreducible smooth representation of a p -adic group G is said to be dis-tinguished with respect to a subgroup H if it admits a non-trivial H -invariant linearform. When H is the fixed group of an involution on G it is suggested by the worksof Herv´e Jacquet from the nineties that distinction can be characterized in terms of theprinciple of functoriality. If the involution is the Galois involution then a recent conjec-ture of Dipendra Prasad predicts a formula for the dimension of the space of invariantlinear forms which once again involves base change. We will describe the proof of thisconjecture (in the generic case) for SL ( n ) which is joint work with Dipendra Prasad.Then we describe one more newly discovered connection between distinction and basechange which is that base change information appears in the constant of proportional-ity between two natural invariant linear forms on a distinguished representation. Thislatter result is for discrete series for GL ( n ) and is joint with Nadir Matringe. Thispaper is a report on the author’s talk in the International Colloquium on ArithmeticGeometry held in January 2020 at TIFR Mumbai.
1. I ntroduction
The so called relative Langlands program has got a major boost in the last decadethanks to the fundamental works of Gan-Gross-Prasad [GGP12] and Sakellaridis-Venkatesh [SV17]. Distinguished representations are the basic objects of study inthe relative Langlands program and they are studied both locally and globally. In thelocal setting, given a p -adic group G , a subgroup H of G , and a character χ of H , anirreducible admissible representation π of G is said to be ( H , χ ) -distinguished ifHom H ( π , χ ) = { } .In the global setting, where H < G are adelic groups, and π is a cuspidal represen-tation of G , the interest is not in any non-trivial ( H , χ ) -invariant linear form, insteaddistinction is defined in terms of the non-vanishing of a specific ( H , χ ) -invariant linearform which is the ( H , χ ) -period integral. When χ is the trivial character of H , it is of-ten omitted from the definition and we say that the representation is H -distinguished.In situations such as in [GGP12], local distinction is connected to the sign of certainlocal root numbers and global distinction is connected to the special values of certainautomorphic L -functions. In situations such as in [SV17], it is expected that distinctioncan often be characterized in terms of the principle of functoriality.The notion of a distinguished representation, in both its local and global avatars,goes back to the pioneering work of Harder-Langlands-Rapoport about the Tate con-jecture on algebraic cycles in the context of Hilbert-Blumenthal surfaces [HLR86]. Inthis work they are led to study distinction for the pair ( R F / Q GL ( ) , GL ( )) , where F is a real quadratic number field. Several of the connections that distinction has Mathematics Subject Classification. with other objects of interest are already there in [HLR86]. In particular, it is provedthat a cuspidal representation of GL ( A F ) of trivial central character is distinguishedprecisely when its Asai L -function has a pole at s = ( A Q ) withnon-trivial central character. Roughly around the same time, the work of Jacquet-Laiintroduced the relative trace formula to investigate distinction [JL85].In the nineties, a series of papers of Jacquet and his collaborators investigated dis-tinction for a number of symmetric pairs such as ( GL ( m + n ) , GL ( m ) × GL ( n )) , ( GL ( n ) , Sp ( n )) & ( R E / F U ( n , E / F ) , U ( n , E / F )) where E / F is a quadratic extension of p -adic fields (or number fields) and U ( n , E / F ) denotes a unitary group defined with respect to E / F . It was proposed that distinctionfor a symmetric pair ( G , H ) should have a simple characterization in terms of theprinciple of functoriality (for instance, see [JY96]).An instance of this proposal is that an irreducible admissible generic representa-tion of GL n ( E ) is distinguished with respect to U ( n , E / F ) if and only if it is a basechange lift from GL n ( F ) . This is now known by the works of Jacquet [Jac04, Jac05]and Feigon-Lapid-Offen [FLO12] (and [AC89]). Similarly, an irreducible admissiblegeneric representation of GL n ( E ) is ( GL n ( F ) , ω n − E / F ) -distinguished if and only if it is abase change lift from U ( n , E / F ) , where ω E / F is the quadratic character of F × associ-ated to E / F . This suggestion is sometimes referred to as the Flicker-Rallis conjecture(it is the local version of the global conjecture stated in [Fli91, p. 143]). This too isnow known by combining [Mat11, Theorem 5.2] and [Mok15, Lemma 2.2.1].This paper is an informal exposition of two results of the author which bring tolight the close relationship between distinction and base change; one of these is jointwork with Dipendra Prasad [AP18, Theorem 5.6] and the other is joint work withNadir Matringe [AM20b, Theorem 6.1]. The first is Theorem 4.1 and the secondis Theorem 5.1 of this paper. Theorem 4.1 proves a conjecture of Dipendra Prasad[Pra20, Conjecture 13.3] for G = SL ( n ) . We start by recalling a few aspects of Prasad’sconjecture in Section 2 and then summarize a number of facts connecting distinctionfor ( GL n ( E ) , GL n ( F )) and base change from U ( n , E / F ) to GL n ( E ) (cf. [Mok15]) inSection 3.2. In Section 4, we take up Theorem 4.1, and we discuss Theorem 5.1 inSection 5. 2. P rasad ’ s C onjecture Prasad’s conjecture is for p -adic fields and for a Galois pair [Pra20, Conjecture 13.3].As in Section 1, E / F is a quadratic extension of p -adic fields. Let G be a connected re-ductive group defined over F . Let G = G ( E ) and H = G ( F ) . The Galois involution σ acts on G and hence on representations of G . For an irreducible admissible represen-tation π of G its Galois conjugate is denoted by π σ . The contragredient representationof π is denoted by π ∨ . If π is such that π ∨ ∼ = π σ , we say that π is conjugate self-dual.There are two objects associated to the data ( G, E / F ) which appear in the formu-lation of the conjecture. One of these is the opposition group denoted by G op con-structed in [Pra20, §5] which in particular has the property that it is isomorphic to ISTINCTION AND BASE CHANGE 3
G over E . The other is a character of H of order ≤
2, denoted by ω G , constructed in[Pra20, §6].The conjecture has three parts. The first part asserts that an irreducible admis-sible representation of G which is ( H , ω G ) -distinguished is conjugate self-dual andmoreover its L-packet arises as the base change lift of an L-packet of G op ( F ) .The two examples considered in Section 1 correspond to • G = GL ( n ) , G op = U ( n , E / F ) , ω G = ω n − E / F ◦ det, • G = U ( n , E / F ) , G op = GL ( n ) , ω G = = GL ( n ) , this is [Fli91, Proposition 12], and for G = U ( n , E / F ) , this is [FLO12,Theorem 0.2].The second and third parts of the conjecture are for generic representations soassume that G is quasi-split over F . Fix a Borel subgroup B of G and let N be itsunipotent radical.The second part of the conjecture probes for distinction inside a generic L-packetand proposes a simple recipe to detect distinction. It says that an irreducible admis-sible representation which is generic for a non-degenerate character of N ( E ) / N ( F ) is ( H , ω G ) -distinguished provided its L-packet is a base change lift of an L-packet ofG op ( F ) .The third and perhaps the most important part of the conjecture is the multiplicityformula. We do not state it as it involves more technical terms and refer to [Pra20,Conjecture 13.3] for the precise statement. It suffices to say that a key ingredient inthe conjectural formula for the multiplicity - which is dim C Hom H ( π , ω G ) - is thecardinality of the fiber of the base change map from L-packets of G op ( F ) to L-packetsof G . Thus the connection between distinction and base change is indeed quite deep.For G = GL ( n ) , the corresponding Galois pair is a Gelfand pair [Fli91, Proposition11]; i.e., dim C Hom H ( π , ω G ) = π of G which is ( H , ω G ) -distinguished.This fits well with the conjectural multiplicity formula which in this case equals thecardinality of the fiber of the base change map from U ( n , E / F ) to GL n ( E ) as this mapis known to be injective (cf. [Pra20, Proposition 7.10]).For G = U ( n , E / F ) , the multiplicity can be more than 1. For instance, it is easy tosee that the principal series representation Ps ( χ , χ − ) of GL ( E ) with χ = χ σ is bothGL ( F ) -distinguished and ( GL ( F ) , ω E / F ) -distinguished and that the correspondinginvariant linear functionals are U ( E / F ) -invariant. It can be seen thatdim C Hom H ( π , ω G ) = π is an irreducible admissible generic representation ofGL n ( E ) then it is parabolically induced from a number of essentially square-integrablerepresentations π i of GL n i ( E ) , 1 ≤ i ≤ t , n = n + · · · + n t , say π = π × . . . × π t . U. K. ANANDAVARDHANAN
Suppose π and r many of these π i ’s are Galois invariant. The conjectural multiplicityformula would then predictdim C Hom H ( π , ω G ) = ( r − r ≥ r = π i ’s are distinct. Recently, R. Beuzart-Plessis has proved this multiplicity formula ingeneral [BP20, Theorem 3].3. B ase change from U ( n , E / F ) to GL n ( E ) In Section 2, we have briefly mentioned the connection between distinction for theGalois pair ( GL n ( E ) , GL n ( F )) and base change from a unitary group. We make itmore precise in this section. We closely follow [Mok15, §2.1, 2.2].In order to discuss base change we need to introduce the notion of a Langlands pa-rameter of a group G defined over a p -adic field k . It is an admissible homomorphismfrom the Weil-Deligne group of kW ′ k = W k × SL ( C ) to the Langlands dual group L G = G ∨ ⋊ W k of G, where G ∨ is the complex dual group of G. Thus, a Langlands parameter is acontinuous homomorphism ϕ : W ′ k → L Gthat commutes with the natural projections W ′ k → W k , L G → W k , it sends W k tosemisimple elements and its restriction to SL ( C ) is algebraic. Two Langlands param-eters are equivalent if they are conjugate by G ∨ . We denote by Φ ( G ) (or sometimesby Φ ( G ( k )) ) the set of equivalence classes of Langlands parameters of G.The groups of interest to us are GL ( n ) and the quasi-split unitary group defined byU ( n ) = { g ∈ GL n ( E ) | t g σ Jg = J } ,where J = anti-diag (( − ) n − , . . . , 1 )) is the anti-diagonal matrix with ± ( n ) is GL n ( C ) . The Langlands dual group of U ( n ) is given by L U ( n ) = GL n ( C ) ⋊ W F ,where the Weil group W F of F acts by projection to Gal ( E / F ) , and w σ ∈ W F r W E actsas the automorphism g J t g − J − .The base change map from U ( n ) to GL n ( E ) at the level of Langlands parameters isa map from Φ ( U ( n )) = { ϕ : W ′ F → L U ( n ) } / ∼ to Φ ( GL n ( E )) = { ρ : W ′ E → GL n ( C ) } / ∼ ISTINCTION AND BASE CHANGE 5 given by BC : Φ ( U ( n )) → Φ ( GL n ( E )) ϕ ϕ | W ′ E .Now the Flicker-Rallis conjecture (see [Fli91, p. 143] where it is stated in the globalcontext) is the assertion that, for an irreducible admissible generic representation π ofGL n ( E ) with Langlands parameter ρ π , ρ π ∈ Image(BC) ⇐⇒ ( π is GL n ( F ) -distinguished for n odd, π is ( GL n ( F ) , ω E / F ) -distinguished for n even. Remark . What we described above is what is called the stable base change mapfrom U ( n ) to GL n ( E ) . There is also an unstable base change map with respect to anextension κ of ω E / F to E × . A representation π of GL n ( E ) is in the image of the stablebase change map if and only if π ⊗ κ is in the image of the unstable base change mapwith respect to κ .As mentioned in Section 2, this is now proved thanks to [Mat11, Theorem 5.2]and [Mok15, Lemma 2.2.1]. We end this section by stating these two results and byparaphrasing the Flicker-Rallis conjecture in the language of Prasad’s conjecture.To this end, we introduce the notion of parity of a conjugate self-dual Langlandsparameter for GL n ( E ) . So let ρ : W ′ E → GL n ( C ) be such that ρ σ ∼ = ρ ∨ , where ρ σ ( g ) = ρ ( w − σ gw σ ) for w σ ∈ W F r W E . Then ρ is said tobe of parity η ( ρ ) ∈ {± } if there is a non-degenerate bilinear form B on ρ with • B ( ρ ( g ) v , ρ σ ( g ) w ) = B ( v , w ) , • B ( v , w ) = η ( ρ ) · B ( w , ρ ( w σ ) v ) .Thus the Flicker-Rallis conjecture follows from combining Theorem 3.1 and Theo-rem 3.2. Theorem 3.1 (Matringe) . An irreducible admissible generic representation of GL n ( E ) is GL n ( F ) -distinguished if and only if its Langlands parameter is conjugate self-dual of parity + . Theorem 3.1 is [Mat11, Theorem 5.2] which builds on earlier results in [Kab04,AKT04, AR05, Mat09]. Theorem 3.2 is [Mok15, Lemma 2.2.1].
Theorem 3.2 (Mok) . A Langlands parameter for GL n ( E ) is in the image of the base changemap from U ( n ) -parameters if and only if it is conjugate self-dual of parity ( − ) n − . Now we state all these together in the language of Prasad’s conjecture.
Theorem 3.3 (Flicker-Matringe-Mok) . An irreducible admissible generic representation e π of GL n ( E ) is ( GL n ( F ) , ω n − E / F ) -distinguished if and only if its Langlands parameter e ρ e π is inthe image of BC : Φ ( U ( n )) → Φ ( GL n ( E )) , and moreover, dim C Hom GL n ( F ) ( e π , ω n − E / F ) = | BC − ( e ρ e π ) | . U. K. ANANDAVARDHANAN ( SL n ( E ) , SL n ( F )) Prasad’s conjecture for SL ( n ) is proved in [AP18] (see also [AP03, Ana05] for someof the early works in this direction). In this section, following [AP18], we summarisethe key steps involved in its proof.The opposition group for G = SL ( n ) over E / F is G op = SU ( n , E / F ) and Prasad’scharacter ω G is the trivial character. Thus, we are interested in the space of SL n ( F ) -invariant linear forms on an irreducible admissible generic representation π of SL n ( E ) and also in the base change map from SU ( n ) to SL n ( E ) .Base change for SU ( n ) fits into the commutative diagram Φ ( SU ( n )) p BC / / Φ ( SL n ( E )) Φ ( U ( n )) p F O O BC / / Φ ( GL n ( E )) p E O O where p F and p E are the natural projections induced by the homomorphismGL n ( C ) → PGL n ( C ) .The key observation is that the maps p F , p E are surjective and BC is injective. Thesurjectivity of p E follows from Tate’s theorem according to which H ( W ′ E , C × ) = W ′ E acts trivially on C × . The surjectivity of p F also follows from a similarsecond cohomology vanishing result (cf. [AP18, Lemma 5.1]). The map BC is injectiveis the result [Pra20, Proposition 7.10].To proceed, let ρ ∈ Φ ( SU ( n )) . Let e ρ ∈ p − F ( ρ ) . Now observe thatBC ( p − F ( ρ )) = { BC ( e ρ ) ⊗ χ | χ ∈ \ E × / F × } ,and p − E ( p BC ( ρ )) = { BC ( e ρ ) ⊗ χ | χ ∈ c E × } .The above observation leads us to make the following crucial definitions (see also[Ana05, Remark 2]). Two members of BC ( Φ ( U ( n ))) are weakly (resp. strongly) equiva-lent if they differ by a character of E × (resp. E × / F × ). Strong and weak equivalenceclasses are similarly defined in the set of ( GL n ( F ) , ω n − E / F ) -distinguished representa-tions.It follows that the cardinality of { µ ∈ Φ ( SU ( n )) | p BC ( µ ) = p BC ( ρ ) } is the number of strong equivalence classes in the weak equivalence class of BC ( e ρ ) .We now state the main theorem of [AP18] which is the exact analogue of Theorem3.3 for SL ( n ) with an additional condition to probe for distinction inside an L-packet[AP18, Theorem 5.6]. Theorem 4.1 (Anandavardhanan-Prasad) . An irreducible admissible generic representa-tion π of SL n ( E ) is distinguished by SL n ( F ) if and only if (1) its Langlands parameter ρ π is in the image ofp BC : Φ ( SU ( n )) → Φ ( SL n ( E )) , ISTINCTION AND BASE CHANGE 7 (2) π has a Whittaker model for a non-degenerate character of N ( E ) / N ( F ) .Further, if Hom SL n ( F ) ( π , 1 ) = , dim C Hom SL n ( F ) ( π , 1 ) = | p BC − ( ρ π ) | .Part (1) of Theorem 4.1 follows from the commutative diagram above for basechange from SU ( n ) together with the fact that π ∈ e π | SL n ( E ) for some irreducible admis-sible generic representation e π of GL n ( E ) which can be taken to be ( GL n ( F ) , ω n − E / F ) -distinguished.The key ingredient in proving (2) is Theorem 4.2 below, which follows from com-bining a number of results. Firstly, we have a result due to Flicker by which foran irreducible admissible unitary generic representation π of GL n ( E ) and for a non-degenerate character ψ of N ( E ) / N ( F ) , the integral Z N ( F ) \ P ( F ) W ( p ) dp is absolutely convergent for W in the ψ -Whittaker model W ( π , ψ ) of π , where P ( F ) denotes the mirabolic subgroup of GL n ( F ) [Fli88, Lemma in §4]. Also, by [Fli88,Proposition in §4], the linear form defined on W ( π , ψ ) by such an integral is non-trivial. If π is non-unitary but generic and also GL n ( F ) -distinguished then it is shownin [AM17, Section 7] that the integral Z N ( F ) \ P ( F ) W ( p ) | det p | s − dp (which is convergent for Re ( s ) large and admitting a meromorphic continuation to C by the Rankin-Selberg theory of the Asai L -function and in particular by [Mat09,Lemma 3.5]) is holomorphic at s =
1. We denote this regularized integral by Z ∗ N ( F ) \ P ( F ) W ( p ) dp which is not identically zero as a linear form on W ( π , ψ ) (see, for instance, [AM17,Theorem 7.2]). Secondly, we have a result due to Youngbin Ok by whichHom P ( F ) ( π , 1 ) = Hom GL n ( F ) ( π , 1 ) for any irreducible admissible GL n ( F ) -distinguished representation π (see [Mat10,Proposition 2.1] and [Off11, Theorem 3.1]). We remark here that Ok’s result in thetempered case is proved independently in [AKT04, Theorem 1.1]. Theorem 4.2.
The unique, up to multiplication by scalars, GL n ( F ) -invariant linear form ona GL n ( F ) -distinguished irreducible admissible generic representation e π of GL n ( E ) is givenon its ψ -Whittaker model W ( π , ψ ) by ℓ ( W ) = Z ∗ N ( F ) \ P ( F ) W ( p ) dp , where ψ is a non-degenerate character of N ( E ) / N ( F ) . In order to prove the multiplicity formuladim C Hom SL n ( F ) ( π , 1 ) = | p BC − ( ρ π ) | , U. K. ANANDAVARDHANAN choose an irreducible admissible generic representation e π of GL n ( E ) containing therepresentation π which is ( GL n ( F ) , ω E / F ) -distinguished and e ρ such that BC ( e ρ ) = e ρ e π . The right hand side is the number of strong equivalence classes in the weakequivalence class of e ρ e π (inside Image(BC)). The left hand side can be shown to be thenumber of strong equivalence classes in the weak equivalence class of e π (among the ( GL n ( F ) , ω n − E / F ) -distinguished irreducible generic representations of GL n ( E ) ). Remark . A consequence of Theorem 4.1 (2) is that an SL n ( F ) -distinguished irre-ducible admissible generic representation of SL n ( E ) is conjugate self-dual. This fol-lows from the uniqueness of non-degenerate Whittaker models for GL n ( E ) . Indeed,for an irreducible ψ -generic representation π of SL n ( E ) which is SL n ( F ) -distinguishedlet e π be an irreducible admissible generic representation of GL n ( E ) such that π ap-pears in its restriction to SL n ( E ) . By multiplicity one of Whittaker functionals on e π we see that π is the unique ψ -generic representation in its L-packet. A character ψ of N ( E ) / N ( F ) has the property that ψ − = ψ σ and therefore it follows that both π ∨ and π σ are generic with respect to the same character of N ( E ) and thus they areisomorphic. Remark . In [AM20a], Theorem 4.1 (2) is proved for any unitary representation ofSL n ( E ) and therefore distinction by SL n ( F ) implies conjugate self-duality for any uni-tary representation of SL n ( E ) by the multiplicity one result for degenerate Whittakermodels by the argument outlined in Remark 2. Thus we have established one of theassertions in Prasad’s conjecture for all the unitary representations of SL n ( E ) . The roleplayed by Theorem 4.2 in the proof of Theorem 4.1 (2) is played by [Mat14, Proposi-tions 2.2 and 2.5] in the unitary context. Remark . The assertions in Remark 2 are true also over finite fields; i.e., for the pair ( SL n ( F q ) , SL n ( F q )) . This is [AM20b, Theorem 5.1].5. A new connection between distinction and base change In Section 2, we saw Prasad’s conjecture which related distinction and base changein a precise way and in particular the cardinality of the fiber of the base change fromthe opposition group G op ( F ) goes into the conjectural formula for the dimension ofthe space of ( G ( F ) , ω G ) -invariant forms on an irreducible admissible generic repre-sentation of G ( E ) . In Section 4, we saw the main ideas behind the proof of Prasad’sconjecture for SL ( n ) in [AP18].In this section, we present a result, joint with Nadir Matringe, which illustratesthe connection between distinction and base change in yet another way which is thatbase change information appears in the constant of proportionality between two nat-ural invariant linear forms on a distinguished representation. The result is for thepair ( GL n ( E ) , GL n ( F )) and for GL n ( F ) -distinguished discrete series representationsof GL n ( E ) and it is contained in [AM20b, Section 6]. We give an informal introduc-tion to the result and its proof.The main points to keep in mind are:(1) The pair ( GL n ( E ) , GL n ( F )) is of multiplicity one [Fli91, Proposition 11].(2) There are two natural GL n ( F ) -invariant forms on a GL n ( F ) -distinguished dis-crete series representation of GL n ( E ) . One due to Flicker [Fli88, Lemma in §4], ISTINCTION AND BASE CHANGE 9 denoted by ℓ , which we saw in Theorem 4.2, and the other due to Kable, say λ [Kab04, Theorem 4].(3) The distinguishing linear forms λ and ℓ differ by a constant by Flicker’s mul-tiplicity one result.(4) Flicker-Rallis conjecture, recalled in Theorem 3.3, according to which distinc-tion for ( GL n ( E ) , GL ( n ( F )) is related to base change from U ( n , E / F ) .Our result evaluates the proportionality constant in (3) above and it involves theformal degrees of the base changed and base changing representations. We state theresult towards the end of this section (cf. Theorem 5.1).Thus the two inputs for the statement of Theorem 5.1 are base change and formaldegree. We have introduced base change from U ( n ) to GL n ( E ) in Section 3.2. Let usnow recall the definition of the formal degree of a discrete series representation.If π is a discrete series representation of a p -adic group, there exists d µ ( π ) ∈ R > such that Z G / Z h π ( g ) v , v ′ ih π ( g ) w , w ′ i d µ ( g ) = d µ ( π ) h v , w ih v ′ , w ′ i .This d µ ( π ) is the formal degree of π , which of course depends on the choice of theHaar measure µ . Remark . Recall the orthogonality relations for matrix coefficients for a finite group.A matrix coefficient of a (unitary) representation π is a function on G given by g
7→ h π ( g ) v , v ′ i where v , v ′ ∈ π . If π is irreducible, we have1 | G | ∑ g ∈ G h π ( g ) v , v ′ ih π ( g ) w , w ′ i = π h v , w ih v ′ , w ′ i .Thus, formal degree, for infinite dimensional representations, plays the role of thedimension of a finite dimensional representation.The Hiraga-Ichino-Ikeda conjecture gives a formula for the formal degree of a dis-crete series representation of a p -adic group in terms of its adjoint gamma functionevaluated at s = ( n ) in[HII08] itself (cf. [HII08, Theorem 3.1]) and it is proved for U ( n , E / F ) by Beuzart-Plessis [BP18b, Corollary 5.5.4].With respect to the specific choice of Haar measure as in [HII ’08], we have:(1) For a discrete series representation π of GL n ( E ) , its formal degree is given by[HII08, Theorem 3.1] d ( π ) = − q − n (cid:12)(cid:12)(cid:12)(cid:12) lim s → γ ( s , π × π ∨ , ψ ) − q − s (cid:12)(cid:12)(cid:12)(cid:12) ,where the gamma factor is the Rankin-Selberg gamma factor.(2) For a discrete series representation ρ of U ( n , E / F ) , its formal degree is givenby [BP18b, Corollary 5.5.4] (see also [HII08, Lemma 8.1]) d ( ρ ) = (cid:12)(cid:12) γ ( π , r ′ , ψ ) (cid:12)(cid:12) ,where the gamma factor is the twisted Asai gamma factor. Remark . There are three ways of defining a gamma factor. Via the Langlands formal-ism, via the Langlands-Shahidi method, and via the Rankin-Selberg integral method.All these three definitions coincide in the cases considered in this section. For pairsof representations of GL ( n ) , this follows from the local Langlands conjecture and by[Sha84, Theorem 5.1]. In case of U ( n ) , this follows from [Sha18, Theorem 1] and[BP18a, Theorem 3.4.1] (see also [AR05, Remark 3.5]). In the latter case, an appropri-ate normalization is required for the definition via the integral method (cf. [AKM18,Definition 9.10] and [BP18a, Theorem 3.4.1]).Let ψ be a non-degenerate character of N ( E ) / N ( F ) . Consider ℓ ( W ) = Z N ( F ) \ P ( F ) W ( p ) dp on the ψ -Whittaker model W ( π , ψ ) of an irreducible unitary generic representation π of GL n ( E ) (cf. Theorem 4.2). As mentioned earlier, this linear form is first consideredby Flicker who also proved the absolute convergence of the above integral [Fli88,§4]. The form ℓ is always non-zero and clearly P ( F ) -invariant. It is GL n ( F ) -invariantprecisely when π is GL n ( F ) -distinguished (see Section 4).Now consider λ ( W ) = Z F × N ( F ) \ GL n ( F ) W ( g ) dg on W ( π , ψ ) which is obviously GL n ( F ) -invariant but this integral is convergent onlywhen π is a discrete series representation [Kab04, Lemma 2]. So assume π is a dis-crete series representation. The form λ is non-zero precisely when π is GL n ( F ) -distinguished [Kab04, Theorem 4].Now [AM20b, Theorem 6.1] is the following. Theorem 5.1 (Anandavardhanan-Matringe) . Let π be a discrete series representation of GL n ( E ) which is GL n ( F ) -distinguished. Let ρ be the (discrete series) representation of U ( n , E / F ) that base changes to π (stably or unstably depending on the parity of n). Letd ( ρ ) (resp. d ( π ) ) denote the formal degree of ρ . Then, λ = c · d ( ρ ) d ( π ) · ℓ , where c is a positive constant that does not depend on the representations ρ and π . For the proof of Theorem 5.1, we refer to [AM20b, Section 6]. However, we indicatethe key ingredients in its proof in a sort of informal way.The starting point of the proof of Theorem 5.1 is the functional equation for theAsai L -function defined via the Rankin-Selberg integral method. This is due to [Fli93,Appendix] and [Kab04, Proposition 2]. Thus, if Z ( s , W , Φ ) = Z N n ( E ) \ GL n ( E ) W ( g ) Φ ((
0, . . . , 0, 1 ) g ) | det g | s dg for W ∈ W ( π , ψ ) and for a Schwartz-Bruhat function on E n , we have Z ( − s , e W , b Φ ) = γ ( s , π , r , ψ ) Z ( s , W , Φ ) (1)where e W ∈ W ( π ∨ , ψ − ) is given by W ( J t g − ) and b Φ is the Fourier transform of Φ . ISTINCTION AND BASE CHANGE 11
Now by the proof of [Kab04, Theorem 4], the right hand side of (1) is connected tothe linear form λ .On the other hand, by [AKT04, Theorem 1.4], the left hand side of ( ) is connectedto the linear form ℓ . This step is subtle and involves a new functional equation [AM17,Theorem 6.3] which in turn requires knowledge of the sign of the local root numberwhich is a particular case of [MO18, Theorem 3.6] (cf. [Ana08, Conjecture 5.1]).Thus, the linear forms λ and ℓ get connected via γ ( π , r , ψ ) , and via the factoriza-tion γ ( s , π × π ∨ , ψ ) = γ ( s , π , r , ψ ) γ ( s , π , r ′ , ψ ) they get connected to the formal degrees d ( π ) and d ( ρ ) by the Hiraga-Ichino-Ikedaconjecture for GL ( n ) as well as U ( n ) .This analysis finally leads to the relation λ = c · ǫ ( π , r , ψ ) · d ( ρ ) d ( π ) · ℓ ,where c is a measure theoretic positive constant which does not depend on the repre-sentations π and ρ .Theorem 5.1 follows since in the situation at hand it is known that ǫ ( π , r , ψ ) = Remark . In [AM20b], we were led to the formulation of Theorem 5.1 by a result overfinite fields showing that, for G = GL ( n ) or G = U ( n , q ) , the Bessel function B π , ψ is a test vector for the natural G ( F q ) -invariant linear form on an irreducible genericrepresentation π of G ( F q ) which is a base change from G op ( F q ) . Here, ψ is a non-degenerate character of N ( F q ) / N ( F q ) , where N is the unipotent radical of a (fixed)Borel subgroup of G. In fact, by [AM20b, Theorem 1.1],1G ( F q ) ∑ h ∈ G ( F q ) B π , ψ ( h ) = (cid:12)(cid:12)(cid:12) G ( F q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( F q ) (cid:12)(cid:12) (cid:12)(cid:12) G op ( F q ) (cid:12)(cid:12) · dim ρ dim π ,where ρ is the irreducible generic representation of G op ( F q ) that base changes to π .This identity for finite fields is generalized to all irreducible generic uniform rep-resentations for any connected quasi-split reductive group by Chang Yang [Yan20,Theorem 1.1]. It will be of interest to look for a p -adic analogue as in Theorem 5.1 forG = GL ( n ) . A cknowledgements My own results described in this paper are from joint work with Nadir Matringeand Dipendra Prasad. It has been a pleasure to work together and I thank bothof them heartily. Also, I am grateful to both of them for their suggestions on thefirst draft of this manuscript. I would like to thank Dipendra Prasad for constantsupport and encouragement over the last two decades. Indeed, our work on SL ( n ) originated when I was a postdoc at TIFR Mumbai during 2003-2005. I would like to thank C. S. Rajan, my initial work with him, again during my postdoc period, hashad a major impact on several of the results presented in this paper. Thanks are dueto Rajat Tandon for introducing me to the beautiful area of representation theory of p -adic groups and in particular to distinguished representations. Finally I thank theorganizers for the invitation to speak at the International Colloquium.R eferences [AC89] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of thetrace formula , Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton,NJ, 1989. MR 1007299[AKM18] U. K. Anandavardhanan, R. Kurinczuk, N. Matringe, V. S´echerre, and S. Stevens,
Galois self-dual cuspidal types and Asai local factors , J. Eur. Math. Soc. (JEMS), to appear, arXiv:1807.07755(2018).[AKT04] U. K. Anandavardhanan, Anthony C. Kable, and R. Tandon,
Distinguished representationsand poles of twisted tensor L-functions , Proc. Amer. Math. Soc. (2004), no. 10, 2875–2883.MR 2063106 (2005g:11080)[AM17] U. K. Anandavardhanan and Nadir Matringe,
Test vectors for local periods , Forum Math. (2017), no. 6, 1245–1260. MR 3719298[AM20a] , Distinction inside L -packets of SL ( n ) , arXiv:2010.05678 (2020).[AM20b] , Test vectors for finite periods and base change , Adv. Math. (2020), 106915, 27.MR 4035951[Ana05] U. K. Anandavardhanan,
Distinguished non-Archimedean representations , Algebra and numbertheory, Hindustan Book Agency, Delhi, 2005, pp. 183–192. MR 2193352[Ana08] ,
Root numbers of Asai L-functions , Int. Math. Res. Not. IMRN (2008), Art. ID rnn125,25. MR 2448081[AP03] U. K. Anandavardhanan and Dipendra Prasad,
Distinguished representations for SL ( ) , Math.Res. Lett. (2003), no. 5-6, 867–878. MR 2025061[AP18] , Distinguished representations for SL ( n ) , Math. Res. Lett. (2018), no. 6, 1695–1717.MR 3934841[AR05] U. K. Anandavardhanan and C. S. Rajan, Distinguished representations, base change, and re-ducibility for unitary groups , Int. Math. Res. Not. (2005), no. 14, 841–854. MR 2146859[BP18a] Rapha¨el Beuzart-Plessis,
Archimedean theory and ε -factors for the Asai Rankin-Selberg integrals ,arXiv:1812.00053 (2018).[BP18b] , Plancherel formula for GL n ( F ) \ GL n ( E ) and applications to the Ichino-Ikeda and formaldegree conjectures for unitary groups , arXiv:1812.00047 (2018).[BP20] , Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices ,arXiv:2008.05036 (2020).[Fli88] Yuval Z. Flicker,
Twisted tensors and Euler products , Bull. Soc. Math. France (1988), no. 3,295–313. MR 984899[Fli91] ,
On distinguished representations , J. Reine Angew. Math. (1991), 139–172.MR 1111204[Fli93] ,
On zeroes of the twisted tensor L-function , Math. Ann. (1993), no. 2, 199–219.MR 1241802[FLO12] Brooke Feigon, Erez Lapid, and Omer Offen,
On representations distinguished by unitary groups ,Publ. Math. Inst. Hautes ´Etudes Sci. (2012), 185–323. MR 2930996[GGP12] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad,
Symplectic local root numbers, centralcritical L values, and restriction problems in the representation theory of classical groups , Ast´erisque(2012), no. 346, 1–109, Sur les conjectures de Gross et Prasad. I. MR 3202556[HII08] Kaoru Hiraga, Atsushi Ichino, and Tamotsu Ikeda,
Formal degrees and adjoint γ -factors , J.Amer. Math. Soc. (2008), no. 1, 283–304. MR 2350057[HLR86] G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Fl¨achen , J. Reine Angew. Math. (1986), 53–120. MR 833013
ISTINCTION AND BASE CHANGE 13 [Jac04] Herv´e Jacquet,
Kloosterman identities over a quadratic extension , Ann. of Math. (2) (2004),no. 2, 755–779. MR 2123938[Jac05] ,
Kloosterman identities over a quadratic extension. II , Ann. Sci. ´Ecole Norm. Sup. (4) (2005), no. 4, 609–669. MR 2172953[JL85] H. Jacquet and K. F. Lai, A relative trace formula , Compositio Math. (1985), no. 2, 243–310.MR 783512[JY96] Herv´e Jacquet and Yangbo Ye, Distinguished representations and quadratic base change for GL ( ) ,Trans. Amer. Math. Soc. (1996), no. 3, 913–939. MR 1340178[Kab04] Anthony C. Kable, Asai L-functions and Jacquet’s conjecture , Amer. J. Math. (2004), no. 4,789–820. MR 2075482[Mat09] Nadir Matringe,
Conjectures about distinction and local Asai L-functions , Int. Math. Res. Not.IMRN (2009), no. 9, 1699–1741. MR 2500974[Mat10] ,
Distinguished representations and exceptional poles of the Asai-L-function , ManuscriptaMath. (2010), no. 3-4, 415–426. MR 2592088[Mat11] ,
Distinguished generic representations of GL ( n ) over p-adic fields , Int. Math. Res. Not.IMRN (2011), no. 1, 74–95. MR 2755483[Mat14] , Unitary representations of GL ( n , K ) distinguished by a Galois involution for a p-adic fieldK , Pacific J. Math. (2014), no. 2, 445–460. MR 3267536[MO18] Nadir Matringe and Omer Offen, Gamma factors, root numbers, and distinction , Canad. J. Math. (2018), no. 3, 683–701. MR 3785415[Mok15] Chung Pang Mok, Endoscopic classification of representations of quasi-split unitary groups , Mem.Amer. Math. Soc. (2015), no. 1108, vi+248. MR 3338302[Off11] Omer Offen,
On local root numbers and distinction , J. Reine Angew. Math. (2011), 165–205.MR 2787356 (2012c:22025)[Pra20] Dipendra Prasad,
A relative local Langlands correspondence and geometry of parameter spaces , https://drive.google.com/file/d/1UeLzLGwjwYFuRsSd4GFzx40amSw1-BGV/view (2020).[Sha84] Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for GL ( n ) ,Amer. J. Math. (1984), no. 1, 67–111. MR 729755[Sha18] Daniel Shankman, Local Langlands correspondence for Asai L-functions and epsilon factors ,arXiv:1810.11852 (2018).[SV17] Yiannis Sakellaridis and Akshay Venkatesh,
Periods and harmonic analysis on spherical varieties ,Ast´erisque (2017), no. 396, viii+360. MR 3764130[Yan20] Chang Yang,
Distinguished representations, Shintani base change and a finite field analogue of aconjecture of Prasad , Adv. Math. (2020), 107087, 27. MR 4070311D epartment of M athematics , I ndian I nstitute of T echnology B ombay , M umbai - 400076, I ndia . Email address ::