SOME STRONG TWISTED BASE CHANGES FOR UNITARY SIMILITUDE GROUPS SHRENIK SHAH

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1 SOME STRONG TWISTED BASE CHANGES FOR UNITARY SIMILITUDE GROUPS SHRENIK SHAH Abstract. Building upon the work of Morel [3], Skinner [48] and Shin [45] associate to certain cohomological cuspidal automorphic representations π on a unitary similitude group GU(a, b) (defined ia a quadratic imaginary field E of discriminant d E) a base change τ to G m GL n such that the expected local-global compatibility relations hold at all places except those in a set Σ(π) containing those primes l such that either π l is ramified or l d E. In this paper we proe compatibility between π and τ at an odd prime l d E under the condition that GU(a, b) l is quasi-split and π l has a ector fixed under a certain special maximal compact subgroup of GU(a, b) l. This allows us to construct the first strong twisted base changes for unitary similitude groups, which are needed in work of Skinner-Urban on the Bloch-Kato conjecture [56]. We obtain new cases of the generalized Ramanujan conjecture in this setting. The proof uses the doubling method, the local theory of p-adic representations, ariation in p-adic families, and p-adic analytic contination of periods. Our methods act as a supplement to the type of compatibility that should eentually result from the Arthur program. 1. Introduction Let F/F be the compositum of a totally real field F with an imaginary quadratic field E and let G = GU(J) /Q denote the unitary similitude group acting on the Hermitian space (F n, J) with similitudes in Q, where J denotes the Hermitian pairing. If J = J a,b = ( 1 a ) b, where a, b are nonzero positie integers, and F = Q, then Skinner [48] and Morel [3] associate to a regular cuspidal automorphic representation π on G a weak base change τ to Res E Q G m Res F Q GL n and a Galois representation ρ π : G F GL n (Q p ) with compatibility at any l such that π l is unramified and l does not diide the discriminant d E. For general J and F, Shin [45] gies a similar construction that also generalizes work of Labesse [4] for the case [F : Q] > 1. Moreoer, his weak base change τ has full compatibility with π at eery place of Q that splits in E. We improe on these results by proing cases of local-global compatibility for π with τ and ρ π at primes p d E with p >. We require that p is unramified in F. We study the situation where G(Q p ) is quasi-split and π p is spherical with respect to a certain special maximal compact subgroup K G(Q p ), which is defined as follows. (See Section 4 for more details.) By our hypothesis that G is quasi-split at p, the group G /Qp is isomorphic to the unitary similitude group oer Q p stabilizing the form 1 ϖ 1 J = or ϖ ϖ ϖ in the case where n is odd or een, respectiely. Here ϖ is a uniformizer at the place of E oer p such that ϖ Q p. Then K is the group of integral matrices in the group defined with respect to the form J. 1

2 The first step, carried out in Sections and 3, is to establish compatibility up to monodromy at p. This uses the theory of γ-factors, and follows the strategy of the aforementioned work of Skinner [48]. The key ingredients are the classical theory of γ-factors on GL n ia the doubling method of Piatetski-Shapiro and Rallis [37, 14], Lapid and Rallis s adaptation of the doubling method [5] to construct γ-factors for unitary groups and proe their fundamental properties, and the stability of γ-factors on unitary groups and on GL n, proed by Brenner [6] and Jacquet-Shalika [18], respectiely. To improe this to full compatibility, we introduce a new tool: ariation in a p-adic family. Loosely speaking, the argument is three steps. (1) By assumption, π p has triial monodromy, so we hae compatibility if τ has triial monodromy. If instead, τ has non-triial monodromy, the Satake parameters of π are of a particular form. We use this to construct in Sections 4 and 5 a pathological p-stabilized oerconergent automorphic representation σ attached to π. () We show in Section 6 that if σ moes in a family of full dimension oer weight space (or, more precisely, in a family that moes in a particular direction in weight space), we can construct a crystalline period in ρ π that iolates its known purity. (3) In Section 7, we explain how to construct a suitable family using Urban s eigenarieties [55] under a non-criticality hypothesis on σ. Then in Section 8, we substantially weaken this hypothesis using a closer examination of the automorphic multiplicities appearing in Urban s work. We first show that wheneer there exists w W, where W is the Weyl group of G, such that a certain oerconergent cuspidal automorphic multiplicity m G,0 (σw,λ, w λ) is nonzero, the argument of Section 7 applies. We then check that such a nonanishing multiplicity exists using a combination of the regularity of π, the temperedness of π at unramified places, and the existence of an inductie formula for the classical multiplicity in terms of twisted oerconergent Eisenstein multiplicities. We also gie a simpler argument for the case [F : Q] using a triial bound on the defect of the Leopoldt conjecture. A precise statement of the main theorem is gien in Theorem.3 below. As a consequence of this result, one can produce the first examples of strong base changes τ of cuspidal automorphic representations π on G. (Since E/Q always has a place of ramification, there is no way to produce a strong base change by een applying, for instance, Shin s result [45].) To obtain such examples, one selects E so that it is split at places of ramification of F and selects the form J so that G(Q p ) is quasi-split at all p ramified in E. One then considers cuspidal automorphic representations π with leel K at each ramified place of E, hyperspecial leel at each inert place of E, and arbitrary leel at split places. Then a strong base of τ exists. In practice, one often is able to choose E and the form J of the unitary group, so it is not too difficult to satisfy the hypotheses aboe. One key application is to constructing elements of Selmer groups using the strategy of Skinner-Urban [51, 50]. Since these elements are constructed as extensions of Galois representations and must satisfy local conditions at eery place, it is essential to hae a strong base change. In fact, our result is applied in their recent work [56]. Another application is to the generalized Ramanujan conjecture for the group G. This conjecture states that any globally generic cuspidal automorphic representation should be eerywhere tempered (up to the central character); we proe temperedness at p under the hypotheses aboe. Our arguments apply without modification to cuspidal representations π on G that are possibly irregular discrete series at infinity if they satisfy two additional hypotheses. Namely, we require that the classical Euler-Poincaré characteristic of π is nonanishing and that the weak base change τ of π constructed by Shin [45] is tempered at finite places. (See Remark 1.) We hope to relax the latter hypothesis in future work. Using the lifting from unitary groups to similitude groups 1, 1 I learned about this lifting from Stefan Patrikis.

3 compatibility for unitary groups follows in the limited setting F = EF considered aboe; see Corollary.4. The strategy to use interpolation results for crystalline periods in order to proe local-global compatibility for Galois representations is due to Skinner [49], and our method can be seen as a higher rank generalization of his strategy. Jorza [19] and Luu [30] hae also used strategies based on Skinner s for GSp 4 and GL n. Three differences in our work are as follows. Instead of constructing a crystalline period in order to identify a Satake parameter or show a representation is crystalline, we use the interpolation in a negatie way we assume a representation fails local-global compatibility and use that failure to construct a crystalline period that we know cannot exist. Since we are working on a ramified group, a substantial part of the argument deals with the local theory required to produce a suitable family. We are considering completely general unitary groups (rather than ones that are anisotropic oer Q), which makes it more challenging to produce a suitable family due to the presence of Eisenstein cohomology classes. In the remainder of this introduction, we sketch some of the ideas that go into the proof. For compatibility up to monodromy, first write π 0 for an irreducible subrepresentation of the restriction of π to the unitary group G 0 = U(J) /Q and write τ 0 for the restriction of τ to H 0 = Res F Q GL n. Then τ 0 is tempered at p [46]. Skinner shows that the γ-factors of π 0,p and τ 0,p are equal. We check that if π p,0 is a subquotient of the unramified principal series, the Satake parameters are of a particular form, and then τ 0,p is the unique tempered representation that agrees with π p,0 up to monodromy; this uses a numerical inariant attached to the γ-factors and the Bernstein-Zeleinsky classification. If τ 0,p has non-triial monodromy, then for step (1) aboe we need the action of a certain Hecke operator U on the p-stabilized oerconergent automorphic representation σ to hae an eigenalue that is too large for a tempered representation. We require a ery precise understanding of π 0,p to produce such a σ. Since G 0 (Q p ) is ramified, its structure theory is somewhat sophisticated. Howeer, it turns out that the Iwahori-Hecke algebra of G 0 (Q p ) can be identified with that of a split group G 0 (Q p). This obseration is a case of a technique employed by Lusztig [8] to reduce the study of unipotent representations of certain possibly ramified groups to unramified cases, and was used to study Steinberg representations of G 0 (Q p ) by Clozel-Thorne [11, 1]. To study the representation π 0,p of G 0 (Q p) corresponding to π 0,p, we apply results of Reeder [39] that translates the structure of π 0,p into questions about orbits on a certain prehomogeneous ector space. Assuming that σ can be p-adically deformed in a suitable family, in step () we construct a crystalline period in ρ π that iolates its known purity. For this, we apply a result of Kisin [1] and Nakamura [33] to analytically continue a period from points of ery regular weight to ρ π. We need to carefully choose a line such that it contains a dense set of points with arbitrarily regular algebraic weight, the eigenalue of U aries analytically oer the family, the eigenalue of U is equal to a crystalline Frobenius eigenalue in ery regular weight, and the attached Galois representations hae a fixed Hodge-Tate weight. Finally, in step (3), to actually produce the required family of Galois representations, we apply the technique of pseudorepresentations due to Wiles [57] and Taylor [5] to the p-adic families constructed by Urban [55]. See Sections 7 and 8 for the subtleties inoled here. Urban s main theorem applies when the multiplicity m G,0 (σ, λ) 0, which is true if σ has non-critical slope and I learned of this identification from Jack Thorne. 3

4 regular weight. To construct the p-adic families in critical slope cases, we first study oerconergent automorphic multiplicities and look at aatars σ w,λ of classical representations in non-dominant weight w λ in Urban s eigenariety, where w W is an element of the Weyl group of G and is a normalized action on the weights. To our knowledge, this is the first time such a technique has been applied. If any of these σ w,λ can be p-adically interpolated in Urban s eigenariety, the same argument as in the non-critical case applies. If all of their automorphic multiplicities anish, we use Urban s work to produce a finite slope form on a proper Lei subgroup of G such that π is a subquotient of its parabolic induction to G. This Lei subgroup must hae at least one GL 1/F factor. Although this form may not be classical, we check that the restriction to any of its GL 1/F factors is a Hecke character, and use a comparison of the weight of that character with the weight enforced by the temperedness of π at a split place to find a contradiction. Our technique is not special to unitary groups, though many of the calculations here are only carried out for such groups. In subsequent work we will gie many additional cases of local-global compatibility.. Weak base change, L-functions, and γ-factors In this section, we use an approach similar to Skinner [48, 3] in order to compare the local Langlands parameters at a finite place of π and its weak base change τ..1. Unitary groups and base change. Let F be a totally real field, let E C be a quadratic imaginary field (regarded as haing a fixed embedding into C), and let F = EF. Let J be a Hermitian form on F n. We define G = GU(J) /Q as follows. For a Q-algebra R, G(R) = { g GL n (R Q F ) gj t g = µ(g)j, µ(g) R }, where we use for the action of the nontriial element of Gal(E/Q). This also defines a homomorphism of Q-groups µ : G G m. Let G 0 denote the kernel of this homomorphism; it is precisely the restriction of scalars to Q of the usual unitary group of J oer F. We define H = Res E/Q G /E and H 0 = Res E/Q G 0/E. If R is an E-algebra, we may identify R Q E with R R by sending r e (er, er). We use this to identify G /E with G m Res F/E GL n as follows. For any E-algebra R, we send an element g = (g 1, g ) G(R) GL n (R F ) = GL n (R F ) GL n (R F ) to the element (µ(g), g 1 ) R GL n (R Q F ). Since J = t J, the conditions in each factor for g = (g 1, g ) G(R) are equialent to one another under transposition, and the condition on the first factor is g 1 J t g = µ(g)j, or g = t (J 1 g1 1 µ(g)j), so the (g 1, g ) are in bijection with the (µ(g), g 1 ). (We hae µ(g) R by definition.) Thus we hae H = Res E/Q G m Res F/Q GL n and H 0 = Res F/Q GL n. Using this identification, we define a map θ H : H(R) H(R) by (1) θ H ((x, g)) = (x, x t g 1 ). We also write θ H0 for the restriction to H 0. It follows from the preceding discussion that we hae an identification Ĝ = C ν:f R GL n(c), where Ĝ denotes the dual. Define an action of the nontriial element c Gal(E/Q) to be the unique outer automorphism presering the standard splitting; its action is gien by (x, (g ν ) ν ) (x ν det g ν, (Φ 1 n t gν 1 Φ n ) ν ), where the n n-matrix Φ n is defined by (Φ n ) ij = ( 1) i1 δ i,n1 j and δ ij is the Kronecker δ function. There is also an action of G Q by precomposition on the ν : F R, so that it permutes the GL n (C) factors. This action commutes with that of c. Then G Q acts by the product of the action ia its quotient Gal(E/Q) and its permutation action on the ν. We similarly define an 4

5 action of G Q on Ĝ0 = ν:f R GL n(c) by the same product of actions, but where c Gal(E/Q) now acts by c((g ν ) ν ) = (Φ 1 n t gν 1 Φ n ) ν. We then define L G = Ĝ W Q and L G 0 = Ĝ0 W Q, where W Q acts by its quotient G Q. There is a natural L-homomorphism L G L G 0 defined by (x, g) σ g σ. Since H = Res E/Q G /E, we hae Ĥ = Ĝ Ĝ with the action of c Gal(E/Q) gien by c(g, h) = (c(h), c(g)), and the action of W Q again factoring through G Q and equal to the product of the Gal(E/Q) action just gien with the permutation action in each factor. A similar definition holds for H 0. This defines L H and L H 0, and the diagonal embeddings (using the identity map on W Q ) yield L-homomorphisms BC : L G L H and BC 0 : L G 0 L H 0. In the usual way, these global Langlands dual groups gie rise to local Langlands dual groups using the embeddings W Qp W Q. Example 1. We discuss one particular case for G(J) that sered as the focus of Morel s work [3]. Let a b = n with a b and define J a,b by () J a,b = A a 1 b a where 1 is the identity matrix and A m is the m m matrix defined by 1 (3) A m = Then G(J a,b ), sometimes called GU(a, b), is an example in the class of unitary groups under consideration. In the following, we write eerything with respect to the form J a,b. We fix a maximal torus defined by λ 1 T (R) =... λ 1 λ n = = λ a λ b1 GL n (R Q F ) = λ a1 λ a1 = = λ b λ b. λ n A maximal Q-split subtorus of T is (λ 1) diag(λ 1 1,..., λ a 1, 1 1,..., 1 1,λ 1 a 1,..., λ 1 }{{} 1 1) S(R) = b a GL n (R Q F ) if b > a or S(R) = { diag(λλ 1 1,..., λλ a 1, λ 1 a A a, 1,..., λ 1 1 1) GL n (R Q F ) } if b = a. We let B n denote the standard Borel subgroup of GL n. Then a minimal parabolic subgroup of G is gien by P (R) = M 1 M G(R) M 3 M 1, M 3 B a (R Q F ), M GL b a (R Q F ). For any m 1,..., m k Z >0 with m 1 m k = m a, we obtain a standard parabolic of G by intersecting G inside Res F/Q GL n with the standard parabolic of type (m 1,..., m k, b a m, m k,..., m 1 ). 5

6 .. Weak base changes and Galois representations attached to automorphic representations of G. It is a conjecture of Langlands and Clozel that certain automorphic representations those that are algebraic should be attached to moties (and thus compatible families of Galois representations). In this work we will be concerned only with cuspidal automorphic representations π on G that are regular discrete series at infinity. Such π satisfy Clozel s hypothesis. In fact, the Galois representation attached to π is the one attached to its weak base change τ to H. Since the Galois representations ρ τ attached to such a τ hae been constructed thanks to the work of many mathematicians, and nearly all the expected properties of ρ τ are known, the relationship between π and τ is the main focus of this article. When the unitary group is noncompact at infinity and F = Q, it is only possible to say anything about the group G because of the intricate study of the intersection cohomology of the Shimura ariety attached to G by Morel [3], who attaches a ery weak base change τ to a π on G. (By ery weak here, we refer to the indeterminacy of the set of places where π is compatible with τ.) This case is the most important one for applications to the Bloch-Kato conjecture and Iwasawa theory for elliptic cures oer Q. Using different methodologies, Skinner [48] and Shin [45] describe additional compatibility for τ, yielding a weak base change, i.e. a base change that has compatibility at an explicit set of places including all the ones where the data is unramified. The former work requires F = Q, ab 0, and gies a slightly weaker form of compatibility than the latter work at places of Q that split in E. For this reason, the statement below is based on Shin s result [45]. Howeer, in what follows, we largely follow the notations and conentions of Skinner s paper [48]. Skinner and Shin hae opposite conentions for the L-packet attached to the algebraic representation V λ of weight λ namely, Skinner asks for nonanishing of the cohomology of π Vλ while Shin uses π V λ instead (as does Morel s work [3]). We later rely on Urban s work [55], which uses Skinner s conention as we do. We write BC(π p ) for the representation of H(Q p ) with L-parameter BC ψ πp, where ψ πp denotes the Langlands parameter of π p. If τ is an irreducible admissible representation of H(A Q ), then by using the identification of H with Res E/Q G m Res F/Q GL n in Section.1, we can think of τ as being a pair (ψ, τ 0 ) of representations of Res E/Q G m (A Q ) and H 0 = Res F/Q GL n (A Q ). Recall the inolution θ H defined in (1). We say that a τ as aboe is θ H -stable if τ θ H = τ, which is equialent to τ0 = τ0 c and ψ = ψc χ c τ 0, where denotes the contragredient, χ τ0 is the central character of τ 0, and c denotes the conjugate, i.e. the composition with the inolution on Res E/Q G m Res F/Q GL n induced by the non-triial element of Gal(E/Q). Theorem.1 ([3, 48, 46, 45]). Let F be a totally real field and let F/F be the compositum of F with an imaginary quadratic field E. Let J be a Hermitian form on F n, suppose that π is a cuspidal automorphic representation on the unitary similitude group G = GU(J), and let G 0, H, and H 0 be defined as in Section.1. Recall that ia Weil restriction of scalars we regard all of these groups as being oer Q. Moreoer, assume that there exists an algebraic representation V λ of G /F such that π is a regular discrete series representation inside the L-packet attached to V λ. Then there exists a possibly non-cuspidal automorphic representation τ = (ψ, τ 0 ) on H with the following properties. (1) We hae τ p = BC(π p ) for any prime p of Q that either (a) splits in E, or (b) is inert in E with π p unramified and p not a prime of ramification of F. () The infinitesimal character of τ is associated to the algebraic representation V λ V θ λ of H /F, where this is regarded as a representation of H /F ia the identifications (from Section.1) of G /F with G m ν:f R GL n and H /F with G /F G /F ia the map R Q E R R (also defined in Section.1). 6

7 (3) The representation τ is θ H -stable. Moreoer, ψ = χ c π and χ τ0 = χ π /χ c π, where χ π denotes the central character of π. (4) The representation τ 0 is tempered at all finite places. Shin requires only that π is discrete series, and the resulting τ is an isobaric sum of discrete conjugate self-dual representations (rather than cuspidal ones). Howeer, since we are assuming π is regular discrete series, these discrete representations are cuspidal and τ is tempered this is proed in [47, Corollary 4.16]. We can be more precise about the possibilities for the representation V λ mentioned aboe. If T G is the diagonal torus, then T /F G /F is identified with G m F R Gn m. Writing k for [F : Q] and ν i, i = 1,..., k, for the distinct maps F R, we can identify the character group X(T ) with Z 1kn as follows. We write c = (c, c ν1,..., c νk ) Z 1kn, where c νi = (c νi,1,..., c νi,n). Then λ(c) X(T ) is gien by (t, diag(t i,1,..., t i,n )) t c k n i=1 j=1 tc ν i,j i,j, where the group of elements of the form diag(t i,1,..., t i,n ) is the torus of the factor G n m indexed by ν i. Using the upper-triangular Borel, the dominant characters are exactly those satisfying c νi,1 c νi,n and regular dominant characters hae strict inequalities. Then a algebraic representation V λ = V λ(c) as considered in Theorem.1 is determined by such a c. We will also need to know what the representation V λ Vλ θ is in this context. Letting T H H be the usual maximal torus, we hae X(T H ) = X(T ) X(T ), where the first factor corresponds to the chosen embedding E C and the second is the conjugate. So an algebraic representation of H /F = G/F G /F is determined by a pair (c, c ) of data in the format of c aboe. For V λ Vλ θ, c 1 = c. The algebraic representation Vλ θ = V λθ can be calculated by examining the effect of θ((x, g)) = (x, x t g 1 ) using the explicit form of λ on the torus. In particular, for each i, we hae (t, diag(t i,1,..., t i,n )) θ 1 1 (t, diag(tt i,1,..., tti,n )), so that c corresponds to the map (t, diag(t i,1,..., t i,n )) t c k n i=1 j=1 t c ν i,j t c ν i,j i,j. Also note that θ takes the Borel to its opposite, so we need to conjugate by the longest element of the Weyl group, which interchanges t i with t n1 i. We deduce that c = (c, c ν 1,..., c ν k ), where c = c k n i=1 j=1 c ν i,j and c ν i = ( c νi,n,..., c νi,1). We note that if c is dominant or regular dominant, so is c. Definition 1. Suppose that λ = λ(c). We define the weight of the algebraic representation V λ to be c c = c k n i=1 j=1 c ν i,j. This can alternatiely be defined as the weight of the induced algebraic representation of the center of G. We now describe the Galois representation attached to π (which is just the one attached to τ = (ψ, τ 0 )). Its construction is the culmination of works of many authors, including Shin [46, Theorem 1.]. We note that usually one only attaches a Galois representation to τ 0 ; following Skinner [48, Theorem 10] we simply tensor that representation with the one attached to ψ. For the entire paper we will fix an isomorphism ι : C Q p. Theorem.. Suppose that π and τ = (ψ, τ 0 ) are as described in Theorem.1. Then there is a continuous semi-simple representation ρ π : G F GL n (Q p ) satisfying the following properties. (1) At places p, ρ π GF is potentially semi-stable at. The Hodge-Tate weights are gien in terms of the aforementioned data (c 1, c ) attached to V λ Vλ θ as follows. For an embedding ν : F R (which then maps to Q p ia ι), there is a set of Hodge-Tate weights attached to the chosen embedding E C and a set of weights attached to its conjugate. This ordered 7

8 pair of sets is gien by HT ν (ρ π GF ) = ({ c j 1 c ν,j }, { c j 1 c ν,n1 j } ). () The representation WD(ρ π GF ) is pure of weight n 1 w for eery place F, where V λ has weight w, and moreoer WD(ρ π GF ) Fr ss = ιrec (τ ψ 1 n ). Here WD denotes the Weil-Deligne representation and Fr ss denotes Frobenius semisimplification. For the meaning of WD(ρ π GF ) when p (at least in the semi-stable case, which is all we will use), see Section 6.1. We also remark that the weights in each of the two sets in HT ν (ρ π GF ) are all distinct when the data of c is dominant. We now state our main theorem, which is an improement to Theorem.1. It also follows that π has better compatibility with its Galois representation. Theorem.3. Maintain the notation of Theorem.1. Assume that the odd prime p ramifies in E but not in F, G /Qp is quasi-split, and π p is K-spherical for the special maximal compact subgroup named in Section 4.4 or 4.5. Then in addition to the compatibility described in Theorem.1, we also hae τ p = BC(π p ) and π p is tempered (up to its central character). In fact, our argument can apply to certain irregular π as well. Remark 1. If π is discrete series but not regular, Shin still proes Theorem.1 but without the temperedness of τ 0 at all finite places. If we instead take this temperedness as an additional hypothesis and also assume that the classical Euler-Poincare characteristic of π is non-anishing (which is automatic in the regular case), then the entirety of our argument here applies without modification. We can deduce unitary cases as well. Corollary.4. Suppose that E, F, F, and J are as in Theorem.1, but consider the unitary group G 0 in place of the unitary similitude group G. Then gien an automorphic representation π 0 on G 0, there exists a base change τ 0 to H 0 with compatibility as described in Theorems.1 and.3. Proof. The work of Langlands-Labesse [3, 6] shows that since G/G 0 is a torus, there exists an extension of the central character of π 0 to G and a lifting π of π 0 to G with that central character. The precise construction of such an extension of the central character is gien by an argument of Patrikis [36, Proposition 3.1.4]. Then we just apply Theorems.1 and.3 and read off the compatibility between π 0 and τ A relation between Satake parameters. We now regard G 0 and H 0 (as defined in Section.1) as F -groups. Suppose that we are in the situation of Theorem.1. To aoid cluttering the notation, we let π denote the cuspidal automorphic representation of G 0 (A F ) gien by choosing an irreducible subrepresentation of the restriction of the π of Theorem.1 to G 0 and write τ in place of τ 0. Then τ is a weak base change of π to H 0. We write q for the size of the residue field of F. Note that if we study the base change properties of these F groups in place of the Q-groups, the effect on the Langlands dual group is only to ignore the permutation action of G Q on the ν : F R, which is harmless. Suppose that the rational prime p ramifies in E and that F is unramified at p. Fix a place p of F. We also assume that G 0, is quasi-split and that π is a subquotient of the parabolic induction of an unramified character χ of the maximal torus of G 0,. 8

9 Looking ahead, in Section 3 we will classify possibilities for τ and π. A consequence of the main result there is the following, which asserts compatibility as long as a special relationship between the Satake parameters of χ does not occur. We gie a short, self-contained argument for this result at the end of the section. See Definition for the definitions of Satake parameters and unramified principal series representations used below. Theorem.5. Let π be a cuspidal automorphic representation of G 0 (A F ) and let τ be a weak base change of π to H 0 (A F ). Suppose that the rational prime p ramified in E, p is a place of F, and G 0, is quasi-split. Moreoer, assume that π is a subquotient of Ind G 0 B χ, where χ is an unramified character of the diagonal maximal F -rational torus of G 0. Let {α i } i {1,...,n} be the set of Satake parameters of π, and write ψ τ and ψ π for the Langlands parameters. (1) Suppose that π is an unramified principal series representation. Then if α i q α j for i, j {1,..., n}, we hae ψ τ = BC 0 ψ π. () Suppose that α i α j and α i q α j for i, j {1,..., n}. Then we hae ψ τ = BC 0 ψ π. Our first task will be to calculate the standard local L-factor of π ω, where π is assumed to be an almost unramified principal series representation and ω is an unramified character of the group R, where R = Res F F G m. We will then use a result of Skinner [48] to compare the γ-factors of π and τ. Skinner s work employs results of Godemont-Jacquet [18], Lapid-Rallis [5], and Brenner [6] on the construction, properties, and stability of γ-factors for unitary and general linear groups..4. Local L-factors for tori. Let w be the place of F oer. Yu [58] calculates the local Langlands correspondence for an induced torus T = Res Fw/F F w to be the composition Hom(F w, C ) Hom(W Fw, C ) H 1 (W F, Ind W F W Fw C ) = H 1 (W F, T ), where the first map is ia local class field theory and the second is the isomorphism of Shapiro s lemma. (We use geometric normalizations, so a uniformizer in F w maps to a geometric Frobenius element.) Note that the inerse of this second map is restriction to W Fw followed by ealuation at 1 WF. (See, e.g., [44, Proposition 10,.5]). The image of the unramified character χ α : F w C sending a uniformizer ϖ to α C is, under the first map, the unramified character sending Frob Fw to α, where Frob Fw denotes a geometric Frobenius element. We obtain this upon restriction to W Fw and ealuation at 1 WF of the homomorphism ϕ α : W F Ind W F W Fw C defined as follows. Let I F denote the inertia subgroup and set ϕ α (I F ) = 1, so that ϕ α factors through W F /I F. Then define ϕ α (Frob m F ) = α m on W F /I F, where m Z and α denotes the constant function σ α for σ W F. The homomorphism ϕ α is an element of H 1 (W F, Ind W F W Fw C ) since W F acts triially on the constant functions in Ind W F W Fw C. Thus, χ α and ϕ α correspond to each other under the local Langlands correspondence for T. We also note that for the torus T = U(1) oer F, there is only the triial unramified character, so the local Langlands correspondence takes this character to the triial element of H 1 (W F, L T )..5. Unitary groups oer p-adic fields. We summarize some basic facts regarding unitary groups oer p-adic fields. One reference for these is an article of Minguez [31]. We define a unitary group U for a quadratic extension L/L of p-adic fields in the same way as the global case. Howeer, in the p-adic case, if the dimension n of the Hermitian space is odd, there is only one possible unitary group U up to isomorphism, and it is quasi-split. (There are two non-isomorphic 9

10 Hermitian forms, but the associated unitary groups are isomorphic.) If the dimension n is een, there are two possibilities for U, but only one is quasi-split. In both cases, the quasi-split unitary group can be gien by the Hermitian form defined by A n aboe. In Section 4, we will use a different choice of form in order to simplify the discussion of the finer integral structure of U, but for now A n will suffice. Let G = U(A n ) /L. The maximal torus T and Borel B of G can be defined by requiring µ(g) = 1 in the formulas in Example 1 for T and P, where we set a = b or a 1 = b depending on whether n is een or odd. The description of parabolics containing B is also the same as in the global case with these alues of a and b. If n = m is een, T = (Res L/L L ) m, whereas if n = m 1 is odd, T = (Res L/L L ) m U(1). The spherical Weyl group is isomorphic to S m (Z/Z) m, where S m permutes the matrix entries λ 1,..., λ m and their inerses in the description of T in Example 1, and the i th cyclic factor switches λ i with λ 1 i. The L-group is defined using the same action as proided in the global case..6. Local L-factors for ramified unitary groups. Let T be the aforementioned maximal torus in G 0, oer F. We are assuming G 0, is quasi-split, so T m ( m ) = Res Fw/F F w or T = U(1) /F. i=1 In particular, we hae dual groups T = m i=1 Ind W F W Fw C or T = i=1 ( m Res Fw/F i=1 F w Ind W F W Fw C ) C. This defines the L-groups: the action of W F factors through Gal(F w /F ), and the action of the nontriial element c Gal(F w /F ) inerts the U(1) /F factor and acts in the usual way on the induction spaces. Let β = (β 1,..., β m ) (C ) m. Under the local Langlands correspondence calculated in Section.4, the unramified character χ β, defined by sending uniformizers in each non-u(1) /F factor to β 1,..., β m respectiely, maps to ((β 1, β 1 ),..., (β m, β m )) T and ((β 1, β 1 ),..., (β m, β m ), 1) T in the respectie cases aboe. Here, we hae denoted an element of Ind W F W Fw C by the ordered pair giing the alues of a function at 1 WF and an arbitrary fixed lift of c to W F. In order to calculate the local Langlands parameter W F L G 0, of the corresponding unramified principal series representations, we need to determine how these tori embed into the L-group of G 0,. We find that the morphisms T Ĝ0, gien by and ((t 1, t ),..., (t m 1, t m )) diag(t 1, t 3,..., t m 1, t 1 m,..., t 1 ) ((t 1, t ),..., (t m 3, t m ), t m1 ) diag(t 1, t 3,..., t m 1, t m1, t 1 m,..., t 1 ) gie embeddings L T L G 0, in the een and odd cases, respectiely, since these maps are W F - equiariant. If K is a special maximal compact subgroup, these calculations and unramified functoriality determine the local Langlands parameter ψ β : W F SL (C) Ĝ0, attached to a K-spherical subquotient of the normalized induction Ind U B χ β to be the map that kills SL (C) and sends any 10

11 f W F lifting Frob k F to (diag(β 1,..., β m, β 1 m,..., β 1 1 ))k f and (diag(β 1,..., β m, 1, β 1 m,..., β 1 1 ))k f, respectiely. The map ψ β : W F ψ β (I F ) to Ĝ0, is triial. L G 0, is nearly unramified in the sense that the projection of Let R = Res F/F G m, so that ˆR = C C with the action of the nontriial element c Gal(F/F ) defined by c(α, β) = (β, α). The standard representation r st,g0 : L G 0 WF L R GL n (C) is defined by ( αg r st,g0 (g 1, (α, β) 1) = ) ( ) 1 βφ 1 n t g 1, r Φ st,g0 (1 c, 1 c) = n. n 1 n The standard representation r st,h0 : L H 0 L WF R GL n (C) is defined by ( αg1 r st,h0 ((g 1, g ) 1, (α, β) 1) = βφ 1 n t g 1 Φ n ) (, r st,h0 (1 c, 1 c) = ) 1 n. 1 n We hae r st,g0 = r st,h0 (BC 0 1L R). Let π be the K-spherical subquotient of Ind U B χ β as before and let ω be an unramified character mapping uniformizers to β. We consider the representation r st,g0 (ψ π, ψ ω ) : W F GL n (C). (We may ignore the SL (C) factor.) Obsere that the image of I F is the two element subgroup generated by ( 1 n ) 1 n, so that we are interested in the action of FrobF on the subspace V I F of ectors of the form t (, ), C n. Correspondingly, the image of Frob F has the form or diag(ββ 1,..., ββ m, ββ 1 m,..., ββ 1 1, ββ 1,..., ββ m, ββ 1 m,..., ββ 1 1 ) diag(ββ 1,..., ββ m, 1, ββ 1 m,..., ββ 1 1, ββ 1,..., ββ m, 1, ββ 1 m,..., ββ 1 1 ). We calculate in the een rank case L(π ω ) = det(1 q s r st,g0 (ψ β (Frob F ), ψ ω (Frob F )) V I F ) 1 = det(diag(1 q s m = (1 q s i=1 or, in the odd rank case, L(π ω ) = det(diag(1 q s = (1 q s ββ 1,..., 1 q s ββ i ) 1 (1 q s ββi 1 ) 1, β) 1 ββ m, 1 q s ββ 1 ββ 1,..., 1 q s ββ m, 1 q s β, m (1 q s i=1 1 q s ββ 1 m,..., 1 q s m,..., 1 q s ββ i ) 1 (1 q s ββi 1 ) 1. ββ 1 1 )) 1 ββ 1 1 )) 1 Definition. Suppose that χ β is an unramified character of the maximal torus. Then if π is any subquotient of Ind U B χ β, we say that the Satake parameters of π are the multiset { β i, βi 1 } or { βi, 1, βi 1 } in the een or odd case, respectiely. If additionally π is the K-spherical subquotient for a special maximal compact K, we say π is an unramified principal series representation with Satake parameters {α i }. 11

12 Using the same approach as aboe, once can carry out the calculation of L-factors for an unramified principal series representation τ with Satake parameters {γ 1,..., γ n } and ω sending a uniformizer to β. (By unramified principal series, we again mean that it is the subquotient of the releant parabolic induction that is spherical for a hyperspecial maximal compact.) Using the diagonal torus, the embedding m Ind W F W Fw C Ĥ0, sends i=1 diag((t 1, t ),..., (t n 1, t n )) (diag(t 1, t 3,..., t n 1 ), diag(t 1 n,..., t 1 )). Thus ψ τ sends Frob F to (diag(γ 1, γ,..., γ n ), diag(γn 1,..., γ1 1 )). In particular, the image of Frob F under the composite map r st,h0 (ψ τ, ψ ω ) : W F GL n (C) is gien by which yields the L-factor diag(βγ 1,..., βγ n, βγ 1,..., βγ n ), L(τ ω ) = n (1 q s γ i β) 1. i=1 The obseration aboe that r st,g0 = r st,h0 (BC 0 1L R) implies that L(s, π ) = L(s, τ ) if BC 0 ψ π = ψ τ. We can gie a conerse to this as follows. Proposition.6. Suppose that π and τ are unramified principal series representations. Then if L(s, π ) = L(s, τ ), we hae BC 0 ψ π = ψ τ. Proof. Indeed, the image of the map ψ π defined aboe under the base change map BC 0 sends Frob F to (diag(β 1,..., β m, βm 1,..., β1 1 ), diag(β 1,..., β m, βm 1,..., β1 1 )) Ĥ or (diag(β 1,..., β m, 1, βm 1,..., β1 1 ), diag(β 1,..., β m, 1, βm 1,..., β1 1 )) Ĥ in the respectie cases. If {γ 1,..., γ n } = { β 1,..., β m, βm 1,..., β 1 } { or β1,..., β m, 1, βm 1,..., β1 1 }, 1 respectiely, then BC 0 ψ π is immediately seen to be equialent to ψ τ..7. γ-factors. Lapid and Rallis [5] define γ-factors γ(s, π ω, ψ ), where is a place of F and ψ is an additie character of F. We fix ψ to be the standard additie character of F, denoting the corresponding γ-factor by γ(s, π ω ). Skinner [48, 3] proes the following result using the results of Lapid and Rallis [5, Theorem 4] on γ-factors of unitary groups, their analogues in the general linear case [15], and stability of γ-factors in both the general linear [18] and unitary [6] cases. Lemma.7. Let π and τ be as in the statement of Theorem.5, ignoring the condition on Satake parameters. Then γ(s, π ω ) = γ(s, τ ω ) for all and ω. We may now proe Theorem.5. We use arious facts about the Bernstein-Zeleinsky classification that will be introduced in greater detail in Section 3.1. Proof of Theorem.5. For now, assume only that π is a subquotient of the induction of an unramified character. We write γ(s, σ) = γ(s, σ 1) for any admissible representation σ of a local 1

13 group, and we let {α i } n i=1 denote the multiset of Satake parameters of π. (This multiset has the form {β i } { βi 1 } or {βi } { βi 1 } {1} in the notation aboe.) We first note that γ(s, π ) = L(1 s, π ) L(s, π ) i=1 = n i=1 m i=1 (1 q s α i ) (1 q 1s αi 1 ) by multiplicatiity of γ-factors [5, Theorem 4.]. Let { Ω = z π im(z) < π } C. log q log q The hypotheses of the proposition imply that this function has n roots and n poles as a function of s in Ω, counted with multiplicity, and moreoer, these roots and poles determine the function γ(s, π ). There exist {n i } k i=1 such that n = i n i and supercuspidal representations τ i on GL ni such that τ is a subrepresentation of Ind H 0 P iτ i, where P is the parabolic subgroup associated to the decomposition (n 1,..., n k ). By multiplicatiity and additiity of γ factors, γ(s, τ ω ) = i γ(s, τ i ω ). Obsere that γ(s, τ i ) = L(1 s, τ i) ɛ(s, τ i ) L(s, τ i ) where ɛ(s, τ ) is holomorphic and nonanishing as a function of s. In particular, Lemma.7 implies that L(1 s, τ ) = γ(s, τ )ɛ(s, τ ) 1 = γ(s, π )ɛ(s, τ ) 1 L(s, τ ) has n roots and n poles as a function of s in Ω. Each γ(s, τ i ) has at most one root and one pole in Γ, as follows, for example, from the calculations in [15, Proposition 5.11]. Thus k = n and τ is a subrepresentation of the parabolic induction from a Borel of the character defined by the τ i. In fact, γ(s, τ i ) is holomorphic for a ramified character τ i [15, Proposition 5.11], so each τ i must be unramified. As a consequence, n L(1 s, τ i ) n i=1 (1 q s γ i ) γ(s, τ i ) = = L(s, τ i ) m i=1 (1 q 1s γi 1 ), where the multiset {γ i } n i=1 is the set of Satake parameters of τ i. In this situation, the Satake parameters of both π and τ are determined by the poles and roots of their γ-factors in Ω, so this shows that {γ i } = {α i }. Finally, τ is an unramified principal series representation by Theorem 4. of [5] and the condition α i q α j of the proposition, which now applies to the γ i. Thus, if π is an unramified principal series representation, by Proposition.6, the Langlands parameter of τ is the base change of that of π, proing part (1). For part (), it suffices to show that π is an unramified principal series representation under the additional hypothesis that α i α j for all i, j. For this, we first note that τ is a tempered unramified principal series representation by Theorem.1, so that the γ i hae complex absolute alue 1. In this situation, since the character χ β is both unitary and regular, we may apply Bruhat irreducibility [9, Theorem 6.6.1] to see that the full parabolic induction with Satake parameters {α i } is already irreducible. It follows that π is an unramified principal series representation. 3. Possibilities for τ and π In this section we use the Bernstein-Zeleinsky classification of automorphic representations and the calculation by Godemont-Jacquet of their L-factors in order to derie consequences for π and τ in case the conditions of Theorem.5 fail to hold. In particular, we proe the following result. 13

14 Theorem 3.1. Let π be a cuspidal automorphic representation of G 0 (A F ) and let τ be a weak base change of π to H 0 (A F ). Suppose that G 0 (Q p ) is quasi-split and ramified, and we assume that π is a subquotient of the parabolic induction of an unramified character of the Borel. Then the multiset of Satake parameters of π are the disjoint union of any number of sets of the form m 1 i=0 { q any number of sets of the form (4) m 1 i m 1 i=0 { q γ } m 1 i m 1 i=0 } { } m 1 i q γ 1, γ S 1 \ ±1, or m 1 i=0 { q m 1 i with m een, an een (if n is een) or odd (if n is odd) number of sets of the form m 1 { } m 1 i (5) q i=0 with m odd, and an een number of sets of the form m 1 { (6) q with m odd. i=0 m 1 i 3.1. Local L-factors and γ-factors of representations of GL n. In this section, we state some results that calculate local L-factors for representations of GL n (L), where L/Q p is a finite extension. Let q denote the order of the residue field of L. The essential facts regarding the classification of representations are due to Bernstein and Zeleinsky [4], while the results on L-factors we need were originally computed by Godemont and Jacquet [15]. We will use as our primary reference a recent paper of Cogdell and Piatetski-Shapiro [13]. For P GL n (L) a parabolic subgroup of type (n 1,..., n k ) and representations ρ i of GL ni (L), we write Ind GLn(L) P (ρ 1 ρ k ) for the normalized induction. By the Bernstein-Zeleinsky classification, eery irreducible admissible tempered representation ρ can be expressed as the unique irreducible quotient of Ind GLn(L) P (ρ 1 ρ k ), where each ρ i is square-integrable. We write ρ = Q(ρ 1 ρ k ), where the right hand side denotes the unique irreducible quotient of the normalized induction. Theorem 3. ([13, 4]). Let ρ = Q(ρ 1 ρ k ) be any irreducible admissible tempered representation of GL n (L), and let ω be any character of GL 1 (L). Then we hae k k L(s, ρ ω) = L(s, ρ i ω) and γ(s, ρ ω) = γ(s, ρ i ω). i=1 Now let ρ be any irreducible square-integrable representation, and let ν denote the determinant character det on GL n (L) for any n. Then } } i=1 ρ = Q(ρ ν m 1 ρ ν m 1 1 ρ ν m 1 ), where ρ is a supercuspidal representation of GL n m (L). Theorem 3.3 ([13, 4]). Let ρ = Q(ρ ν m 1 ρ ν m 1 1 ρ ν m 1 ) be any square-integrable representation of GL n (L), and let ω be any character of GL 1 (L). Then we hae ( L(s, ρ ω) = L s m 1 ), ρ ω. 14

15 Lemma 3.4. In the situation of Theorem 3.3, we hae γ(s, ρ ω) = m 1 i=0 γ(s, ρ m 1 i ν ω). Proof. Let P be the parabolic used in the induction defining ρ, and let ρ be the representation of the Lei factor gien by the ordered sequence Using Theorem 4. of [5], we find that as needed. ρ ν m 1, ρ ν m 1 1,..., ρ ν m 1. γ(s, ρ ω) = γ(s, ρ ω) = m 1 i=0 γ(s, ρ m 1 i ν ω), We are reduced to L-functions and γ-factors for supercuspidal representations. Theorem 3.5 ([13, ]). Let ρ be a supercuspidal representation of GL n (L) and ω a character of GL 1 (L). Then if n > 1, L(s, ρ ω) = 1. If n = 1, where the product is taken oer α = q s 0 L(s, ρ ω) = α (1 αq s ) 1, such that ρ = ων s 0, where denotes the contragredient. In the case where ω is unramified, the calculation is the familiar one used aboe. In particular, L(s, ρ ω) = 1 unless ρ also unramified, and if ρ(ϖ) = α and ω(ϖ) = β, then L(s, ρ ω) = (1 αβq s ) 1. We also hae γ(s, ρ ω) = L(1 s, ρ ω) L(s, ρ ω) = 1 αβq s 1 α 1 β 1 q 1s. 3.. Roots and poles of γ-factors. We are interested in the roots and poles of the function n i=1 (1 q s α i ) γ(s, π ) = n i=1 (1 q 1s αi 1 ), where {α i } is the multiset of Satake parameters of π. We always consider alues of log q in the region Ω defined in the proof of Theorem.5. The denominator anishes when 1 = q 1s αi 1, or s = 1 log q α i, while the numerator anishes at s = log q α i. Let S N and S D denote the multisets of zeros of the numerator and denominator, respectiely, counted with multiplicity. Obsere that (7) s s = n, s S D s S N and that this quantity depends on the function γ(s, π ) only up to multiplication by a nowhere anishing holomorphic function. We hae some additional information coming from the fact that where τ is tempered. Suppose that τ = Q(ρ 1,..., ρ m ), where ρ i = Q(ρ iν m i 1 ρ iν m i 1 ). n i=1 (1 q s α i ) n i=1 (1 q 1s α 1 i ) = γ(s, τ ), Then γ(s, τ ) = m i=1 γ(s, ρ i), and γ(s, ρ i ) is holomorphic unless ρ i is an unramified character. So let J {1,..., m} denote the set of indices j J where ρ j is an unramified character, and write 15

16 γ j for ρ j (ϖ). Then γ j = 1 by temperedness, and we may compute the γ-factor as follows, using F 1 (s) and F (s) to denote nonanishing holomorphic functions of s. We hae γ(s, τ ) = F 1 (s) j J m j 1 k=0 m j 1 = F 1 (s) j J k=0 = F 1 (s) j J = F 1 (s) j J = F (s) j J γ(s, ρ jν k m k 1 ) = F 1 (s) m j 1 j J k=0 1 q s k m j 1 γ j 1 q 1sk m j 1 γj 1 1 q s m j 1 γ j 1 q 1s m j 1 γj 1 1 q s m j 1 γ j 1 q 1s m j 1 γj 1 1 q s m j 1 γ j 1 q 1s m j 1 γ 1 j. m j 1 k=1 m j 1 k=1 L(1 s, ρ j ν k m k 1 ) 1 q s k1 m j 1 γ j 1 q 1sk m j 1 L(s, ρ j νk m k 1 ) γ 1 j ) ( γ j q 1 s k m j 1 Note that in the fourth equality, we pulled out the k = m j 1 term of the numerator and the k = 0 term of the denominator, shifting the indices appropriately. The roots of the numerator (with multiplicity) are m j 1 log q γ j for j J, and the roots of the denominator are 1 m j 1 log q γ j. Using the equality γ(s, π ) = γ(s, τ ) and the quantity calculated in (7), we calculate n = (( 1 m ) ( j 1 m )) j 1 = m j. j J j J From this we immediately deduce that J = {1,..., m}, so that in fact τ is a tempered representation appearing as a subquotient inside an induction from a character of the Borel Possibilities for π. Using the calculations aboe, we obsere that the roots and poles of γ(s, τ ) completely determine all of the Satake parameters of π, since for each j {1,..., n} and each factor of the form 1 q s m j 1 γ j 1 q 1s m j 1 each alue (taken with multiplicities across different j) s = m j 1 log q γ j, m j 1 γ 1 j, 1 log q γ j,..., m j 1 log q γ j must appear as a root of the numerator of γ(s, π ). If τ is a unitary principal series, we cannot hae α i = q α j for some i j (and so compatibility follows from Theorem.5. Otherwise, we hae the following constraint on the Satake parameters α i : the set { α1,..., α m, α1 1,..., } α 1 m or 16 { α1,..., α m, 1, α1 1,..., } α 1 m

17 (if n is een or odd, respectiely) is equal to m j=1 m j 1 i=0 {q i m j 1 } γ j. Obsere that the inerses of } m j 1 i=0 {q i m j 1 γ j are m j 1 i=0 {q i m j 1 } γj 1. No Satake parameters of the form q i m j 1 γ j can be inerse to one another unless the m j s hae the same parity, so we may consider them separately, grouping the extra Satake parameter 1 into the odd m j case if n is odd. In either case, each m j 1 i=0 {q i m j 1 } γ j is either exactly equal to its set of inerses (if γ j = ±1) or is disjoint from its set of inerses. Now fix a Satake parameter γ ±1, γ = 1. Claim 3.6. We can pair up each j with γ j = γ with a j with γ j = γ 1 such that m j = m j, using each j {1,..., m} with γ j = γ ±1 exactly once. Proof. The key obseration is that for j with γ j = γ, if we consider the term of largest absolute alue, we see that there must be a j such that γ j = γj 1 and m j m j. Now note that by considering C/q Z -equialence classes, the number of Satake parameters of the form q 1 k γ j or q k γ j, k Z (depending on whether m j is een or odd), is equal to the number of the form q 1 k γj 1 or q k γj 1, respectiely. Let j 1 be such that m j1 is maximal among those j 1 with γ j1 = γ j ±1. By the key obseration and maximality, there exists j 1 with m j 1 = m j 1 and γ j 1 = γj 1 1. We now throw out these alues and find j, j, j 3, j 3,... iteratiely in this way, yielding the claim. We may now complete the proof of the main result of the section. Proof of Theorem 3.1. The classification of Satake parameters associated to j with γ j ±1 follows. Namely, these Satake parameters are the disjoint union of sets of the form m 1 { q i m 1 } m 1 { γ q i m 1 } γ 1, i=0 i=0 again with een m ; the latter use two of the alues j. Now consider j such that γ j = ±1. Note that each set of the form m i=0 { q i m 1 γ = ±1 for een m has each element matched to a distinct inerse, and if m is odd, each element is matched to a distinct inerse except for γ itself. We may thus include any number of sets of this form if m is een. If m is odd, and n is een, we require that there be an een number of sets of this form with γ = 1 and an een number with γ = 1. If n is odd, we require an odd number of sets of this form with γ = 1 and an een number with γ = Iwahori-Hecke algebras of quasi-split ramified unitary groups In this section, we perform some calculations to identify the Iwahori-Hecke algebras of quasi-split ramified unitary groups with those of split groups. None of the results in this section are noel. Iwahori-Matsumoto and Bernstein hae gien different presentations of these algebras. Recently Rostami has generalized these constructions so that they are alid for any reductie group [43]. The identification with algebras for split groups is based on an obseration of Lusztig [9, 10.13]. 17 } γ with